3488:
3058:
3483:{\displaystyle {\begin{aligned}\Delta (v_{1}\otimes \cdots \otimes v_{m})&=\Delta (v_{1})\otimes \cdots \otimes \Delta (v_{m})\\&=\sum _{p=0}^{m}\left(v_{1}\otimes \cdots \otimes v_{p}\right)\;\omega \;\left(v_{p+1}\otimes \cdots \otimes v_{m}\right)\\&=\sum _{p=0}^{m}\;\sum _{\sigma \in \mathrm {Sh} (p,m-p)}\;\left(v_{\sigma (1)}\otimes \dots \otimes v_{\sigma (p)}\right)\boxtimes \left(v_{\sigma (p+1)}\otimes \dots \otimes v_{\sigma (m)}\right)\end{aligned}}}
9085:
3796:
7633:
4996:
7351:
3539:
2867:
4232:
7388:
4794:
7026:
1549:
obey the required consistency conditions for the definition of a bialgebra and Hopf algebra; this can be explicitly checked in the manner below. Whenever one has a product obeying these consistency conditions, the construction goes through; insofar as such a product gave rise to a quotient space, the
8041:
6308:
3791:{\displaystyle {\begin{aligned}\operatorname {Sh} (p,q)=\{\sigma :\{1,\dots ,p+q\}\to \{1,\dots ,p+q\}\;\mid \;&\sigma {\text{ is bijective}},\;\sigma (1)<\sigma (2)<\cdots <\sigma (p),\\&{\text{and }}\;\sigma (p+1)<\sigma (p+2)<\cdots <\sigma (m)\}.\end{aligned}}}
7170:
2451:
7133:
2686:
1359:; a sign must also be kept track of, when permuting elements of the exterior algebra. This correspondence also lasts through the definition of the bialgebra, and on to the definition of a Hopf algebra. That is, the exterior algebra can also be given a Hopf algebra structure.
4026:
7628:{\displaystyle {\begin{aligned}(\nabla \circ (S\boxtimes \mathrm {id} )\circ \Delta )(v)&=(\nabla \circ (S\boxtimes \mathrm {id} ))(v\boxtimes 1+1\boxtimes v)\\&=\nabla (-v\boxtimes 1+1\boxtimes v)\\&=-v\otimes 1+1\otimes v\\&=-v+v\\&=0\end{aligned}}}
5797:
5626:
4783:
486:
2253:
4991:{\displaystyle {\begin{aligned}((\mathrm {id} \boxtimes \epsilon )\circ \Delta )(x)&=(\mathrm {id} \boxtimes \epsilon )(1\boxtimes x+x\boxtimes 1)\\&=1\boxtimes \epsilon (x)+x\boxtimes \epsilon (1)\\&=0+x\boxtimes 1\\&\cong x\end{aligned}}}
6856:
8435:
909:
7882:
6217:
4403:
6112:
3043:
7346:{\displaystyle {\begin{aligned}(\nabla \circ (S\boxtimes \mathrm {id} )\circ \Delta )(k)&=(\nabla \circ (S\boxtimes \mathrm {id} ))(1\boxtimes k)\\&=\nabla (1\boxtimes k)\\&=1\otimes k\\&=k\end{aligned}}}
1255:
Because of the generality of the tensor algebra, many other algebras of interest can be constructed by starting with the tensor algebra and then imposing certain relations on the generators, i.e. by constructing certain
6029:
6484:
1490:
293:
2862:{\displaystyle {\begin{aligned}\Delta (v\otimes w)&=(v\boxtimes 1+1\boxtimes v)\otimes (w\boxtimes 1+1\boxtimes w)\\&=(v\otimes w)\boxtimes 1+v\boxtimes w+w\boxtimes v+1\boxtimes (v\otimes w)\end{aligned}}}
2675:
2334:
7042:
3949:
2126:
5908:
3879:
1821:
symbol. This is not done below, and the two symbols are used independently and explicitly, so as to show the proper location of each. The result is a bit more verbose, but should be easier to comprehend.
4227:{\displaystyle \Delta (v_{1}\otimes \cdots \otimes v_{n})=\sum _{S\subseteq \{1,\dots ,n\}}\left(\prod _{k=1 \atop k\in S}^{n}v_{k}\right)\boxtimes \left(\prod _{k=1 \atop k\notin S}^{n}v_{k}\right)\!,}
1931:
6845:
568:
5858:
5157:
4694:
5695:
5524:
4702:
7769:
7701:
362:
7393:
7175:
6861:
4799:
3544:
3063:
2691:
5210:
6358:
6209:
1308:
structure. The other structure, although simpler, cannot be extended to a bialgebra. The first structure is developed immediately below; the second structure is given in the section on the
8247:
2303:
1975:
5107:
2166:
1660:
7832:
One may proceed in a similar manner, by homomorphism, verifying that the antipode inserts the appropriate cancellative signs in the shuffle, starting with the compatibility condition on
1547:
1419:
2029:
2943:
5037:
8102:
6765:
4633:
4536:
4478:
2610:
5943:
8300:
1241:
7021:{\displaystyle {\begin{aligned}S(v_{1}\otimes \cdots \otimes v_{m})&=S(v_{m})\otimes \cdots \otimes S(v_{1})\\&=(-1)^{m}v_{m}\otimes \cdots \otimes v_{1}\end{aligned}}}
4446:
5418:
1878:
4296:
6679:
6596:
5687:
5658:
5476:
5380:
4578:
765:
8134:
648:
6387:
5330:
5230:
2896:
1819:
1737:
symbol is used to denote the "external" tensor product, needed for the definition of a coalgebra. It is being used to distinguish it from the "internal" tensor product
1735:
6522:
8470:
5512:
5441:
5290:
5250:
4598:
1799:
1779:
1755:
1384:
1357:
1161:
859:
8345:
7801:
6720:
5310:
5270:
3511:
2568:
2474:
2158:
1843:
1512:
1337:
8337:
7860:
7380:
7162:
6634:
4504:
1195:
4011:
1715:
1125:
6413:
7828:
4258:
3984:
2520:
2497:
2326:
1686:
6554:
6143:
2544:
2049:
8036:{\displaystyle \Delta (v_{1}\otimes \dots \otimes v_{k}):=\sum _{j=0}^{k}(v_{0}\otimes \dots \otimes v_{j})\boxtimes (v_{j+1}\otimes \dots \otimes v_{k+1})}
6303:{\displaystyle \Delta \circ \nabla =(\nabla \boxtimes \nabla )\circ (\mathrm {id} \boxtimes \tau \boxtimes \mathrm {id} )\circ (\Delta \boxtimes \Delta )}
4307:
6043:
2953:
4017:
in the riffle shuffle: the riffle shuffle merely splits the ordered sequence into two ordered sequences, one on the left, and one on the right.
5951:
5810:
The unit and counit, and multiplication and comultiplication, all have to satisfy compatibility conditions. It is straightforward to see that
2446:{\displaystyle ((\mathrm {id} _{TV}\boxtimes \Delta )\circ \Delta )(v)=v\boxtimes 1\boxtimes 1+1\boxtimes v\boxtimes 1+1\boxtimes 1\boxtimes v}
7128:{\displaystyle \nabla \circ (S\boxtimes \mathrm {id} )\circ \Delta =\eta \circ \epsilon =\nabla \circ (\mathrm {id} \boxtimes S)\circ \Delta }
6421:
1424:
225:
1600:
to each of these. Verifying that quotienting preserves the Hopf algebra structure is the same as verifying that the maps are indeed natural.
2618:
3884:
2057:
5869:
3820:
9478:
1761:, below, for further clarification on this issue). In order to avoid confusion between these two symbols, most texts will replace
1886:
6773:
9627:
5792:{\displaystyle \nabla \circ (\mathrm {id} _{TV}\boxtimes \eta )=\mathrm {id} _{TV}\otimes \eta =\mathrm {id} _{TV}\cdot \eta }
5621:{\displaystyle \nabla \circ (\eta \boxtimes \mathrm {id} _{TV})=\eta \otimes \mathrm {id} _{TV}=\eta \cdot \mathrm {id} _{TV}}
4778:{\displaystyle (\mathrm {id} \boxtimes \epsilon )\circ \Delta =\mathrm {id} =(\epsilon \boxtimes \mathrm {id} )\circ \Delta .}
505:
8571:
5816:
481:{\displaystyle T(V)=\bigoplus _{k=0}^{\infty }T^{k}V=K\oplus V\oplus (V\otimes V)\oplus (V\otimes V\otimes V)\oplus \cdots .}
5118:
4641:
3048:
Continuing in this fashion, one can obtain an explicit expression for the coproduct acting on a homogenous element of order
1366:
can also be given the structure of a Hopf algebra, in exactly the same fashion, by replacing everywhere the tensor product
7712:
7644:
1781:
by a plain dot, or even drop it altogether, with the understanding that it is implied from context. This then allows the
9662:
9341:
5165:
6316:
9121:
8601:
8536:
2248:{\displaystyle (\mathrm {id} _{TV}\boxtimes \Delta )\circ \Delta =(\Delta \boxtimes \mathrm {id} _{TV})\circ \Delta }
6148:
908:
9543:
8178:
2261:
1942:
8524:
5065:
1618:
8171:, the corresponding coalgebra is termed cocomplete co-free. With the usual product this is not a bialgebra. It
1517:
1389:
7876:
One may define a different coproduct on the tensor algebra, simpler than the one given above. It is given by
9394:
9326:
9039:
8559:
1983:
958:
a unique isomorphism), but this definition requires to prove that an object satisfying this property exists.
