11884:
2418:
15112:
11709:
905:
9902:
8502:
has a unique left-invariant extension. We then identify the vector in the tangent space with the associated left-invariant vector field. Now, the commutator (as differential operators) of two left-invariant vector fields is again a vector field and again left-invariant. We can then define the bracket
2473:
is the field over which the Lie algebra is defined. From here, through to the remainder of this article, the tensor product is always explicitly shown. Many authors omit it, since, with practice, its location can usually be inferred from context. Here, a very explicit approach is adopted, to minimize
6612:
One can also state the theorem in a coordinate-free fashion, avoiding the use of total orders and basis elements. This is convenient when there are difficulties in defining the basis vectors, as there can be for infinite-dimensional Lie algebras. It also gives a more natural form that is more easily
2477:
The first step in the construction is to "lift" the Lie bracket from the Lie algebra (where it is defined) to the tensor algebra (where it is not), so that one can coherently work with the Lie bracket of two tensors. The lifting is done as follows. First, recall that the bracket operation on a Lie
1403:
In general, elements of the universal enveloping algebra are linear combinations of products of the generators in all possible orders. Using the defining relations of the universal enveloping algebra, we can always re-order those products in a particular order, say with all the factors of
5555:
10533:
3328:
That is, the Lie bracket defines the equivalence relation used to perform the quotienting. The result is still a unital associative algebra, and one can still take the Lie bracket of any two members. Computing the result is straight-forward, if one keeps in mind that each element of
4046:
The above construction focuses on Lie algebras and on the Lie bracket, and its skewness and antisymmetry. To some degree, these properties are incidental to the construction. Consider instead some (arbitrary) algebra (not a Lie algebra) over a vector space, that is, a vector space
2298:
2041:
The formal construction of the universal enveloping algebra takes the above ideas, and wraps them in notation and terminology that makes it more convenient to work with. The most important difference is that the free associative algebra used in the above is narrowed to the
12614:
6785:
15850:
is in many ways analogous to constructing the universal enveloping algebra for a given Lie algebra. Both constructions are universal and translate representation theory into module theory. Furthermore, both group algebras and universal enveloping algebras carry natural
7033:
10344:
6304:
3381:; the Poisson algebra is likewise rather complicated, with many peculiar properties. It is compatible with the tensor algebra, and so the modding can be performed. The Hopf algebra structure is conserved; this is what leads to its many novel applications, e.g. in
756:
6874:
7642:. This is perhaps not immediately obvious: to get this result, one must repeatedly apply the commutation relations, and turn the crank. The essence of the PoincarĂ©âBirkhoffâWitt theorem is that it is always possible to do this, and that the result is unique.
15411:
68:. Because Casimir operators commute with all elements of a Lie algebra, they can be used to classify representations. The precise definition also allows the importation of Casimir operators into other areas of mathematics, specifically, those that have a
14955:
10593:
is just the canonical embedding (with subscripts, respectively for algebras one and two). It is straightforward to verify that this embedding lifts, given the prescription above. See, however, the discussion of the bialgebra structure in the article on
11879:{\displaystyle {\begin{aligned}\delta (v\otimes w\otimes \cdots \otimes u)=&\,\delta (v)\otimes w\otimes \cdots \otimes u\\&+v\otimes \delta (w)\otimes \cdots \otimes u\\&+\cdots +v\otimes w\otimes \cdots \otimes \delta (u).\end{aligned}}}
4563:, any homomorphism on its generating set can be extended to the entire algebra. Everything else proceeds as described above: upon completion, one has a unital associative algebra; one can take a quotient in either of the two ways described above.
3529:
4340:
3636:
9729:
13169:
12235:
3028:
It is straightforward to verify that the above definition is bilinear, and is skew-symmetric; one can also show that it obeys the Jacobi identity. The final result is that one has a Lie bracket that is consistently defined on all of
9102:, that places the algebraic structure of the Lie algebra onto what is otherwise a standard associative algebra. That is, what the PBW theorem obscures (the commutation relations) the algebra of symbols restores into the spotlight.
14589:
14409:
2603:
14853:
If the Lie algebra acts on a differentiable manifold, then each
Casimir operator corresponds to a higher-order differential on the cotangent manifold, the second-order differential being the most common and most important.
7312:
13306:
12400:
7983:
6534:
and the symmetric algebra are isomorphic, and it is the PBW theorem that shows that this is so. See, however, the section on the algebra of symbols, below, for a more precise statement of the nature of the isomorphism.
2776:
15863:: the tensor algebra has a Hopf algebra structure on it, and because the Lie bracket is consistent with (obeys the consistency conditions for) that Hopf structure, it is inherited by the universal enveloping algebra.
11172:
3970:
2654:
5399:
5231:
4823:
2520:
10403:
1746:, discussed below, asserts that these elements are linearly independent and thus form a basis for the universal enveloping algebra. In particular, the universal enveloping algebra is always infinite dimensional.
14097:
is of rank one, and thus has one
Casimir operator. It is three-dimensional, and thus the Casimir operator must have order (3 â 1) = 2 i.e. be quadratic. Of course, this is the Lie algebra of
13900:
13536:
13790:
4926:
710:
16009:
9398:
4550:
4449:
6441:
2413:{\displaystyle T({\mathfrak {g}})=K\,\oplus \,{\mathfrak {g}}\,\oplus \,({\mathfrak {g}}\otimes {\mathfrak {g}})\,\oplus \,({\mathfrak {g}}\otimes {\mathfrak {g}}\otimes {\mathfrak {g}})\,\oplus \,\cdots }
1650:
16086:
5305:
4667:
5116:
3238:
11525:
5841:
Although the canonical construction, given above, can be applied to other algebras, the result, in general, does not have the universal property. Thus, for example, when the construction is applied to
1947:; the universal enveloping algebra is always infinite dimensional. Thus, in the case of sl(2,C), if we identify our Lie algebra as a subspace of its universal enveloping algebra, we must not interpret
12467:
2151:
8257:
10588:
10395:
9994:
7863:
6694:
6114:
5025:
4036:
1166:
12777:
8300:
6956:
4973:
14947:
13095:
11714:
4570:
is constructed. One need only to carefully keep track of the sign, when permuting elements. In this case, the (anti-)commutator of the superalgebra lifts to an (anti-)commuting
Poisson bracket.
4262:
16214:
16136:
10240:
7898:. It is constructed by appeal to the same notion of naturality as before. One starts with the same tensor algebra, and just uses a different ideal, the ideal that makes all elements commute:
6169:
14735:
9594:
3702:
6796:
6542:: this is what one gets, if one mods out by all commutators, without specifying what the values of the commutators are. The second step is to apply the specific commutation relations from
4710:
551:
14026:
11305:
9543:
2718:
This is a consistent, coherent definition, because both sides are bilinear, and both sides are skew symmetric (the Jacobi identity will follow shortly). The above defines the bracket on
13693:
11405:
3747:
13623:
14458:
8527:
as the commutator on the associated left-invariant vector fields. This definition agrees with any other standard definition of the bracket structure on the Lie algebra of a Lie group.
6482:
15107:{\displaystyle H={\begin{pmatrix}-1&0\\0&1\end{pmatrix}},{\text{ }}E={\begin{pmatrix}0&1\\0&0\end{pmatrix}},{\text{ }}F={\begin{pmatrix}0&0\\1&0\end{pmatrix}}}
4153:
3815:
16311:
15255:
13981:
13199:
9076:
5651:
1393:
15929:
14084:
11269:
7717:
7640:
7532:
4862:
3454:
3323:
2816:
2713:
13342:
12921:
11951:
11205:
11109:
11064:
10894:
10759:
6029:
5612:
5391:
14258:
12699:
12350:
12294:
8093:
7826:
7785:
7751:
7677:
7600:
7566:
7492:
7455:
7417:
7383:
7349:
7189:
6944:
6648:
3023:
2931:
12051:
11478:
11372:
10990:
10930:
10852:
8336:
8205:
8129:
7225:
7155:
7080:
6684:
5774:
3063:
1793:
16268:
15782:
15736:
15625:
15546:
15482:
14809:
14776:
13942:
12015:
11442:
10792:
10696:
10221:
10152:
9710:
9673:
9337:
9136:
9036:
9003:
8970:
8929:
8867:
8805:
8659:
8169:
8052:
8019:
7896:
7119:
6532:
6379:
6340:
5902:
5827:
5731:
5338:
4345:
This is consistent precisely because the tensor product is bilinear, and the multiplication is bilinear. The rest of the lift is performed so as to preserve multiplication as a
3360:
3148:
3096:
2255:
900:{\displaystyle E={\begin{pmatrix}0&1\\0&0\end{pmatrix}}\qquad F={\begin{pmatrix}0&0\\1&0\end{pmatrix}}\qquad H={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}~,}
597:
412:
14877:, respectively, one can then contract upper and lower indices (with the metric tensor) to obtain more interesting structures. For the quadratic Casimir invariant, this is the
11336:
10702:, of all orders. (See, for example, the realization of the universal enveloping algebra as left-invariant differential operators on the associated group, as discussed above.)
9640:
6910:
6594:
6567:
4103:
3882:
15690:
15659:
15583:
15513:
15441:
14840:
11982:
11918:
11678:
11549:
11014:
10954:
10816:
10657:
10184:
10115:
8743:
8686:
8525:
8500:
8456:
8432:
8408:
5678:
5583:
5362:
5069:
3847:
3465:
3172:
2287:
2215:
2179:
1945:
1921:
1871:
741:
369:
337:
289:
185:
137:
313:
265:
241:
161:
12994:
12846:
14169:
12810:
5977:
11651:
1849:
of the original Lie algebra. That is to say, we identify the original Lie algebra as the subspace of its universal enveloping algebra spanned by the generators. Although
9897:{\displaystyle p(t)\star q(t)=\left.\exp \left(t_{i}m^{i}\left({\frac {\partial }{\partial u}},{\frac {\partial }{\partial v}}\right)\right)p(u)q(v)\right\vert _{u=v=t}}
4257:
1530:
1493:
719:. Below we will make this "generators and relations" construction more precise by constructing the universal enveloping algebra as a quotient of the tensor algebra over
6490:. The proof of the theorem involves noting that, if one starts with out-of-order basis elements, these can always be swapped by using the commutator (together with the
3546:
2031:
1897:
1225:
1199:
13445:
13400:
6065:
5251:
2441:
13813:
12879:
2461:
1713:
1331:
1278:
445:
14126:
13486:
10075:
9475:
9100:
3773:
14196:
13845:
12115:
12084:
10024:
9455:
9428:
9300:
9273:
9190:
9163:
8550:
can be expressed (non-uniquely) as a linear combination of products of left-invariant vector fields. The collection of all left-invariant differential operators on
6141:
3264:
1847:
1820:
1740:
1680:
1561:
1456:
1429:
1063:
217:
15162:
8896:
8834:
8772:
8719:
8626:
8597:
4205:
993:
15244:
15203:
13122:
949:
14647:
12459:
2839:
1031:
12314:
12258:
12143:
12135:
11698:
10044:
9246:
9226:
8568:
8548:
8476:
8384:
6161:
5794:
5698:
5136:
5045:
4245:
4225:
4176:
4065:
3975:
where each arrow is a linear map, and the kernel of that map is given by the image of the previous map. The universal enveloping algebra can then be defined as
2005:
1985:
1965:
1298:
1245:
5853:. Likewise, the PoincarĂ©âBirkhoffâWitt theorem, below, constructs a basis for an enveloping algebra; it just won't be universal. Similar remarks hold for the
6116:
be the injection into the tensor algebra; this is used to give the tensor algebra a basis as well. This is done by lifting: given some arbitrary sequence of
3388:
The construction can be performed in a slightly different (but ultimately equivalent) way. Forget, for a moment, the above lifting, and instead consider the
14470:
1333:
because we do not impose this relation in the construction of the enveloping algebra. Indeed, it follows from the
PoincarĂ©âBirkhoffâWitt theorem (discussed
14266:
3117:
such algebra, however; it contains far more elements than needed. One can get something smaller by projecting back down. The universal enveloping algebra
1563:'s. Doing this sort of thing repeatedly eventually converts any element into a linear combination of terms in ascending order. Thus, elements of the form
2537:
15698:
as abelian Lie algebra, the left-invariant differential operators are the constant coefficient operators, which are indeed a polynomial algebra in the
7235:
16223:
essentially states that every commutative Hopf algebra is isomorphic to the Hopf algebra of continuous functions on some compact topological group
13213:
12355:
7904:
2721:
1923:
does not consist of (finite-dimensional) matrices. In particular, there is no finite-dimensional algebra that contains the universal enveloping of
5561:
The point is that because there are no other relations in the universal enveloping algebra besides those coming from the commutation relations of
8434:
with the space of left-invariant vector fields (i.e., first-order left-invariant differential operators). Specifically, if we initially think of
5550:{\displaystyle {\widehat {\varphi }}(X_{i_{1}}\cdots X_{i_{N}})=\varphi (X_{i_{1}})\cdots \varphi (X_{i_{N}}),\quad X_{i_{j}}\in {\mathfrak {g}}}
11114:
16740:
8721:
of left-invariant differential operators is generated by elements (the left-invariant vector fields) that satisfy the commutation relations of
3897:
2615:
10528:{\displaystyle i({\mathfrak {g}}_{1}\oplus {\mathfrak {g}}_{2})=i_{1}({\mathfrak {g}}_{1})\otimes 1\oplus 1\otimes i_{2}({\mathfrak {g}}_{2})}
6494:). The hard part of the proof is establishing that the final result is unique and independent of the order in which the swaps were performed.
5141:
4733:
2481:
64:
can be constructed as quotients of the universal enveloping algebra. In addition, the enveloping algebra gives a precise definition for the
7457:
of the filtration, as one might naively surmise. It is not constructed through a set subtraction mechanism associated with the filtration.
13850:
13491:
12406:, and so an explicit indexed form of the above equations can be given, in terms of the Lie algebra basis; this is originally a theorem of
13704:
4877:
3884:
Establishing that this is an ideal is important, because ideals are precisely those things that one can quotient with; ideals lie in the
14035:
by the above, these clearly corresponds to the roots of the characteristic equation. One concludes that the roots form a space of rank
605:
16229:âthe theory of compact topological groups and the theory of commutative Hopf algebras are the same. For Lie groups, this implies that
15937:
14846:, then one can construct Casimir invariants on the corresponding space of operators. The quadratic Casimir operator corresponds to an
9345:
4460:
4359:
103:
From an analytic viewpoint, the universal enveloping algebra of the Lie algebra of a Lie group may be identified with the algebra of
8689:
6387:
5866:
1743:
1569:
16020:
9717:
5256:
4623:
12609:{\displaystyle f_{ij}^{\;\;k}\kappa ^{jl\cdots m}+f_{ij}^{\;\;l}\kappa ^{kj\cdots m}+\cdots +f_{ij}^{\;\;m}\kappa ^{kl\cdots j}=0}
8688:
arises as the Lie algebra of a real Lie group, one can use left-invariant differential operators to give an analytic proof of the
4573:
Another possibility is to use something other than the tensor algebra as the covering algebra. One such possibility is to use the
13107:
5928:
of a totally ordered set. Recall that a free vector space is defined as the space of all finitely supported functions from a set
5085:
3377:
structure. (This is a non-trivial statement; the tensor algebra has a rather complicated structure: it is, among other things, a
3183:
11490:
6780:{\displaystyle T_{m}{\mathfrak {g}}=K\oplus {\mathfrak {g}}\oplus T^{2}{\mathfrak {g}}\oplus \cdots \oplus T^{m}{\mathfrak {g}}}
2064:
14897:
are able to evade the premises of the
ColemanâMandula theorem, and can be used to mix together space and internal symmetries.
8210:
7028:{\displaystyle K\subset {\mathfrak {g}}\subset T_{2}{\mathfrak {g}}\subset \cdots \subset T_{m}{\mathfrak {g}}\subset \cdots }
16680:
16640:
16584:
10544:
10352:
9913:
7831:
6070:
4981:
3981:
1071:
12710:
8354:, rather than the symmetric algebra. In essence, the construction zeros out the anti-commutators. The resulting algebra is
8266:
4937:
14908:
14778:
above was that it was a differential algebra, by dint of the fact that any derivation on the Lie algebra can be lifted to
13018:
10339:{\displaystyle U({\mathfrak {g}}_{1}\oplus {\mathfrak {g}}_{2})\cong U({\mathfrak {g}}_{1})\otimes U({\mathfrak {g}}_{2})}
6299:{\displaystyle h(e_{a}\otimes e_{b}\otimes \cdots \otimes e_{c})=h(e_{a})\otimes h(e_{b})\otimes \cdots \otimes h(e_{c})}
16147:
16097:
16610:
15792:
6869:{\displaystyle T^{m}{\mathfrak {g}}=T^{\otimes m}{\mathfrak {g}}={\mathfrak {g}}\otimes \cdots \otimes {\mathfrak {g}}}
14661:
13902:, one concludes that this tensor must be completely symmetric. This tensor is exactly the Casimir invariant of order
9551:
3644:
3113:
with a Lie bracket that is compatible with the Lie algebra bracket; it is compatible by construction. It is not the
8358:
enveloping algebra, but is not universal. As mentioned above, it fails to envelop the exceptional Jordan algebras.
4679:
457:
13986:
11274:
9483:
16572:
13634:
12963:
11377:
10159:
10090:
5076:
4158:
the multiplication is bilinear, then the same construction and definitions can go through. One starts by lifting
3713:
3175:
2051:
104:
50:
15406:{\displaystyle U({\mathfrak {sl}}_{2})={\frac {\mathbb {C} \langle x,y,z\rangle }{(xy-yx+2y,xz-zx-2z,yz-zy+x)}}}
13548:
15827:
for this; one must take a quotient, so that the central elements of the Lie algebra act as prescribed scalars.
14417:
6449:
4620:
The universal enveloping algebra, or rather the universal enveloping algebra together with the canonical map
4108:
3778:
16672:
16624:
16576:
16316:
16273:
13947:
13177:
11408:
9048:
5617:
1340:
15893:
14044:
11210:
7682:
7605:
7497:
4831:
3403:
3272:
2781:
2662:
14812:
13314:
12884:
11923:
11177:
11072:
11027:
10857:
10722:
10661:
5982:
5588:
5367:
4593:
coming from the grading on the exterior algebra. (The
Gerstenhaber algebra should not be confused with the
2527:
14201:
12625:
12319:
12263:
8064:
7797:
7756:
7722:
7648:
7571:
7537:
7463:
7426:
7388:
7354:
7320:
7160:
6915:
6619:
2942:
2850:
16345:
16220:
14890:
14874:
12020:
11447:
11341:
10959:
10899:
10821:
8305:
8174:
8098:
7753:
has the effect of setting all commutators to zero. What PBW states is that the commutator of elements in
7194:
7124:
7049:
6653:
5736:
3032:
1752:
85:
16666:
16240:
15754:
15708:
15597:
15518:
15454:
14781:
14748:
13914:
11987:
11414:
10764:
10668:
10193:
10124:
9682:
9675:, as written, and that one must first perform a tedious reshuffling of the basis elements (applying the
9645:
9309:
9108:
9008:
8975:
8942:
8901:
8839:
8777:
8631:
8141:
8024:
7991:
7868:
7091:
6504:
6351:
6312:
5874:
5799:
5703:
5310:
3524:{\displaystyle {\mathfrak {g}}\oplus ({\mathfrak {g}}\otimes {\mathfrak {g}})\subset T({\mathfrak {g}})}
3332:
3120:
3068:
2227:
559:
374:
16546:
15789:
14866:
12927:
10711:
7568:
to zero. One can see this by observing that the commutator of a pair of elements whose products lie in
5776:.) This observation is important because it allows (as discussed below) the Casimir elements to act on
5079:
to the universal enveloping algebra. This map is an embedding, by the
PoincarĂ©âBirkhoffâWitt theorem.)
3110:
2154:
556:
Then the universal enveloping algebra is the associative algebra (with identity) generated by elements
20:
11314:
9601:
6888:
6572:
6545:
4070:
3852:
3370:
such algebra; one cannot find anything smaller that still obeys the axioms of an associative algebra.
3366:: one just takes the bracket as usual, and searches for the coset that contains the result. It is the
15671:
15640:
15564:
15494:
15422:
14821:
14461:
14041:
and that the
Casimir invariants span this space. That is, the Casimir invariants generate the center
13003:
11956:
11892:
11659:
11530:
10995:
10935:
10797:
10638:
10187:
10165:
10096:
8724:
8667:
8506:
8481:
8437:
8413:
8389:
8132:
5659:
5564:
5343:
5050:
3820:
3153:
2268:
2196:
2160:
1926:
1902:
1852:
1795:
themselves are linearly independent. It is therefore commonâif potentially confusingâto identify the
722:
350:
318:
270:
166:
118:
97:
93:
294:
246:
222:
142:
16340:
14882:
12969:
12818:
10618:
8530:
We may then consider left-invariant differential operators of arbitrary order. Every such operator
6947:
6614:
4335:{\displaystyle {\begin{aligned}m:V\otimes V&\to V\\a\otimes b&\mapsto m(a,b)\end{aligned}}}
2033:
matrices, but rather as symbols with no further properties (other than the commutation relations).
77:
14131:
12785:
6538:
It is useful, perhaps, to split the process into two steps. In the first step, one constructs the
5943:
5656:
The universal property of the enveloping algebra immediately implies that every representation of
3631:{\displaystyle c\otimes d\otimes \cdots \otimes (a\otimes b-b\otimes a-)\otimes f\otimes g\cdots }
12926:
The center of the universal enveloping algebra of a simple Lie algebra is given in detail by the
12055:
From the PBW theorem, it is clear that all central elements are linear combinations of symmetric
11561:
9712:
in the properly ordered basis. An explicit expression for this product can be given: this is the
7043:
3885:
72:. They also play a central role in some recent developments in mathematics. In particular, their
5940:(finitely supported means that only finitely many values are non-zero); it can be given a basis
1498:
1461:
16735:
15485:
14893:
restricts the form that these can take, when one considers ordinary Lie algebras. However, the
13448:
12939:
11308:
11017:
10227:
7085:
5846:
4590:
2010:
1876:
1204:
1178:
46:
13414:
13369:
6688:
First, a notation is needed for an ascending sequence of subspaces of the tensor algebra. Let
6038:
5236:
2426:
1033:. The universal enveloping algebra of sl(2,C) is then the algebra generated by three elements
16708:
15628:
13798:
12851:
12056:
10699:
10231:
3373:
The universal enveloping algebra is what remains of the tensor algebra after modding out the
2446:
1685:
1303:
1250:
417:
57:
16701:
15819:
variables with polynomial coefficients may be obtained starting with the Lie algebra of the
14101:
13465:
13164:{\displaystyle \operatorname {ad} :{\mathfrak {g}}\to \operatorname {End} ({\mathfrak {g}})}
12090:
are the irreducible homogenous polynomials of a given, fixed degree. That is, given a basis
10060:
9460:
9085:
8259:
is a projection, and one then gets PBW-type theorems for the associated graded algebra of a
7787:
is necessarily zero. What is left are the elements that are not expressible as commutators.
3752:
16730:
16650:
16594:
14174:
14094:
13818:
12093:
12062:
11481:
10717:
10118:
10002:
9433:
9406:
9278:
9251:
9168:
9141:
9079:
6119:
5904:. This can be done in either one of two different ways: either by reference to an explicit
4869:
4594:
4582:
3889:
3249:
3244:
1825:
1798:
1718:
1658:
1539:
1434:
1407:
1036:
190:
69:
16690:
15123:
12230:{\displaystyle C_{(m)}=\kappa ^{ab\cdots c}e_{a}\otimes e_{b}\otimes \cdots \otimes e_{c}}
8872:
8810:
8748:
8695:
8602:
8573:
7042:, since the filtration preserves the algebraic properties of the subspaces. Note that the
4181:
3385:. However, for the purposes of the formal definition, none of this particularly matters.)
954:
8:
16320:
16315:
These ideas can then be extended to the non-commutative case. One starts by defining the
15847:
15208:
15167:
14862:
14653:
12403:
11307:
Thus, a technique is needed for computing that kernel. What we have is the action of the
9676:
6491:
4609:
913:
448:
291:. There may be many ways to make such an embedding, but there is a unique "largest" such
42:
15738:
consists of the left- and right- invariant differential operators; this, in the case of
14600:
12413:
2821:
998:
16654:
16528:
15699:
12407:
12299:
12243:
12120:
11683:
10050:
10029:
9231:
9195:
8553:
8533:
8461:
8369:
7794:. This is the algebra where all commutators vanish. It can be defined as a filtration
6146:
6032:
5779:
5683:
5121:
5030:
4670:
4230:
4210:
4161:
4050:
1990:
1970:
1950:
1533:
1283:
1230:
16402:
14584:{\displaystyle C_{(2)}=L^{2}=e_{1}\otimes e_{1}+e_{2}\otimes e_{2}+e_{3}\otimes e_{3}}
16676:
16658:
16636:
16606:
16580:
16532:
15831:
14886:
14847:
14404:{\displaystyle \det \left(xL_{1}+yL_{2}+zL_{3}-tI\right)=-t^{3}-(x^{2}+y^{2}+z^{2})t}
12087:
8836:âwhich one can establish analyticallyâthey must certainly be linearly independent in
7791:
6498:
5925:
3098:
in the conventional sense of a "lift" from a base space (here, the Lie algebra) to a
73:
13360:
such roots; this is the rank of the algebra. This implies that the highest value of
16696:
16686:
16628:
16520:
16444:
16439:
16427:
16406:
16398:
15830:
The universal enveloping algebra of a finite-dimensional Lie algebra is a filtered
15820:
15796:
15745:
14894:
12923:
follows from the fact that the
Killing form is invariant under the adjoint action.