2901:
1880:
and then by homomorphically extending it to the whole algebra. A suitable choice for the coproduct is then
9770:
9419:
8887:
5443:
is "trivial": it is just part of the standard definition of the tensor product of vector spaces. That is,
5004:
1285:
1257:
126:
98:
8049:
6728:
4606:
4509:
4451:
2573:
9657:
5916:
5332:
is the one required in the definition of comultiplication in a coalgebra. These two tensor products are
3986:). The shuffle follows directly from the first axiom of a co-algebra: the relative order of the elements
342:
8256:
4696:
It is a straightforward matter to verify that this counit satisfies the needed axiom for the coalgebra:
2160:
It is straightforward to verify that this definition satisfies the axioms of a coalgebra: that is, that
1200:
9468:
9288:
9060:
8818:
4416:
5391:
1848:
9140:
972:
9622:
6415:
The verification is verbose but straightforward; it is not given here, except for the final result:
5272:; and notational sloppiness here would lead to utter chaos. To strengthen this: the tensor product
4263:
9724:
9642:
9596:
9303:
9065:
9055:
8764:
8140:
6642:
6559:
5663:
5634:
5446:
5350:
4541:
1066:
735:
8107:
611:
136:
structures; one simple one, which does not make it a bialgebra, but does lead to the concept of a
9694:
9381:
9298:
9268:
9070:
8774:
8664:
8303:
6363:
2547:
8430:{\displaystyle \Delta (v\otimes w)=1\boxtimes (v\otimes w)+v\boxtimes w+(v\otimes w)\boxtimes 1}
5315:
5215:
2898:
as this is just plain-old scalar multiplication in the algebra; that is, one trivially has that
2875:
1804:
1720:
9780:
9652:
9508:
9463:
8952:
8877:
8798:
8788:
8754:
8704:
6492:
4448:
is given by the projection of the field component out from the algebra. This can be written as
1591:
776:
8443:
5485:
5426:
5275:
5235:
4583:
1784:
1764:
1757:, which is already being used to denote multiplication in the tensor algebra (see the section
1740:
1369:
1342:
1130:
832:
9734:
9689:
9169:
9114:
8947:
8941:
8714:
8594:
7777:
6687:
5295:
5255:
3496:
2553:
2459:
2134:
1828:
1497:
1322:
8312:
7835:
7359:
7141:
6604:
4483:
1170:
9709:
9637:
9523:
9389:
9351:
9283:
8911:
8834:
8792:
8735:
8657:
8486:
8481:
3989:
880:
676:
658:
51:
6211:
The most difficult to verify is the compatibility of multiplication and comultiplication:
1691:
1101:
954:
can be defined as the unique algebra satisfying this property (specifically, it is unique
8:
9586:
9409:
9399:
9248:
9233:
9189:
8968:
8892:
8838:
8824:
8814:
8758:
8699:
8684:
8674:
8496:
6392:
5802:
where the right-hand side of these equations should be understood as the scalar product.
5515:
901:
784:
183:
159:
7810:
4240:
3966:
2502:
2479:
2308:
1668:
818:, which formally expresses the statement that it is the most general algebra containing
9719:
9576:
9429:
9243:
9179:
8927:
8844:
8748:
8679:
8647:
8626:
8621:
6539:
6128:
5051:
defines both multiplication, and comultiplication, and requires them to be compatible.
2529:
2456:
and likewise for the other side. At this point, one could invoke a lemma, and say that
2034:
1247:, as they take in a vector and give out a scalar (the given coordinate of the vector).
815:
87:
2570:
is a homomorphism. However, it is insightful to provide explicit expressions. So, for
1049:, another way of looking at the tensor algebra is as the "algebra of polynomials over
657:
The construction generalizes in a straightforward manner to the tensor algebra of any
9765:
9714:
9483:
9458:
9273:
9184:
9164:
9018:
8988:
8858:
8768:
8719:
8652:
8567:
8532:
8491:
8156:
7036:
Compatibility of the antipode with multiplication and comultiplication requires that
1363:
1300:
structures. One is compatible with the tensor product, and thus can be extended to a
1273:
802:
114:
79:
9775:
9729:
9404:
9371:
9356:
9238:
9107:
9088:
8973:
8642:
8587:
8514:
7871:
1316:
1309:
1277:
1269:
1244:
1084:
665:
137:
118:
110:
5112:
which, in this case, was already given as the "internal" tensor product. That is,
9699:
9647:
9591:
9571:
9473:
9361:
9228:
9199:
9008:
8998:
8917:
8744:
8669:
8563:
8528:
8518:
3514:
1554:
8309:
The difference between this, and the other coalgebra is most easily seen in the
4398:{\displaystyle \Delta :T^{m}V\to \bigoplus _{k=0}^{m}T^{k}V\boxtimes T^{(m-k)}V}
9739:
9704:
9601:
9434:
9424:
9414:
9336:
9308:
9293:
9278:
9012:
8931:
8782:
8694:
590:
208:
155:
63:
9684:
6107:{\displaystyle (\epsilon \circ \nabla )(x\boxtimes y)=\epsilon (x\otimes y)=0}
3038:{\displaystyle \Delta :T^{2}V\to \bigoplus _{k=0}^{2}T^{k}V\boxtimes T^{2-k}V}
9759:
9676:
9581:
9493:
9366:
8937:
8902:
6360:
exchanges elements. The compatibility condition only needs to be verified on
3518:
1080:
923:
82:
from algebras to vector spaces: it is the "most general" algebra containing
9744:
9548:
9533:
9498:
9346:
9331:
8921:
8854:
8828:
8709:
8168:
8139:
This coproduct gives rise to a coalgebra. It describes a coalgebra that is
6533:
6524:
an explicit expression for this was given in the coalgebra section, above.
1305:
1281:
1058:
798:
718:
573:
given by the tensor product, which is then extended by linearity to all of
324:
310:
179:
145:
122:
75:
67:
28:
9632:
9606:
9528:
9217:
9156:
8992:
8978:
8896:
8882:
8848:
6024:{\displaystyle (\Delta \circ \eta )(k)=\Delta (k)=k(1\boxtimes 1)\cong k}
2523:
1584:
taking vector spaces to the category of exterior algebras, and a functor
788:
20:
6479:{\displaystyle (\Delta \circ \nabla )(v\boxtimes w)=\Delta (v\otimes w)}
1485:{\displaystyle v\otimes _{\mathrm {Sym} }w=w\otimes _{\mathrm {Sym} }v.}
288:{\displaystyle T^{k}V=V^{\otimes k}=V\otimes V\otimes \cdots \otimes V.}
9513:
9002:
8982:
8907:
8551:
8501:
827:
768:
6389:; the full compatibility follows as a homomorphic extension to all of
2670:{\displaystyle \Delta :v\otimes w\mapsto \Delta (v)\otimes \Delta (w)}
9488:
9439:
5945:
in order to work; without this, one loses linearity. Component-wise,
5048:
3517:. This is expressed in the second summation, which is taken over all
1845:
is most easily built up in stages, first by defining it for elements
1609:
1301:
1297:
163:
141:
133:
97:
The tensor algebra is important because many other algebras arise as
3944:{\displaystyle v_{\sigma (p+1)}\otimes \dots \otimes v_{\sigma (m)}}
2121:{\displaystyle \Delta (k)=k(1\boxtimes 1)=k\boxtimes 1=1\boxtimes k}
9518:
9503:
8778:
8740:
8689:
8472:, which is clearly missing a shuffled term, as compared to before.
687:
5903:{\displaystyle \Delta \circ \eta =\eta \boxtimes \eta \cong \eta }
3874:{\displaystyle v_{\sigma (1)}\otimes \dots \otimes v_{\sigma (p)}}
1315:
The development provided below can be equally well applied to the
9212:
9174:
8610:
5312:
used in the definition of an algebra, whereas the tensor product
1558:
968:
729:
697:-modules because the iterated tensor products cannot be formed.)
190:
9538:
9130:
1494:
In each case, this is possible because the alternating product
55:
6639:
This is sometimes called the "anti-identity". The antipode on
5001:
where, for the last step, one has made use of the isomorphism
6145:, and otherwise, one has scalar multiplication on the field:
955:
3817:}}. It is also convenient to take the pure tensor products
2947:
The extension above preserves the algebra grading. That is,
581:). This multiplication rule implies that the tensor algebra
9099:
8579:
5039:, as is appropriate for the defining axiom of the counit.
1926:{\displaystyle \Delta :v\mapsto v\boxtimes 1+1\boxtimes v}
6840:{\displaystyle S(v\otimes w)=S(w)\otimes S(v)=w\otimes v}
5863:
Similarly, the unit is compatible with comultiplication:
6536:
adds an antipode to the bialgebra axioms. The antipode
6034:
with the right-hand side making use of the isomorphism.
5292:
of the tensor algebra corresponds to the multiplication
943:. As for other universal properties, the tensor algebra
563:{\displaystyle T^{k}V\otimes T^{\ell }V\to T^{k+\ell }V}
144:, and can be extended by giving an antipode to create a
5853:{\displaystyle \epsilon \circ \eta =\mathrm {id} _{K}.}
5518:
require the two homomorphisms (or commuting diagrams):
3513:
symbol, which should appear as ш, the sha, denotes the
2872:
In the above expansion, there is no need to ever write
1590:
taking vector spaces to symmetric algebras. There is a
1304:, and can be further be extended with an antipode to a
814:
Explicitly, the tensor algebra satisfies the following
8261:
5152:{\displaystyle \nabla :x\boxtimes y\mapsto x\otimes y}
4689:{\displaystyle x\in T^{1}V\oplus T^{2}V\oplus \cdots }
91:
8446:
8348:
8315:
8259:
8181:
8110:
8052:
7885:
7838:
7813:
7780:
7715:
7647:
7391:
7362:
7173:
7144:
7045:
6859:
6776:
6731:
6690:
6645:
6607:
6562:
6542:
6495:
6424:
6395:
6366:
6319:
6220:
6151:
6131:
6046:
5954:
5919:
5872:
5819:
5698:
5666:
5637:
5527:
5488:
5449:
5429:
5394:
5353:
5318:
5298:
5278:
5258:
5238:
5218:
5168:
5121:
5068:
5007:
4797:
4705:
4644:
4609:
4586:
4544:
4512:
4486:
4454:
4419:
4310:
4266:
4243:
4029:
3992:
3969:
3887:
3823:
3542:
3499:
3061:
2956:
2904:
2878:
2689:
2621:
2576:
2556:
2532:
2505:
2482:
2462:
2337:
2311:
2264:
2169:
2137:
2060:
2037:
1986:
1945:
1889:
1851:
1831:
1807:
1787:
1767:
1743:
1723:
1694:
1671:
1621:
1520:
1500:
1427:
1392:
1372:
1345:
1325:
1203:
1173:
1133:
1104:
835:
738:
614:
508:
365:
228:
7764:{\displaystyle (\eta \circ \epsilon )(x)=\eta (0)=0}
7696:{\displaystyle (\eta \circ \epsilon )(k)=\eta (k)=k}
5423:
That the unit is compatible with the tensor product
1550:
quotient space inherits the Hopf algebra structure.
700:
7138:This is straightforward to verify componentwise on
1578:-associative algebras. But there is also a functor
8464:
8429:
8331:
8294:
8241:
8128:
8096:
8035:
7854:
7822:
7795:
7763:
7695:
7627:
7374:
7345:
7156:
7127:
7020:
6839:
6759:
6714:
6673:
6628:
6590:
6548:
6516:
6478:
6407:
6381:
6352:
6302:
6203:
6137:
6106:
6023:
5937:
5902:
5852:
5791:
5681:
5652:
5620:
5506:
5470:
5435:
5412:
5374:
5324:
5304:
5284:
5264:
5244:
5224:
5204:
5151:
5101:
5031:
4990:
4777:
4688:
4627:
4592:
4572:
4530:
4498:
4472:
4440:
4397:
4290:
4252:
4226:
4005:
3978:
3943:
3873:
3790:
3505:
3482:
3037:
2937:
2890:
2861:
2669:
2604:
2562:
2538:
2514:
2491:
2468:
2445:
2320:
2297:
2247:
2152:
2120:
2043:
2023:
1969:
1925:
1872:
1837:
1813:
1793:
1773:
1749:
1729:
1709:
1680:
1654:
1541:
1506:
1484:
1413:
1378:
1351:
1331:
1235:
1189:
1155:
1119:
987:-algebras. This means that any linear map between
853:
759:
642:
562:
480:
287:
154:: In this article, all algebras are assumed to be
5205:{\displaystyle \nabla (x\boxtimes y)=x\otimes y.}
4220:
679:, one can still perform the construction for any
330:(as a one-dimensional vector space over itself).
9757:
6353:{\displaystyle \tau (x\boxtimes y)=y\boxtimes x}
162:. The unit is explicitly required to define the
8167:. In the same way that the tensor algebra is a
8046:Here, as before, one uses the notational trick
7865:
6204:{\displaystyle k_{1}\otimes k_{2}=k_{1}k_{2}.}
6037:Multiplication and the counit are compatible:
5913:The above requires the use of the isomorphism
1036:
9115:
8595:
8285:
8264:
8242:{\displaystyle v_{i}\cdot v_{j}=(i,j)v_{i+j}}
4301:As before, the algebra grading is preserved:
2298:{\displaystyle \mathrm {id} _{TV}:x\mapsto x}
1970:{\displaystyle \Delta :1\mapsto 1\boxtimes 1}
499:) is determined by the canonical isomorphism
140:, and a more complicated one, which yields a
16:Universal construction in multilinear algebra
8175:be turned into a bialgebra with the product
5102:{\displaystyle \nabla :TV\boxtimes TV\to TV}
4580:. By homomorphism under the tensor product
4285:
4267:
4100:
4082:
3778:
3634:
3610:
3604:
3580:
3571:
1655:{\displaystyle \Delta :TV\to TV\boxtimes TV}
637:
631:
604:subspace. This grading can be extended to a
62:(of any rank) with multiplication being the
4260:, and where the sum is over all subsets of
2612:, one has (by definition) the homomorphism
2476:extends trivially, by linearity, to all of
1065:, those become non-commuting variables (or
961:The above universal property implies that
9122:
9108:
8602:
8588:
8550:
3717:
3655:
3641:
3637:
3352:
3309:
3232:
3228:
1717:to avoid an explosion of parentheses. The
1542:{\displaystyle \otimes _{\mathrm {Sym} }}
1414:{\displaystyle \otimes _{\mathrm {Sym} }}
811:-algebra to its underlying vector space.
9479:Covariance and contravariance of vectors
8513:
1608:The coalgebra is obtained by defining a
1094:Note that the algebra of polynomials on
912:Universal property of the tensor algebra
5252:was actually one and the same thing as
5212:The above should make it clear why the
5059:Multiplication is given by an operator
2024:{\displaystyle 1\in K=T^{0}V\subset TV}
9758:
2938:{\displaystyle 1\otimes v=1\cdot v=v.}
9103:
8583:
8253:denotes the binomial coefficient for
5032:{\displaystyle TV\boxtimes K\cong TV}
3963:, respectively (the empty product in
1296:The tensor algebra has two different
1163:: a (homogeneous) linear function on
1057:non-commuting variables". If we take
8097:{\displaystyle v_{0}=v_{k+1}=1\in K}
6760:{\displaystyle v\otimes w\in T^{2}V}
4628:{\displaystyle \epsilon :x\mapsto 0}
4531:{\displaystyle \epsilon :k\mapsto k}
4473:{\displaystyle \epsilon :v\mapsto 0}
2605:{\displaystyle v\otimes w\in T^{2}V}
1079:), subject to no constraints beyond
86:, in the sense of the corresponding
5938:{\displaystyle K\boxtimes K\cong K}
13:
9342:Tensors in curvilinear coordinates
8349:
8295:{\displaystyle {\tbinom {i+j}{i}}}
8268:
7886:
7519:
7472:
7469:
7453:
7428:
7418:
7415:
7399:
7289:
7254:
7251:
7235:
7210:
7200:
7197:
7181:
7122:
7106:
7103:
7093:
7075:
7065:
7062:
7046:
7031:
6458:
6434:
6428:
6294:
6288:
6275:
6272:
6258:
6255:
6242:
6236:
6227:
6221:
6056:
5982:
5958:
5873:
5837:
5834:
5770:
5767:
5743:
5740:
5713:
5710:
5699:
5605:
5602:
5578:
5575:
5548:
5545:
5528:
5514:More verbosely, the axioms for an
5299:
5259:
5169:
5122:
5069:
4857:
4854:
4828:
4812:
4809:
4769:
4759:
4756:
4739:
4736:
4729:
4713:
4710:
4311:
4174:
4115:
4030:
3325:
3322:
3139:
3111:
3066:
2957:
2694:
2655:
2640:
2622:
2557:
2463:
2374:
2365:
2349:
2346:
2270:
2267:
2242:
2223:
2220:
2212:
2203:
2194:
2178:
2175:
2061:
2051:. By linearity, one obviously has
1946:
1890:
1832:
1801:symbol to be used in place of the
1622:
1572:-vector spaces to the category of
1533:
1530:
1527:
1470:
1467:
1464:
1443:
1440:
1437:
1405:
1402:
1399:
1386:by the symmetrized tensor product
1236:{\displaystyle x^{1},\dots ,x^{n}}
907:
783:-vector spaces to the category of
397:
14:
9792:
8302:. This bialgebra is known as the
5054:
4788:Working this explicitly, one has
4441:{\displaystyle \epsilon :TV\to K}
3801:By convention, one takes that Sh(
701:Adjunction and universal property
693:. (It does not work for ordinary
9084:
9083:
8562:, vol. 211 (3rd ed.),
6850:This extends homomorphically to
5805:
5413:{\displaystyle \eta :k\mapsto k}
1873:{\displaystyle v\in V\subset TV}
608:-grading by appending subspaces
132:The tensor algebra also has two
6527:
5385:is just the embedding, so that
1825:The definition of the operator
879:can be uniquely extended to an
169:
8418:
8406:
8388:
8376:
8364:
8352:
8220:
8208:
8030:
7986:
7980:
7948:
7921:
7889:
7752:
7746:
7737:
7731:
7728:
7716:
7684:
7678:
7669:
7663:
7660:
7648:
7549:
7522:
7506:
7482:
7479:
7476:
7459:
7450:
7440:
7434:
7431:
7422:
7405:
7396:
7304:
7292:
7276:
7264:
7261:
7258:
7241:
7232:
7222:
7216:
7213:
7204:
7187:
7178:
7116:
7099:
7069:
7052:
6976:
6966:
6953:
6940:
6925:
6912:
6899:
6867:
6822:
6816:
6807:
6801:
6792:
6780:
6700:
6694:
6617:
6611:
6473:
6461:
6452:
6440:
6437:
6425:
6335:
6323:
6297:
6285:
6279:
6251:
6245:
6233:
6095:
6083:
6074:
6062:
6059:
6047:
6012:
6000:
5991:
5985:
5976:
5970:
5967:
5955:
5732:
5705:
5561:
5534:
5404:
5363:
5184:
5172:
5137:
5090:
4943:
4937:
4922:
4916:
4894:
4870:
4867:
4850:
4840:
4834:
4831:
4822:
4805:
4802:
4763:
4746:
4723:
4706:
4619:
4522:
4464:
4432:
4387:
4375:
4330:
4291:{\displaystyle \{1,\dots ,n\}}
4065:
4033:
3936:
3930:
3908:
3896:
3866:
3860:
3838:
3832:
3775:
3769:
3754:
3742:
3733:
3721:
3701:
3695:
3680:
3674:
3665:
3659:
3638:
3607:
3565:
3553:
3466:
3460:
3438:
3426:
3400:
3394:
3372:
3366:
3347:
3329:
3155:
3142:
3127:
3114:
3101:
3069:
2976:
2852:
2840:
2798:
2786:
2773:
2749:
2743:
2719:
2709:
2697:
2664:
2658:
2649:
2643:
2637:
2386:
2380:
2377:
2368:
2341:
2338:
2289:
2236:
2209:
2197:
2170:
2091:
2079:
2070:
2064:
1955:
1899:
1704:
1698:
1634:
1339:in place of the tensor symbol
1150:
1137:
1114:
1108:
900:as indicated by the following
845:
754:
748:
742:
538:
466:
448:
442:
430:
375:
369:
1:
9395:Exterior covariant derivative
9327:Tensor (intrinsic definition)
8560:Graduate Texts in Mathematics
8507:
7862:and proceeding by induction.