11021:
10614:
10155:
8351:
8260:
7039:
6539:
5854:
4586:
4578:
4574:
4567:
65:
16389:
Perez-Izquierdo, J.M.; Shestakov, I.P. (2004). "An envelope for Malcev algebras".
15744:
not commutative, is often not generated by first-order operators (see for example
12938:
The number of algebraically independent
Casimir operators of a finite-dimensional
10992:
Because of this, the center is directly useful for classifying representations of
5829:) act as scalars and provide important information about the representations. The
2598:{\displaystyle T({\mathfrak {g}})\otimes T({\mathfrak {g}})\to T({\mathfrak {g}})}
16646:
16590:
16566:
16562:
14843:
12943:
10599:
8138:
The construction above, due to its use of quotienting, implies that the limit of
5921:
5909:
5830:
3389:
3374:
3106:
2531:
2058:
7307:{\displaystyle G_{m}{\mathfrak {g}}=U_{m}{\mathfrak {g}}/U_{m-1}{\mathfrak {g}}}
15860:
14858:
14128:
As an elementary exercise, one can compute this directly. Changing notation to
10629:
10595:
8347:
6309:
The PoincarĂ©âBirkhoffâWitt theorem then states that one can obtain a basis for
5850:
5842:
5072:
4601:
3099:
2262:
2222:
2047:
2043:
89:
39:
16411:
1280:, as is easily verified. But in the universal enveloping algebra, the element
16724:
16508:
16328:
13539:
13344:
The non-zero roots of this characteristic polynomial (that are roots for all
13301:{\displaystyle \det(tI-\operatorname {ad} _{x})=\sum _{n=0}^{d}p_{n}(x)t^{n}}
12395:{\displaystyle \left(\operatorname {ad} _{\mathfrak {g}}\right)^{\otimes m}.}
10078:
7978:{\displaystyle S({\mathfrak {g}})=T({\mathfrak {g}})/(a\otimes b-b\otimes a)}
6613:
extended to other kinds of algebras. This is accomplished by constructing a
3382:
2771:{\displaystyle T^{2}({\mathfrak {g}})={\mathfrak {g}}\otimes {\mathfrak {g}}}
2523:
1749:
The PoincarĂ©âBirkhoffâWitt theorem implies, in particular, that the elements
92:
of the corresponding Lie group. This relationship generalizes to the idea of
81:
7157:
This also works naturally on the subspaces, and so one obtains a filtration
15856:
15824:
15662:
15489:
12813:
11552:
10625:
10054:
9248:. Indeed, the correspondence is trivial: one simply substitutes the symbol
6604:; these are explicitly constructed to behave appropriately as commutators.
5905:
4560:
4346:
3378:
2258:
2218:
1171:
and no other relations. We emphasize that the universal enveloping algebra
61:
11167:{\displaystyle z=v\otimes w\otimes \cdots \otimes u\in U({\mathfrak {g}})}
10089:
The universal enveloping algebra preserves the representation theory: the
16605:, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,
16603:
Lie Groups, Lie Algebras, and Representations: An Elementary Introduction
15890:. This algebra has a natural Hopf algebra structure: given two functions
15785:
13351:
12959:
11700:, the Lie bracket is that of vector fields.) The lifting is performed by
10598:
for a review of some of the finer points of doing so: in particular, the
6601:
4556:
3965:{\displaystyle 0\to I\to T({\mathfrak {g}})\to T({\mathfrak {g}})/I\to 0}
2649:{\displaystyle {\mathfrak {g}}\otimes {\mathfrak {g}}\to {\mathfrak {g}}}
2182:
35:
27:
15249:
this shows us that the universal enveloping algebra has the presentation
5226:{\displaystyle \varphi ()=\varphi (X)\varphi (Y)-\varphi (Y)\varphi (X)}
4818:{\displaystyle \varphi ()=\varphi (X)\varphi (Y)-\varphi (Y)\varphi (X)}
2515:{\displaystyle {\mathfrak {g}}\times {\mathfrak {g}}\to {\mathfrak {g}}}
371:
be a Lie algebra, assumed finite-dimensional for simplicity, with basis
16524:
16511:(1967). "Some remarks about the associated envelope of a Lie algebra".
15887:
15843:
15549:
12942:
is equal to the rank of that algebra, i.e. is equal to the rank of the
3710:
are obtained as linear combinations of elements of this form. Clearly,
2464:
1682:'s being non-negative integers, span the enveloping algebra. (We allow
115:
The idea of the universal enveloping algebra is to embed a Lie algebra
16632:
13207:-dimensional endomorphism, and so one has the characteristic equation
10602:
employed there corresponds to the Wigner-Racah coefficients, i.e. the
7988:
Thus, one can view the PoincarĂ©âBirkhoffâWitt theorem as stating that
7084:
It was already established, above, that quotienting by the ideal is a
15852:
15586:
14878:
10607:
10603:
6597:
4727:
given by the commutator). More explicitly, this means that we assume
2605:
that is also bilinear, skew symmetric and obeys the Jacobi identity.
16671:, Graduate Studies in Mathematics, vol. 131, Providence, R.I.:
13895:{\displaystyle \operatorname {ad} _{x+y}=\operatorname {ad} _{y+x},}
13531:{\displaystyle \operatorname {ad} _{kx}=k\,\operatorname {ad} _{x}.}
10230:
rests on the observation that there is an isomorphism, known as the
9457:. The algebraic structure is obtained by requiring that the product
4577:; that is, to replace every occurrence of the tensor product by the
1495:(in the "wrong" order), we can use the relations to rewrite this as
16623:, Graduate Studies in Mathematics, vol. 34, Providence, R.I.:
16270:; more precisely, it is isomorphic to a subspace of the dual space
13785:{\displaystyle p_{n}(x)=x_{a}x_{b}\cdots x_{c}\kappa ^{ab\cdots c}}
10624:
Construction of representations typically proceeds by building the
8207:
In more general settings, with loosened conditions, one finds that
4921:{\displaystyle {\widehat {\varphi }}\colon U({\mathfrak {g}})\to A}
4605:
163:
with identity in such a way that the abstract bracket operation in
15117:
which satisfy the following identities under the standard bracket:
6596:
It can also be precisely defined: the basis elements are given by
5871:
The PoincarĂ©âBirkhoffâWitt theorem gives a precise description of
5849:, but not the exceptional ones: that is, it does not envelope the
1395:
are all linearly independent in the universal enveloping algebra.
6569:
The first step is universal, and does not depend on the specific
4247:
does â symmetry or antisymmetry or whatever. The lifting is done
747:
705:{\displaystyle x_{i}x_{j}-x_{j}x_{i}=\sum _{k=1}^{n}c_{ijk}x_{k}}
16004:{\displaystyle (\nabla (\varphi ,\psi ))(x)=\varphi (x)\psi (x)}
12402:
Recall that the adjoint representation is given directly by the
4581:. If the base algebra is a Lie algebra, then the result is the
9393:{\displaystyle w:\star ({\mathfrak {g}})\to U({\mathfrak {g}})}
5614:
is well defined, independent of how one writes a given element
4545:{\displaystyle m(a,b\otimes c)=m(a,b)\otimes c+b\otimes m(a,c)}
4444:{\displaystyle m(a\otimes b,c)=a\otimes m(b,c)+m(a,c)\otimes b}
16547:
Universal enveloping algebras and some applications in physics
8807:. Thus, if the PBW basis elements are linearly independent in
6436:{\displaystyle e_{a}\otimes e_{b}\otimes \cdots \otimes e_{c}}
4566:
The above is exactly how the universal enveloping algebra for
1645:{\displaystyle x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{n}^{k_{n}}}
16081:{\displaystyle (\Delta (\varphi ))(x\otimes y)=\varphi (xy),}
14870:
10613:
Also important is that the universal enveloping algebra of a
10057:(modulo the relation that the center be the unit); here, the
8745:. Thus, by the universal property of the enveloping algebra,
5653:
as a linear combination of products of Lie algebra elements.
5300:{\displaystyle {\widehat {\varphi }}:U({\mathfrak {g}})\to A}
4662:{\displaystyle h\colon {\mathfrak {g}}\to U({\mathfrak {g}})}
3363:
19:
For the universal enveloping W* algebra of a C* algebra, see
13847:. By linearity and the commutativity of addition, i.e. that
7645:
Since commutators of elements whose products are defined in
4589:
of the corresponding Lie group. As before, it has a grading
16712:
14198:
belonging to the adjoint rep, a general algebra element is
9765:
5111:{\displaystyle \varphi \colon {\mathfrak {g}}\rightarrow A}
3233:{\displaystyle U({\mathfrak {g}})=T({\mathfrak {g}})/\sim }
14464:
for the rotation group is that Casimir operator. That is,
11520:{\displaystyle \delta :{\mathfrak {g}}\to {\mathfrak {g}}}
10397:. The isomorphism follows from a lifting of the embedding
5920:
One way is to suppose that the Lie algebra can be given a
4041:
2474:
any possible confusion about the meanings of expressions.
2050:. The commutation relations are imposed by constructing a
1458:, etc. For example, whenever we have a term that contains
14740:
7534:
has the effect of setting all Lie commutators defined in
6497:
This basis should be easily recognized as the basis of a
2289:, without any restrictions whatsoever on those products.
2146:{\displaystyle x_{i}x_{j}-x_{j}x_{i}-\Sigma c_{ijk}x_{k}}
2046:, so that the product of symbols is understood to be the
76:
provides a commutative example of the objects studied in
16388:
11680:
is the space of left invariant vector fields on a group
8361:
8252:{\displaystyle S({\mathfrak {g}})\to G({\mathfrak {g}})}
16621:
Differential geometry, Lie groups, and symmetric spaces
10583:{\displaystyle i:{\mathfrak {g}}\to U({\mathfrak {g}})}
10390:{\displaystyle {\mathfrak {g}}_{1},{\mathfrak {g}}_{2}}
9989:{\displaystyle m(A,B)=\log \left(e^{A}e^{B}\right)-A-B}
7858:{\displaystyle \operatorname {Sym} ^{m}{\mathfrak {g}}}
6109:{\displaystyle h:{\mathfrak {g}}\to T({\mathfrak {g}})}
5020:{\displaystyle h:{\mathfrak {g}}\to U({\mathfrak {g}})}
4597:; both invoke anticommutation, but in different ways.)
4031:{\displaystyle U({\mathfrak {g}})=T({\mathfrak {g}})/I}
1161:{\displaystyle he-eh=2e,\quad hf-fh=-2f,\quad ef-fe=h,}
15073:
15023:
14970:
12772:{\displaystyle C_{(2)}=\kappa ^{ij}e_{i}\otimes e_{j}}
11962:
11898:
11383:
11280:
11234:
857:
814:
771:
16551:
Lecture, Modave Summer School in Mathematical Physics
16276:
16243:
16150:
16100:
16023:
15940:
15896:
15757:
15711:
15674:
15643:
15627:
can be identified with the algebra of left-invariant
15600:
15567:
15521:
15497:
15457:
15425:
15258:
15211:
15170:
15126:
14958:
14911:
14824:
14784:
14751:
14664:
14603:
14473:
14420:
14269:
14204:
14177:
14134:
14104:
14047:
13989:
13950:
13917:
13853:
13821:
13801:
13707:
13637:
13551:
13494:
13468:
13417:
13372:
13317:
13216:
13180:
13125:
13021:
12972:
12887:
12854:
12821:
12788:
12713:
12628:
12470:
12416:
12358:
12322:
12302:
12266:
12246:
12146:
12123:
12096:
12065:
12023:
11990:
11959:
11926:
11895:
11712:
11686:
11662:
11564:
11533:
11493:
11450:
11417:
11411:, and that the space of derivations can be lifted to
11380:
11344:
11317:
11277:
11213:
11180:
11117:
11075:
11030:
10998:
10962:
10938:
10902:
10860:
10824:
10800:
10767:
10725:
10671:
10641:
10547:
10406:
10355:
10243:
10196:
10168:
10127:
10099:
10063:
10032:
10005:
9916:
9732:
9685:
9648:
9604:
9554:
9486:
9463:
9436:
9409:
9348:
9312:
9281:
9254:
9234:
9198:
9171:
9144:
9111:
9088:
9051:
9011:
8978:
8945:
8904:
8875:
8842:
8813:
8780:
8751:
8727:
8698:
8670:
8634:
8605:
8576:
8556:
8536:
8509:
8484:
8464:
8440:
8416:
8392:
8372:
8308:
8295:{\displaystyle \operatorname {gr} U({\mathfrak {g}})}
8269:
8213:
8177:
8144:
8101:
8067:
8027:
7994:
7907:
7871:
7834:
7800:
7759:
7725:
7685:
7651:
7608:
7574:
7540:
7500:
7466:
7429:
7391:
7357:
7323:
7238:
7197:
7163:
7127:
7094:
7052:
6959:
6918:
6891:
6799:
6697:
6656:
6622:
6575:
6548:
6507:
6452:
6390:
6354:
6315:
6172:
6149:
6122:
6073:
6041:
5985:
5946:
5877:
5802:
5782:
5739:
5706:
5686:
5662:
5620:
5591:
5567:
5402:
5370:
5346:
5313:
5259:
5239:
5144:
5124:
5088:
5053:
5033:
4984:
4968:{\displaystyle \varphi ={\widehat {\varphi }}\circ h}
4940:
4880:
4834:
4736:
4682:
4626:
4555:
This extension is consistent by appeal to a lemma on
4463:
4362:
4260:
4233:
4213:
4184:
4164:
4111:
4073:
4053:
3984:
3900:
3855:
3823:
3781:
3755:
3716:
3647:
3549:
3468:
3406:
3335:
3275:
3252:
3186:
3156:
3123:
3071:
3035:
2945:
2853:
2824:
2784:
2724:
2665:
2618:
2540:
2484:
2449:
2429:
2301:
2271:
2230:
2199:
2163:
2067:
2013:
1993:
1973:
1953:
1929:
1905:
1879:
1855:
1828:
1801:
1755:
1721:
1715:, meaning that we allow terms in which no factors of
1688:
1661:
1572:
1542:
1501:
1464:
1437:
1410:
1343:
1306:
1286:
1253:
1233:
1207:
1181:
1074:
1039:
1001:
957:
916:
759:
725:
608:
562:
460:
420:
377:
353:
321:
297:
273:
249:
225:
193:
169:
145:
121:
14942:{\displaystyle {\mathfrak {g}}={\mathfrak {sl}}_{2}}
14881:. Quartic Casimir operators allow one to square the
13106:-dimensional Lie algebra, that is, an algebra whose
13090:{\displaystyle \det(tI-M)=\sum _{n=0}^{d}p_{n}t^{n}}
6484:, the ordering being that of total order on the set
5393:
must be uniquely determined by the requirement that
2153:. The universal enveloping algebra is the "largest"
15837:
15751:Another characterization in Lie group theory is of
14815:, from which one can construct Casimir invariants.
13462:This can be seen in several ways: Given a constant
11111:corresponds to linear combinations of all elements
8338:serving to remind that it is the filtered algebra.
1334:
16305:
16262:
16209:{\displaystyle (S(\varphi ))(x)=\varphi (x^{-1}).}
16208:
16131:{\displaystyle \varepsilon (\varphi )=\varphi (e)}
16130:
16080:
16003:
15923:
15776:
15730:
15684:
15653:
15619:
15577:
15540:
15507:
15476:
15435:
15405:
15238:
15197:
15156:
15106:
14941:
14834:
14803:
14770:
14729:
14641:
14583:
14452:
14403:
14252:
14190:
14163:
14120:
14078:
14020:
13975:
13936:
13894:
13839:
13807:
13784:
13687:
13617:
13530:
13480:
13439:
13394:
13336:
13300:
13193:
13163:
13089:
12988:
12915:
12873:
12840:
12804:
12771:
12693:
12608:
12453:
12394:
12344:
12316:belonging to the adjoint representation. That is,
12308:
12288:
12252:
12229:
12129:
12109:
12078:
12045:
12009:
11976:
11945:
11912:
11878:
11692:
11672:
11645:
11543:
11519:
11472:
11436:
11399:
11366:
11330:
11299:
11263:
11199:
11166:
11103:
11058:
11008:
10984:
10948:
10932:and in particular with the canonical embedding of
10924:
10888:
10846:
10810:
10786:
10753:
10690:
10651:
10582:
10527:
10389:
10338:
10215:
10178:
10146:
10109:
10069:
10038:
10018:
9988:
9896:
9704:
9667:
9634:
9588:
9537:
9469:
9449:
9422:
9392:
9331:
9294:
9267:
9240:
9220:
9184:
9157:
9130:
9094:
9070:
9030:
8997:
8964:
8923:
8890:
8861:
8828:
8799:
8766:
8737:
8713:
8680:
8653:
8628:is isomorphic to the universal enveloping algebra
8620:
8591:
8562:
8542:
8519:
8494:
8470:
8450:
8426:
8402:
8378:
8330:
8294:
8251:
8199:
8163:
8123:
8087:
8046:
8013:
7977:
7890:
7857:
7820:
7779:
7745:
7711:
7671:
7634:
7594:
7560:
7526:
7486:
7449:
7411:
7377:
7343:
7306:
7219:
7183:
7149:
7113:
7074:
7027:
6938:
6904:
6868:
6779:
6678:
6642:
6588:
6561:
6526:
6476:
6435:
6373:
6334:
6298:
6155:
6135:
6108:
6059:
6023:
5971:
5896:
5821:
5788:
5768:
5725:
5692:
5672:
5645:
5606:
5577:
5549:
5385:
5356:
5332:
5299:
5245:
5225:
5130:
5110:
5063:
5039:
5019:
4967:
4920:
4856:
4817:
4704:
4661:
4544:
4443:
4334:
4239:
4219:
4199:
4170:
4147:
4097:
4059:
4030:
3964:
3876:
3841:
3809:
3767:
3741:
3696:
3630:
3523:
3448:
3354:
3317:
3258:
3232:
3166:
3142:
3090:
3057:
3017:
2925:
2833:
2810:
2770:
2707:
2648:
2597:
2514:
2455:
2435:
2412:
2281:
2249:
2209:
2173:
2145:
2025:
1999:
1979:
1959:
1939:
1915:
1891:
1865:
1841:
1814:
1787:
1734:
1707:
1674:
1644:
1555:
1524:
1487:
1450:
1423:
1387:
1325:
1292:
1272:
1239:
1219:
1193:
1160:
1057:
1025:
987:
943:
899:
735:
704:
591:
545:
439:
406:
363:
331:
307:
283:
259:
235:
211:
179:
155:
131:
12704:As an example, the quadratic Casimir operator is
10190:to the abelian category of all left modules over
9598:The primary issue with this construction is that
8410:. Following the modern approach, we may identify
7385:of strictly smaller filtration degree. Note that
5860:
2608:The lifting can be done grade by grade. Begin by
2534:. We wish to define a Lie bracket that is a map
16722:
14842:acts on a space of linear operators, such as in
14270:
14088:
13217:
13022:
9720:for the product of two elements of a Lie group.
7191:whose limit is the universal enveloping algebra
6650:whose limit is the universal enveloping algebra
6342:from the above, by enforcing the total order of
5845:, the resulting enveloping algebra contains the
16425:
14900:
14730:{\displaystyle =\varepsilon _{ij}^{\;\;k}e_{k}}
12352:can be (should be) thought of as an element of
9589:{\displaystyle p,q\in \star ({\mathfrak {g}}).}
4600:The construction has also been generalized for
4227:obeys all of the same properties that the base
3697:{\displaystyle a,b,c,d,f,g\in {\mathfrak {g}}.}
14869:endowed with a metric and the symmetry groups
9105:The algebra is obtained by taking elements of
8972:may be given a new algebra structure so that
56:Universal enveloping algebras are used in the
15859:. This is made precise in the article on the
14745:A key observation during the construction of
9192:to obtain the space of symmetric polynomials
4705:{\displaystyle \varphi :{\mathfrak {g}}\to A}
3888:of the quotienting map. That is, one has the
3065:one says that it has been "lifted" to all of
1175:the same as (or contained in) the algebra of
546:{\displaystyle =\sum _{k=1}^{n}c_{ijk}X_{k}.}
315:, called the universal enveloping algebra of
60:of Lie groups and Lie algebras. For example,
15314:
15296:
14021:{\displaystyle \operatorname {ad} _{x}(z)=0}
11300:{\displaystyle {\mbox{ad}}_{\mathfrak {g}}.}
9538:{\displaystyle w(p\star q)=w(p)\otimes w(q)}
7790:In this way, one is lead immediately to the
5833:is of particular importance in this regard.
13688:{\displaystyle x=\sum _{i=1}^{d}x_{i}e_{i}}
13628:By linearity, if one expands in the basis,
13354:of the algebra. In general, there are only
11400:{\displaystyle {\mbox{ad}}_{\mathfrak {g}}}
11024:form a distinguished basis from the center
8095:also form a filtered algebra; its limit is
6501:. That is, the underlying vector spaces of
3742:{\displaystyle I\subset T({\mathfrak {g}})}
3105:The result of this lifting is explicitly a
342:
15880:of continuous complex-valued functions on
14711:
14710:
13618:{\displaystyle p_{n}(kx)=k^{d-n}p_{n}(x).}
12675:
12674:
12575:
12574:
12527:
12526:
12485:
12484:
12296:is a completely symmetric tensor of order
10794:can be identified with the centralizer of
8869:. (And, at this point, the isomorphism of
3749:is a subspace. It is an ideal, in that if
2185:compatible with the original Lie algebra.
16668:Lie Superalgebras and Enveloping Algebras
16443:
16410:
15813:The algebra of differential operators in
15292:
14453:{\displaystyle \kappa ^{ij}=\delta ^{ij}}
13514:
11752:
10117:correspond in a one-to-one manner to the
9642:is not trivially, inherently a member of
9302:. The resulting polynomial is called the
7046:of this filtration is the tensor algebra
6477:{\displaystyle a\leq b\leq \cdots \leq c}
2406:
2402:
2368:
2364:
2340:
2336:
2328:
2324:
16618:
16495:
16483:
15585:is the Lie algebra corresponding to the
10084:
10049:The universal enveloping algebra of the
9477:act as an isomorphism, that is, so that
9165:by an indeterminate, commuting variable
5700:extends uniquely to a representation of
4148:{\displaystyle a\times b\mapsto m(a,b).}
3810:{\displaystyle x\in T({\mathfrak {g}}),}
2054:of the tensor algebra quotiented by the
910:which satisfy the commutation relations
746:Consider, for example, the Lie algebra
110:
16561:
16507:
16319:, and then performing what is called a
16306:{\displaystyle U^{*}({\mathfrak {g}}).}
15665:as first-order differential operators.
15558:, with one variable per basis element.
13976:{\displaystyle z\in Z({\mathfrak {g}})}
13194:{\displaystyle \operatorname {ad} _{x}}
11066:. These may be constructed as follows.