6674:{\displaystyle v\in V=T^{1}V}
6591:{\displaystyle k\in K=T^{0}V}
5682:{\displaystyle TV\boxtimes K}
5660:, and that symmetrically, on
5653:{\displaystyle K\boxtimes TV}
5471:{\displaystyle k\otimes x=kx}
5375:{\displaystyle \eta :K\to TV}
5232:symbol needs to be used: the
4573:{\displaystyle k\in K=T^{0}V}
1286:universal enveloping algebras
1268:). Examples of this are the
760:{\displaystyle V\mapsto T(V)}
127:universal enveloping algebras
9420:Raising and lowering indices
8888:Eigenvalues and eigenvectors
8143:to the algebra structure on
8129:{\displaystyle v\otimes 1=v}
5042:
1688:is used as a short-hand for
1603:
1291:
1250:
805:that sends each associative
643:{\displaystyle T^{k}V=\{0\}}
7:
9658:Gluon field strength tensor
9129:
8609:
8475:
7866:Cofree cocomplete coalgebra
6382:{\displaystyle V\subset TV}
1557:, one says that there is a
1037:Non-commutative polynomials
1011:-algebra homomorphism from
305:consists of all tensors on
10:
9797:
9469:Cartan formalism (physics)
9289:Penrose graphical notation
8339:term. Here, one has that
8304:divided power Hopf algebra
7869:
5325:{\displaystyle \boxtimes }
5225:{\displaystyle \boxtimes }
4237:where the products are in
2891:{\displaystyle 1\otimes v}
1814:{\displaystyle \boxtimes }
1730:{\displaystyle \boxtimes }
1514:and the symmetric product
1421:, i.e. that product where
867:to an associative algebra
732:; this means that the map
9675:
9615:
9564:
9557:
9449:
9380:
9317:
9261:
9208:
9155:
9148:
9141:Glossary of tensor theory
9137:
9079:
9048:
9032:
8961:
8868:
8807:
8728:
8635:
8617:
6517:{\displaystyle v,w\in V,}
5344:The unit for the algebra
4408:
2550:of the free algebra, and
2031:is the unit of the field
1319:, using the wedge symbol
973:category of vector spaces
9725:Gregorio Ricci-Curbastro
9597:Riemann curvature tensor
9304:Van der Waerden notation
8465:{\displaystyle v,w\in V}
5507:{\displaystyle x\in TV.}
5436:{\displaystyle \otimes }
5285:{\displaystyle \otimes }
5245:{\displaystyle \otimes }
4593:{\displaystyle \otimes }
1794:{\displaystyle \otimes }
1774:{\displaystyle \otimes }
1750:{\displaystyle \otimes }
1379:{\displaystyle \otimes }
1352:{\displaystyle \otimes }
1197:for example coordinates
1156:{\displaystyle T(V^{*})}
854:{\displaystyle f:V\to A}
74:, in the sense of being
9695:Elwin Bruno Christoffel
9628:Angular momentum tensor
9299:Tetrad (index notation)
9269:Abstract index notation
8962:Algebraic constructions
8665:Algebraic number theory
8525:Elements of Mathematics
8520:Algebra I. Chapters 1-3
7796:{\displaystyle x\in TV}
6715:{\displaystyle S(v)=-v}
5339:
5305:{\displaystyle \nabla }
5265:{\displaystyle \nabla }
3809:) equals {id: {1, ...,
3506:{\displaystyle \omega }
2563:{\displaystyle \Delta }
2469:{\displaystyle \Delta }
2305:is the identity map on
2153:{\displaystyle k\in K.}
1838:{\displaystyle \Delta }
1507:{\displaystyle \wedge }
1332:{\displaystyle \wedge }
787:. Similarly with other
9509:Levi-Civita connection
8705:Noncommutative algebra
8466:
8431:
8333:
8332:{\displaystyle T^{2}V}
8296:
8243:
8130:
8098:
8037:
7947:
7856:
7855:{\displaystyle T^{2}V}
7824:
7797:
7765:
7697:
7629:
7376:
7375:{\displaystyle v\in V}
7347:
7158:
7157:{\displaystyle k\in K}
7129:
7022:
6841:
6761:
6716:
6675:
6630:
6629:{\displaystyle S(k)=k}
6592:
6550:
6518:
6480:
6409:
6383:
6354:
6304:
6205:
6139:
6108:
6025:
5939:
5904:
5854:
5793:
5683:
5654:
5622:
5508:
5472:
5437:
5414:
5376:
5326:
5306:
5286:
5266:
5246:
5226:
5206:
5153:
5103:
5033:
4992:
4779:
4690:
4629:
4594:
4574:
4532:
4500:
4499:{\displaystyle v\in V}
4474:
4442:
4399:
4353:
4292:
4254:
4228:
4204:
4145:
4007:
3980:
3945:
3875:
3792:
3507:
3484:
3308:
3188:
3039:
2999:
2939:
2892:
2863:
2671:
2606:
2564:
2540:
2516:
2493:
2470:
2447:
2322:
2299:
2249:
2154:
2122:
2045:
2025:
1971:
1927:
1874:
1839:
1815:
1795:
1775:
1751:
1731:
1711:
1682:
1656:
1543:
1508:
1486:
1415:
1380:
1353:
1333:
1243:on a vector space are
1237:
1191:
1190:{\displaystyle V^{*},}
1157:
1121:
1005:extends uniquely to a
913:
855:
761:
650:for negative integers
644:
564:
491:The multiplication in
482:
401:
289:
189:. For any nonnegative
109:). These include the
9735:Jan Arnoldus Schouten
9690:Augustin-Louis Cauchy
9170:Differential geometry
8942:Orthogonal complement
8715:Representation theory
8467:
8432:
8334:
8297:
8244:
8131:
8099:
8038:
7927:
7857:
7825:
7798:
7766:
7698:
7630:
7377:
7348:
7159:
7130:
7023:
6842:
6762:
6717:
6676:
6631:
6593:
6551:
6519:
6481:
6410:
6384:
6355:
6305:
6206:
6140:
6109:
6026:
5940:
5905:
5855:
5794:
5684:
5655:
5623:
5509:
5473:
5438:
5415:
5377:
5327:
5307:
5287:
5267:
5247:
5227:
5207:
5154:
5104:
5034:
4993:
4780:
4691:
4630:
4595:
4575:
4533:
4501:
4475:
4443:
4400:
4333:
4293:
4255:
4229:
4169:
4110:
4008:
4006:{\displaystyle v_{k}}
3981:
3946:
3876:
3793:
3508:
3485:
3288:
3168:
3040:
2979:
2940:
2893:
2864:
2672:
2607:
2565:
2541:
2517:
2494:
2471:
2448:
2323:
2300:
2250:
2155:
2123:
2046:
2026:
1972:
1928:
1875:
1840:
1816:
1796:
1776:
1752:
1732:
1712:
1683:
1657:
1612:or diagonal operator
1566:from the category of
1544:
1509:
1487:
1416:
1381:
1354:
1334:
1238:
1192:
1158:
1122:
1045:has finite dimension
981:, to the category of
911:
856:
762:
645:
600:serving as the grade-
565:
483:
381:
290:
9710:Carl Friedrich Gauss
9643:stress–energy tensor
9638:Cauchy stress tensor
9390:Covariant derivative
9352:Antisymmetric tensor
9284:Multi-index notation
9040:Algebraic structures
8808:Algebraic structures
8793:Equivalence relation
8736:Algebraic expression
8487:Braided Hopf algebra
8482:Braided vector space
8444:
8346:
8313:
8257:
8179:
8108:
8050:
7883:
7836:
7811:
7778:
7713:
7645:
7389:
7360:
7171:
7142:
7043:
6857:
6774:
6729:
6688:
6643:
6605:
6560:
6540:
6493:
6422:
6393:
6364:
6317:
6218:
6149:
6129:
6125:are not elements of
6044:
5952:
5917:
5870:
5817:
5696:
5664:
5635:
5525:
5486:
5447:
5427:
5392:
5351:
5316:
5296:
5276:
5256:
5236:
5216:
5166:
5119:
5066:
5005:
4795:
4703:
4642:
4607:
4584:
4542:
4510:
4484:
4452:
4417:
4308:
4264:
4241:
4027:
3990:
3967:
3885:
3821:
3540:
3497:
3059:
2954:
2902:
2876:
2687:
2619:
2574:
2554:
2530:
2503:
2480:
2460:
2335:
2328:. Indeed, one gets
2309:
2262:
2167:
2135:
2058:
2035:
1984:
1943:
1887:
1849:
1829:
1805:
1785:
1765:
1741:
1721:
1710:{\displaystyle T(V)}
1692:
1669:
1619:
1518:
1498:
1425:
1390:
1370:
1343:
1323:
1201:
1171:
1131:
1120:{\displaystyle T(V)}
1102:
881:algebra homomorphism
833:
785:associative algebras
736:
722:on the vector space
677:non-commutative ring
612:
506:
363:
226:
9771:Multilinear algebra
9587:Nonmetricity tensor
9442:(2nd-order tensors)
9410:Hodge star operator
9400:Exterior derivative
9249:Transport phenomena
9234:Continuum mechanics
9190:Multilinear algebra
8969:Composition algebra
8893:Inner product space
8871:multilinear algebra
8759:Polynomial function
8700:Multilinear algebra
8685:Homological algebra
8675:Commutative algebra
8497:Multilinear algebra
6408:{\displaystyle TV.}
5516:associative algebra
4600:, this extends to