9071:{\displaystyle \star ({\mathfrak {g}})}
8021:is isomorphic to the symmetric algebra
5915:
5796:. These operators (from the center of
5646:{\displaystyle x\in U({\mathfrak {g}})}
4042:Superalgebras and other generalizations
1388:{\displaystyle 1,e,e^{2},e^{3},\ldots }
16723:
16664:
15924:{\displaystyle \varphi ,\psi \in C(G)}
15661:lying inside it as the left-invariant
14741:Example: Pseudo-differential operators
14414:The quadratic term can be read off as
14079:{\displaystyle Z(U({\mathfrak {g}})).}
11264:{\displaystyle ={\mbox{ad}}_{x}(z)=0.}
7712:{\displaystyle U_{m-1}{\mathfrak {g}}}
7635:{\displaystyle U_{m-1}{\mathfrak {g}}}
7527:{\displaystyle U_{m-1}{\mathfrak {g}}}
5253:extends to an algebra homomorphism of
5118:is a linear map into a unital algebra
4857:{\displaystyle X,Y\in {\mathfrak {g}}}
4673:. Suppose we have any Lie algebra map
3449:{\displaystyle a\otimes b-b\otimes a-}
3318:{\displaystyle a\otimes b-b\otimes a=}
2811:{\displaystyle T^{n}({\mathfrak {g}})}
2708:{\displaystyle a\otimes b-b\otimes a=}
1899:matrices, the universal enveloping of
16741:Representation theory of Lie algebras
15872:, one can construct the vector space
13337:{\displaystyle x\in {\mathfrak {g}}.}
12946:. This may be seen as follows. For a
12916:{\displaystyle Z(U({\mathfrak {g}}))}
11946:{\displaystyle x\in {\mathfrak {g}},}
11200:{\displaystyle x\in {\mathfrak {g}};}
11104:{\displaystyle Z(U({\mathfrak {g}}))}
11059:{\displaystyle Z(U({\mathfrak {g}}))}
10889:{\displaystyle Z(U({\mathfrak {g}}))}
10754:{\displaystyle Z(U({\mathfrak {g}}))}
9723:A closed form expression is given by
8934:
8478:at the identity, then each vector in
8386:is a real Lie group with Lie algebra
8362:Left-invariant differential operators
7865:. Its limit is the symmetric algebra
6024:{\displaystyle e_{a}(b)=\delta _{ab}}
5607:{\displaystyle {\widehat {\varphi }}}
5386:{\displaystyle {\widehat {\varphi }}}
4615:
2221:. Thus, one is free to construct the
105:left-invariant differential operators
16600:
16470:
16458:
16376:
16364:
16325:quantum universal enveloping algebra
14594:and explicit computation shows that
14253:{\displaystyle xL_{1}+yL_{2}+zL_{3}}
12694:{\displaystyle =f_{ij}^{\;\;k}e_{k}}
12345:{\displaystyle \kappa ^{ab\cdots c}}
12289:{\displaystyle \kappa ^{ab\cdots c}}
11480:This implies that both of these are
10705:
8088:{\displaystyle G_{m}{\mathfrak {g}}}
7821:{\displaystyle S_{m}{\mathfrak {g}}}
7780:{\displaystyle G_{m}{\mathfrak {g}}}
7746:{\displaystyle G_{m}{\mathfrak {g}}}
7672:{\displaystyle U_{m}{\mathfrak {g}}}
7595:{\displaystyle U_{m}{\mathfrak {g}}}
7561:{\displaystyle U_{m}{\mathfrak {g}}}
7487:{\displaystyle U_{m}{\mathfrak {g}}}
7450:{\displaystyle U^{m}{\mathfrak {g}}}
7412:{\displaystyle G_{m}{\mathfrak {g}}}
7378:{\displaystyle U_{n}{\mathfrak {g}}}
7344:{\displaystyle U_{m}{\mathfrak {g}}}
7184:{\displaystyle U_{m}{\mathfrak {g}}}
6939:{\displaystyle T_{m}{\mathfrak {g}}}
6643:{\displaystyle U_{m}{\mathfrak {g}}}
3018:{\displaystyle =\otimes c+b\otimes }
2926:{\displaystyle =a\otimes +\otimes b}
2188:
16292:
16252:
15766:
15720:
15677:
15646:
15609:
15570:
15530:
15500:
15466:
15428:
15271:
15268:
14928:
14925:
14914:
14827:
14793:
14760:
14062:
13965:
13926:
13326:
13153:
13134:
12902:
12370:
12046:{\displaystyle U({\mathfrak {g}}).}
12032:
11999:
11935:
11665:
11536:
11512:
11502:
11473:{\displaystyle U({\mathfrak {g}}).}
11459:
11426:
11391:
11367:{\displaystyle U({\mathfrak {g}}).}
11353:
11320:
11288:
11271:That is, they are in the kernel of
11189:
11156:
11090:
11045:
11001:
10985:{\displaystyle U({\mathfrak {g}}).}
10971:
10941:
10925:{\displaystyle U({\mathfrak {g}}),}
10911:
10875:
10847:{\displaystyle U({\mathfrak {g}}).}
10833:
10803:
10776:
10740:
10680:
10644:
10572:
10556:
10511:
10466:
10433:
10416:
10376:
10359:
10322:
10296:
10270:
10253:
10205:
10171:
10136:
10102:
9694:
9679:as needed) to obtain an element of
9657:
9575:
9382:
9363:
9321:
9120:
9060:
9042:. This leads to the concept of the
9020:
8987:
8954:
8913:
8851:
8789:
8730:
8673:
8643:
8512:
8487:
8443:
8419:
8395:
8341:
8331:{\displaystyle G({\mathfrak {g}}),}
8317:
8284:
8241:
8222:
8200:{\displaystyle U({\mathfrak {g}}).}
8186:
8153:
8124:{\displaystyle G({\mathfrak {g}}).}
8110:
8080:
8036:
8003:
7935:
7916:
7880:
7850:
7813:
7772:
7738:
7704:
7664:
7627:
7587:
7553:
7519:
7479:
7442:
7404:
7370:
7336:
7299:
7271:
7251:
7220:{\displaystyle U({\mathfrak {g}}).}
7206:
7176:
7150:{\displaystyle U({\mathfrak {g}}).}
7136:
7103:
7075:{\displaystyle T({\mathfrak {g}}).}
7061:
7014:
6988:
6968:
6931:
6894:
6861:
6845:
6835:
6812:
6772:
6746:
6726:
6710:
6679:{\displaystyle U({\mathfrak {g}}).}
6665:
6635:
6578:
6551:
6516:
6363:
6324:
6098:
6082:
5886:
5811:
5769:{\displaystyle A=\mathrm {End} (V)}
5715:
5665:
5635:
5570:
5542:
5349:
5322:
5283:
5097:
5056:
5009:
4993:
4904:
4849:
4691:
4651:
4635:
4012:
3993:
3940:
3921:
3796:
3731:
3686:
3513:
3494:
3484:
3471:
3344:
3214:
3195:
3159:
3132:
3080:
3058:{\displaystyle T({\mathfrak {g}});}
3044:
2800:
2763:
2753:
2740:
2641:
2631:
2621:
2587:
2568:
2549:
2507:
2497:
2487:
2394:
2384:
2374:
2356:
2346:
2331:
2310:
2274:
2239:
2202:
2166:
1932:
1908:
1858:
1788:{\displaystyle x_{1},\ldots ,x_{n}}
728:
356:
324:
276:
172:
124:
13:
16575:, vol. 11, Providence, R.I.:
16263:{\displaystyle U({\mathfrak {g}})}
16027:
15944:
15777:{\displaystyle U({\mathfrak {g}})}
15731:{\displaystyle Z({\mathfrak {g}})}
15668:To relate the above two cases: if
15620:{\displaystyle U({\mathfrak {g}})}
15541:{\displaystyle U({\mathfrak {g}})}
15477:{\displaystyle U({\mathfrak {g}})}
14804:{\displaystyle U({\mathfrak {g}})}
14771:{\displaystyle U({\mathfrak {g}})}
13937:{\displaystyle Z({\mathfrak {g}})}
13116:-dimensional, the linear operator
12619:where the structure constants are
12010:{\displaystyle T({\mathfrak {g}})}
11437:{\displaystyle T({\mathfrak {g}})}
11374:The easiest route is to note that
10787:{\displaystyle U({\mathfrak {g}})}
10691:{\displaystyle U({\mathfrak {g}})}
10216:{\displaystyle U({\mathfrak {g}})}
10147:{\displaystyle U({\mathfrak {g}})}
9828:
9824:
9810:
9806:
9716:. It follows essentially from the
9705:{\displaystyle U({\mathfrak {g}})}
9668:{\displaystyle U({\mathfrak {g}})}
9332:{\displaystyle S({\mathfrak {g}})}
9131:{\displaystyle S({\mathfrak {g}})}
9031:{\displaystyle S({\mathfrak {g}})}
8998:{\displaystyle U({\mathfrak {g}})}
8965:{\displaystyle S({\mathfrak {g}})}
8924:{\displaystyle U({\mathfrak {g}})}
8862:{\displaystyle U({\mathfrak {g}})}
8800:{\displaystyle U({\mathfrak {g}})}
8654:{\displaystyle U({\mathfrak {g}})}
8263:. To emphasize this, the notation
8164:{\displaystyle G({\mathfrak {g}})}
8047:{\displaystyle S({\mathfrak {g}})}
8014:{\displaystyle G({\mathfrak {g}})}
7891:{\displaystyle S({\mathfrak {g}})}
7114:{\displaystyle T({\mathfrak {g}})}
6607:
6600:, a special case of which are the
6527:{\displaystyle U({\mathfrak {g}})}
6374:{\displaystyle U({\mathfrak {g}})}
6335:{\displaystyle U({\mathfrak {g}})}
5897:{\displaystyle U({\mathfrak {g}})}
5822:{\displaystyle U({\mathfrak {g}})}
5753:
5750:
5747:
5726:{\displaystyle U({\mathfrak {g}})}
5333:{\displaystyle U({\mathfrak {g}})}
3397:generated by elements of the form
3355:{\displaystyle U({\mathfrak {g}})}
3143:{\displaystyle U({\mathfrak {g}})}
3091:{\displaystyle T({\mathfrak {g}})}
2292:That is, one constructs the space
2261:: it simply contains all possible
2250:{\displaystyle T({\mathfrak {g}})}
2114:
1398:
592:{\displaystyle x_{1},\ldots x_{n}}
407:{\displaystyle X_{1},\ldots X_{n}}
300:
252:
228:
148:
14:
16752:
14949:, then it has a basis of matrices
13698:then the polynomial has the form
12260:terms in the tensor product, and
5836:
2257:from it. The tensor algebra is a
84:. This dual can be shown, by the
15931:, one defines multiplication as
15838:Hopf algebras and quantum groups
15447:(that is, the bracket is always
14857:If the action of the algebra is
14811:. Thus, one is led to a ring of
11331:{\displaystyle {\mathfrak {g}};}
9718:BakerâCampbellâHausdorff formula
9635:{\displaystyle w(p)\otimes w(q)}
9306:of the corresponding element of
6905:{\displaystyle {\mathfrak {g}}.}
6589:{\displaystyle {\mathfrak {g}}.}
6562:{\displaystyle {\mathfrak {g}}.}
4715:to a unital associative algebra
4559:: since the tensor algebra is a
4098:{\displaystyle m:V\times V\to V}
3877:{\displaystyle x\otimes j\in I.}
3459:This generator is an element of
2061:containing elements of the form
267:is generated by the elements of
16573:Graduate Studies in Mathematics
16539:
15748:of a semi-simple Lie algebra).
15685:{\displaystyle {\mathfrak {g}}}
15654:{\displaystyle {\mathfrak {g}}}
15578:{\displaystyle {\mathfrak {g}}}
15508:{\displaystyle {\mathfrak {g}}}
15436:{\displaystyle {\mathfrak {g}}}
14835:{\displaystyle {\mathfrak {g}}}
13944:corresponded to those elements
13542:in the above, one obtains that
12964:completely antisymmetric tensor
11977:{\displaystyle {\mbox{ad}}_{x}}
11913:{\displaystyle {\mbox{ad}}_{x}}
11673:{\displaystyle {\mathfrak {g}}}
11544:{\displaystyle {\mathfrak {g}}}
11174:that commute with all elements
11009:{\displaystyle {\mathfrak {g}}}
10949:{\displaystyle {\mathfrak {g}}}
10811:{\displaystyle {\mathfrak {g}}}
10652:{\displaystyle {\mathfrak {g}}}
10179:{\displaystyle {\mathfrak {g}}}
10110:{\displaystyle {\mathfrak {g}}}
8939:The underlying vector space of
8738:{\displaystyle {\mathfrak {g}}}
8681:{\displaystyle {\mathfrak {g}}}
8520:{\displaystyle {\mathfrak {g}}}
8495:{\displaystyle {\mathfrak {g}}}
8451:{\displaystyle {\mathfrak {g}}}
8427:{\displaystyle {\mathfrak {g}}}
8403:{\displaystyle {\mathfrak {g}}}
7719:, the quotienting that defines
6143:, one defines the extension of
5673:{\displaystyle {\mathfrak {g}}}
5578:{\displaystyle {\mathfrak {g}}}
5519:
5357:{\displaystyle {\mathfrak {g}}}
5064:{\displaystyle {\mathfrak {g}}}
5027:is the canonical map. (The map
3842:{\displaystyle j\otimes x\in I}
3167:{\displaystyle {\mathfrak {g}}}
2282:{\displaystyle {\mathfrak {g}}}
2210:{\displaystyle {\mathfrak {g}}}
2174:{\displaystyle {\mathfrak {g}}}
1940:{\displaystyle {\mathfrak {g}}}
1916:{\displaystyle {\mathfrak {g}}}
1866:{\displaystyle {\mathfrak {g}}}
1133:
1102:
845:
802:
736:{\displaystyle {\mathfrak {g}}}
364:{\displaystyle {\mathfrak {g}}}
332:{\displaystyle {\mathfrak {g}}}
284:{\displaystyle {\mathfrak {g}}}
180:{\displaystyle {\mathfrak {g}}}
132:{\displaystyle {\mathfrak {g}}}
16501:
16489:
16476:
16464:
16452:
16445:10.1016/j.jalgebra.2004.09.038
16428:"An envelope for Bol algebras"
16426:Perez-Izquierdo, J.M. (2005).
16419:
16382:
16370:
16358:
16317:quasi-triangular Hopf algebras
16297:
16287:
16257:
16247:
16200:
16184:
16175:
16169:
16166:
16163:
16157:
16151:
16125:
16119:
16110:
16104:
16072:
16063:
16054:
16042:
16039:
16036:
16030:
16024:
15998:
15992:
15986:
15980:
15971:
15965:
15962:
15959:
15947:
15941:
15918:
15912:
15771:
15761:
15725:
15715:
15614:
15604:
15535:
15525:
15471:
15461:
15397:
15319:
15282:
15262:
15224:
15212:
15183:
15171:
15139:
15127:
14798:
14788:
14765:
14755:
14691:
14665:
14630:
14604:
14485:
14479:
14395:
14356:
14070:
14067:
14057:
14051:
14009:
14003:
13970:
13960:
13931:
13921:
13724:
13718:
13609:
13603:
13571:
13562:
13434:
13428:
13389:
13383:
13285:
13279:
13242:
13220:
13158:
13148:
13139:
13040:
13025:
12910:
12907:
12897:
12891:
12866:
12860:
12725:
12719:
12655:
12629:
12442:
12437:
12431:
12417:
12158:
12152:
12117:, a Casimir operator of order
12037:
12027:
12004:
11994:
11866:
11860:
11808:
11802:
11762:
11756:
11744:
11720:
11640:
11637:
11631:
11619:
11613:
11604:
11598:
11592:
11586:
11583:
11571:
11568:
11507:
11464:
11454:
11431:
11421:
11358:
11348:
11252:
11246:
11226:
11214:
11161:
11151:
11098:
11095:
11085:
11079:
11053:
11050:
11040:
11034:
10976:
10966:
10916:
10906:
10883:
10880:
10870:
10864:
10838:
10828:
10781:
10771:
10748:
10745:
10735:
10729:
10685:
10675:
10577:
10567:
10561:
10522:
10505:
10477:
10460:
10444:
10410:
10333:
10316:
10307:
10290:
10281:
10247:
10210:
10200:
10154:. In more abstract terms, the
10141:
10131:
9932:
9920:
9868:
9862:
9856:
9850:
9757:
9751:
9742:
9736:
9699:
9689:
9662:
9652:
9629:
9623:
9614:
9608:
9580:
9570:
9532:
9526:
9517:
9511:
9502:
9490:
9387:
9377:
9371:
9368:
9358:
9326:
9316:
9215:
9202:
9125:
9115:
9082:, endowed with a product, the
9065:
9055:
9025:
9015:
8992:
8982:
8959:
8949:
8918:
8908:
8885:
8879:
8856:
8846:
8823:
8817:
8794:
8784:
8761:
8755:
8708:
8702:
8690:PoincarĂ©âBirkhoffâWitt theorem
8648:
8638:
8615:
8609:
8586:
8580:
8322:
8312:
8289:
8279:
8246:
8236:
8230:
8227:
8217:
8191:
8181:
8158:
8148:
8115:
8105:
8041:
8031:
8008:
7998:
7972:
7948:
7940:
7930:
7921:
7911:
7885:
7875:
7211:
7201:
7141:
7131:
7108:
7098:
7066:
7056:
6670:
6660:
6521:
6511:
6368:
6358:
6329:
6319:
6293:
6280:
6265:
6252:
6243:
6230:
6221:
6176:
6103:
6093:
6087:
6002:
5996:
5963:
5891:
5881:
5867:PoincarĂ©âBirkhoffâWitt theorem
5861:PoincarĂ©âBirkhoffâWitt theorem
5816:
5806:
5763:
5757:
5720:
5710:
5640:
5630:
5513:
5493:
5484:
5464:
5455:
5415:
5327:
5317:
5291:
5288:
5278:
5220:
5214:
5208:
5202:
5193:
5187:
5181:
5175:
5166:
5163:
5151:
5148:
5102:
5014:
5004:
4998:
4912:
4909:
4899:
4812:
4806:
4800:
4794:
4785:
4779:
4773:
4767:
4758:
4755:
4743:
4740:
4696:
4656:
4646:
4640:
4539:
4527:
4506:
4494:
4485:
4467:
4432:
4420:
4411:
4399:
4384:
4366:
4325:
4313:
4307:
4284:
4194:
4188:
4139:
4127:
4121:
4089:
4017:
4007:
3998:
3988:
3956:
3945:
3935:
3929:
3926:
3916:
3910:
3904:
3801:
3791:
3736:
3726:
3610:
3607:
3595:
3568:
3534:A general member of the ideal
3518:
3508:
3499:
3479:
3443:
3431:
3349:
3339:
3312:
3300:
3219:
3209:
3200:
3190:
3137:
3127:
3085:
3075:
3049:
3039:
3012:
3000:
2982:
2970:
2964:
2946:
2914:
2902:
2896:
2884:
2872:
2854:
2805:
2795:
2745:
2735:
2702:
2690:
2636:
2592:
2582:
2576:
2573:
2563:
2554:
2544:
2502:
2399:
2369:
2361:
2341:
2315:
2305:
2244:
2234:
2193:Recall that every Lie algebra
2036:
1744:PoincarĂ©âBirkhoffâWitt theorem
1014:
1002:
970:
958:
929:
917:
487:
461:
308:{\displaystyle {\mathcal {A}}}
260:{\displaystyle {\mathcal {A}}}
236:{\displaystyle {\mathcal {A}}}
187:corresponds to the commutator
156:{\displaystyle {\mathcal {A}}}
1:
16673:American Mathematical Society
16625:American Mathematical Society
16577:American Mathematical Society
16403:10.1016/s0021-8693(03)00389-2
16351:
14813:pseudo-differential operators
14260:and direct computation gives
14089:Example: Rotation group SO(3)
12989:{\displaystyle V^{\otimes d}}
12841:{\displaystyle \kappa _{ij}.}
12812:is the inverse matrix of the
10662:infinitesimal transformations
10226:The representation theory of
9138:and replacing each generator
7828:of symmetric tensor products
7602:actually gives an element in
7423:the same as the leading term
2841:This is done recursively, by
16709:Universal enveloping algebra
14901:Examples in particular cases
14164:{\displaystyle e_{i}=L_{i},}
12805:{\displaystyle \kappa ^{ij}}
8692:. Specifically, the algebra
7351:modulo all of the subspaces
5972:{\displaystyle e_{a}:X\to K}
5908:on the Lie algebra, or in a
5340:is generated by elements of
5075:and then composing with the
4067:endowed with multiplication
3102:(here, the tensor algebra).
139:into an associative algebra
49:correspond precisely to the
32:universal enveloping algebra
7:
16619:Helgason, Sigurdur (2001),
16498:Chapter II, Proposition 1.9
16346:Harish-Chandra homomorphism
16334:
15548:can be identified with the
15416:as a non-commutative ring.
14867:pseudo-Riemannian manifolds
14861:, as would be the case for
11646:{\displaystyle \delta ()=+}
11016:. For a finite-dimensional
10635:In a typical context where
6348:onto the algebra. That is,
2778:; it must now be lifted to
2443:is the tensor product, and
2265:of all possible vectors in
1201:matrices. For example, the
100:and their representations.
10:
16757:
16237:is isomorphically dual to
12952:-dimensional vector space
12928:Harish-Chandra isomorphism
12848:That the Casimir operator
10712:Harish-Chandra isomorphism
10709:
9403:that replaces each symbol
8570:forms an algebra, denoted
8058:as a commutative algebra.
7038:More precisely, this is a
5924:basis, that is, it is the
5864:
5082:To put it differently, if
3111:unital associative algebra
2155:unital associative algebra
1525:{\displaystyle x_{1}x_{2}}
1488:{\displaystyle x_{2}x_{1}}
750:, spanned by the matrices
98:compact topological groups
18:
15484:is commutative; and if a
14462:angular momentum operator
13004:characteristic polynomial
10896:must commute with all of
8133:associated graded algebra
8054:, both as a vector space
6885:-times tensor product of
5831:quadratic Casimir element
5680:acting on a vector space
5047:is obtained by embedding
4610:left alternative algebras
4251:as before, starting with
2157:generated by elements of
2026:{\displaystyle 2\times 2}
1892:{\displaystyle n\times n}
1220:{\displaystyle 2\times 2}
1194:{\displaystyle 2\times 2}
1065:subject to the relations
599:subject to the relations
16014:and comultiplication as
15842:The construction of the
14652:after making use of the
13488:, ad is linear, so that
13440:{\displaystyle p_{n}(x)}
13395:{\displaystyle p_{n}(x)}
12086:of the Lie algebra. The
11920:is a derivation for any
10619:free associative algebra
8458:as the tangent space to
8346:The theorem, applied to
6060:{\displaystyle a,b\in X}
5246:{\displaystyle \varphi }
2467:of vector spaces. Here,
2436:{\displaystyle \otimes }
451:for this basis, so that
343:Generators and relations
78:non-commutative geometry
16665:Musson, Ian M. (2012),
16601:Hall, Brian C. (2015),
16221:GelfandâNaimark theorem
14891:ColemanâMandula theorem
13808:{\displaystyle \kappa }
13449:homogeneous polynomials
12933:
12874:{\displaystyle C_{(2)}}
10228:semisimple Lie algebras
9040:as associative algebras
8599:. It can be shown that
7229:Next, define the space
5847:special Jordan algebras
3362:can be understood as a
2456:{\displaystyle \oplus }
1822:'s with the generators
1708:{\displaystyle k_{j}=0}
1431:first, then factors of
1326:{\displaystyle e^{2}=0}
1273:{\displaystyle E^{2}=0}
440:{\displaystyle c_{ijk}}
86:GelfandâNaimark theorem
16307:
16264:
16210:
16132:
16082:
16005:
15925:
15778:
15732:
15686:
15655:
15629:differential operators
15621:
15579:
15542:
15515:has been chosen, then
15509:
15478:
15437:
15414:
15407:
15247:
15240:
15199:
15158:
15115:
15108:
14943:
14836:
14805:
14772:
14731:
14643:
14585:
14454:
14405:
14254:
14192:
14165:
14122:
14121:{\displaystyle A_{1}.}
14080:
14022:
13977:
13938:
13896:
13841:
13809:
13786:
13689:
13664:
13619:
13532:
13482:
13481:{\displaystyle k\in K}
13441:
13396:
13338:
13302:
13268:
13195:
13165:
13108:adjoint representation
13091:
13066:
12990:
12940:semisimple Lie algebra
12917:
12881:belongs to the center
12875:
12842:
12806:
12773:
12695:
12610:
12455:
12396:
12346:
12310:
12290:
12254:
12231:
12131:
12111:
12080:
12059:in the basis elements
12057:homogenous polynomials
12047:
12011:
11978:
11947:
11914:
11880:
11694:
11674:
11647:
11545:
11521:
11474:
11438:
11401:
11368:
11332:
11309:adjoint representation
11301:
11265:
11201:
11168:
11105:
11060:
11018:semisimple Lie algebra
11010:
10986:
10950:
10926:
10890:
10848:
10812:
10788:
10755:
10700:differential operators
10692:
10653:
10584:
10529:
10391:
10340:
10217:
10180:
10148:
10111:
10077:product is called the
10071:
10070:{\displaystyle \star }
10040:
10020:
9990:
9898:
9706:
9669:
9636:
9590:
9539:
9471:
9470:{\displaystyle \star }
9451:
9424:
9394:
9333:
9296:
9269:
9242:
9222:
9186:
9159:
9132:
9096:
9095:{\displaystyle \star }
9072:
9032:
8999:
8966:
8925:
8892:
8863:
8830:
8801:
8768:
8739:
8715:
8682:
8655:
8622:
8593:
8564:
8544:
8521:
8496:
8472:
8452:
8428:
8404:
8380:
8332:
8302:is sometimes used for
8296:
8253:
8201:
8165:
8125:
8089:
8048:
8015:
7979:
7892:
7859:
7822:
7781:
7747:
7713:
7673:
7636:
7596:
7562:
7528:
7488:
7451:
7413:
7379:
7345:
7308:
7221:
7185:
7151:
7115:
7086:natural transformation
7076:
7029:
6940:
6906:
6870:
6781:
6680:
6644:
6590:
6563:
6528:
6478:
6437:
6375:
6336:
6300:
6157:
6137:
6110:
6061:
6025:
5973:
5898:
5823:
5790:
5770:
5727:
5694:
5674:
5647:
5608:
5579:
5551:
5387:
5358:
5334:
5301:
5247:
5227:
5132:
5112:
5065:
5041:
5021:
4969:
4922:
4864:. Then there exists a
4858:
4819:
4706:
4663:
4546:
4445:
4336:
4241:
4221:
4201:
4172:
4149:
4099:
4061:
4032:
3966:
3878:
3843:
3811:
3769:
3768:{\displaystyle j\in I}
3743:
3698:
3632:
3525:
3450:
3356:
3319:
3260:
3234:
3168:
3144:
3092:
3059:
3019:
2927:
2835:
2812:
2772:
2709:
2650:
2599:
2516:
2457:
2437:
2414:
2283:
2251:
2211:
2175:
2147:
2027:
2001:
1981:
1961:
1941:
1917:
1893:
1867:
1843:
1816:
1789:
1736:
1709:
1676:
1646:
1557:
1526:
1489:
1452:
1425:
1389:
1327:
1294:
1274:
1241:
1221:
1195:
1162:
1059:
1027:
989:
945:
901:
737:
706:
675:
593:
547:
513:
441:
408:
365:
333:
309:
285:
261:
237:
213:
181:
157:
133:
21:ShermanâTakeda theorem
16:Concept in mathematics
16705:, Harvard University.