2680:Expanding, one has
1553:In the language of
924:canonical inclusion
902:commutative diagram
716:is also called the
705:The tensor algebra
9720:Tullio Levi-Civita
9663:Metric tensor (GR)
9577:Levi-Civita symbol
9430:Tensor contraction
9244:General relativity
9180:Euclidean geometry
8749:Quadratic equation
8680:Elementary algebra
8648:Algebraic geometry
8544:(See Chapter 3 §5)
8462:
8427:
8329:
8292:
8290:
8239:
8126:
8094:
8033:
7852:
7823:{\displaystyle K.}
7820:
7793:
7761:
7693:
7625:
7623:
7372:
7343:
7341:
7154:
7125:
7018:
7016:
6837:
6757:
6712:
6671:
6626:
6588:
6546:
6514:
6476:
6405:
6379:
6350:
6300:
6201:
6135:
6104:
6021:
5935:
5900:
5850:
5789:
5679:
5650:
5618:
5504:
5478:for field element
5468:
5433:
5410:
5372:
5322:
5302:
5282:
5262:
5242:
5222:
5202:
5149:
5099:
5029:
4988:
4986:
4775:
4686:
4625:
4590:
4570:
4528:
4496:
4470:
4438:
4395:
4288:
4253:{\displaystyle TV}
4250:
4224:
4104:
4003:
3979:{\displaystyle TV}
3976:
3941:
3871:
3788:
3786:
3649: is bijective
3503:
3480:
3478:
3351:
3035:
2935:
2888:
2859:
2857:
2667:
2602:
2560:
2536:
2515:{\displaystyle TV}
2512:
2492:{\displaystyle TV}
2489:
2466:
2443:
2321:{\displaystyle TV}
2318:
2295:
2245:
2150:
2118:
2041:
2021:
1967:
1923:
1870:
1835:
1811:
1791:
1771:
1747:
1727:
1707:
1681:{\displaystyle TV}
1678:
1652:
1539:
1504:
1482:
1411:
1376:
1349:
1329:
1233:
1187:
1153:
1117:
914:
851:
816:universal property
789:free constructions
757:
640:
560:
478:
333:We then construct
285:
88:universal property
9753:
9752:
9715:Hermann Grassmann
9671:
9670:
9623:Moment of inertia
9484:Differential form
9459:Affine connection
9274:Einstein notation
9257:
9256:
9185:Exterior calculus
9165:Coordinate system
9097:
9096:
9019:Symmetric algebra
8989:Geometric algebra
8769:Linear inequality
8720:Universal algebra
8653:Algebraic variety
8573:978-0-387-95385-4
8515:Bourbaki, Nicolas
8492:Monoidal category
8283:
8157:dual vector space
6549:{\displaystyle S}
6138:{\displaystyle K}
4197:
4138:
4071:
3715:
3650:
3533:. The shuffle is
3310:
2539:{\displaystyle V}
2044:{\displaystyle K}
1364:symmetric algebra
1278:Clifford algebras
1274:symmetric algebra
1258:quotient algebras
1167:is an element of
803:forgetful functor
589:) is naturally a
119:Clifford algebras
115:symmetric algebra
99:quotient algebras
80:forgetful functor
9788:
9730:Bernhard Riemann
9562:
9561:
9405:Exterior product
9372:Two-point tensor
9357:Symmetric tensor
9239:Electromagnetism
9153:
9152:
9124:
9117:
9110:
9101:
9100:
9087:
9086:
8974:Exterior algebra
8643:Abstract algebra
8604:
8597:
8590:
8581:
8580:
8576:
8542:
8471:
8469:
8468:
8463:
8436:
8434:
8433:
8428:
8338:
8336:
8335:
8330:
8325:
8324:
8301:
8299:
8298:
8293:
8291:
8289:
8288:
8279:
8267:
8248:
8246:
8245:
8240:
8238:
8237:
8204:
8203:
8191:
8190:
8135:
8133:
8132:
8127:
8104:(recalling that
8103:
8101:
8100:
8095:
8081:
8080:
8062:
8061:
8042:
8040:
8039:
8034:
8029:
8028:
8004:
8003:
7979:
7978:
7960:
7959:
7946:
7941:
7920:
7919:
7901:
7900:
7872:Cofree coalgebra
7861:
7859:
7858:
7853:
7848:
7847:
7829:
7827:
7826:
7821:
7802:
7800:
7799:
7794:
7770:
7768:
7767:
7762:
7702:
7700:
7699:
7694:
7634:
7632:
7631:
7626:
7624:
7611:
7589:
7555:
7512:
7475:
7421:
7381:
7379:
7378:
7373:
7352:
7350:
7349:
7344:
7342:
7329:
7310:
7282:
7257:
7203:
7163:
7161:
7160:
7155:
7134:
7132:
7131:
7126:
7109:
7068:
7027:
7025:
7024:
7019:
7017:
7013:
7012:
6994:
6993:
6984:
6983:
6959:
6952:
6951:
6924:
6923:
6898:
6897:
6879:
6878:
6846:
6844:
6843:
6838:
6766:
6764:
6763:
6758:
6753:
6752:
6721:
6719:
6718:
6713:
6680:
6678:
6677:
6672:
6667:
6666:
6635:
6633:
6632:
6627:
6597:
6595:
6594:
6589:
6584:
6583:
6555:
6553:
6552:
6547:
6523:
6521:
6520:
6515:
6485:
6483:
6482:
6477:
6414:
6412:
6411:
6406:
6388:
6386:
6385:
6380:
6359:
6357:
6356:
6351:
6309:
6307:
6306:
6301:
6278:
6261:
6210:
6208:
6207:
6202:
6197:
6196:
6187:
6186:
6174:
6173:
6161:
6160:
6144:
6142:
6141:
6136:
6113:
6111:
6110:
6105:
6030:
6028:
6027:
6022:
5944:
5942:
5941:
5936:
5909:
5907:
5906:
5901:
5859:
5857:
5856:
5851:
5846:
5845:
5840:
5798:
5796:
5795:
5790:
5782:
5781:
5773:
5755:
5754:
5746:
5725:
5724:
5716:
5688:
5686:
5685:
5680:
5659:
5657:
5656:
5651:
5627:
5625:
5624:
5619:
5617:
5616:
5608:
5590:
5589:
5581:
5560:
5559:
5551:
5513:
5511:
5510:
5505:
5477:
5475:
5474:
5469:
5442:
5440:
5439:
5434:
5419:
5417:
5416:
5411:
5381:
5379:
5378:
5373:
5336:the same thing!
5331:
5329:
5328:
5323:
5311:
5309:
5308:
5303:
5291:
5289:
5288:
5283:
5271:
5269:
5268:
5263:
5251:
5249:
5248:
5243:
5231:
5229:
5228:
5223:
5211:
5209:
5208:
5203:
5158:
5156:
5155:
5150:
5108:
5106:
5105:
5100:
5038:
5036:
5035:
5030:
4997:
4995:
4994:
4989:
4987:
4974:
4949:
4900:
4860:
4815:
4784:
4782:
4781:
4776:
4762:
4742:
4716:
4695:
4693:
4692:
4687:
4676:
4675:
4660:
4659:
4634:
4632:
4631:
4626:
4599:
4597:
4596:
4591:
4579:
4577:
4576:
4571:
4566:
4565:
4537:
4535:
4534:
4529:
4505:
4503:
4502:
4497:
4479:
4477:
4476:
4471:
4447:
4445:
4444:
4439:
4404:
4402:
4401:
4396:
4391:
4390:
4363:
4362:
4352:
4347:
4326:
4325:
4297:
4295:
4294:
4289:
4259:
4257:
4256:
4251:
4233:
4231:
4230:
4225:
4219:
4215:
4214:
4213:
4203:
4198:
4196:
4185:
4160:
4156:
4155:
4154:
4144:
4139:
4137:
4126:
4103:
4064:
4063:
4045:
4044:
4012:
4010:
4009:
4004:
4002:
4001:
3985:
3983:
3982:
3977:
3950:
3948:
3947:
3942:
3940:
3939:
3912:
3911:
3880:
3878:
3877:
3872:
3870:
3869:
3842:
3841:
3797:
3795:
3794:
3789:
3787:
3716:
3713:
3710:
3651:
3648:
3512:
3510:
3509:
3504:
3489:
3487:
3486:
3481:
3479:
3475:
3471:
3470:
3469:
3442:
3441:
3409:
3405:
3404:
3403:
3376:
3375:
3350:
3328:
3307:
3302:
3281:
3277:
3273:
3272:
3271:
3253:
3252:
3227:
3223:
3222:
3221:
3203:
3202:
3187:
3182:
3161:
3154:
3153:
3126:
3125:
3100:
3099:
3081:
3080:
3044:
3042:
3041:
3036:
3031:
3030:
3009:
3008:
2998:
2993:
2972:
2971:
2944:
2942:
2941:
2936:
2897:
2895:
2894:
2889:
2868:
2866:
2865:
2860:
2858:
2779:
2676:
2674:
2673:
2668:
2611:
2609:
2608:
2603:
2598:
2597:
2569:
2567:
2566:
2561:
2545:
2543:
2542:
2537:
2521:
2519:
2518:
2513:
2498:
2496:
2495:
2490:
2475:
2473:
2472:
2467:
2452:
2450:
2449:
2444:
2361:
2360:
2352:
2327:
2325:
2324:
2319:
2304:
2302:
2301:
2296:
2282:
2281:
2273:
2254:
2252:
2251:
2246:
2235:
2234:
2226:
2190:
2189:
2181:
2159:
2157:
2156:
2151:
2127:
2125:
2124:
2119:
2050:
2048:
2047:
2042:
2030:
2028:
2027:
2022:
2008:
2007:
1976:
1974:
1973:
1968:
1932:
1930:
1929:
1924:
1879:
1877:
1876:
1871:
1844:
1842:
1841:
1836:
1820:
1818:
1817:
1812:
1800:
1798:
1797:
1792:
1780:
1778:
1777:
1772:
1756:
1754:
1753:
1748:
1736:
1734:
1733:
1728:
1716:
1714:
1713:
1708:
1687:
1685:
1684:
1679:
1661:
1659:
1658:
1653:
1599:
1589:
1583:
1577:
1571:
1565:
1548:
1546:
1545:
1540:
1538:
1537:
1536:
1513:
1511:
1510:
1505:
1491:
1489:
1488:
1483:
1475:
1474:
1473:
1448:
1447:
1446:
1420:
1418:
1417:
1412:
1410:
1409:
1408:
1385:
1383:
1382:
1377:
1358:
1356:
1355:
1350:
1338:
1336:
1335:
1330:
1317:exterior algebra
1312:, further down.