16486:Chapter II, Section 1
16308:
16265:
16211:
16133:
16083:
16006:
15926:
15886:, and turn it into a
15779:
15733:
15687:
15656:
15622:
15580:
15543:
15510:
15479:
15438:
15408:
15251:
15241:
15200:
15159:
15119:
15109:
14951:
14944:
14885:, giving rise to the
14837:
14806:
14773:
14732:
14644:
14586:
14460:, and so the squared
14455:
14406:
14255:
14193:
14191:{\displaystyle L_{i}}
14166:
14123:
14081:
14023:
13978:
13939:
13897:
13842:
13840:{\displaystyle m=d-n}
13810:
13787:
13690:
13644:
13620:
13540:plugging and chugging
13533:
13483:
13442:
13397:
13339:
13303:
13248:
13196:
13166:
13092:
13046:
12991:
12918:
12876:
12843:
12807:
12774:
12696:
12611:
12456:
12397:
12347:
12311:
12291:
12255:
12232:
12132:
12112:
12110:{\displaystyle e_{a}}
12081:
12079:{\displaystyle e_{a}}
12048:
12012:
11979:
11948:
11915:
11881:
11695:
11675:
11648:
11546:
11522:
11482:differential algebras
11475:
11439:
11402:
11369:
11333:
11302:
11266:
11202:
11169:
11106:
11061:
11011:
10987:
10951:
10927:
10891:
10849:
10813:
10789:
10756:
10693:
10654:
10617:is isomorphic to the
10585:
10530:
10392:
10341:
10218:
10181:
10149:
10112:
10085:Representation theory
10072:
10046:in the chosen basis.
10041:
10021:
10019:{\displaystyle m^{i}}
9991:
9899:
9707:
9670:
9637:
9591:
9540:
9472:
9452:
9450:{\displaystyle e_{i}}
9425:
9423:{\displaystyle t_{i}}
9395:
9339:. The inverse map is
9334:
9297:
9295:{\displaystyle e_{i}}
9270:
9268:{\displaystyle t_{i}}
9243:
9223:
9187:
9185:{\displaystyle t_{i}}
9160:
9158:{\displaystyle e_{i}}
9133:
9097:
9080:symmetric polynomials
9073:
9033:
9000:
8967:
8926:
8893:
8864:
8831:
8802:
8769:
8740:
8716:
8683:
8656:
8623:
8594:
8565:
8545:
8522:
8497:
8473:
8453:
8429:
8405:
8381:
8333:
8297:
8254:
8202:
8166:
8126:
8090:
8049:
8016:
7980:
7893:
7860:
7823:
7782:
7748:
7714:
7674:
7637:
7597:
7563:
7529:
7489:
7452:
7414:
7380:
7346:
7309:
7222:
7186:
7152:
7116:
7077:
7030:
6941:
6907:
6871:
6782:
6681:
6645:
6591:
6564:
6529:
6479:
6438:
6376:
6337:
6301:
6158:
6138:
6136:{\displaystyle e_{a}}
6111:
6062:
6026:
5974:
5899:
5824:
5791:
5771:
5728:
5695:
5675:
5648:
5609:
5580:
5552:
5388:
5359:
5335:
5302:
5248:
5228:
5133:
5113:
5066:
5042:
5022:
4970:
4923:
4859:
4820:
4721:(with Lie bracket in
4707:
4664:
4547:
4446:
4337:
4242:
4222:
4202:
4173:
4150:
4100:
4062:
4033:
3967:
3879:
3844:
3812:
3770:
3744:
3699:
3633:
3526:
3451:
3357:
3320:
3261:
3259:{\displaystyle \sim }
3235:
3169:
3145:
3093:
3060:
3020:
2928:
2836:
2813:
2773:
2710:
2651:
2600:
2517:
2458:
2438:
2415:
2284:
2252:
2212:
2176:
2148:
2028:
2002:
1982:
1962:
1942:
1918:
1894:
1873:may be an algebra of
1868:
1844:
1842:{\displaystyle X_{j}}
1817:
1815:{\displaystyle x_{j}}
1790:
1737:
1735:{\displaystyle x_{j}}
1710:
1677:
1675:{\displaystyle k_{j}}
1647:
1558:
1556:{\displaystyle x_{j}}
1527:
1490:
1453:
1451:{\displaystyle x_{2}}
1426:
1424:{\displaystyle x_{1}}
1390:
1328:
1295:
1275:
1242:
1222:
1196:
1163:
1060:
1058:{\displaystyle e,f,h}
1028:
990:
946:
902:
738:
707:
655:
594:
548:
493:
442:
409:
366:
334:
310:
286:
262:
238:
214:
212:{\displaystyle xy-yx}
182:
158:
134:
111:Informal construction
94:TannakaâKrein duality
58:representation theory
53:of that Lie algebra.
16341:MilnorâMoore theorem
16274:
16241:
16148:
16141:and the antipode as
16098:
16021:
15938:
15894:
15855:that turn them into
15755:
15709:
15672:
15641:
15598:
15565:
15519:
15495:
15455:
15423:
15256:
15209:
15168:
15157:{\displaystyle =-2E}
15124:
14956:
14909:
14883:stressâenergy tensor
14822:
14782:
14749:
14662:
14601:
14471:
14418:
14267:
14202:
14175:
14132:
14102:
14095:rotation group SO(3)
14045:
13987:
13948:
13915:
13851:
13819:
13815:is a tensor of rank
13799:
13705:
13635:
13549:
13492:
13466:
13415:
13402:is non-vanishing is
13370:
13315:
13214:
13178:
13123:
13019:
13002:, one may write the
12970:
12885:
12852:
12819:
12786:
12711:
12626:
12468:
12414:
12356:
12320:
12300:
12264:
12244:
12144:
12121:
12094:
12063:
12021:
11988:
11957:
11924:
11893:
11710:
11684:
11660:
11562:
11531:
11491:
11448:
11415:
11378:
11342:
11315:
11275:
11211:
11178:
11115:
11073:
11028:
10996:
10960:
10936:
10900:
10858:
10822:
10798:
10765:
10723:
10669:
10639:
10545:
10404:
10353:
10241:
10194:
10166:
10125:
10097:
10061:
10030:
10003:
9914:
9730:
9683:
9646:
9602:
9552:
9484:
9461:
9434:
9407:
9346:
9310:
9279:
9252:
9232:
9196:
9169:
9142:
9109:
9086:
9049:
9009:
8976:
8943:
8902:
8891:{\displaystyle D(G)}
8873:
8840:
8829:{\displaystyle D(G)}
8811:
8778:
8767:{\displaystyle D(G)}
8749:
8725:
8714:{\displaystyle D(G)}
8696:
8668:
8632:
8621:{\displaystyle D(G)}
8603:
8592:{\displaystyle D(G)}
8574:
8554:
8534:
8507:
8482:
8462:
8438:
8414:
8390:
8370:
8306:
8267:
8211:
8175:
8142:
8099:
8065:
8025:
7992:
7905:
7869:
7832:
7798:
7757:
7723:
7683:
7649:
7606:
7572:
7538:
7498:
7464:
7427:
7389:
7355:
7321:
7236:
7195:
7161:
7125:
7092:
7088:that takes one from
7050:
6957:
6916:
6889:
6797:
6695:
6654:
6620:
6573:
6546:
6505:
6450:
6388:
6352:
6313:
6170:
6147:
6120:
6071:
6039:
5983:
5944:
5916:Using basis elements
5875:
5800:
5780:
5737:
5704:
5684:
5660:
5618:
5589:
5565:
5400:
5368:
5344:
5311:
5257:
5237:
5142:
5122:
5086:
5051:
5031:
4982:
4938:
4878:
4870:algebra homomorphism
4832:
4734:
4680:
4624:
4595:Poisson superalgebra
4583:Gerstenhaber algebra
4461:
4360:
4258:
4231:
4211:
4200:{\displaystyle T(V)}
4182:
4162:
4109:
4105:that takes elements
4071:
4051:
3982:
3898:
3890:short exact sequence
3853:
3821:
3779:
3753:
3714:
3645:
3547:
3466:
3404:
3333:
3273:
3250:
3245:equivalence relation
3184:
3154:
3121:
3069:
3033:
2943:
2851:
2822:
2782:
2722:
2663:
2616:
2538:
2482:
2447:
2427:
2299:
2269:
2228:
2197:
2161:
2065:
2011:
1991:
1971:
1951:
1927:
1903:
1877:
1853:
1826:
1799:
1753:
1719:
1686:
1659:
1570:
1540:
1499:
1462:
1435:
1408:
1341:
1337:) that the elements
1304:
1284:
1251:
1231:
1205:
1179:
1072:
1037:
999:
988:{\displaystyle =-2F}
955:
914:
757:
723:
606:
560:
458:
418:
375:
351:
319:
295:
271:
247:
223:
191:
167:
143:
119:
70:differential algebra
16568:Enveloping algebras
16321:quantum deformation
15700:partial derivatives
15631:(of all orders) on
15239:{\displaystyle =-H}
15198:{\displaystyle =2F}
14818:If the Lie algebra
14716:
14654:structure constants
12680:
12580:
12532:
12490:
12404:structure constants
11527:is a derivation on
11207:that is, for which
9677:structure constants
8135:of the filtration.
6492:structure constants
4207:so that the lifted
3540:will have the form
2217:is in particular a
1641:
1616:
1594:
944:{\displaystyle =2E}
449:structure constants
16525:10.1007/bf01076082
16432:Journal of Algebra
16412:10338.dmlcz/140108
16391:Journal of Algebra
16303:
16260:
16206:
16128:
16078:
16001:
15921:
15866:Given a Lie group
15774:
15728:
15692:is a vector space
15682:
15651:
15617:
15575:
15538:
15505:
15474:
15433:
15403:
15236:
15195:
15154:
15104:
15098:
15048:
14998:
14939:
14832:
14801:
14768:
14727:
14697:
14642:{\displaystyle =0}
14639:
14581:
14450:
14401:
14250:
14188:
14161:
14118:
14076:
14018:
13973:
13934:
13892:
13837:
13805:
13782:
13685:
13615:
13528:
13478:
13437:
13392:
13334:
13298:
13191:
13161:
13087:
12986:
12958:, recall that the
12913:
12871:
12838:
12802:
12769:
12691:
12661:
12606:
12561:
12513:
12471:
12461:, it follows that
12454:{\displaystyle =0}
12451:
12392:
12342:
12306:
12286:
12250:
12227:
12127:
12107:
12088:Casimir invariants
12076:
12043:
12007:
11974:
11966:
11953:the above defines
11943:
11910:
11902:
11876:
11874:
11690:
11670:
11643:
11541:
11517:
11470:
11434:
11397:
11387:
11364:
11328:
11297:
11284:
11261:
11238:
11197:
11164:
11101:
11056:
11006:
10982:
10946:
10922:
10886:
10844:
10808:
10784:
10751:
10688:
10665:, the elements of
10649:
10580:
10525:
10387:
10336:
10213:
10176:
10144:
10107:
10067:
10051:Heisenberg algebra
10036:
10016:
9986:
9894:
9702:
9665:
9632:
9586:
9535:
9467:
9447:
9420:
9390:
9329:
9292:
9265:
9238:
9218:
9182:
9155:
9128:
9092:
9068:
9044:algebra of symbols
9028:
8995:
8962:
8935:Algebra of symbols
8921:
8888:
8859:
8826:
8797:
8764:
8735:
8711:
8678:
8651:
8618:
8589:
8560:
8540:
8517:
8492:
8468:
8448:
8424:
8400:
8376:
8328:
8292:
8249:
8197:
8161:
8121:
8085:
8044:
8011:
7975:
7888:
7855:
7818:
7777:
7743:
7709:
7669:
7632:
7592:
7558:
7524:
7484:
7447:
7409:
7375:
7341:
7317:This is the space
7304:
7217:
7181:
7147:
7111:
7072:
7025:
6936:
6902:
6866:
6777:
6676:
6640:
6586:
6559:
6524:
6474:
6433:
6371:
6332:
6296:
6153:
6133:
6106:
6057:
6033:indicator function
6021:
5969:
5894:
5819:
5786:
5766:
5723:
5690:
5670:
5643:
5604:
5575:
5547:
5383:
5354:
5330:
5297:
5243:
5223:
5128:
5108:
5061:
5037:
5017:
4965:
4918:
4854:
4815:
4702:
4671:universal property
4659:
4616:Universal property
4542:
4441:
4332:
4330:
4237:
4217:
4197:
4168:
4145:
4095:
4057:
4028:
3962:
3874:
3839:
3807:
3765:
3739:
3694:
3628:
3521:
3446:
3352:
3315:
3256:
3230:
3174:is defined as the
3164:
3140:
3088:
3055:
3015:
2923:
2834:{\displaystyle n.}
2831:
2808:
2768:
2705:
2646:
2595:
2530:and satisfies the
2512:
2453:
2433:
2410:
2279:
2247:
2207:
2171:
2143:
2023:
1997:
1977:
1957:
1937:
1913:
1889:
1863:
1839:
1812:
1785:
1732:
1705:
1672:
1642:
1620:
1595:
1573:
1553:
1534:linear combination
1522:
1485:
1448:
1421:
1385:
1323:
1290:
1270:
1237:
1217:
1191:
1158:
1055:
1026:{\displaystyle =H}
1023:
985:
941:
897:
885:
839:
796:
733:
717:no other relations
702:
589:
543:
437:
404:
361:
329:
305:
281:
257:
233:
209:
177:
153:
129:
16682:978-0-8218-6867-6
16642:978-0-8218-2848-9
16586:978-0-8218-0560-2
16545:Xavier Bekaert, "
16513:Funct. Anal. Appl
15853:comultiplications
15832:quadratic algebra
15401:
15060:
15010:
14895:Lie superalgebras
14887:Yang-Mills action
14848:elliptic operator
12996:. Given a matrix
12944:CartanâWeyl basis
12309:{\displaystyle m}
12253:{\displaystyle m}
12130:{\displaystyle m}
11965:
11901:
11693:{\displaystyle G}
11386:
11283:
11237:
11022:Casimir operators
10706:Casimir operators
10349:for Lie algebras
10232:Kronecker product
10039:{\displaystyle m}
9835:
9817:
9241:{\displaystyle K}
9221:{\displaystyle K}
8774:is a quotient of
8664:In the case that
8563:{\displaystyle G}
8543:{\displaystyle A}
8471:{\displaystyle G}
8379:{\displaystyle G}
8171:is isomorphic to
7792:symmetric algebra
6499:symmetric algebra
6156:{\displaystyle h}
5926:free vector space
5855:Lie superalgebras
5789:{\displaystyle V}
5693:{\displaystyle V}
5601:
5412:
5380:
5269:
5131:{\displaystyle A}
5040:{\displaystyle h}
4956:
4890:
4568:Lie superalgebras
4240:{\displaystyle m}
4220:{\displaystyle m}
4171:{\displaystyle m}
4060:{\displaystyle V}
2478:algebra is a map
2189:Formal definition
2000:{\displaystyle H}
1980:{\displaystyle F}
1960:{\displaystyle E}
1300:does not satisfy
1293:{\displaystyle x}
1240:{\displaystyle E}
893:
88:, to contain the
66:Casimir operators
16748:
16697:Shlomo Sternberg
16693:
16661:
16615:
16597:
16563:Dixmier, Jacques
16554:
16543:
16537:
16536:
16505:
16499:
16493:
16487:
16480:
16474:
16468:
16462:
16456:
16450:
16449:
16447:
16423:
16417:
16416:
16414:
16386:
16380:
16374:
16368:
16362:
16312:
16310:
16309:
16304:
16296:
16295:
16286:
16285:
16269:
16267:
16266:
16261:
16256:
16255:
16236:
16228:
16215:
16213:
16212:
16207:
16199:
16198:
16137:
16135:
16134:
16129:
16087:
16085:
16084:
16079:
16010:
16008:
16007:
16002:
15930:
15928:
15927:
15922:
15885:
15879:
15871:
15821:Heisenberg group
15818:
15809:
15803:
15797:identity element
15783:
15781:
15780:
15775:
15770:
15769:
15746:Casimir operator
15743:
15737:
15735:
15734:
15729:
15724:
15723:
15702:of first order.
15697:
15691:
15689:
15688:
15683:
15681:
15680:
15660:
15658:
15657:
15652:
15650:
15649:
15636:
15626:
15624:
15623:
15618:
15613:
15612:
15593:
15584:
15582:
15581:
15576:
15574:
15573:
15557:
15547:
15545:
15544:
15539:
15534:
15533:
15514:
15512:
15511:
15506:
15504:
15503:
15483:
15481:
15480:
15475:
15470:
15469:
15450:
15442:
15440:
15439:
15434:
15432:
15431:
15412:
15410:
15409:
15404:
15402:
15400:
15317:
15295:
15289:
15281:
15280:
15275:
15274:
15245:
15243:
15242:
15237:
15204:
15202:
15201:
15196:
15163:
15161:
15160:
15155:
15113:
15111:
15110:
15105:
15103:
15102:
15061:
15058:
15053:
15052:
15011:
15008:
15003:
15002:
14948:
14946:
14945:
14940:
14938:
14937:
14932:
14931:
14918:
14917:
14841:
14839:
14838:
14833:
14831:
14830:
14810:
14808:
14807:
14802:
14797:
14796:
14777:
14775:
14774:
14769:
14764:
14763:
14736:
14734:
14733:
14728:
14726:
14725:
14715:
14708:
14690:
14689:
14677:
14676:
14648:
14646:
14645:
14640:
14629:
14628:
14616:
14615:
14590:
14588:
14587:
14582:
14580:
14579:
14567:
14566:
14554:
14553:
14541:
14540:
14528:
14527:
14515:
14514:
14502:
14501:
14489:
14488:
14459:
14457:
14456:
14451:
14449:
14448:
14433:
14432:
14410:
14408:
14407:
14402:
14394:
14393:
14381:
14380:
14368:
14367:
14352:
14351:
14336:
14332:
14322:
14321:
14306:
14305:
14290:
14289:
14259:
14257:
14256:
14251:
14249:
14248:
14233:
14232:
14217:
14216:
14197:
14195:
14194:
14189:
14187:
14186:
14170:
14168:
14167:
14162:
14157:
14156:
14144:
14143:
14127:
14125:
14124:
14119:
14114:
14113:
14085:
14083:
14082:
14077:
14066:
14065:
14040:
14034:
14027:
14025:
14024:
14019:
13999:
13998:
13982:
13980:
13979:
13974:
13969:
13968:
13943:
13941:
13940:
13935:
13930:
13929:
13908:
13901:
13899:
13898:
13893:
13888:
13887:
13869:
13868:
13846:
13844:
13843:
13838:
13814:
13812:
13811:
13806:
13791:
13789:
13788:
13783:
13781:
13780:
13762:
13761:
13749:
13748:
13739:
13738:
13717:
13716:
13694:
13692:
13691:
13686:
13684:
13683:
13674:
13673:
13663:
13658:
13624:
13622:
13621:
13616:
13602:
13601:
13592:
13591:
13561:
13560:
13537:
13535:
13534:
13529:
13524:
13523:
13507:
13506:
13487:
13485:
13484:
13479:
13461:
13446:
13444:
13443:
13438:
13427:
13426:
13408:
13401:
13399:
13398:
13393:
13382:
13381:
13365:
13359:
13349:
13343:
13341:
13340:
13335:
13330:
13329:
13307:
13305:
13304:
13299:
13297:
13296:
13278:
13277:
13267:
13262:
13241:
13240:
13206:
13200:
13198:
13197:
13192:
13190:
13189:
13170:
13168:
13167:
13162:
13157:
13156:
13138:
13137:
13115:
13105:
13096:
13094:
13093:
13088:
13086:
13085:
13076:
13075:
13065:
13060:
13011:
13001:
12995:
12993:
12992:
12987:
12985:
12984:
12957:
12951:
12922:
12920:
12919:
12914:
12906:
12905:
12880:
12878:
12877:
12872:
12870:
12869:
12847:
12845:
12844:
12839:
12834:
12833:
12811:
12809:
12808:
12803:
12801:
12800:
12778:
12776:
12775:
12770:
12768:
12767:
12755:
12754:
12745:
12744:
12729:
12728:
12700:
12698:
12697:
12692:
12690:
12689:
12679:
12672:
12654:
12653:
12641:
12640:
12615:
12613:
12612:
12607:
12599:
12598:
12579:
12572:
12551:
12550:
12531:
12524:
12509:
12508:
12489:
12482:
12460:
12458:
12457:
12452:
12441:
12440:
12410:. That is, from
12401:
12399:
12398:
12393:
12388:
12387:
12379:
12375:
12374:
12373:
12351:
12349:
12348:
12343:
12341:
12340:
12315:
12313:
12312:
12307:
12295:
12293:
12292:
12287:
12285:
12284:
12259:
12257:
12256:
12251:
12240:where there are
12236:
12234:
12233:
12228:
12226:
12225:
12207:
12206:
12194:
12193:
12184:
12183:
12162:
12161:
12136:
12134:
12133:
12128:
12116:
12114:
12113:
12108:
12106:
12105:
12085:
12083:
12082:
12077:
12075:
12074:
12052:
12050:
12049:
12044:
12036:
12035:
12016:
12014:
12013:
12008:
12003:
12002:
11983:
11981:
11980:
11975:
11973:
11972:
11967:
11963:
11952:
11950:
11949:
11944:
11939:
11938:
11919:
11917:
11916:
11911:
11909:
11908:
11903:
11899:
11885:
11883:
11882:
11877:
11875:
11826:
11786:
11699:
11697:
11696:
11691:
11679:
11677:
11676:
11671:
11669:
11668:
11652:
11650:
11649:
11644:
11550:
11548:
11547:
11542:
11540:
11539:
11526:
11524:
11523:
11518:
11516:
11515:
11506:
11505:
11479:
11477:
11476:
11471:
11463:
11462:
11443:
11441:
11440:
11435:
11430:
11429:
11406:
11404:
11403:
11398:
11396:
11395:
11394:
11388:
11384:
11373:
11371:
11370:
11365:
11357:
11356:
11337:
11335:
11334:
11329:
11324:
11323:
11306:
11304:
11303:
11298:
11293:
11292:
11291:
11285:
11281:
11270:
11268:
11267:
11262:
11245:
11244:
11239:
11235:
11206:
11204:
11203:
11198:
11193:
11192:
11173:
11171:
11170:
11165:
11160:
11159:
11110:
11108:
11107:
11102:
11094:
11093:
11065:
11063:
11062:
11057:
11049:
11048:
11015:
11013:
11012:
11007:
11005:
11004:
10991:
10989:
10988:
10983:
10975:
10974:
10955:
10953:
10952:
10947:
10945:
10944:
10931:
10929:
10928:
10923:
10915:
10914:
10895:
10893:
10892:
10887:
10879:
10878:
10853:
10851:
10850:
10845:
10837:
10836:
10817:
10815:
10814:
10809:
10807:
10806:
10793:
10791:
10790:
10785:
10780:
10779:
10760:
10758:
10757:
10752:
10744:
10743:
10697:
10695:
10694:
10689:
10684:
10683:
10658:
10656:
10655:
10650:
10648:
10647:
10615:free Lie algebra
10589:
10587:
10586:
10581:
10576:
10575:
10560:
10559:
10534:
10532:
10531:
10526:
10521:
10520:
10515:
10514:
10504:
10503:
10476:
10475:
10470:
10469:
10459:
10458:
10443:
10442:
10437:
10436:
10426:
10425:
10420:
10419:
10396:
10394:
10393:
10388:
10386:
10385:
10380:
10379:
10369:
10368:
10363:
10362:
10345:
10343:
10342:
10337:
10332:
10331:
10326:
10325:
10306:
10305:
10300:
10299:
10280:
10279:
10274:
10273:
10263:
10262:
10257:
10256:
10222:
10220:
10219:
10214:
10209:
10208:
10185:
10183:
10182:
10177:
10175:
10174:
10156:abelian category
10153:
10151:
10150:
10145:
10140:
10139:
10116:
10114:
10113:
10108:
10106:
10105:
10076:
10074:
10073:
10068:
10045:
10043:
10042:
10037:
10025:
10023:
10022:
10017:
10015:
10014:
9995:
9993:
9992:
9987:
9973:
9969:
9968:
9967:
9958:
9957:
9903:
9901:
9900:
9895:
9893:
9892:
9875:
9871:
9846:
9842:
9841:
9837:
9836:
9834:
9823:
9818:
9816:
9805:
9798:
9797:
9788:
9787:
9711:
9709:
9708:
9703:
9698:
9697:
9674:
9672:
9671:
9666:
9661:
9660:
9641:
9639:
9638:
9633:
9595:
9593:
9592:
9587:
9579:
9578:
9548:for polynomials
9544:
9542:
9541:
9536:
9476:
9474:
9473:
9468:
9456:
9454:
9453:
9448:
9446:
9445:
9429:
9427:
9426:
9421:
9419:
9418:
9399:
9397:
9396:
9391:
9386:
9385:
9367:
9366:
9338:
9336:
9335:
9330:
9325:
9324:
9301:
9299:
9298:
9293:
9291:
9290:
9274:
9272:
9271:
9266:
9264:
9263:
9247:
9245:
9244:
9239:
9227:
9225:
9224:
9219:
9214:
9213:
9191:
9189:
9188:
9183:
9181:
9180:
9164:
9162:
9161:
9156:
9154:
9153:
9137:
9135:
9134:
9129:
9124:
9123:
9101:
9099:
9098:
9093:
9077:
9075:
9074:
9069:
9064:
9063:
9037:
9035:
9034:
9029:
9024:
9023:
9004:
9002:
9001:
8996:
8991:
8990:
8971:
8969:
8968:
8963:
8958:
8957:
8930:
8928:
8927:
8922:
8917:
8916:
8897:
8895:
8894:
8889:
8868:
8866:
8865:
8860:
8855:
8854:
8835:
8833:
8832:
8827:
8806:
8804:
8803:
8798:
8793:
8792:
8773:
8771:
8770:
8765:
8744:
8742:
8741:
8736:
8734:
8733:
8720:
8718:
8717:
8712:
8687:
8685:
8684:
8679:
8677:
8676:
8660:
8658:
8657:
8652:
8647:
8646:
8627:
8625:
8624:
8619:
8598:
8596:
8595:
8590:
8569:
8567:
8566:
8561:
8549:
8547:
8546:
8541:
8526:
8524:
8523:
8518:
8516:
8515:
8501:
8499:
8498:
8493:
8491:
8490:
8477:
8475:
8474:
8469:
8457:
8455:
8454:
8449:
8447:
8446:
8433:
8431:
8430:
8425:
8423:
8422:
8409:
8407:
8406:
8401:
8399:
8398:
8385:
8383:
8382:
8377:
8352:exterior algebra
8337:
8335:
8334:
8329:
8321:
8320:
8301:
8299:
8298:
8293:
8288:
8287:
8261:filtered algebra
8258:
8256:
8255:
8250:
8245:
8244:
8226:
8225:
8206:
8204:
8203:
8198:
8190:
8189:
8170:
8168:
8167:
8162:
8157:
8156:
8130:
8128:
8127:
8122:
8114:
8113:
8094:
8092:
8091:
8086:
8084:
8083:
8077:
8076:
8053:
8051:
8050:
8045:
8040:
8039:
8020:
8018:
8017:
8012:
8007:
8006:
7984:
7982:
7981:
7976:
7947:
7939:
7938:
7920:
7919:
7897:
7895:
7894:
7889:
7884:
7883:
7864:
7862:
7861:
7856:
7854:
7853:
7844:
7843:
7827:
7825:
7824:
7819:
7817:
7816:
7810:
7809:
7786:
7784:
7783:
7778:
7776:
7775:
7769:
7768:
7752:
7750:
7749:
7744:
7742:
7741:
7735:
7734:
7718:
7716:
7715:
7710:
7708:
7707:
7701:
7700:
7678:
7676:
7675:
7670:
7668:
7667:
7661:
7660:
7641:
7639:
7638:
7633:
7631:
7630:
7624:
7623:
7601:
7599:
7598:
7593:
7591:
7590:
7584:
7583:
7567:
7565:
7564:
7559:
7557:
7556:
7550:
7549:
7533:
7531:
7530:
7525:
7523:
7522:
7516:
7515:
7493:
7491:
7490:
7485:
7483:
7482:
7476:
7475:
7456:
7454:
7453:
7448:
7446:
7445:
7439:
7438:
7418:
7416:
7415:
7410:
7408:
7407:
7401:
7400:
7384:
7382:
7381:
7376:
7374:
7373:
7367:
7366:
7350:
7348:
7347:
7342:
7340:
7339:
7333:
7332:
7313:
7311:
7310:
7305:
7303:
7302:
7296:
7295:
7280:
7275:
7274:
7268:
7267:
7255:
7254:
7248:
7247:
7226:
7224:
7223:
7218:
7210:
7209:
7190:
7188:
7187:
7182:
7180:
7179:
7173:
7172:
7156:
7154:
7153:
7148:
7140:
7139:
7120:
7118:
7117:
7112:
7107:
7106:
7081:
7079:
7078:
7073:
7065:
7064:
7040:filtered algebra
7034:
7032:
7031:
7026:
7018:
7017:
7011:
7010:
6992:
6991:
6985:
6984:
6972:
6971:
6945:
6943:
6942:
6937:
6935:
6934:
6928:
6927:
6911:
6909:
6908:
6903:
6898:
6897:
6884:
6875:
6873:
6872:
6867:
6865:
6864:
6849:
6848:
6839:
6838:
6832:
6831:
6816:
6815:
6809:
6808:
6786:
6784:
6783:
6778:
6776:
6775:
6769:
6768:
6750:
6749:
6743:
6742:
6730:
6729:
6714:
6713:
6707:
6706:
6685:
6683:
6682:
6677:
6669:
6668:
6649:
6647:
6646:
6641:
6639:
6638:
6632:
6631:
6595:
6593:
6592:
6587:
6582:
6581:
6568:
6566:
6565:
6560:
6555:
6554:
6540:free Lie algebra
6533:
6531:
6530:
6525:
6520:
6519:
6489:
6483:
6481:
6480:
6475:
6442:
6440:
6439:
6434:
6432:
6431:
6413:
6412:
6400:
6399:
6380:
6378:
6377:
6372:
6367:
6366:
6347:
6341:
6339:
6338:
6333:
6328:
6327:
6305:
6303:
6302:
6297:
6292:
6291:
6264:
6263:
6242:
6241:
6220:
6219:
6201:
6200:
6188:
6187:
6162:
6160:
6159:
6154:
6142:
6140:
6139:
6134:
6132:
6131:
6115:
6113:
6112:
6107:
6102:
6101:
6086:
6085:
6066:
6064:
6063:
6058:
6030:
6028:
6027:
6022:
6020:
6019:
5995:
5994:
5978:
5976:
5975:
5970:
5956:
5955:
5939:
5933:
5903:
5901:
5900:
5895:
5890:
5889:
5828:
5826:
5825:
5820:
5815:
5814:
5795:
5793:
5792:
5787:
5775:
5773:
5772:
5767:
5756:
5732:
5730:
5729:
5724:
5719:
5718:
5699:
5697:
5696:
5691:
5679:
5677:
5676:
5671:
5669:
5668:
5652:
5650:
5649:
5644:
5639:
5638:
5613:
5611:
5610:
5605:
5603:
5602:
5594:
5584:
5582:
5581:
5576:
5574:
5573:
5556:
5554:
5553:
5548:
5546:
5545:
5536:
5535:
5534:
5533:
5512:
5511:
5510:
5509:
5483:
5482:
5481:
5480:
5454:
5453:
5452:
5451:
5434:
5433:
5432:
5431:
5414:
5413:
5405:
5392:
5390:
5389:
5384:
5382:
5381:
5373:
5363:
5361:
5360:
5355:
5353:
5352:
5339:
5337:
5336:
5331:
5326:
5325:
5306:
5304:
5303:
5298:
5287:
5286:
5271:
5270:
5262:
5252:
5250:
5249:
5244:
5232:
5230:
5229:
5224:
5137:
5135:
5134:
5129:
5117:
5115:
5114:
5109:
5101:
5100:
5070:
5068:
5067:
5062:
5060:
5059:
5046:
5044:
5043:
5038:
5026:
5024:
5023:
5018:
5013:
5012:
4997:
4996:
4974:
4972:
4971:
4966:
4958:
4957:
4949:
4927:
4925:
4924:
4919:
4908:
4907:
4892:
4891:
4883:
4863:
4861:
4860:
4855:
4853:
4852:
4824:
4822:
4821:
4816:
4726:
4720:
4711:
4709:
4708:
4703:
4695:
4694:
4668:
4666:
4665:
4660:
4655:
4654:
4639:
4638:
4587:exterior algebra
4579:exterior product
4575:exterior algebra
4551:
4549:
4548:
4543:
4450:
4448:
4447:
4442:
4341:
4339:
4338:
4333:
4331:
4246:
4244:
4243:
4238:
4226:
4224:
4223:
4218:
4206:
4204:
4203:
4198:
4177:
4175:
4174:
4169:
4154:
4152:
4151:
4146:
4104:
4102:
4101:
4096:
4066:
4064:
4063:
4058:
4037:
4035:
4034:
4029:
4024:
4016:
4015:
3997:
3996:
3971:
3969:
3968:
3963:
3952:
3944:
3943:
3925:
3924:
3883:
3881:
3880:
3875:
3848:
3846:
3845:
3840:
3816:
3814:
3813:
3808:
3800:
3799:
3774:
3772:
3771:
3766:
3748:
3746:
3745:
3740:
3735:
3734:
3709:
3704:All elements of
3703:
3701:
3700:
3695:
3690:
3689:
3637:
3635:
3634:
3629:
3539:
3530:
3528:
3527:
3522:
3517:
3516:
3498:
3497:
3488:
3487:
3475:
3474:
3455:
3453:
3452:
3447:
3396:
3361:
3359:
3358:
3353:
3348:
3347:
3324:
3322:
3321:
3316:
3265:
3263:
3262:
3257:
3239:
3237:
3236:
3231:
3226:
3218:
3217:
3199:
3198:
3173:
3171:
3170:
3165:
3163:
3162:
3149:
3147:
3146:
3141:
3136:
3135:
3097:
3095:
3094:
3089:
3084:
3083:
3064:
3062:
3061:
3056:
3048:
3047:
3024:
3022:
3021:
3016:
2932:
2930:
2929:
2924:
2840:
2838:
2837:
2832:
2817:
2815:
2814:
2809:
2804:
2803:
2794:
2793:
2777:
2775:
2774:
2769:
2767:
2766:
2757:
2756:
2744:
2743:
2734:
2733:
2714:
2712:
2711:
2706:
2655:
2653:
2652:
2647:
2645:
2644:
2635:
2634:
2625:
2624:
2604:
2602:
2601:
2596:
2591:
2590:
2572:
2571:
2553:
2552:
2521:
2519:
2518:
2513:
2511:
2510:
2501:
2500:
2491:
2490:
2472:
2462:
2460:
2459:
2454:
2442:
2440:
2439:
2434:
2419:
2417:
2416:
2411:
2398:
2397:
2388:
2387:
2378:
2377:
2360:
2359:
2350:
2349:
2335:
2334:
2314:
2313:
2288:
2286:
2285:
2280:
2278:
2277:
2256:
2254:
2253:
2248:
2243:
2242:
2216:
2214:
2213:
2208:
2206:
2205:
2180:
2178:
2177:
2172:
2170:
2169:
2152:
2150:
2149:
2144:
2142:
2141:
2132:
2131:
2110:
2109:
2100:
2099:
2087:
2086:
2077:
2076:
2032:
2030:
2029:
2024:
2006:
2004:
2003:
1998:
1986:
1984:
1983:
1978:
1966:
1964:
1963:
1958:
1946:
1944:
1943:
1938:
1936:
1935:
1922:
1920:
1919:
1914:
1912:
1911:
1898:
1896:
1895:
1890:
1872:
1870:
1869:
1864:
1862:
1861:
1848:
1846:
1845:
1840:
1838:
1837:
1821:
1819:
1818:
1813:
1811:
1810:
1794:
1792:
1791:
1786:
1784:
1783:
1765:
1764:
1741:
1739:
1738:
1733:
1731:
1730:
1714:
1712:
1711:
1706:
1698:
1697:
1681:
1679:
1678:
1673:
1671:
1670:
1651:
1649:
1648:
1643:
1640:
1639:
1638:
1628:
1615:
1614:
1613:
1603:
1593:
1592:
1591:
1581:
1562:
1560:
1559:
1554:
1552:
1551:
1531:
1529:
1528:
1523:
1521:
1520:
1511:
1510:
1494:
1492:
1491:
1486:
1484:
1483:
1474:
1473:
1457:
1455:
1454:
1449:
1447:
1446:
1430:
1428:
1427:
1422:
1420:
1419:
1394:
1392:
1391:
1386:
1378:
1377:
1365:
1364:
1332:
1330:
1329:
1324:
1316:
1315:
1299:
1297:
1296:
1291:
1279:
1277:
1276:
1271:
1263:
1262:
1246:
1244:
1243:
1238:
1226:
1224:
1223:
1218:
1200:
1198:
1197:
1192:
1167:
1165:
1164:
1159:
1064:
1062:
1061:
1056:
1032:
1030:
1029:
1024:
994:
992:
991:
986:
950:
948:
947:
942:
906:
904:
903:
898:
891:
890:
889:
844:
843:
801:
800:
742:
740:
739:
734:
732:
731:
711:
709:
708:
703:
701:
700:
691:
690:
674:
669:
651:
650:
641:
640:
628:
627:
618:
617:
598:
596:
595:
590:
588:
587:
572:
571:
552:
550:
549:
544:
539:
538:
529:
528:
512:
507:
486:
485:
473:
472:
446:
444:
443:
438:
436:
435:
413:
411:
410:
405:
403:
402:
387:
386:
370:
368:
367:
362:
360:
359:
338:
336:
335:
330:
328:
327:
314:
312:
311:
306:
304:
303:
290:
288:
287:
282:
280:
279:
266:
264:
263:
258:
256:
255:
243:and the algebra
242:
240:
239:
234:
232:
231:
218:
216:
215:
210:
186:
184:
183:
178:
176:
175:
162:
160:
159:
154:
152:
151:
138:
136:
135:
130:
128:
127:
16756:
16755:
16751:
16750:
16749:
16747:
16746:
16745:
16721:
16720:
16683:
16643:
16633:10.1090/gsm/034
16613:
16587:
16558:
16557:
16544:
16540:
16506:
16502:
16494:
16490:
16481:
16477:
16469:
16465:
16457:
16453:
16424:
16420:
16387:
16383:
16375:
16371:
16363:
16359:
16354:
16337:
16291:
16290:
16281:
16277:
16275:
16272:
16271:
16251:
16250:
16242:
16239:
16238:
16230:
16224:
16191:
16187:
16149:
16146:
16145:
16099:
16096:
16095:
16022:
16019:
16018:
15939:
15936:
15935:
15895:
15892:
15891:
15881:
15873:
15867:
15840:
15814:
15805:
15799:
15765:
15764:
15756:
15753:
15752:
15739:
15719:
15718:
15710:
15707:
15706:
15693:
15676:
15675:
15673:
15670:
15669:
15645:
15644:
15642:
15639:
15638:
15632:
15608:
15607:
15599:
15596:
15595:
15589:
15569:
15568:
15566:
15563:
15562:
15553:
15529:
15528:
15520:
15517:
15516:
15499:
15498:
15496:
15493:
15492:
15465:
15464:
15456:
15453:
15452:
15448:
15427:
15426:
15424:
15421:
15420:
15318:
15291:
15290:
15288:
15276:
15267:
15266:
15265:
15257:
15254:
15253:
15210:
15207:
15206:
15169:
15166:
15165:
15125:
15122:
15121:
15097:
15096:
15091:
15085:
15084:
15079:
15069:
15068:
15057:
15047:
15046:
15041:
15035:
15034:
15029:
15019:
15018:
15007:
14997:
14996:
14991:
14985:
14984:
14979:
14966:
14965:
14957:
14954:
14953:
14933:
14924:
14923:
14922:
14913:
14912:
14910:
14907:
14906:
14903:
14844:Fredholm theory
14826:
14825:
14823:
14820:
14819:
14792:
14791:
14783:
14780:
14779:
14759:
14758:
14750:
14747:
14746:
14743:
14721:
14717:
14709:
14701:
14685:
14681:
14672:
14668:
14663:
14660:
14659:
14624:
14620:
14611:
14607:
14602:
14599:
14598:
14575:
14571:
14562:
14558:
14549:
14545:
14536:
14532:
14523:
14519:
14510:
14506:
14497:
14493:
14478:
14474:
14472:
14469:
14468:
14441:
14437:
14425:
14421:
14419:
14416:
14415:
14389:
14385:
14376:
14372:
14363:
14359:
14347:
14343:
14317:
14313:
14301:
14297:
14285:
14281:
14277:
14273:
14268:
14265:
14264:
14244:
14240:
14228:
14224:
14212:
14208:
14203:
14200:
14199:
14182:
14178:
14176:
14173:
14172:
14152:
14148:
14139:
14135:
14133:
14130:
14129:
14109:
14105:
14103:
14100:
14099:
14091:
14061:
14060:
14046:
14043:
14042:
14036:
14029:
13994:
13990:
13988:
13985:
13984:
13964:
13963:
13949:
13946:
13945:
13925:
13924:
13916:
13913:
13912:
13903:
13877:
13873:
13858:
13854:
13852:
13849:
13848:
13820:
13817:
13816:
13800:
13797:
13796:
13767:
13763:
13757:
13753:
13744:
13740:
13734:
13730:
13712:
13708:
13706:
13703:
13702:
13679:
13675:
13669:
13665:
13659:
13648:
13636:
13633:
13632:
13597:
13593:
13581:
13577:
13556:
13552:
13550:
13547:
13546:
13519:
13515:
13499:
13495:
13493:
13490:
13489:
13467:
13464:
13463:
13452:
13422:
13418:
13416:
13413:
13412:
13403:
13377:
13373:
13371:
13368:
13367:
13361:
13355:
13345:
13325:
13324:
13316:
13313:
13312:
13292:
13288:
13273:
13269:
13263:
13252:
13236:
13232:
13215:
13212:
13211:
13202:
13185:
13181:
13179:
13176:
13175:
13152:
13151:
13133:
13132:
13124:
13121:
13120:
13111:
13101:
13081:
13077:
13071:
13067:
13061:
13050:
13020:
13017:
13016:
13007:
12997:
12977:
12973:
12971:
12968:
12967:
12953:
12947:
12936:
12901:
12900:
12886:
12883:
12882:
12859:
12855:
12853:
12850:
12849:
12826:
12822:
12820:
12817:
12816:
12793:
12789:
12787:
12784:
12783:
12763:
12759:
12750:
12746:
12737:
12733:
12718:
12714:
12712:
12709:
12708:
12685:
12681:
12673:
12665:
12649:
12645:
12636:
12632:
12627:
12624:
12623:
12585:
12581:
12573:
12565:
12537:
12533:
12525:
12517:
12495:
12491:
12483:
12475:
12469:
12466:
12465:
12430:
12426:
12415:
12412:
12411:
12408:Israel Gel'fand
12380:
12369:
12368:
12364:
12360:
12359:
12357:
12354:
12353:
12327:
12323:
12321:
12318:
12317:
12301:
12298:
12297:
12271:
12267:
12265:
12262:
12261:
12245:
12242:
12241:
12221:
12217:
12202:
12198:
12189:
12185:
12170:
12166:
12151:
12147:
12145:
12142:
12141:
12122:
12119:
12118:
12101:
12097:
12095:
12092:
12091:
12070:
12066:
12064:
12061:
12060:
12031:
12030:
12022:
12019:
12018:
11998:
11997:
11989:
11986:
11985:
11968:
11961:
11960:
11958:
11955:
11954:
11934:
11933:
11925:
11922:
11921:
11904:
11897:
11896:
11894:
11891:
11890:
11873:
11872:
11824:
11823:
11784:
11783:
11750:
11713:
11711:
11708:
11707:
11685:
11682:
11681:
11664:
11663:
11661:
11658:
11657:
11563:
11560:
11559:
11535:
11534:
11532:
11529:
11528:
11511:
11510:
11501:
11500:
11492:
11489:
11488:
11487:By definition,
11458:
11457:
11449:
11446:
11445:
11425:
11424:
11416:
11413:
11412:
11390:
11389:
11382:
11381:
11379:
11376:
11375:
11352:
11351:
11343:
11340:
11339:
11319:
11318:
11316:
11313:
11312:
11287:
11286:
11279:
11278:
11276:
11273:
11272:
11240:
11233:
11232:
11212:
11209:
11208:
11188:
11187:
11179:
11176:
11175:
11155:
11154:
11116:
11113:
11112:
11089:
11088:
11074:
11071:
11070:
11044:
11043:
11029:
11026:
11025:
11000:
10999:
10997:
10994:
10993:
10970:
10969:
10961:
10958:
10957:
10940:
10939:
10937:
10934:
10933:
10910:
10909:
10901:
10898:
10897:
10874:
10873:
10859:
10856:
10855:
10854:Any element of
10832:
10831:
10823:
10820:
10819:
10802:
10801:
10799:
10796:
10795:
10775:
10774:
10766:
10763:
10762:
10739:
10738:
10724:
10721:
10720:
10714:
10708:
10679:
10678:
10670:
10667:
10666:
10643:
10642:
10640:
10637:
10636:
10630:highest weights
10600:shuffle product
10596:tensor algebras
10571:
10570:
10555:
10554:
10546:
10543:
10542:
10516:
10510:
10509:
10508:
10499:
10495:
10471:
10465:
10464:
10463:
10454:
10450:
10438:
10432:
10431:
10430:
10421:
10415:
10414:
10413:
10405:
10402:
10401:
10381:
10375:
10374:
10373:
10364:
10358:
10357:
10356:
10354:
10351:
10350:
10327:
10321:
10320:
10319:
10301:
10295:
10294:
10293:
10275:
10269:
10268:
10267:
10258:
10252:
10251:
10250:
10242:
10239:
10238:
10204:
10203:
10195:
10192:
10191:
10170:
10169:
10167:
10164:
10163:
10160:representations
10135:
10134:
10126:
10123:
10122:
10101:
10100:
10098:
10095:
10094:
10091:representations
10087:
10062:
10059:
10058:
10031:
10028:
10027:
10010:
10006:
10004:
10001:
10000:
9963:
9959:
9953:
9949:
9948:
9944:
9915:
9912:
9911:
9876:
9827:
9822:
9809:
9804:
9803:
9799:
9793:
9789:
9783:
9779:
9778:
9774:
9767:
9764:
9763:
9731:
9728:
9727:
9714:Berezin formula
9693:
9692:
9684:
9681:
9680:
9656:
9655:
9647:
9644:
9643:
9603:
9600:
9599:
9574:
9573:
9553:
9550:
9549:
9485:
9482:
9481:
9462:
9459:
9458:
9441:
9437:
9435:
9432:
9431:
9414:
9410:
9408:
9405:
9404:
9381:
9380:
9362:
9361:
9347:
9344:
9343:
9320:
9319:
9311:
9308:
9307:
9286:
9282:
9280:
9277:
9276:
9259:
9255:
9253:
9250:
9249:
9233:
9230:
9229:
9228:over the field
9209:
9205:
9197:
9194:
9193:
9176:
9172:
9170:
9167:
9166:
9149:
9145:
9143:
9140:
9139:
9119:
9118:
9110:
9107:
9106:
9087:
9084:
9083:
9078:: the space of
9059:
9058:
9050:
9047:
9046:
9038:are isomorphic
9019:
9018:
9010:
9007:
9006:
8986:
8985:
8977:
8974:
8973:
8953:
8952:
8944:
8941:
8940:
8937:
8912:
8911:
8903:
8900:
8899:
8874:
8871:
8870:
8850:
8849:
8841:
8838:
8837:
8812:
8809:
8808:
8788:
8787:
8779:
8776:
8775:
8750:
8747:
8746:
8729:
8728:
8726:
8723:
8722:
8697:
8694:
8693:
8672:
8671:
8669:
8666:
8665:
8642:
8641:
8633:
8630:
8629:
8604:
8601:
8600:
8575:
8572:
8571:
8555:
8552:
8551:
8535:
8532:
8531:
8511:
8510:
8508:
8505:
8504:
8486:
8485:
8483:
8480:
8479:
8463:
8460:
8459:
8442:
8441:
8439:
8436:
8435:
8418:
8417:
8415:
8412:
8411:
8394:
8393:
8391:
8388:
8387:
8371:
8368:
8367:
8364:
8348:Jordan algebras
8344:
8316:
8315:
8307:
8304:
8303:
8283:
8282:
8268:
8265:
8264:
8240:
8239:
8221:
8220:
8212:
8209:
8208:
8185:
8184:
8176:
8173:
8172:
8152:
8151:
8143:
8140:
8139:
8109:
8108:
8100:
8097:
8096:
8079:
8078:
8072:
8068:
8066:
8063:
8062:
8035:
8034:
8026:
8023:
8022:
8002:
8001:
7993:
7990:
7989:
7943:
7934:
7933:
7915:
7914:
7906:
7903:
7902:
7879:
7878:
7870:
7867:
7866:
7849:
7848:
7839:
7835:
7833:
7830:
7829:
7812:
7811:
7805:
7801:
7799:
7796:
7795:
7771:
7770:
7764:
7760:
7758:
7755:
7754:
7737:
7736:
7730:
7726:
7724:
7721:
7720:
7703:
7702:
7690:
7686:
7684:
7681:
7680:
7663:
7662:
7656:
7652:
7650:
7647:
7646:
7626:
7625:
7613:
7609:
7607:
7604:
7603:
7586:
7585:
7579:
7575:
7573:
7570:
7569:
7552:
7551:
7545:
7541:
7539:
7536:
7535:
7518:
7517:
7505:
7501:
7499:
7496:
7495:
7478:
7477:
7471:
7467:
7465:
7462:
7461:
7441:
7440:
7434:
7430:
7428:
7425:
7424:
7403:
7402:
7396:
7392:
7390:
7387:
7386:
7369:
7368:
7362:
7358:
7356:
7353:
7352:
7335:
7334:
7328:
7324:
7322:
7319:
7318:
7298:
7297:
7285:
7281:
7276:
7270:
7269:
7263:
7259:
7250:
7249:
7243:
7239:
7237:
7234:
7233:
7205:
7204:
7196:
7193:
7192:
7175:
7174:
7168:
7164:
7162:
7159:
7158:
7135:
7134:
7126:
7123:
7122:
7102:
7101:
7093:
7090:
7089:
7060:
7059:
7051:
7048:
7047:
7013:
7012:
7006:
7002:
6987:
6986:
6980:
6976:
6967:
6966:
6958:
6955:
6954:
6930:
6929:
6923:
6919:
6917:
6914:
6913:
6893:
6892:
6890:
6887:
6886:
6880:
6860:
6859:
6844:
6843:
6834:
6833:
6824:
6820:
6811:
6810:
6804:
6800:
6798:
6795:
6794:
6771:
6770:
6764:
6760:
6745:
6744:
6738:
6734:
6725:
6724:
6709:
6708:
6702:
6698:
6696:
6693:
6692:
6664:
6663:
6655:
6652:
6651:
6634:
6633:
6627:
6623:
6621:
6618:
6617:
6610:
6608:Coordinate-free
6577:
6576:
6574:
6571:
6570:
6550:
6549:
6547:
6544:
6543:
6515:
6514:
6506:
6503:
6502:
6485:
6451:
6448:
6447:
6427:
6423:
6408:
6404:
6395:
6391:
6389:
6386:
6385:
6362:
6361:
6353:
6350:
6349:
6343:
6323:
6322:
6314:
6311:
6310:
6287:
6283:
6259:
6255:
6237:
6233:
6215:
6211:
6196:
6192:
6183:
6179:
6171:
6168:
6167:
6148:
6145:
6144:
6127:
6123:
6121:
6118:
6117:
6097:
6096:
6081:
6080:
6072:
6069:
6068:
6040:
6037:
6036:
6012:
6008:
5990:
5986:
5984:
5981:
5980:
5951:
5947:
5945:
5942:
5941:
5935:
5929:
5922:totally ordered
5918:
5910:coordinate-free
5885:
5884:
5876:
5873:
5872:
5869:
5863:
5851:Albert algebras
5843:Jordan algebras
5839:
5810:
5809:
5801:
5798:
5797:
5781:
5778:
5777:
5746:
5738:
5735:
5734:
5714:
5713:
5705:
5702:
5701:
5685:
5682:
5681:
5664:
5663:
5661:
5658:
5657:
5634:
5633:
5619:
5616:
5615:
5593:
5592:
5590:
5587:
5586:
5569:
5568:
5566:
5563:
5562:
5541:
5540:
5529:
5525:
5524:
5520:
5505:
5501:
5500:
5496:
5476:
5472:
5471:
5467:
5447:
5443:
5442:
5438:
5427:
5423:
5422:
5418:
5404:
5403:
5401:
5398:
5397:
5372:
5371:
5369:
5366:
5365:
5348:
5347:
5345:
5342:
5341:
5321:
5320:
5312:
5309:
5308:
5282:
5281:
5261:
5260:
5258:
5255:
5254:
5238:
5235:
5234:
5143:
5140:
5139:
5123:
5120:
5119:
5096:
5095:
5087:
5084:
5083:
5055:
5054:
5052:
5049:
5048:
5032:
5029:
5028:
5008:
5007:
4992:
4991:
4983:
4980:
4979:
4948:
4947:
4939:
4936:
4935:
4903:
4902:
4882:
4881:
4879:
4876:
4875:
4848:
4847:
4833:
4830:
4829:
4735:
4732:
4731:
4722:
4716:
4690:
4689:
4681:
4678:
4677:
4650:
4649:
4634:
4633:
4625:
4622:
4621:
4618:
4602:Malcev algebras
4462:
4459:
4458:
4361:
4358:
4357:
4329:
4328:
4303:
4291:
4290:
4280:
4261:
4259:
4256:
4255:
4232:
4229:
4228:
4212:
4209:
4208:
4183:
4180:
4179:
4163:
4160:
4159:
4110:
4107:
4106:
4072:
4069:
4068:
4052:
4049:
4048:
4044:
4020:
4011:
4010:
3992:
3991:
3983:
3980:
3979:
3948:
3939:
3938:
3920:
3919:
3899:
3896:
3895:
3854:
3851:
3850:
3822:
3819:
3818:
3795:
3794:
3780:
3777:
3776:
3754:
3751:
3750:
3730:
3729:
3715:
3712:
3711:
3705:
3685:
3684:
3646:
3643:
3642:
3548:
3545:
3544:
3535:
3512:
3511:
3493:
3492:
3483:
3482:
3470:
3469:
3467:
3464:
3463:
3405:
3402:
3401:
3392:
3390:two-sided ideal
3375:Poisson algebra
3343:
3342:
3334:
3331:
3330:
3274:
3271:
3270:
3251:
3248:
3247:
3222:
3213:
3212:
3194:
3193:
3185:
3182:
3181:
3158:
3157:
3155:
3152:
3151:
3131:
3130:
3122:
3119:
3118:
3107:Poisson algebra
3079:
3078:
3070:
3067:
3066:
3043:
3042:
3034:
3031:
3030:
2944:
2941:
2940:
2852:
2849:
2848:
2823:
2820:
2819:
2799:
2798:
2789:
2785:
2783:
2780:
2779:
2762:
2761:
2752:
2751:
2739:
2738:
2729:
2725:
2723:
2720:
2719:
2664:
2661:
2660:
2640:
2639:
2630:
2629:
2620:
2619:
2617:
2614:
2613:
2612:the bracket on
2586:
2585:
2567:
2566:
2548:
2547:
2539:
2536:
2535:
2532:Jacobi identity
2506:
2505:
2496:
2495:
2486:
2485:
2483:
2480:
2479:
2468:
2448:
2445:
2444:
2428:
2425:
2424:
2393:
2392:
2383:
2382:
2373:
2372:
2355:
2354:
2345:
2344:
2330:
2329:
2309:
2308:
2300:
2297:
2296:
2273:
2272:
2270:
2267:
2266:
2263:tensor products
2238:
2237:
2229:
2226:
2225:
2201:
2200:
2198:
2195:
2194:
2191:
2165:
2164:
2162:
2159:
2158:
2137:
2133:
2121:
2117:
2105:
2101:
2095:
2091:
2082:
2078:
2072:
2068:
2066:
2063:
2062:
2059:two-sided ideal
2039:
2012:
2009:
2008:
1992:
1989:
1988:
1972:
1969:
1968:
1952:
1949:
1948:
1931:
1930:
1928:
1925:
1924:
1907:
1906:
1904:
1901:
1900:
1878:
1875:
1874:
1857:
1856:
1854:
1851:
1850:
1833:
1829:
1827:
1824:
1823:
1806:
1802:
1800:
1797:
1796:
1779:
1775:
1760:
1756:
1754:
1751:
1750:
1726:
1722:
1720:
1717:
1716:
1693:
1689:
1687:
1684:
1683:
1666:
1662:
1660:
1657:
1656:
1634:
1630:
1629:
1624:
1609:
1605:
1604:
1599:
1587:
1583:
1582:
1577:
1571:
1568:
1567:
1547:
1543:
1541:
1538:
1537:
1516:
1512:
1506:
1502:
1500:
1497:
1496:
1479:
1475:
1469:
1465:
1463:
1460:
1459:
1442:
1438:
1436:
1433:
1432:
1415:
1411:
1409:
1406:
1405:
1401:
1399:Finding a basis
1373:
1369:
1360:
1356:
1342:
1339:
1338:
1311:
1307:
1305:
1302:
1301:
1285:
1282:
1281:
1258:
1254:
1252:
1249:
1248:
1232:
1229:
1228:
1206:
1203:
1202:
1180:
1177:
1176:
1073:
1070:
1069:
1038:
1035:
1034:
1000:
997:
996:
956:
953:
952:
915:
912:
911:
884:
883:
875:
869:
868:
863:
853:
852:
838:
837:
832:
826:
825:
820:
810:
809:
795:
794:
789:
783:
782:
777:
767:
766:
758:
755:
754:
727:
726:
724:
721:
720:
696:
692:
680:
676:
670:
659:
646:
642:
636:
632:
623:
619:
613:
609:
607:
604:
603:
583:
579:
567:
563:
561:
558:
557:
534:
530:
518:
514:
508:
497:
481:
477:
468:
464:
459:
456:
455:
425:
421:
419:
416:
415:
398:
394:
382:
378:
376:
373:
372:
355:
354:
352:
349:
348:
345:
323:
322:
320:
317:
316:
299:
298:
296:
293:
292:
275:
274:
272:
269:
268:
251:
250:
248:
245:
244:
227:
226:
224:
221:
220:
192:
189:
188:
171:
170:
168:
165:
164:
147:
146:
144:
141:
140:
123:
122:
120:
117:
116:
113:
51:representations
47:representations
24:
17:
12:
11:
5:
16754:
16744:
16743:
16738:
16733:
16719:
16718:
16706:
16694:
16681:
16662:
16641:
16616:
16612:978-3319134666
16611:
16598:
16585:
16556:
16555:
16538:
16500:
16488:
16475:
16463:
16451:
16438:(2): 480â493.
16418:
16381:
16369:
16356:
16355:
16353:
16350:
16349:
16348:
16343:
16336:
16333:
16323:to obtain the
16302:
16299:
16294:
16289:
16284:
16280:
16259:
16254:
16249:
16246:
16217:
16216:
16205:
16202:
16197:
16194:
16190:
16186:
16183:
16180:
16177:
16174:
16171:
16168:
16165:
16162:
16159:
16156:
16153:
16139:
16138:
16127:
16124:
16121:
16118:
16115:
16112:
16109:
16106:
16103:
16091:the counit as
16089:
16088:
16077:
16074:
16071:
16068:
16065:
16062:
16059:
16056:
16053:
16050:
16047:
16044:
16041:
16038:
16035:
16032:
16029:
16026:
16012:
16011:
16000:
15997:
15994:
15991:
15988:
15985:
15982:
15979:
15976:
15973:
15970:
15967:
15964:
15961:
15958:
15955:
15952:
15949:
15946:
15943:
15920:
15917:
15914:
15911:
15908:
15905:
15902:
15899:
15861:tensor algebra
15839:
15836:
15773:
15768:
15763:
15760:
15727:
15722:
15717:
15714:
15679:
15648:
15616:
15611:
15606:
15603:
15572:
15537:
15532:
15527:
15524:
15502:
15473:
15468:
15463:
15460:
15430:
15399:
15396:
15393:
15390:
15387:
15384:
15381:
15378:
15375:
15372:
15369:
15366:
15363:
15360:
15357:
15354:
15351:
15348:
15345:
15342:
15339:
15336:
15333:
15330:
15327:
15324:
15321:
15316:
15313:
15310:
15307:
15304:
15301:
15298:
15294:
15287:
15284:
15279:
15273:
15270:
15264:
15261:
15235:
15232:
15229:
15226:
15223:
15220:
15217:
15214:
15194:
15191:
15188:
15185:
15182:
15179:
15176:
15173:
15153:
15150:
15147:
15144:
15141:
15138:
15135:
15132:
15129:
15101:
15095:
15092:
15090:
15087:
15086:
15083:
15080:
15078:
15075:
15074:
15072:
15067:
15064:
15056:
15051:
15045:
15042:
15040:
15037:
15036:
15033:
15030:
15028:
15025:
15024:
15022:
15017:
15014:
15006:
15001:
14995:
14992:
14990:
14987:
14986:
14983:
14980:
14978:
14975:
14972:
14971:
14969:
14964:
14961:
14936:
14930:
14927:
14921:
14916:
14902:
14899:
14829:
14800:
14795:
14790:
14787:
14767:
14762:
14757:
14754:
14742:
14739:
14738:
14737:
14724:
14720:
14714:
14707:
14704:
14700:
14696:
14693:
14688:
14684:
14680:
14675:
14671:
14667:
14650:
14649:
14638:
14635:
14632:
14627:
14623:
14619:
14614:
14610:
14606:
14592:
14591:
14578:
14574:
14570:
14565:
14561:
14557:
14552:
14548:
14544:
14539:
14535:
14531:
14526:
14522:
14518:
14513:
14509:
14505:
14500:
14496:
14492:
14487:
14484:
14481:
14477:
14447:
14444:
14440:
14436:
14431:
14428:
14424:
14412:
14411:
14400:
14397:
14392:
14388:
14384:
14379:
14375:
14371:
14366:
14362:
14358:
14355:
14350:
14346:
14342:
14339:
14335:
14331:
14328:
14325:
14320:
14316:
14312:
14309:
14304:
14300:
14296:
14293:
14288:
14284:
14280:
14276:
14272:
14247:
14243:
14239:
14236:
14231:
14227:
14223:
14220:
14215:
14211:
14207:
14185:
14181:
14160:
14155:
14151:
14147:
14142:
14138:
14117:
14112:
14108:
14090:
14087:
14075:
14072:
14069:
14064:
14059:
14056:
14053:
14050:
14017:
14014:
14011:
14008:
14005:
14002:
13997:
13993:
13972:
13967:
13962:
13959:
13956:
13953:
13933:
13928:
13923:
13920:
13891:
13886:
13883:
13880:
13876:
13872:
13867:
13864:
13861:
13857:
13836:
13833:
13830:
13827:
13824:
13804:
13793:
13792:
13779:
13776:
13773:
13770:
13766:
13760:
13756:
13752:
13747:
13743:
13737:
13733:
13729:
13726:
13723:
13720:
13715:
13711:
13696:
13695:
13682:
13678:
13672:
13668:
13662:
13657:
13654:
13651:
13647:
13643:
13640:
13626:
13625:
13614:
13611:
13608:
13605:
13600:
13596:
13590:
13587:
13584:
13580:
13576:
13573:
13570:
13567:
13564:
13559:
13555:
13527:
13522:
13518:
13513:
13510:
13505:
13502:
13498:
13477:
13474:
13471:
13436:
13433:
13430:
13425:
13421:
13391:
13388:
13385:
13380:
13376:
13366:for which the
13333:
13328:
13323:
13320:
13309:
13308:
13295:
13291:
13287:
13284:
13281:
13276:
13272:
13266:
13261:
13258:
13255:
13251:
13247:
13244:
13239:
13235:
13231:
13228:
13225:
13222:
13219:
13188:
13184:
13172:
13171:
13160:
13155:
13150:
13147:
13144:
13141:
13136:
13131:
13128:
13098:
13097:
13084:
13080:
13074:
13070:
13064:
13059:
13056:
13053:
13049:
13045:
13042:
13039:
13036:
13033:
13030:
13027:
13024:
12983:
12980:
12976:
12935:
12932:
12912:
12909:
12904:
12899:
12896:
12893:
12890:
12868:
12865:
12862:
12858:
12837:
12832:
12829:
12825:
12799:
12796:
12792:
12780:
12779:
12766:
12762:
12758:
12753:
12749:
12743:
12740:
12736:
12732:
12727:
12724:
12721:
12717:
12702:
12701:
12688:
12684:
12678:
12671:
12668:
12664:
12660:
12657:
12652:
12648:
12644:
12639:
12635:
12631:
12617:
12616:
12605:
12602:
12597:
12594:
12591:
12588:
12584:
12578:
12571:
12568:
12564:
12560:
12557:
12554:
12549:
12546:
12543:
12540:
12536:
12530:
12523:
12520:
12516:
12512:
12507:
12504:
12501:
12498:
12494:
12488:
12481:
12478:
12474:
12450:
12447:
12444:
12439:
12436:
12433:
12429:
12425:
12422:
12419:
12391:
12386:
12383:
12378:
12372:
12367:
12363:
12339:
12336:
12333:
12330:
12326:
12305:
12283:
12280:
12277:
12274:
12270:
12249:
12238:
12237:
12224:
12220:
12216:
12213:
12210:
12205:
12201:
12197:
12192:
12188:
12182:
12179:
12176:
12173:
12169:
12165:
12160:
12157:
12154:
12150:
12126:
12104:
12100:
12073:
12069:
12042:
12039:
12034:
12029:
12026:
12006:
12001:
11996:
11993:
11971:
11942:
11937:
11932:
11929:
11907:
11887:
11886:
11871:
11868:
11865:
11862:
11859:
11856:
11853:
11850:
11847:
11844:
11841:
11838:
11835:
11832:
11829:
11827:
11825:
11822:
11819:
11816:
11813:
11810:
11807:
11804:
11801:
11798:
11795:
11792:
11789:
11787:
11785:
11782:
11779:
11776:
11773:
11770:
11767:
11764:
11761:
11758:
11755:
11751:
11749:
11746:
11743:
11740:
11737:
11734:
11731:
11728:
11725:
11722:
11719:
11716:
11715:
11689:
11667:
11654:
11653:
11642:
11639:
11636:
11633:
11630:
11627:
11624:
11621:
11618:
11615:
11612:
11609:
11606:
11603:
11600:
11597:
11594:
11591:
11588:
11585:
11582:
11579:
11576:
11573:
11570:
11567:
11538:
11514:
11509:
11504:
11499:
11496:
11469:
11466:
11461:
11456:
11453:
11433:
11428:
11423:
11420:
11393:
11363:
11360:
11355:
11350:
11347:
11338:we need it on
11327:
11322:
11296:
11290:
11260:
11257:
11254:
11251:
11248:
11243:
11231:
11228:
11225:
11222:
11219:
11216:
11196:
11191:
11186:
11183:
11163:
11158:
11153:
11150:
11147:
11144:
11141:
11138:
11135:
11132:
11129:
11126:
11123:
11120:
11100:
11097:
11092:
11087:
11084:
11081:
11078:
11055:
11052:
11047:
11042:
11039:
11036:
11033:
11003:
10981:
10978:
10973:
10968:
10965:
10943:
10921:
10918:
10913:
10908:
10905:
10885:
10882:
10877:
10872:
10869:
10866:
10863:
10843:
10840:
10835:
10830:
10827:
10805:
10783:
10778:
10773:
10770:
10750:
10747:
10742:
10737:
10734:
10731:
10728:
10707:
10704:
10687:
10682:
10677:
10674:
10646:
10591:
10590:
10579:
10574:
10569:
10566:
10563:
10558:
10553:
10550:
10536:
10535:
10524:
10519:
10513:
10507:
10502:
10498:
10494:
10491:
10488:
10485:
10482:
10479:
10474:
10468:
10462:
10457:
10453:
10449:
10446:
10441:
10435:
10429:
10424:
10418:
10412:
10409:
10384:
10378:
10372:
10367:
10361:
10347:
10346:
10335:
10330:
10324:
10318:
10315:
10312:
10309:
10304:
10298:
10292:
10289:
10286:
10283:
10278:
10272:
10266:
10261:
10255:
10249:
10246:
10212:
10207:
10202:
10199:
10173:
10143:
10138:
10133:
10130:
10104:
10086:
10083:
10066:
10035:
10013:
10009:
9997:
9996:
9985:
9982:
9979:
9976:
9972:
9966:
9962:
9956:
9952:
9947:
9943:
9940:
9937:
9934:
9931:
9928:
9925:
9922:
9919:
9905:
9904:
9891:
9888:
9885:
9882:
9879:
9874:
9870:
9867:
9864:
9861:
9858:
9855:
9852:
9849:
9845:
9840:
9833:
9830:
9826:
9821:
9815:
9812:
9808:
9802:
9796:
9792:
9786:
9782:
9777:
9773:
9770:
9766:
9762:
9759:
9756:
9753:
9750:
9747:
9744:
9741:
9738:
9735:
9701:
9696:
9691:
9688:
9664:
9659:
9654:
9651:
9631:
9628:
9625:
9622:
9619:
9616:
9613:
9610:
9607:
9585:
9582:
9577:
9572:
9569:
9566:
9563:
9560:
9557:
9546:
9545:
9534:
9531:
9528:
9525:
9522:
9519:
9516:
9513:
9510:
9507:
9504:
9501:
9498:
9495:
9492:
9489:
9466:
9444:
9440:
9417:
9413:
9401:
9400:
9389:
9384:
9379:
9376:
9373:
9370:
9365:
9360:
9357:
9354:
9351:
9328:
9323:
9318:
9315:
9289:
9285:
9262:
9258:
9237:
9217:
9212:
9208:
9204:
9201:
9179:
9175:
9152:
9148:
9127:
9122:
9117:
9114:
9091:
9067:
9062:
9057:
9054:
9027:
9022:
9017:
9014:
8994:
8989:
8984:
8981:
8961:
8956:
8951:
8948:
8936:
8933:
8931:is apparent.)