1310:cofree coalgebra
1270:exterior algebra
1242:
1240:
1239:
1234:
1232:
1231:
1213:
1212:
1196:
1194:
1193:
1188:
1183:
1182:
1162:
1160:
1159:
1154:
1149:
1148:
1126:
1124:
1123:
1118:
1085:distributive law
1032:
1021:
1010:
1004:
998:
992:
986:
980:
966:
953:
942:
931:
921:
899:
893:
878:
872:
866:
860:
858:
857:
852:
810:
796:
782:
766:
764:
763:
758:
727:
715:
649:
647:
646:
641:
624:
623:
569:
567:
566:
561:
556:
555:
534:
533:
518:
517:
487:
485:
484:
479:
411:
410:
400:
395:
316:. By convention
294:
292:
291:
286:
257:
256:
238:
237:
196:, we define the
138:cofree coalgebra
111:exterior algebra
9796:
9795:
9791:
9790:
9789:
9787:
9786:
9785:
9756:
9755:
9754:
9749:
9700:Albert Einstein
9667:
9648:Einstein tensor
9611:
9592:Ricci curvature
9572:Kronecker delta
9558:Notable tensors
9553:
9474:Connection form
9451:
9445:
9376:
9362:Tensor operator
9319:
9313:
9253:
9229:Computer vision
9222:
9204:
9200:Tensor calculus
9144:
9133:
9128:
9098:
9093:
9075:
9044:
9028:
9009:Quotient object
8999:Polynomial ring
8957:
8918:Linear subspace
8870:
8864:
8803:
8745:Linear equation
8724:
8670:Category theory
8631:
8613:
8608:
8574:
8564:Springer Verlag
8539:
8529:Springer-Verlag
8510:
8478:
8445:
8442:
8441:
8347:
8344:
8343:
8320:
8316:
8314:
8311:
8310:
8284:
8269:
8263:
8262:
8260:
8258:
8255:
8254:
8227:
8223:
8199:
8195:
8186:
8182:
8180:
8177:
8176:
8159:of linear maps
8109:
8106:
8105:
8070:
8066:
8057:
8053:
8051:
8048:
8047:
8018:
8014:
7993:
7989:
7974:
7970:
7955:
7951:
7942:
7931:
7915:
7911:
7896:
7892:
7884:
7881:
7880:
7874:
7868:
7843:
7839:
7837:
7834:
7833:
7812:
7809:
7808:
7779:
7776:
7775:
7714:
7711:
7710:
7646:
7643:
7642:
7622:
7621:
7609:
7608:
7587:
7586:
7553:
7552:
7510:
7509:
7468:
7443:
7414:
7392:
7390:
7387:
7386:
7361:
7358:
7357:
7340:
7339:
7327:
7326:
7308:
7307:
7280:
7279:
7250:
7225:
7196:
7174:
7172:
7169:
7168:
7143:
7140:
7139:
7102:
7061:
7044:
7041:
7040:
7034:
7015:
7014:
7008:
7004:
6989:
6985:
6979:
6975:
6957:
6956:
6947:
6943:
6919:
6915:
6902:
6893:
6889:
6874:
6870:
6860:
6858:
6855:
6854:
6775:
6772:
6771:
6748:
6744:
6730:
6727:
6726:
6689:
6686:
6685:
6662:
6658:
6644:
6641:
6640:
6606:
6603:
6602:
6579:
6575:
6561:
6558:
6557:
6541:
6538:
6537:
6530:
6494:
6491:
6490:
6423:
6420:
6419:
6394:
6391:
6390:
6365:
6362:
6361:
6318:
6315:
6314:
6271:
6254:
6219:
6216:
6215:
6192:
6188:
6182:
6178:
6169:
6165:
6156:
6152:
6150:
6147:
6146:
6130:
6127:
6126:
6045:
6042:
6041:
5953:
5950:
5949:
5918:
5915:
5914:
5871:
5868:
5867:
5841:
5833:
5832:
5818:
5815:
5814:
5808:
5774:
5766:
5765:
5747:
5739:
5738:
5717:
5709:
5708:
5697:
5694:
5693:
5665:
5662:
5661:
5636:
5633:
5632:
5609:
5601:
5600:
5582:
5574:
5573:
5552:
5544:
5543:
5526:
5523:
5522:
5487:
5484:
5483:
5448:
5445:
5444:
5428:
5425:
5424:
5393:
5390:
5389:
5352:
5349:
5348:
5342:
5317:
5314:
5313:
5297:
5294:
5293:
5277:
5274:
5273:
5257:
5254:
5253:
5237:
5234:
5233:
5217:
5214:
5213:
5167:
5164:
5163:
5120:
5117:
5116:
5067:
5064:
5063:
5057:
5045:
5006:
5003:
5002:
4985:
4984:
4972:
4971:
4947:
4946:
4898:
4897:
4853:
4843:
4808:
4798:
4796:
4793:
4792:
4755:
4735:
4709:
4704:
4701:
4700:
4671:
4667:
4655:
4651:
4643:
4640:
4639:
4608:
4605:
4604:
4585:
4582:
4581:
4561:
4557:
4543:
4540:
4539:
4511:
4508:
4507:
4485:
4482:
4481:
4453:
4450:
4449:
4418:
4415:
4414:
4411:
4374:
4370:
4358:
4354:
4348:
4337:
4321:
4317:
4309:
4306:
4305:
4265:
4262:
4261:
4242:
4239:
4238:
4209:
4205:
4199:
4186:
4175:
4173:
4168:
4164:
4150:
4146:
4140:
4127:
4116:
4114:
4109:
4105:
4075:
4059:
4055:
4040:
4036:
4028:
4025:
4024:
3997:
3993:
3991:
3988:
3987:
3968:
3965:
3964:
3951:to equal 1 for
3926:
3922:
3892:
3888:
3886:
3883:
3882:
3856:
3852:
3828:
3824:
3822:
3819:
3818:
3785:
3784:
3712:
3708:
3707:
3647:
3642:
3543:
3541:
3538:
3537:
3515:shuffle product
3498:
3495:
3494:
3477:
3476:
3456:
3452:
3422:
3418:
3417:
3413:
3390:
3386:
3362:
3358:
3357:
3353:
3321:
3314:
3303:
3292:
3279:
3278:
3267:
3263:
3242:
3238:
3237:
3233:
3217:
3213:
3198:
3194:
3193:
3189:
3183:
3172:
3159:
3158:
3149:
3145:
3121:
3117:
3104:
3095:
3091:
3076:
3072:
3062:
3060:
3057:
3056:
3020:
3016:
3004:
3000:
2994:
2983:
2967:
2963:
2955:
2952:
2951:
2903:
2900:
2899:
2877:
2874:
2873:
2856:
2855:
2777:
2776:
2712:
2690:
2688:
2685:
2684:
2620:
2617:
2616:
2593:
2589:
2575:
2572:
2571:
2555:
2552:
2551:
2531:
2528:
2527:
2504:
2501:
2500:
2481:
2478:
2477:
2461:
2458:
2457:
2353:
2345:
2344:
2336:
2333:
2332:
2310:
2307:
2306:
2274:
2266:
2265:
2263:
2260:
2259:
2227:
2219:
2218:
2182:
2174:
2173:
2168:
2165:
2164:
2136:
2133:
2132:
2059:
2056:
2055:
2036:
2033:
2032:
2003:
1999:
1985:
1982:
1981:
1944:
1941:
1940:
1888:
1885:
1884:
1850:
1847:
1846:
1830:
1827:
1826:
1806:
1803:
1802:
1786:
1783:
1782:
1766:
1763:
1762:
1742:
1739:
1738:
1722:
1719:
1718:
1693:
1690:
1689:
1670:
1667:
1666:
1620:
1617:
1616:
1606:
1595:
1585:
1579:
1573:
1567:
1561:
1555:category theory
1526:
1525:
1521:
1519:
1516:
1515:
1499:
1496:
1495:
1463:
1462:
1458:
1436:
1435:
1431:
1426:
1423:
1422:
1398:
1397:
1393:
1391:
1388:
1387:
1371:
1368:
1367:
1362:Similarly, the
1344:
1341:
1340:
1324:
1321:
1320:
1294:
1253:
1227:
1223:
1208:
1204:
1202:
1199:
1198:
1178:
1174:
1172:
1169:
1168:
1144:
1140:
1132:
1129:
1128:
1103:
1100:
1099:
1039:
1023:
1012:
1006:
1000:
994:
993:-vector spaces
988:
982:
976:
962:
944:
933:
927:
917:
895:
884:
874:
868:
862:
834:
831:
830:
806:
792:
780:
737:
734:
733:
723:
706:
703:
619:
615:
613:
610:
609:
545:
541:
529:
525:
513:
509:
507:
504:
503:
406:
402:
396:
385:
364:
361:
360:
249:
245:
233:
229:
227:
224:
223:
201:th tensor power
172:
17:
12:
11:
5:
9794:
9784:
9783:
9778:
9773:
9768:
9751:
9750:
9748:
9747:
9742:
9740:Woldemar Voigt
9737:
9732:
9727:
9722:
9717:
9712:
9707:
9705:Leonhard Euler
9702:
9697:
9692:
9687:
9681:
9679:
9677:Mathematicians
9673:
9672:
9669:
9668:
9666:
9665:
9660:
9655:
9650:
9645:
9640:
9635:
9630:
9625:
9619:
9617:
9613:
9612:
9610:
9609:
9604:
9602:Torsion tensor
9599:
9594:
9589:
9584:
9579:
9574:
9568:
9566:
9559:
9555:
9554:
9552:
9551:
9546:
9541:
9536:
9531:
9526:
9521:
9516:
9511:
9506:
9501:
9496:
9491:
9486:
9481:
9476:
9471:
9466:
9461:
9455:
9453:
9447:
9446:
9444:
9443:
9437:
9435:Tensor product
9432:
9427:
9425:Symmetrization
9422:
9417:
9415:Lie derivative
9412:
9407:
9402:
9397:
9392:
9386:
9384:
9378:
9377:
9375:
9374:
9369:
9364:
9359:
9354:
9349:
9344:
9339:
9337:Tensor density
9334:
9329:
9323:
9321:
9315:
9314:
9312:
9311:
9309:Voigt notation
9306:
9301:
9296:
9294:Ricci calculus
9291:
9286:
9281:
9279:Index notation