8920:
8915:
8910:
8907:
8887:
8884:
8881:
8878:
8858:
8853:
8848:
8845:
8825:
8822:
8819:
8816:
8796:
8791:
8786:
8783:
8763:
8760:
8757:
8754:
8732:
8710:
8707:
8704:
8701:
8675:
8650:
8645:
8640:
8637:
8617:
8614:
8611:
8608:
8588:
8585:
8582:
8579:
8559:
8539:
8514:
8489:
8467:
8445:
8421:
8397:
8375:
8363:
8360:
8343:
8342:Other algebras
8340:
8327:
8324:
8319:
8314:
8311:
8291:
8286:
8281:
8278:
8275:
8272:
8248:
8243:
8238:
8235:
8232:
8229:
8224:
8219:
8216:
8196:
8193:
8188:
8183:
8180:
8160:
8155:
8150:
8147:
8120:
8117:
8112:
8107:
8104:
8082:
8075:
8071:
8043:
8038:
8033:
8030:
8010:
8005:
8000:
7997:
7986:
7985:
7974:
7971:
7968:
7965:
7962:
7959:
7956:
7953:
7950:
7946:
7942:
7937:
7932:
7929:
7926:
7923:
7918:
7913:
7910:
7887:
7882:
7877:
7874:
7852:
7847:
7842:
7838:
7815:
7808:
7804:
7774:
7767:
7763:
7740:
7733:
7729:
7706:
7699:
7696:
7693:
7689:
7666:
7659:
7655:
7629:
7622:
7619:
7616:
7612:
7589:
7582:
7578:
7555:
7548:
7544:
7521:
7514:
7511:
7508:
7504:
7481:
7474:
7470:
7444:
7437:
7433:
7406:
7399:
7395:
7372:
7365:
7361:
7338:
7331:
7327:
7315:
7314:
7301:
7294:
7291:
7288:
7284:
7279:
7273:
7266:
7262:
7258:
7253:
7246:
7242:
7216:
7213:
7208:
7203:
7200:
7178:
7171:
7167:
7146:
7143:
7138:
7133:
7130:
7110:
7105:
7100:
7097:
7071:
7068:
7063:
7058:
7055:
7036:
7035:
7024:
7021:
7016:
7009:
7005:
7001:
6998:
6995:
6990:
6983:
6979:
6975:
6970:
6965:
6962:
6933:
6926:
6922:
6901:
6896:
6877:
6876:
6863:
6858:
6855:
6852:
6847:
6842:
6837:
6830:
6827:
6823:
6819:
6814:
6807:
6803:
6788:
6787:
6774:
6767:
6763:
6759:
6756:
6753:
6748:
6741:
6737:
6733:
6728:
6723:
6720:
6717:
6712:
6705:
6701:
6675:
6672:
6667:
6662:
6659:
6637:
6630:
6626:
6609:
6606:
6585:
6580:
6558:
6553:
6523:
6518:
6513:
6510:
6473:
6470:
6467:
6464:
6461:
6458:
6455:
6444:
6443:
6430:
6426:
6422:
6419:
6416:
6411:
6407:
6403:
6398:
6394:
6370:
6365:
6360:
6357:
6331:
6326:
6321:
6318:
6307:
6306:
6295:
6290:
6286:
6282:
6279:
6276:
6273:
6270:
6267:
6262:
6258:
6254:
6251:
6248:
6245:
6240:
6236:
6232:
6229:
6226:
6223:
6218:
6214:
6210:
6207:
6204:
6199:
6195:
6191:
6186:
6182:
6178:
6175:
6152:
6130:
6126:
6105:
6100:
6095:
6092:
6089:
6084:
6079:
6076:
6056:
6053:
6050:
6047:
6044:
6018:
6015:
6011:
6007:
6004:
6001:
5998:
5993:
5989:
5968:
5965:
5962:
5959:
5954:
5950:
5917:
5914:
5893:
5888:
5883:
5880:
5865:Main article:
5862:
5859:
5838:
5837:Other algebras
5835:
5818:
5813:
5808:
5805:
5785:
5765:
5762:
5759:
5755:
5752:
5749:
5745:
5742:
5722:
5717:
5712:
5709:
5689:
5667:
5642:
5637:
5632:
5629:
5626:
5623:
5600:
5597:
5572:
5559:
5558:
5544:
5539:
5532:
5528:
5523:
5518:
5515:
5508:
5504:
5499:
5495:
5492:
5489:
5486:
5479:
5475:
5470:
5466:
5463:
5460:
5457:
5450:
5446:
5441:
5437:
5430:
5426:
5421:
5417:
5411:
5408:
5379:
5376:
5351:
5329:
5324:
5319:
5316:
5296:
5293:
5290:
5285:
5280:
5277:
5274:
5268:
5265:
5242:
5222:
5219:
5216:
5213:
5210:
5207:
5204:
5201:
5198:
5195:
5192:
5189:
5186:
5183:
5180:
5177:
5174:
5171:
5168:
5165:
5162:
5159:
5156:
5153:
5150:
5147:
5127:
5107:
5104:
5099:
5094:
5091:
5073:tensor algebra
5058:
5036:
5016:
5011:
5006:
5003:
5000:
4995:
4990:
4987:
4976:
4975:
4964:
4961:
4955:
4952:
4946:
4943:
4929:
4928:
4917:
4914:
4911:
4906:
4901:
4898:
4895:
4889:
4886:
4851:
4846:
4843:
4840:
4837:
4826:
4825:
4814:
4811:
4808:
4805:
4802:
4799:
4796:
4793:
4790:
4787:
4784:
4781:
4778:
4775:
4772:
4769:
4766:
4763:
4760:
4757:
4754:
4751:
4748:
4745:
4742:
4739:
4713:
4712:
4701:
4698:
4693:
4688:
4685:
4669:, possesses a
4658:
4653:
4648:
4645:
4642:
4637:
4632:
4629:
4617:
4614:
4553:
4552:
4541:
4538:
4535:
4532:
4529:
4526:
4523:
4520:
4517:
4514:
4511:
4508:
4505:
4502:
4499:
4496:
4493:
4490:
4487:
4484:
4481:
4478:
4475:
4472:
4469:
4466:
4454:and also that
4452:
4451:
4440:
4437:
4434:
4431:
4428:
4425:
4422:
4419:
4416:
4413:
4410:
4407:
4404:
4401:
4398:
4395:
4392:
4389:
4386:
4383:
4380:
4377:
4374:
4371:
4368:
4365:
4343:
4342:
4327:
4324:
4321:
4318:
4315:
4312:
4309:
4306:
4304:
4302:
4299:
4296:
4293:
4292:
4289:
4286:
4283:
4281:
4279:
4276:
4273:
4270:
4267:
4264:
4263:
4236:
4216:
4196:
4193:
4190:
4187:
4167:
4144:
4141:
4138:
4135:
4132:
4129:
4126:
4123:
4120:
4117:
4114:
4094:
4091:
4088:
4085:
4082:
4079:
4076:
4056:
4043:
4040:
4039:
4038:
4027:
4023:
4019:
4014:
4009:
4006:
4003:
4000:
3995:
3990:
3987:
3973:
3972:
3961:
3958:
3955:
3951:
3947:
3942:
3937:
3934:
3931:
3928:
3923:
3918:
3915:
3912:
3909:
3906:
3903:
3873:
3870:
3867:
3864:
3861:
3858:
3838:
3835:
3832:
3829:
3826:
3806:
3803:
3798:
3793:
3790:
3787:
3784:
3764:
3761:
3758:
3738:
3733:
3728:
3725:
3722:
3719:
3693:
3688:
3683:
3680:
3677:
3674:
3671:
3668:
3665:
3662:
3659:
3656:
3653:
3650:
3639:
3638:
3627:
3624:
3621:
3618:
3615:
3612:
3609:
3606:
3603:
3600:
3597:
3594:
3591:
3588:
3585:
3582:
3579:
3576:
3573:
3570:
3567:
3564:
3561:
3558:
3555:
3552:
3532:
3531:
3520:
3515:
3510:
3507:
3504:
3501:
3496:
3491:
3486:
3481:
3478:
3473:
3457:
3456:
3445:
3442:
3439:
3436:
3433:
3430:
3427:
3424:
3421:
3418:
3415:
3412:
3409:
3351:
3346:
3341:
3338:
3326:
3325:
3314:
3311:
3308:
3305:
3302:
3299:
3296:
3293:
3290:
3287:
3284:
3281:
3278:
3255:
3241:
3240:
3229:
3225:
3221:
3216:
3211:
3208:
3205:
3202:
3197:
3192:
3189:
3176:quotient space
3161:
3139:
3134:
3129:
3126:
3100:covering space
3087:
3082:
3077:
3074:
3054:
3051:
3046:
3041:
3038:
3026:
3025:
3014:
3011:
3008:
3005:
3002:
2999:
2996:
2993:
2990:
2987:
2984:
2981:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2934:
2933:
2922:
2919:
2916:
2913:
2910:
2907:
2904:
2901:
2898:
2895:
2892:
2889:
2886:
2883:
2880:
2877:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2830:
2827:
2818:for arbitrary
2807:
2802:
2797:
2792:
2788:
2765:
2760:
2755:
2750:
2747:
2742:
2737:
2732:
2728:
2716:
2715:
2704:
2701:
2698:
2695:
2692:
2689:
2686:
2683:
2680:
2677:
2674:
2671:
2668:
2643:
2638:
2633:
2628:
2623:
2594:
2589:
2584:
2581:
2578:
2575:
2570:
2565:
2562:
2559:
2556:
2551:
2546:
2543:
2528:skew-symmetric
2509:
2504:
2499:
2494:
2489:
2452:
2432:
2421:
2420:
2409:
2405:
2401:
2396:
2391:
2386:
2381:
2376:
2371:
2367:
2363:
2358:
2353:
2348:
2343:
2339:
2333:
2327:
2323:
2320:
2317:
2312:
2307:
2304:
2276:
2246:
2241:
2236:
2233:
2223:tensor algebra
2204:
2190:
2187:
2168:
2140:
2136:
2130:
2127:
2124:
2120:
2116:
2113:
2108:
2104:
2098:
2094:
2090:
2085:
2081:
2075:
2071:
2052:quotient space
2048:tensor product
2044:tensor algebra
2038:
2035:
2022:
2019:
2016:
1996:
1976:
1956:
1934:
1910:
1888:
1885:
1882:
1860:
1836:
1832:
1809:
1805:
1782:
1778:
1774:
1771:
1768:
1763:
1759:
1729:
1725:
1704:
1701:
1696:
1692:
1669:
1665:
1653:
1652:
1637:
1633:
1627:
1623:
1619:
1612:
1608:
1602:
1598:
1590:
1586:
1580:
1576:
1550:
1546:
1519:
1515:
1509:
1505:
1482:
1478:
1472:
1468:
1445:
1441:
1418:
1414:
1400:
1397:
1384:
1381:
1376:
1372:
1368:
1363:
1359:
1355:
1352:
1349:
1346:
1322:
1319:
1314:
1310:
1289:
1269:
1266:
1261:
1257:
1236:
1216:
1213:
1210:
1190:
1187:
1184:
1169:
1168:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1108:
1105:
1101:
1098:
1095:
1092:
1089:
1086:
1083:
1080:
1077:
1054:
1051:
1048:
1045:
1042:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
984:
981:
978:
975:
972:
969:
966:
963:
960:
940:
937:
934:
931:
928:
925:
922:
919:
908:
907:
896:
888:
882:
879:
876:
874:
871:
870:
867:
864:
862:
859:
858:
856:
851:
848:
842:
836:
833:
831:
828:
827:
824:
821:
819:
816:
815:
813:
808:
805:
799:
793:
790:
788:
785:
784:
781:
778:
776:
773:
772:
770:
765:
762:
730:
713:
712:
699:
695:
689:
686:
683:
679:
673:
668:
665:
662:
658:
654:
649:
645:
639:
635:
631:
626:
622:
616:
612:
586:
582:
578:
575:
570:
566:
554:
553:
542:
537:
533:
527:
524:
521:
517:
511:
506:
503:
500:
496:
492:
489:
484:
480:
476:
471:
467:
463:
434:
431:
428:
424:
401:
397:
393:
390:
385:
381:
358:
344:
341:
326:
302:
278:
254:
230:
208:
205:
202:
199:
196:
174:
150:
126:
112:
109:
107:on the group.
82:quantum groups
45:algebra whose
15:
9:
6:
4:
3:
2:
16753:
16742:
16739:
16737:
16736:Hopf algebras
16734:
16732:
16729:
16728:
16726:
16717:
16715:
16710:
16707:
16704:
16703:
16698:
16695:
16692:
16688:
16684:
16678:
16674:
16670:
16669:
16663:
16660:
16656:
16652:
16648:
16644:
16638:
16634:
16630:
16626:
16622:
16617:
16614:
16608:
16604:
16599:
16596:
16592:
16588:
16582:
16578:
16574:
16570:
16569:
16564:
16560:
16559:
16552:
16548:
16542:
16534:
16530:
16526:
16522:
16518:
16514:
16510:
16509:Berezin, F.A.
16504:
16497:
16496:Helgason 2001
16492:
16485:
16484:Helgason 2001
16479:
16472:
16467:
16460:
16455:
16446:
16441:
16437:
16433:
16429:
16422:
16413:
16408:
16404:
16400:
16396:
16392:
16385:
16378:
16373:
16366:
16361:
16357:
16347:
16344:
16342:
16339:
16338:
16332:
16331:, for short.
16330:
16329:quantum group
16326:
16322:
16318:
16313:
16300:
16282:
16278:
16244:
16234:
16227:
16222:
16203:
16195:
16192:
16188:
16181:
16178:
16172:
16160:
16154:
16144:
16143:
16142:
16122:
16116:
16113:
16107:
16101:
16094:
16093:
16092:
16075:
16069:
16066:
16060:
16057:
16051:
16048:
16045:
16033:
16017:
16016:
16015:
15995:
15989:
15983:
15977:
15974:
15968:
15956:
15953:
15950:
15934:
15933:
15932:
15915:
15909:
15906:
15903:
15900:
15897:
15889:
15884:
15877:
15870:
15864:
15862:
15858:
15857:Hopf algebras
15854:
15849:
15845:
15844:group algebra
15835:
15833:
15828:
15826:
15822:
15817:
15811:
15808:
15802:
15798:
15794:
15791:
15790:distributions
15787:
15758:
15749:
15747:
15742:
15712:
15703:
15701:
15696:
15666:
15664:
15663:vector fields
15635:
15630:
15601:
15592:
15588:
15559:
15556:
15552:algebra over
15551:
15522:
15491:
15487:
15458:
15446:
15417:
15413:
15394:
15391:
15388:
15385:
15382:
15379:
15376:
15373:
15370:
15367:
15364:
15361:
15358:
15355:
15352:
15349:
15346:
15343:
15340:
15337:
15334:
15331:
15328:
15325:
15322:
15311:
15308:
15305:
15302:
15299:
15285:
15277:
15259:
15250:
15246:
15233:
15230:
15227:
15221:
15218:
15215:
15192:
15189:
15186:
15180:
15177:
15174:
15151:
15148:
15145:
15142:
15136:
15133:
15130:
15118:
15114:
15099:
15093:
15088:
15081:
15076:
15070:
15065:
15062:
15054:
15049:
15043:
15038:
15031:
15026:
15020:
15015:
15012:
15004:
14999:
14993:
14988:
14981:
14976:
14973:
14967:
14962:
14959:
14950:
14934:
14919:
14898:
14896:
14892:
14888:
14884:
14880:
14876:
14872:
14868:
14864:
14860:
14855:
14851:
14849:
14845:
14816:
14814:
14785:
14752:
14722:
14718:
14712:
14705:
14702:
14698:
14694:
14686:
14682:
14678:
14673:
14669:
14658:
14657:
14656:
14655:
14636:
14633:
14625:
14621:
14617:
14612:
14608:
14597:
14596:
14595:
14576:
14572:
14568:
14563:
14559:
14555:
14550:
14546:
14542:
14537:
14533:
14529:
14524:
14520:
14516:
14511:
14507:
14503:
14498:
14494:
14490:
14482:
14475:
14467:
14466:
14465:
14463:
14445:
14442:
14438:
14434:
14429:
14426:
14422:
14398:
14390:
14386:
14382:
14377:
14373:
14369:
14364:
14360:
14353:
14348:
14344:
14340:
14337:
14333:
14329:
14326:
14323:
14318:
14314:
14310:
14307:
14302:
14298:
14294:
14291:
14286:
14282:
14278:
14274:
14263:
14262:
14261:
14245:
14241:
14237:
14234:
14229:
14225:
14221:
14218:
14213:
14209:
14205:
14183:
14179:
14158:
14153:
14149:
14145:
14140:
14136:
14115:
14110:
14106:
14096:
14086:
14073:
14054:
14048:
14039:
14032:
14015:
14012:
14006:
14000:
13995:
13991:
13957:
13954:
13951:
13918:
13909:
13906:
13889:
13884:
13881:
13878:
13874:
13870:
13865:
13862:
13859:
13855:
13834:
13831:
13828:
13825:
13822:
13802:
13777:
13774:
13771:
13768:
13764:
13758:
13754:
13750:
13745:
13741:
13735:
13731:
13727:
13721:
13713:
13709:
13701:
13700:
13699:
13680:
13676:
13670:
13666:
13660:
13655:
13652:
13649:
13645:
13641:
13638:
13631:
13630:
13629:
13612:
13606:
13598:
13594:
13588:
13585:
13582:
13578:
13574:
13568:
13565:
13557:
13553:
13545:
13544:
13543:
13541:
13525:
13520:
13516:
13511:
13508:
13503:
13500:
13496:
13475:
13472:
13469:
13459:
13456: â
13455:
13450:
13431:
13423:
13419:
13409:
13406:
13386:
13378:
13374:
13364:
13358:
13353:
13348:
13331:
13321:
13318:
13311:for elements
13293:
13289:
13282:
13274:
13270:
13264:
13259:
13256:
13253:
13249:
13245:
13237:
13233:
13229:
13226:
13223:
13210:
13209:
13208:
13205:
13186:
13182:
13174:implies that
13145:
13142:
13129:
13126:
13119:
13118:
13117:
13114:
13109:
13104:
13082:
13078:
13072:
13068:
13062:
13057:
13054:
13051:
13047:
13043:
13037:
13034:
13031:
13028:
13015:
13014:
13013:
13010:
13005:
13000:
12981:
12978:
12974:
12965:
12961:
12956:
12950:
12945:
12941:
12931:
12929:
12924:
12894:
12888:
12863:
12856:
12835:
12830:
12827:
12823:
12815:
12797:
12794:
12790:
12764:
12760:
12756:
12751:
12747:
12741:
12738:
12734:
12730:
12722:
12715:
12707:
12706:
12705:
12686:
12682:
12676:
12669:
12666:
12662:
12658:
12650:
12646:
12642:
12637:
12633:
12622:
12621:
12620:
12603:
12600:
12595:
12592:
12589:
12586:
12582:
12576:
12569:
12566:
12562:
12558:
12555:
12552:
12547:
12544:
12541:
12538:
12534:
12528:
12521:
12518:
12514:
12510:
12505:
12502:
12499:
12496:
12492:
12486:
12479:
12476:
12472:
12464:
12463:
12462:
12448:
12445:
12434:
12427:
12423:
12420:
12409:
12405:
12389:
12384:
12381:
12376:
12365:
12361:
12337:
12334:
12331:
12328:
12324:
12303:
12281:
12278:
12275:
12272:
12268:
12247:
12222:
12218:
12214:
12211:
12208:
12203:
12199:
12195:
12190:
12186:
12180:
12177:
12174:
12171:
12167:
12163:
12155:
12148:
12140:
12139:
12138:
12137:has the form
12124:
12102:
12098:
12089:
12071:
12067:
12058:
12053:
12040:
12024:
11991:
11969:
11940:
11930:
11927:
11905:
11869:
11863:
11857:
11854:
11851:
11848:
11845:
11842:
11839:
11836:
11833:
11830:
11828:
11820:
11817:
11814:
11811:
11805:
11799:
11796:
11793:
11790:
11788:
11780:
11777:
11774:
11771:
11768:
11765:
11759:
11753:
11747:
11741:
11738:
11735:
11732:
11729:
11726:
11723:
11717:
11706:
11705:
11704:
11703:
11687:
11634:
11628:
11625:
11622:
11616:
11610:
11607:
11601:
11595:
11589:
11580:
11577:
11574:
11565:
11558:
11557:
11556:
11554:
11553:Leibniz's law
11497:
11494:
11485:
11483:
11467:
11451:
11418:
11410:
11361:
11345:
11325:
11310:
11294:
11258:
11255:
11249:
11241:
11229:
11223:
11220:
11217:
11194:
11184:
11181:
11148:
11145:
11142:
11139:
11136:
11133:
11130:
11127:
11124:
11121:
11118:
11082:
11076:
11067:
11037:
11031:
11023:
11019:
10979:
10963:
10919:
10903:
10867:
10861:
10841:
10825:
10768:
10732:
10726:
10719:
10713:
10703:
10701:
10672:
10664:
10663:
10659:is acting by
10633:
10631:
10627:
10626:Verma modules
10622:
10620:
10616:
10611:
10609:
10605:
10601:
10597:
10564:
10551:
10548:
10541:
10540:
10539:
10517:
10500:
10496:
10492:
10489:
10486:
10483:
10480:
10472:
10455:
10451:
10447:
10439:
10427:
10422:
10407:
10400:
10399:
10398:
10382:
10370:
10365:
10328:
10313:
10310:
10302:
10287:
10284:
10276:
10264:
10259:
10244:
10237:
10236:
10235:
10233:
10229:
10224:
10197:
10189:
10161:
10157:
10128:
10120:
10092:
10082:
10080:
10079:Moyal product
10064:
10056:
10052:
10047:
10033:
10011:
10007:
9983:
9980:
9977:
9974:
9970:
9964:
9960:
9954:
9950:
9945:
9941:
9938:
9935:
9929:
9926:
9923:
9917:
9910:
9909:
9908:
9889:
9886:
9883:
9880:
9877:
9872:
9865:
9859:
9853:
9847:
9843:
9838:
9831:
9819:
9813:
9800:
9794:
9790:
9784:
9780:
9775:
9771:
9768:
9760:
9754:
9748:
9745:
9739:
9733:
9726:
9725:
9724:
9721:
9719:
9715:
9686:
9678:
9649:
9626:
9620:
9617:
9611:
9605:
9596:
9583:
9567:
9564:
9561:
9558:
9555:
9529:
9523:
9520:
9514:
9508:
9505:
9499:
9496:
9493:
9487:
9480:
9479:
9478:
9464:
9442:
9438:
9415:
9411:
9374:
9355:
9352:
9349:
9342:
9341:
9340:
9313:
9305:
9287:
9283:
9260:
9256:
9235:
9210:
9206:
9199:
9177:
9173:
9150:
9146:
9112:
9103:
9089:
9081:
9052:
9045:
9041:
9012:
8979:
8946:
8932:
8905:
8882:
8876:
8843:
8820:
8814:
8781:
8758:
8752:
8705:
8699:
8691:
8662:
8635:
8612:
8606:
8583:
8577:
8557:
8537:
8528:
8503:operation on
8465:
8373:
8359:
8357:
8353:
8350:, yields the
8349:
8339:
8325:
8309:
8276:
8273:
8270:
8262:
8233:
8214:
8194:
8178:
8145:
8136:
8134:
8118:
8102:
8073:
8069:
8059:
8057:
8028:
7995:
7969:
7966:
7963:
7960:
7957:
7954:
7951:
7944:
7927:
7924:
7908:
7901:
7900:
7899:
7872:
7845:
7840:
7836:
7806:
7802:
7793:
7788:
7765:
7761:
7731:
7727:
7697:
7694:
7691:
7687:
7657:
7653:
7643:
7620:
7617:
7614:
7610:
7580:
7576:
7546:
7542:
7512:
7509:
7506:
7502:
7472:
7468:
7458:
7435:
7431:
7422:
7397:
7393:
7363:
7359:
7329:
7325:
7292:
7289:
7286:
7282:
7277:
7264:
7260:
7256:
7244:
7240:
7232:
7231:
7230:
7227:
7214:
7198:
7169:
7165:
7144:
7128:
7095:
7087:
7082:
7069:
7053:
7045:
7041:
7022:
7019:
7007:
7003:
6999:
6996:
6993:
6981:
6977:
6973:
6963:
6960:
6953:
6952:
6951:
6949:
6924:
6920:
6899:
6883:
6856:
6853:
6850:
6840:
6828:
6825:
6821:
6817:
6805:
6801:
6793:
6792:
6791:
6765:
6761:
6757:
6754:
6751:
6739:
6735:
6731:
6721:
6718:
6715:
6703:
6699:
6691:
6690:
6689:
6686:
6673:
6657:
6628:
6624:
6616:
6605:
6603:
6599:
6583:
6556:
6541:
6536:
6508:
6500:
6495:
6493:
6488:
6471:
6468:
6465:
6462:
6459:
6456:
6453:
6428:
6424:
6420:
6417:
6414:
6409:
6405:
6401:
6396:
6392:
6384:
6383:
6382:
6355:
6346:
6316:
6288:
6284:
6277:
6274:
6271:
6268:
6260:
6256:
6249:
6246:
6238:
6234:
6227:
6224:
6216:
6212:
6208:
6205:
6202:
6197:
6193:
6189:
6184:
6180:
6173:
6166:
6165:
6164:
6150:
6128:
6124:
6090:
6077:
6074:
6054:
6051:
6048:
6045:
6042:
6034:
6016:
6013:
6009:
6005:
5999:
5991:
5987:
5966:
5960:
5957:
5952:
5948:
5938:
5934:to the field
5932:
5927:
5923:
5913:
5911:
5907:
5878:
5868:
5858:
5856:
5852:
5848:
5844:
5834:
5832:
5803:
5783:
5760:
5743:
5740:
5707:
5687:
5654:
5627:
5624:
5621:
5598:
5595:
5537:
5530:
5526:
5521:
5516:
5506:
5502:
5497:
5490:
5487:
5477:
5473:
5468:
5461:
5458:
5448:
5444:
5439:
5435:
5428:
5424:
5419:
5409:
5406:
5396:
5395:
5394:
5377:
5374:
5314:
5294:
5275:
5272:
5266:
5263:
5240:
5217:
5211:
5205:
5199:
5196:
5190:
5184:
5178:
5172:
5169:
5160:
5157:
5154:
5145:
5125:
5105:
5092:
5089:
5080:
5078:
5074:
5034:
5001:
4988:
4985:
4962:
4959:
4953:
4950:
4944:
4941:
4934:
4933:
4932:
4915:
4896:
4893:
4887:
4884:
4874:
4873:
4872:
4871:
4867:
4844:
4841:
4838:
4835:
4809:
4803:
4797:
4791:
4788:
4782:
4776:
4770:
4764:
4761:
4752:
4749:
4746:
4737:
4730:
4729:
4728:
4725:
4719:
4699:
4686:
4683:
4676:
4675:
4674:
4672:
4643:
4630:
4627:
4613:
4611:
4607:
4603:
4598:
4596:
4592:
4588:
4584:
4580:
4576:
4571:
4569:
4564:
4562:
4558:
4536:
4533:
4530:
4524:
4521:
4518:
4515:
4512:
4509:
4503:
4500:
4497:
4491:
4488:
4482:
4479:
4476:
4473:
4470:
4464:
4457:
4456:
4455:
4438:
4435:
4429:
4426:
4423:
4417:
4414:
4408:
4405:
4402:
4396:
4393:
4390:
4387:
4381:
4378:
4375:
4372:
4369:
4363:
4356:
4355:
4354:
4353:, one writes
4352:
4351:By definition
4348:
4322:
4319:
4316:
4310:
4305:
4300:
4297:
4294:
4287:
4282:
4277:
4274:
4271:
4268:
4265:
4254:
4253:
4252:
4250:
4234:
4214:
4191:
4185:
4165:
4157:
4142:
4136:
4133:
4130:
4124:
4118:
4115:
4112:
4092:
4086:
4083:
4080:
4077:
4074:
4054:
4025:
4021:
4004:
4001:
3985:
3978:
3977:
3976:
3959:
3953:
3949:
3932:
3913:
3907:
3901:
3894:
3893:
3892:
3891:
3887:
3871:
3868:
3865:
3862:
3859:
3856:
3836:
3833:
3830:
3827:
3824:
3804:
3788:
3785:
3782:
3762:
3759:
3756:
3723:
3720:
3717:
3708:
3691:
3681:
3678:
3675:
3672:
3669:
3666:
3663:
3660:
3657:
3654:
3651:
3648:
3625:
3622:
3619:
3616:
3613:
3604:
3601:
3598:
3592:
3589:
3586:
3583:
3580:
3577:
3574:
3571:
3565:
3562:
3559:
3556:
3553:
3550:
3543:
3542:
3541:
3538:
3505:
3502:
3489:
3476:
3462:
3461:
3460:
3440:
3437:
3434:
3428:
3425:
3422:
3419:
3416:
3413:
3410:
3407:
3400:
3399:
3398:
3395:
3391:
3386:
3384:
3383:string theory
3380:
3376:
3371:
3369:
3365:
3336:
3309:
3306:
3303:
3297:
3294:
3291:
3288:
3285:
3282:
3279:
3276:
3269:
3268:
3267:
3253:
3246:
3227:
3223:
3206:
3203:
3187:
3180:
3179:
3178:
3177:
3124:
3116:
3112:
3108:
3103:
3101:
3072:
3052:
3036:
3009:
3006:
3003:
2997:
2994:
2991:
2988:
2985:
2979:
2976:
2973:
2967:
2961:
2958:
2955:
2952:
2949:
2939:
2938:
2937:
2936:and likewise
2920:
2917:
2911:
2908:
2905:
2899:
2893:
2890:
2887:
2881:
2878:
2875:
2869:
2866:
2863:
2860:
2857:
2847:
2846:
2845:
2844:
2828:
2825:
2790:
2786:
2758:
2748:
2730:
2726:
2699:
2696:
2693:
2687:
2684:
2681:
2678:
2675:
2672:
2669:
2666:
2659:
2658:
2657:
2626:
2611:
2606:
2579:
2560:
2557:
2541:
2533:
2529:
2525:
2492:
2475:
2471:
2466:
2450:
2430:
2407:
2403:
2389:
2379:
2365:
2351:
2337:
2325:
2321:
2318:
2302:
2295:
2294:
2293:
2290:
2264:
2260:
2231:
2224:
2220:
2186:
2184:
2156:
2138:
2134:
2128:
2125:
2122:
2118:
2111:
2106:
2102:
2096:
2092:
2088:
2083:
2079:
2073:
2069:
2060:
2057:
2053:
2049:
2045:
2034:
2020:
2017:
2014:
1994:
1974:
1954:
1886:
1883:
1880:
1834:
1830:
1807:
1803:
1780:
1776:
1772:
1769:
1766:
1761:
1757:
1747:
1745:
1727:
1723:
1702:
1699:
1694:
1690:
1667:
1663:
1635:
1631:
1625:
1621:
1617:
1610:
1606:
1600:
1596:
1588:
1584:
1578:
1574:
1566:
1565:
1564:
1548:
1544:
1535:
1517:
1513:
1507:
1503:
1480:
1476:
1470:
1466:
1443:
1439:
1416:
1412:
1396:
1382:
1379:
1374:
1370:
1366:
1361:
1357:
1353:
1350:
1347:
1344:
1336:
1320:
1317:
1312:
1308:
1287:
1267:
1264:
1259:
1255:
1234:
1214:
1211:
1208:
1188:
1185:
1182:
1174:
1155:
1152:
1149:
1146:
1143:
1140:
1137:
1134:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1109:
1106:
1103:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1068:
1067:
1066:
1052:
1049:
1046:
1043:
1040:
1020:
1017:
1011:
1008:
1005:
982:
979:
976:
973:
967:
964:
961:
938:
935:
932:
926:
923:
920:
894:
886:
880:
877:
872:
865:
860:
854:
849:
846:
840:
834:
829:
822:
817:
811:
806:
803:
797:
791:
786:
779:
774:
768:
763:
760:
753:
752:
751:
749:
744:
718:
697:
693:
687:
684:
681:
677:
671:
666:
663:
660:
656:
652:
647:
643:
637:
633:
629:
624:
620:
614:
610:
602:
601:
600:
584:
580:
576:
573:
568:
564:
540:
535:
531:
525:
522:
519:
515:
509:
504:
501:
498:
494:
490:
482:
478:
474:
469:
465:
454:
453:
452:
450:
432:
429:
426:
422:
399:
395:
391:
388:
383:
379:
340:
206:
203:
200:
197:
194:
108:
106:
101:
99:
95:
91:
87:
83:
79:
75:
71:
67:
63:
62:Verma modules
59:
54:
52:
48:
44:
41:
37:
33:
29:
22:
16713:
16702:Lie algebras
16700:
16667:
16620:
16602:
16567:
16550:
16541:
16516:
16512:
16503:
16491:
16478:
16473:Theorem 9.10
16466:
16454:
16435:
16431:
16421:
16394:
16390:
16384:
16372:
16360:
16324:
16314:
16232:
16225:
16218:
16140:
16090:
16013:
15882:
15875:
15868:
15865:
15846:for a given
15841:
15829:
15825:Weyl algebra
15815:
15812:
15806:
15800:
15795:only at the
15750:
15740:
15704:
15694:
15667:
15633:
15590:
15560:
15554:
15490:vector space
15444:
15418:
15415:
15252:
15248:
15120:
15116:
14952:
14904:
14856:
14852:
14817:
14744:
14651:
14593:
14413:
14092:
14037:
14030:
13910:
13904:
13794:
13697:
13627:
13457:
13453:
13410:
13404:
13362:
13356:
13346:
13310:
13203:
13173:
13112:
13102:
13099:
13008:
12998:
12954:
12948:
12937:
12925:
12814:Killing form
12781:
12703:
12618:
12239:
12054:
11888:
11701:
11655:
11551:if it obeys
11486:
11444:and thus to
11068:
10715:
10660:
10634:
10623:
10612:
10592:
10537:
10348:
10225:
10088:
10055:Weyl algebra
10048:
9998:
9906:
9722:
9713:
9597:
9547:
9402:
9303:
9104:
9043:
9039:
8938:
8663:
8529:
8365:
8355:
8345:
8137:
8131:This is the
8060:
8055:
7987:
7789:
7644:
7460:Quotienting
7459:
7420:
7316:
7228:
7083:
7037:
6881:
6878:
6789:
6687:
6611:
6602:Lyndon words
6537:
6496:
6486:
6445:
6381:has a basis
6344:
6308:
5936:
5930:
5919:
5906:vector basis
5870:
5840:
5655:
5560:
5081:
5077:quotient map
4977:
4930:
4865:
4827:
4723:
4717:
4714:
4619:
4606:Bol algebras
4599:
4585:; it is the
4572:
4565:
4561:free algebra
4557:free objects
4554:
4453:
4350:
4347:homomorphism
4344:
4248:
4155:
4045:
3974:
3706:
3640:
3536:
3533:
3458:
3393:
3387:
3379:Hopf algebra
3372:
3367:
3327:
3266:is given by
3242:
3114:
3104:
3027:
2935:
2842:
2717:
2609:
2607:
2476:
2469:
2422:
2291:
2259:free algebra
2219:vector space
2192:
2055:
2040:
1748:
1742:occur.) The
1654:
1402:
1172:
1170:
909:
745:
716:
714:
555:
346:
114:
102:
55:
31:
25:
16731:Ring theory
16461:Theorem 9.7
16397:: 379â393.
16379:Section 9.3
16367:Section 9.5
15788:algebra of
15786:convolution
15705:The center
13911:The center
13795:that is, a
13352:root system
13350:) form the
12960:determinant
11069:The center
5138:satisfying
2183:Lie bracket
2037:Formalities
43:associative
36:Lie algebra
28:mathematics
16725:Categories
16691:1255.17001
16352:References
15888:C*-algebra
15550:polynomial
14863:Riemannian
13983:for which
13451:of degree
11984:acting on
11409:derivation
10710:See also:
10608:9j-symbols
10188:isomorphic
7421:not at all
6948:filtration
6615:filtration
6598:Hall words
5979:such that
5585:, the map
5364:, the map
4931:such that
3243:where the
3109:. It is a
2465:direct sum
1247:satisfies
90:C* algebra
16659:120016227
16565:(1996) ,
16549:" (2005)
16533:122356554
16519:(2): 91.
16471:Hall 2015
16459:Hall 2015
16377:Hall 2015
16365:Hall 2015
16283:∗
16219:Now, the
16193:−
16182:φ
16161:φ
16117:φ
16108:φ
16102:ε
16061:φ
16049:⊗
16034:φ
16028:Δ
15990:ψ
15978:φ
15957:ψ
15951:φ
15945:∇
15907:∈
15904:ψ
15898:φ
15793:supported
15587:Lie group
15383:−
15365:−
15356:−
15329:−
15315:⟩
15297:⟨
15231:−
15146:−
14974:−
14879:Laplacian
14875:SO (P, Q)
14859:isometric
14699:ε
14569:⊗
14543:⊗
14517:⊗
14439:δ
14423:κ
14354:−
14341:−
14324:−
14001:
13955:∈
13832:−
13803:κ
13775:⋯
13765:κ
13751:⋯
13646:∑
13586:−
13473:∈
13322:∈
13250:∑
13230:−
13146:
13140:→
13048:∑
13035:−
12979:⊗
12824:κ
12791:κ
12757:⊗
12735:κ
12593:⋯
12583:κ
12556:⋯
12545:⋯
12535:κ
12503:⋯
12493:κ
12382:⊗
12335:⋯
12325:κ
12279:⋯
12269:κ
12215:⊗
12212:⋯
12209:⊗
12196:⊗
12178:⋯
12168:κ
11931:∈
11858:δ
11855:⊗
11852:⋯
11849:⊗
11843:⊗
11834:⋯
11818:⊗
11815:⋯
11812:⊗
11800:δ
11797:⊗
11778:⊗
11775:⋯
11772:⊗
11766:⊗
11754:δ
11739:⊗
11736:⋯
11733:⊗
11727:⊗
11718:δ
11629:δ
11596:δ
11566:δ
11508:→
11495:δ
11185:∈
11146:∈
11140:⊗
11137:⋯
11134:⊗
11128:⊗
10698:act like
10562:→
10493:⊗
10487:⊕
10481:⊗
10428:⊕
10311:⊗
10285:≅
10265:⊕
10065:⋆
9981:−
9975:−
9942:
9829:∂
9825:∂
9811:∂
9807:∂
9772:
9746:⋆
9618:⊗
9568:⋆
9565:∈
9521:⊗
9497:⋆
9465:⋆
9372:→
9356:⋆
9090:⋆
9053:⋆
8274:
8231:→
7967:⊗
7961:−
7955:⊗
7846:
7695:−
7618:−
7510:−
7290:−
7023:⋯
7020:⊂
7000:⊂
6997:⋯
6994:⊂
6974:⊂
6964:⊂
6857:⊗
6854:⋯
6851:⊗
6826:⊗
6758:⊕
6755:⋯
6752:⊕
6732:⊕
6722:⊕
6469:≤
6466:⋯
6463:≤
6457:≤
6421:⊗
6418:⋯
6415:⊗
6402:⊗
6275:⊗
6272:⋯
6269:⊗
6247:⊗
6209:⊗
6206:⋯
6203:⊗
6190:⊗
6088:→
6052:∈
6010:δ
5964:→
5912:fashion.
5625:∈
5599:^
5596:φ
5538:∈
5491:φ
5488:⋯
5462:φ
5436:⋯
5410:^
5407:φ
5378:^
5375:φ
5292:→
5267:^
5264:φ
5241:φ
5212:φ
5200:φ
5197:−
5185:φ
5173:φ
5146:φ
5103:→
5093::
5090:φ
5071:into its
4999:→
4960:∘
4954:^
4951:φ
4942:φ
4913:→
4894::
4888:^
4885:φ
4845:∈
4804:φ
4792:φ
4789:−
4777:φ
4765:φ
4738:φ
4697:→
4684:φ
4641:→
4631::
4591:naturally
4522:⊗
4510:⊗
4480:⊗
4436:⊗
4394:⊗
4373:⊗
4308:↦
4298:⊗
4285:→
4275:⊗
4122:↦
4116:×
4090:→
4084:×
3957:→
3930:→
3911:→
3905:→
3866:∈
3860:⊗
3834:∈
3828:⊗
3786:∈
3760:∈
3721:⊂
3682:∈
3641:for some
3626:⋯
3620:⊗
3614:⊗
3593:−
3587:⊗
3581:−
3575:⊗
3566:⊗
3563:⋯
3560:⊗
3554:⊗
3503:⊂
3490:⊗
3477:⊕
3429:−
3423:⊗
3417:−
3411:⊗
3292:⊗
3286:−
3280:⊗
3254:∼
3228:∼
2998:⊗
2986:⊗
2959:⊗
2918:⊗
2882:⊗
2861:⊗
2759:⊗
2682:⊗
2676:−
2670:⊗
2637:→
2627:⊗
2577:→
2558:⊗
2503:→
2493:×
2451:⊕
2431:⊗
2408:⋯
2404:⊕
2390:⊗
2380:⊗
2366:⊕
2352:⊗
2338:⊕
2326:⊕
2115:Σ
2112:−
2089:−
2018:×
1884:×
1770:…
1655:with the
1618:⋯
1383:…
1212:×
1186:×
1141:−
1122:−
1110:−
1082:−
977:−
878:−
657:∑
630:−
577:…
495:∑
392:…
201:−
16699:(2004),
16335:See also
15451:), then
14028:for all
11702:defining
10026:is just
8366:Suppose
5733:. (Take
5307:. Since
4828:for all
3368:smallest
3115:smallest
2843:defining
2610:defining
2524:bilinear
2522:that is
2056:smallest
96:between
16711:at the
16651:1834454
16595:0498740
15784:as the
15637:; with
15594:, then
15488:of the
15445:abelian
12962:is the
10628:of the
10610:, etc.
10158:of all
10119:modules
10053:is the
7679:lie in
6946:form a
6879:is the
6031:is the
5233:, then
4868:unital
4249:exactly
2463:is the
2181:with a
1536:of the
1532:plus a
1335:§ below
1227:matrix
748:sl(2,C)
447:be the
38:is the
16689:
16679:
16657:
16649:
16639:
16609:
16593:
16583:
16531:
15823:. See
15205:, and
15059:
15009:
14889:. The
13100:For a
12782:where
11889:Since
11656:(When
11020:, the
10718:center
10538:where
9907:where
9304:symbol
6790:where
6446:where
6163:to be
6067:. Let
4978:where
4866:unique
4178:up to
3886:kernel
2423:where
1173:is not
995:, and
892:
414:. Let
80:, the
40:unital
30:, the
16655:S2CID
16529:S2CID
16482:E.g.
16327:, or
15848:group
15486:basis
14871:SO(N)
14171:with
13201:is a
11407:is a
10956:into
10121:over
8898:with
7044:limit
3817:then
3364:coset
34:of a
16677:ISBN
16637:ISBN
16607:ISBN
16581:ISBN
14873:and
14093:The
13447:are
13411:The
12934:Rank
12017:and
10716:The
10606:and
9999:and
9275:for
9005:and
8061:The
6912:The
6035:for
4608:and
3849:and
3775:and
1987:and
715:and
347:Let
74:dual
16716:Lab
16687:Zbl
16629:doi
16521:doi
16440:doi
16436:284
16407:hdl
16399:doi
16395:272
15804:of
15561:If
15443:is
15419:If
14905:If
14865:or
14271:det
13538:By
13218:det
13143:End
13110:is
13023:det
13012:as
13006:of
12966:on
11311:on
10818:in
10761:of
10186:is
10162:of
10093:of
9939:log
9769:exp
9430:by
8056:and
7837:Sym
7494:by
7419:is
7121:to
3150:of
2656:as
2007:as
219:in
26:In
16727::
16685:,
16675:,
16653:,
16647:MR
16645:,
16635:,
16627:,
16591:MR
16589:,
16579:,
16571:,
16527:.
16515:.
16434:.
16430:.
16405:.
16393:.
16231:C(
15874:C(
15834:.
15810:.
15164:,
14850:.
13992:ad
13875:ad
13856:ad
13517:ad
13497:ad
13234:ad
13183:ad
13127:ad
12930:.
12366:ad
11964:ad
11900:ad
11555::
11484:.
11385:ad
11282:ad
11259:0.
11236:ad
10632:.
10621:.
10604:6j
10234::
10223:.
10081:.
8661:.
8356:an
8271:gr
6950::
5857:.
4612:.
4604:,
4349:.
4156:If
2526:,
1967:,
951:,
743:.
339:.
16714:n
16631::
16553:.
16535:.
16523::
16517:1
16448:.
16442::
16415:.
16409::
16401::
16301:.
16298:)
16293:g
16288:(
16279:U
16258:)
16253:g
16248:(
16245:U
16235:)
16233:G
16226:G
16204:.
16201:)
16196:1
16189:x
16185:(
16179:=
16176:)
16173:x
16170:(
16167:)
16164:)
16158:(
16155:S
16152:(
16126:)
16123:e
16120:(
16114:=
16111:)
16105:(
16076:,
16073:)
16070:y
16067:x
16064:(
16058:=
16055:)
16052:y
16046:x
16043:(
16040:)
16037:)
16031:(
16025:(
15999:)
15996:x
15993:(
15987:)
15984:x
15981:(
15975:=
15972:)
15969:x
15966:(
15963:)
15960:)
15954:,
15948:(
15942:(
15919:)
15916:G
15913:(
15910:C
15901:,
15883:G
15878:)
15876:G
15869:G
15816:n
15807:G
15801:e
15772:)
15767:g
15762:(
15759:U
15741:G
15726:)
15721:g
15716:(
15713:Z
15695:V
15678:g
15647:g
15634:G
15615:)
15610:g
15605:(
15602:U
15591:G
15571:g
15555:K
15536:)
15531:g
15526:(
15523:U
15501:g
15472:)
15467:g
15462:(
15459:U
15449:0
15429:g
15398:)
15395:x
15392:+
15389:y
15386:z
15380:z
15377:y
15374:,
15371:z
15368:2
15362:x
15359:z
15353:z
15350:x
15347:,
15344:y
15341:2
15338:+
15335:x
15332:y
15326:y
15323:x
15320:(
15312:z
15309:,
15306:y
15303:,
15300:x
15293:C
15286:=
15283:)
15278:2
15272:l
15269:s
15263:(
15260:U
15234:H
15228:=
15225:]
15222:F
15219:,
15216:E
15213:[
15193:F
15190:2
15187:=
15184:]
15181:F
15178:,
15175:H
15172:[
15152:E
15149:2
15143:=
15140:]
15137:E
15134:,
15131:H
15128:[
15100:)
15094:0
15089:1
15082:0
15077:0
15071:(
15066:=
15063:F
15055:,
15050:)
15044:0
15039:0
15032:1
15027:0
15021:(
15016:=
15013:E
15005:,
15000:)
14994:1
14989:0
14982:0
14977:1
14968:(
14963:=
14960:H
14935:2
14929:l
14926:s
14920:=
14915:g
14828:g
14799:)
14794:g
14789:(
14786:U
14766:)
14761:g
14756:(
14753:U
14723:k
14719:e
14713:k
14706:j
14703:i
14695:=
14692:]
14687:j
14683:e
14679:,
14674:i
14670:e
14666:[
14637:0
14634:=
14631:]
14626:k
14622:e
14618:,
14613:2
14609:L
14605:[
14577:3
14573:e
14564:3
14560:e
14556:+
14551:2
14547:e
14538:2
14534:e
14530:+
14525:1
14521:e
14512:1
14508:e
14504:=
14499:2
14495:L
14491:=
14486:)
14483:2
14480:(
14476:C
14446:j
14443:i
14435:=
14430:j
14427:i
14399:t
14396:)
14391:2
14387:z
14383:+
14378:2
14374:y
14370:+
14365:2
14361:x
14357:(
14349:3
14345:t
14338:=
14334:)
14330:I
14327:t
14319:3
14315:L
14311:z
14308:+
14303:2
14299:L
14295:y
14292:+
14287:1
14283:L
14279:x
14275:(
14246:3
14242:L
14238:z
14235:+
14230:2
14226:L
14222:y
14219:+
14214:1
14210:L
14206:x
14184:i
14180:L
14159:,
14154:i
14150:L
14146:=
14141:i
14137:e
14116:.
14111:1
14107:A
14074:.
14071:)
14068:)
14063:g
14058:(
14055:U
14052:(
14049:Z
14038:r
14033:;
14031:x
14016:0
14013:=
14010:)
14007:z
14004:(
13996:x
13971:)
13966:g
13961:(
13958:Z
13952:z
13932:)
13927:g
13922:(
13919:Z
13907:.
13905:m
13890:,
13885:x
13882:+
13879:y
13871:=
13866:y
13863:+
13860:x
13835:n
13829:d
13826:=
13823:m
13778:c
13772:b
13769:a
13759:c
13755:x
13746:b
13742:x
13736:a
13732:x
13728:=
13725:)
13722:x
13719:(
13714:n
13710:p
13681:i
13677:e
13671:i
13667:x
13661:d
13656:1
13653:=
13650:i
13642:=
13639:x
13613:.
13610:)
13607:x
13604:(
13599:n
13595:p
13589:n
13583:d
13579:k
13575:=
13572:)
13569:x
13566:k
13563:(
13558:n
13554:p
13526:.
13521:x
13512:k
13509:=
13504:x
13501:k
13476:K
13470:k
13460:.
13458:n
13454:d
13435:)
13432:x
13429:(
13424:n
13420:p
13407:.
13405:r
13390:)
13387:x
13384:(
13379:n
13375:p
13363:n
13357:r
13347:x
13332:.
13327:g
13319:x
13294:n
13290:t
13286:)
13283:x
13280:(
13275:n
13271:p
13265:d
13260:0
13257:=
13254:n
13246:=
13243:)
13238:x
13227:I
13224:t
13221:(
13204:d
13187:x
13159:)
13154:g
13149:(
13135:g
13130::
13113:d
13103:d
13083:n
13079:t
13073:n
13069:p
13063:d
13058:0
13055:=
13052:n
13044:=
13041:)
13038:M
13032:I
13029:t
13026:(
13009:M
12999:M
12982:d
12975:V
12955:V
12949:d
12911:)
12908:)
12903:g
12898:(
12895:U
12892:(
12889:Z
12867:)
12864:2
12861:(
12857:C
12836:.
12831:j
12828:i
12798:j
12795:i
12765:j
12761:e
12752:i
12748:e
12742:j
12739:i
12731:=
12726:)
12723:2
12720:(
12716:C
12687:k
12683:e
12677:k
12670:j
12667:i
12663:f
12659:=
12656:]
12651:j
12647:e
12643:,
12638:i
12634:e
12630:[
12604:0
12601:=
12596:j
12590:l
12587:k
12577:m
12570:j
12567:i
12563:f
12559:+
12553:+
12548:m
12542:j
12539:k
12529:l
12522:j
12519:i
12515:f
12511:+
12506:m
12500:l
12497:j
12487:k
12480:j
12477:i
12473:f
12449:0
12446:=
12443:]
12438:)
12435:m
12432:(
12428:C
12424:,
12421:x
12418:[
12390:.
12385:m
12377:)
12371:g
12362:(
12338:c
12332:b
12329:a
12304:m
12282:c
12276:b
12273:a
12248:m
12223:c
12219:e
12204:b
12200:e
12191:a
12187:e
12181:c
12175:b
12172:a
12164:=
12159:)
12156:m
12153:(
12149:C
12125:m
12103:a
12099:e
12072:a
12068:e
12041:.
12038:)
12033:g
12028:(
12025:U
12005:)
12000:g
11995:(
11992:T
11970:x
11941:,
11936:g
11928:x
11906:x
11870:.
11867:)
11864:u
11861:(
11846:w
11840:v
11837:+
11831:+
11821:u
11809:)
11806:w
11803:(
11794:v
11791:+
11781:u
11769:w
11763:)
11760:v
11757:(
11748:=
11745:)
11742:u
11730:w
11724:v
11721:(
11688:G
11666:g
11641:]
11638:)
11635:w
11632:(
11626:,
11623:v
11620:[
11617:+
11614:]
11611:w
11608:,
11605:)
11602:v
11599:(
11593:[
11590:=
11587:)
11584:]
11581:w
11578:,
11575:v
11572:[
11569:(
11537:g
11513:g
11503:g
11498::
11468:.
11465:)
11460:g
11455:(
11452:U
11432:)
11427:g
11422:(
11419:T
11392:g
11362:.
11359:)
11354:g
11349:(
11346:U
11326:;
11321:g
11295:.
11289:g
11256:=
11253:)
11250:z
11247:(
11242:x
11230:=
11227:]
11224:x
11221:,
11218:z
11215:[
11195:;
11190:g
11182:x
11162:)
11157:g
11152:(
11149:U
11143:u
11131:w
11125:v
11122:=
11119:z
11099:)
11096:)
11091:g
11086:(
11083:U
11080:(
11077:Z
11054:)
11051:)
11046:g
11041:(
11038:U
11035:(
11032:Z
11002:g
10980:.
10977:)
10972:g
10967:(
10964:U
10942:g
10920:,
10917:)
10912:g
10907:(
10904:U
10884:)
10881:)
10876:g
10871:(
10868:U
10865:(
10862:Z
10842:.
10839:)
10834:g
10829:(
10826:U
10804:g
10782:)
10777:g
10772:(
10769:U
10749:)
10746:)
10741:g
10736:(
10733:U
10730:(
10727:Z
10686:)
10681:g
10676:(
10673:U
10645:g
10578:)
10573:g
10568:(
10565:U
10557:g
10552::
10549:i
10523:)
10518:2
10512:g
10506:(
10501:2
10497:i
10490:1
10484:1
10478:)
10473:1
10467:g
10461:(
10456:1
10452:i
10448:=
10445:)
10440:2
10434:g
10423:1
10417:g
10411:(
10408:i
10383:2
10377:g
10371:,
10366:1
10360:g
10334:)
10329:2
10323:g
10317:(
10314:U
10308:)
10303:1
10297:g
10291:(
10288:U
10282:)
10277:2
10271:g
10260:1
10254:g
10248:(
10245:U
10211:)
10206:g
10201:(
10198:U
10172:g
10142:)
10137:g
10132:(
10129:U
10103:g
10034:m
10012:i
10008:m
9984:B
9978:A
9971:)
9965:B
9961:e
9955:A
9951:e
9946:(
9936:=
9933:)
9930:B
9927:,
9924:A
9921:(
9918:m
9890:t
9887:=
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5967:K
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5218:X
5215:(
5209:)
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5182:)
5179:X
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5170:=
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5152:[
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4801:)
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4786:)
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4774:)
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4768:(
4762:=
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4744:[
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4724:A
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3301:[
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3277:a
3224:/
3220:)
3215:g
3210:(
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3138:)
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3045:g
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3037:T
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3001:[
2995:b
2992:+
2989:c
2983:]
2980:b
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2971:[
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2965:]
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2950:a
2947:[
2921:b
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2855:[
2829:.
2826:n
2806:)
2801:g
2796:(
2791:n
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2749:=
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2741:g
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2731:2
2727:T
2703:]
2700:b
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2691:[
2688:=
2685:a
2679:b
2673:b
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2642:g
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2593:)
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2583:(
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2370:(
2362:)
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2245:)
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1804:x
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1724:x
1703:0
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1345:1
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971:]
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933:=
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850:=
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830:1
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634:x
625:j
621:x
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491:=
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479:X
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470:i
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462:[
433:k
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325:g
301:A
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229:A
207:x
204:y
198:y
195:x
173:g
149:A
125:g
23:.
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