9276:
9271:
9265:
9263:
9259:
9258:
9255:
9254:
9252:
9251:
9246:
9241:
9236:
9231:
9225:
9223:
9221:
9220:
9215:
9209:
9206:
9205:
9203:
9202:
9197:
9195:Tensor algebra
9192:
9187:
9182:
9177:
9175:Dyadic algebra
9172:
9167:
9161:
9159:
9150:
9146:
9145:
9138:
9135:
9134:
9127:
9126:
9119:
9112:
9104:
9095:
9094:
9092:
9091:
9080:
9077:
9076:
9074:
9073:
9068:
9063:
9061:Linear algebra
9058:
9052:
9050:
9046:
9045:
9043:
9042:
9036:
9034:
9030:
9029:
9027:
9026:
9024:Tensor algebra
9021:
9016:
9013:Quotient group
9006:
8996:
8986:
8976:
8971:
8965:
8963:
8959:
8958:
8956:
8955:
8950:
8945:
8935:
8932:Euclidean norm
8925:
8915:
8905:
8900:
8890:
8885:
8880:
8874:
8872:
8866:
8865:
8863:
8862:
8852:
8842:
8832:
8822:
8811:
8809:
8805:
8804:
8802:
8801:
8796:
8786:
8783:Multiplication
8772:
8762:
8752:
8738:
8732:
8730:
8729:Basic concepts
8726:
8725:
8723:
8722:
8717:
8712:
8707:
8702:
8697:
8695:Linear algebra
8692:
8687:
8682:
8677:
8672:
8667:
8662:
8661:
8660:
8655:
8645:
8639:
8637:
8633:
8632:
8630:
8629:
8624:
8618:
8615:
8614:
8607:
8606:
8599:
8592:
8584:
8578:
8577:
8572:
8547:
8546:
8537:
8509:
8506:
8505:
8504:
8499:
8494:
8489:
8484:
8477:
8474:
8461:
8458:
8455:
8452:
8449:
8438:
8437:
8426:
8423:
8420:
8417:
8414:
8411:
8408:
8405:
8402:
8399:
8396:
8393:
8390:
8387:
8384:
8381:
8378:
8375:
8372:
8369:
8366:
8363:
8360:
8357:
8354:
8351:
8328:
8323:
8319:
8287:
8282:
8278:
8275:
8272:
8266:
8236:
8233:
8230:
8226:
8222:
8219:
8216:
8213:
8210:
8207:
8202:
8198:
8194:
8189:
8185:
8125:
8122:
8119:
8116:
8113:
8093:
8090:
8087:
8084:
8079:
8076:
8073:
8069:
8065:
8060:
8056:
8044:
8043:
8032:
8027:
8024:
8021:
8017:
8013:
8010:
8007:
8002:
7999:
7996:
7992:
7988:
7985:
7982:
7977:
7973:
7969:
7966:
7963:
7958:
7954:
7950:
7945:
7940:
7937:
7934:
7930:
7926:
7923:
7918:
7914:
7910:
7907:
7904:
7899:
7895:
7891:
7888:
7870:Main article:
7867:
7864:
7851:
7846:
7842:
7819:
7816:
7792:
7789:
7786:
7783:
7772:
7771:
7760:
7757:
7754:
7751:
7748:
7745:
7742:
7739:
7736:
7733:
7730:
7727:
7724:
7721:
7718:
7704:
7703:
7692:
7689:
7686:
7683:
7680:
7677:
7674:
7671:
7668:
7665:
7662:
7659:
7656:
7653:
7650:
7636:
7635:
7620:
7617:
7614:
7612:
7610:
7607:
7604:
7601:
7598:
7595:
7592:
7590:
7588:
7585:
7582:
7579:
7576:
7573:
7570:
7567:
7564:
7561:
7558:
7556:
7554:
7551:
7548:
7545:
7542:
7539:
7536:
7533:
7530:
7527:
7524:
7521:
7518:
7515:
7513:
7511:
7508:
7505:
7502:
7499:
7496:
7493:
7490:
7487:
7484:
7481:
7478:
7474:
7471:
7467:
7464:
7461:
7458:
7455:
7452:
7449:
7446:
7444:
7442:
7439:
7436:
7433:
7430:
7427:
7424:
7420:
7417:
7413:
7410:
7407:
7404:
7401:
7398:
7395:
7394:
7371:
7368:
7365:
7356:Similarly, on
7354:
7353:
7338:
7335:
7332:
7330:
7328:
7325:
7322:
7319:
7316:
7313:
7311:
7309:
7306:
7303:
7300:
7297:
7294:
7291:
7288:
7285:
7283:
7281:
7278:
7275:
7272:
7269:
7266:
7263:
7260:
7256:
7253:
7249:
7246:
7243:
7240:
7237:
7234:
7231:
7228:
7226:
7224:
7221:
7218:
7215:
7212:
7209:
7206:
7202:
7199:
7195:
7192:
7189:
7186:
7183:
7180:
7177:
7176:
7153:
7150:
7147:
7136:
7135:
7124:
7121:
7118:
7115:
7112:
7108:
7105:
7101:
7098:
7095:
7092:
7089:
7086:
7083:
7080:
7077:
7074:
7071:
7067:
7064:
7060:
7057:
7054:
7051:
7048:
7033:
7030:
7029:
7028:
7011:
7007:
7003:
7000:
6997:
6992:
6988:
6982:
6978:
6974:
6971:
6968:
6965:
6962:
6960:
6958:
6955:
6950:
6946:
6942:
6939:
6936:
6933:
6930:
6927:
6922:
6918:
6914:
6911:
6908:
6905:
6903:
6901:
6896:
6892:
6888:
6885:
6882:
6877:
6873:
6869:
6866:
6863:
6862:
6848:
6847:
6836:
6833:
6830:
6827:
6824:
6821:
6818:
6815:
6812:
6809:
6806:
6803:
6800:
6797:
6794:
6791:
6788:
6785:
6782:
6779:
6756:
6751:
6747:
6743:
6740:
6737:
6734:
6723:
6722:
6711:
6708:
6705:
6702:
6699:
6696:
6693:
6670:
6665:
6661:
6657:
6654:
6651:
6648:
6637:
6636:
6625:
6622:
6619:
6616:
6613:
6610:
6587:
6582:
6578:
6574:
6571:
6568:
6565:
6545:
6529:
6526:
6513:
6510:
6507:
6504:
6501:
6498:
6487:
6486:
6475:
6472:
6469:
6466:
6463:
6460:
6457:
6454:
6451:
6448:
6445:
6442:
6439:
6436:
6433:
6430:
6427:
6404:
6401:
6398:
6378:
6375:
6372:
6369:
6349:
6346:
6343:
6340:
6337:
6334:
6331:
6328:
6325:
6322:
6311:
6310:
6299:
6296:
6293:
6290:
6287:
6284:
6281:
6277:
6274:
6270:
6267:
6264:
6260:
6257:
6253:
6250:
6247:
6244:
6241:
6238:
6235:
6232:
6229:
6226:
6223:
6200:
6195:
6191:
6185:
6181:
6177:
6172:
6168:
6164:
6159:
6155:
6134:
6115:
6114:
6103:
6100:
6097:
6094:
6091:
6088:
6085:
6082:
6079:
6076:
6073:
6070:
6067:
6064:
6061:
6058:
6055:
6052:
6049:
6032:
6031:
6020:
6017:
6014:
6011:
6008:
6005:
6002:
5999:
5996:
5993:
5990:
5987:
5984:
5981:
5978:
5975:
5972:
5969:
5966:
5963:
5960:
5957:
5934:
5931:
5928:
5925:
5922:
5911:
5910:
5899:
5896:
5893:
5890:
5887:
5884:
5881:
5878:
5875:
5861:
5860:
5849:
5844:
5839:
5836:
5831:
5828:
5825:
5822:
5807:
5804:
5800:
5799:
5788:
5785:
5780:
5777:
5772:
5769:
5764:
5761:
5758:
5753:
5750:
5745:
5742:
5737:
5734:
5731:
5728:
5723:
5720:
5715:
5712:
5707:
5704:
5701:
5678:
5675:
5672:
5669:
5649:
5646:
5643:
5640:
5629:
5628:
5615:
5612:
5607:
5604:
5599:
5596:
5593:
5588:
5585:
5580:
5577:
5572:
5569:
5566:
5563:
5558:
5555:
5550:
5547:
5542:
5539:
5536:
5533:
5530:
5503:
5500:
5497:
5494:
5491:
5467:
5464:
5461:
5458:
5455:
5452:
5432:
5421:
5420:
5409:
5406:
5403:
5400:
5397:
5383:
5382:
5371:
5368:
5365:
5362:
5359:
5356:
5341:
5338:
5321:
5301:
5281:
5261:
5241:
5221:
5201:
5198:
5195:
5192:
5189:
5186:
5183:
5180:
5177:
5174:
5171:
5160:
5159:
5148:
5145:
5142:
5139:
5136:
5133:
5130:
5127:
5124:
5110:
5109:
5098:
5095:
5092:
5089:
5086:
5083:
5080:
5077:
5074:
5071:
5056:
5055:Multiplication
5053:
5044:
5041:
5028:
5025:
5022:
5019:
5016:
5013:
5010:
4999:
4998:
4983:
4980:
4977:
4975:
4973:
4970:
4967:
4964:
4961:
4958:
4955:
4952:
4950:
4948:
4945:
4942:
4939:
4936:
4933:
4930:
4927:
4924:
4921:
4918:
4915:
4912:
4909:
4906:
4903:
4901:
4899:
4896:
4893:
4890:
4887:
4884:
4881:
4878:
4875:
4872:
4869:
4866:
4863:
4859:
4856:
4852:
4849:
4846:
4844:
4842:
4839:
4836:
4833:
4830:
4827:
4824:
4821:
4818:
4814:
4811:
4807:
4804:
4801:
4800:
4786:
4785:
4774:
4771:
4768:
4765:
4761:
4758:
4754:
4751:
4748:
4745:
4741:
4738:
4734:
4731:
4728:
4725:
4722:
4719:
4715:
4712:
4708:
4685:
4682:
4679:
4674:
4670:
4666:
4663:
4658:
4654:
4650:
4647:
4636:
4635:
4624:
4621:
4618:
4615:
4612:
4589:
4569:
4564:
4560:
4556:
4553:
4550:
4547:
4527:
4524:
4521:
4518:
4515:
4495:
4492:
4489:
4469:
4466:
4463:
4460:
4457:
4437:
4434:
4431:
4428:
4425:
4422:
4410:
4407:
4406:
4405:
4394:
4389:
4386:
4383:
4380:
4377:
4373:
4369:
4366:
4361:
4357:
4351:
4346:
4343:
4340:
4336:
4332:
4329:
4324:
4320:
4316:
4313:
4287:
4284:
4281:
4278:
4275:
4272:
4269:
4249:
4246:
4235:
4234:
4223:
4218:
4212:
4208:
4202:
4195:
4192:
4189:
4184:
4181:
4178:
4172:
4167:
4163:
4159:
4153:
4149:
4143:
4136:
4133:
4130:
4125:
4122:
4119:
4113:
4108:
4102:
4099:
4096:
4093:
4090:
4087:
4084:
4081:
4078:
4074:
4070:
4067:
4062:
4058:
4054:
4051:
4048:
4043:
4039:
4035:
4032:
4020:Equivalently,
4000:
3996:
3975:
3972:
3938:
3935:
3932:
3929:
3925:
3921:
3918:
3915:
3910:
3907:
3904:
3901:
3898:
3895:
3891:
3868:
3865:
3862:
3859:
3855:
3851:
3848:
3845:
3840:
3837:
3834:
3831:
3827:
3799:
3798:
3783:
3780:
3777:
3774:
3771:
3768:
3765:
3762:
3759:
3756:
3753:
3750:
3747:
3744:
3741:
3738:
3735:
3732:
3729:
3726:
3723:
3720:
3711:
3709:
3706:
3703:
3700:
3697:
3694:
3691:
3688:
3685:
3682:
3679:
3676:
3673:
3670:
3667:
3664:
3661:
3658:
3654:
3646:
3643:
3640:
3636:
3633:
3630:
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:
3549:
3546:
3545:
3502:
3491:
3490:
3474:
3468:
3465:
3462:
3459:
3455:
3451:
3448:
3445:
3440:
3437:
3434:
3431:
3428:
3425:
3421:
3416:
3412:
3408:
3402:
3399:
3396:
3393:
3389:
3385:
3382:
3379:
3374:
3371:
3368:
3365:
3361:
3356:
3349:
3346:
3343:
3340:
3337:
3334:
3331:
3327:
3324:
3320:
3317:
3313:
3306:
3301:
3298:
3295:
3291:
3287:
3284:
3282:
3280:
3276:
3270:
3266:
3262:
3259:
3256:
3251:
3248:
3245:
3241:
3236:
3231:
3226:
3220:
3216:
3212:
3209:
3206:
3201:
3197:
3192:
3186:
3181:
3178:
3175:
3171:
3167:
3164:
3162:
3160:
3157:
3152:
3148:
3144:
3141:
3138:
3135:
3132:
3129:
3124:
3120:
3116:
3113:
3110:
3107:
3105:
3103:
3098:
3094:
3090:
3087:
3084:
3079:
3075:
3071:
3068:
3065:
3064:
3046:
3045:
3034:
3029:
3026:
3023:
3019:
3015:
3012:
3007:
3003:
2997:
2992:
2989:
2986:
2982:
2978:
2975:
2970:
2966:
2962:
2959:
2934:
2931:
2928:
2925:
2922:
2919:
2916:
2913:
2910:
2907:
2887:
2884:
2881:
2870:
2869:
2854:
2851:
2848:
2845:
2842:
2839:
2836:
2833:
2830:
2827:
2824:
2821:
2818:
2815:
2812:
2809:
2806:
2803:
2800:
2797:
2794:
2791:
2788:
2785:
2782:
2780:
2778:
2775:
2772:
2769:
2766:
2763:
2760:
2757:
2754:
2751:
2748:
2745:
2742:
2739:
2736:
2733:
2730:
2727:
2724:
2721:
2718:
2715:
2713:
2711:
2708:
2705:
2702:
2699:
2696:
2693:
2692:
2678:
2677:
2666:
2663:
2660:
2657:
2654:
2651:
2648:
2645:
2642:
2639:
2636:
2633:
2630:
2627:
2624:
2601:
2596:
2592:
2588:
2585:
2582:
2579:
2559:
2535:
2511:
2508:
2488:
2485:
2465:
2454:
2453:
2442:
2439:
2436:
2433:
2430:
2427:
2424:
2421:
2418:
2415:
2412:
2409:
2406:
2403:
2400:
2397:
2394:
2391:
2388:
2385:
2382:
2379:
2376:
2373:
2370:
2367:
2364:
2359:
2356:
2351:
2348:
2343:
2340:
2317:
2314:
2294:
2291:
2288:
2285:
2280:
2277:
2272:
2269:
2256:
2255:
2244:
2241:
2238:
2233:
2230:
2225:
2222:
2217:
2214:
2211:
2208:
2205:
2202:
2199:
2196:
2193:
2188:
2185:
2180:
2177:
2172:
2149:
2146:
2143:
2140:
2129:
2128:
2117:
2114:
2111:
2108:
2105:
2102:
2099:
2096:
2093:
2090:
2087:
2084:
2081:
2078:
2075:
2072:
2069:
2066:
2063:
2040:
2020:
2017:
2014:
2011:
2006:
2002:
1998:
1995:
1992:
1989:
1978:
1977:
1966:
1963:
1960:
1957:
1954:
1951:
1948:
1934:
1933:
1922:
1919:
1916:
1913:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1869:
1866:
1863:
1860:
1857:
1854:
1834:
1810:
1790:
1770:
1759:Multiplication
1746:
1726:
1706:
1703:
1700:
1697:
1677:
1674:
1663:
1662:
1651:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1605:
1602:
1535:
1532:
1529:
1524:
1503:
1481:
1478:
1472:
1469:
1466:
1461:
1457:
1454:
1451:
1445:
1442:
1439:
1434:
1430:
1407:
1404:
1401:
1396:
1375:
1348:
1328:
1293:
1290:
1252:
1249:
1230:
1226:
1222:
1219:
1216:
1211:
1207:
1186:
1181:
1177:
1152:
1147:
1143:
1139:
1136:
1116:
1113:
1110:
1107:
1068:indeterminates
1038:
1035:
906:
905:
850:
847:
844:
841:
838:
791:, the functor
771:for forming a
756:
753:
750:
747:
744:
741:
702:
699:
639:
636:
633:
630:
627:
622:
618:
591:graded algebra
571:
570:
559:
554:
551:
548:
544:
540:
537:
532:
528:
524:
521:
516:
512:
489:
488:
477:
474:
471:
468:
465:
462:
459:
456:
453:
450:
447:
444:
441:
438:
435:
432:
429:
426:
423:
420:
417:
414:
409:
405:
399:
394:
391:
388:
384:
380:
377:
374:
371:
368:
296:
295:
284:
281:
278:
275:
272:
269:
266:
263:
260:
255:
252:
248:
244:
241:
236:
232:
209:tensor product
171:
168:
64:tensor product
25:tensor algebra
15:
9:
6:
4:
3:
2:
9793:
9782:
9781:Hopf algebras
9779:
9777:
9774:
9772:
9769:
9767:
9764:
9763:
9761:
9746:
9743:
9741:
9738:
9736:
9733:
9731:
9728:
9726:
9723:
9721:
9718:
9716:
9713:
9711:
9708:
9706:
9703:
9701:
9698:
9696:
9693:
9691:
9688:
9686:
9683:
9682:
9680:
9678:
9674:
9664:
9661:
9659:
9656:
9654:
9651:
9649:
9646:
9644:
9641:
9639:
9636:
9634:
9631:
9629:
9626:
9624:
9621:
9620:
9618:
9614:
9608:
9605:
9603:
9600:
9598:
9595:
9593:
9590:
9588:
9585:
9583:
9582:Metric tensor
9580:
9578:
9575:
9573:
9570:
9569:
9567:
9563:
9560:
9556:
9550:
9547:
9545:
9542:
9540:
9537:
9535:
9532:
9530:
9527:
9525:
9522:
9520:
9517:
9515:
9512:
9510:
9507:
9505:
9502:
9500:
9497:
9495:
9494:Exterior form
9492:
9490:
9487:
9485:
9482:
9480:
9477:
9475:
9472:
9470:
9467:
9465:
9462:
9460:
9457:
9456:
9454:
9448:
9441:
9438:
9436:
9433:
9431:
9428:
9426:
9423:
9421:
9418:
9416:
9413:
9411:
9408:
9406:
9403:
9401:
9398:
9396:
9393:
9391:
9388:
9387:
9385:
9383:
9379:
9373:
9370:
9368:
9367:Tensor bundle
9365:
9363:
9360:
9358:
9355:
9353:
9350:
9348:
9345:
9343:
9340:
9338:
9335:
9333:
9330:
9328:
9325:
9324:
9322:
9316:
9310:
9307:
9305:
9302:
9300:
9297:
9295:
9292:
9290:
9287:
9285:
9282:
9280:
9277:
9275:
9272:
9270:
9267:
9266:
9264:
9260:
9250:
9247:
9245:
9242:
9240:
9237:
9235:
9232:
9230:
9227:
9226:
9224:
9219:
9216:
9214:
9211:
9210:
9207:
9201:
9198:
9196:
9193:
9191:
9188:
9186:
9183:
9181:
9178:
9176:
9173:
9171:
9168:
9166:
9163:
9162:
9160:
9158:
9154:
9151:
9147:
9143:
9142:
9136:
9132:
9125:
9120:
9118:
9113:
9111:
9106:
9105:
9102:
9090:
9082:
9081:
9078:
9072:
9069:
9067:
9064:
9062:
9059:
9057:
9054:
9053:
9051:
9047:
9041:
9038:
9037:
9035:
9031:
9025:
9022:
9020:
9017:
9014:
9010:
9007:
9004:
9000:
8997:
8994:
8990:
8987:
8984:
8980:
8977:
8975:
8972:
8970:
8967:
8966:
8964:
8960:
8954:
8951:
8949:
8946:
8943:
8939:
8938:Orthogonality
8936:
8933:
8929:
8926:
8923:
8919:
8916:
8913:
8909:
8906:
8904:
8903:Hilbert space
8901:
8898:
8894:
8891:
8889:
8886:
8884:
8881:
8879:
8876:
8875:
8873:
8867:
8860:
8856:
8853:
8850:
8846:
8843:
8840:
8836:
8833:
8830:
8826:
8823:
8820:
8816:
8813:
8812:
8810:
8806:
8800:
8797:
8794:
8790:
8787:
8784:
8780:
8776:
8773:
8770:
8766:
8763:
8760:
8756:
8753:
8750:
8746:
8742:
8739:
8737:
8734:
8733:
8731:
8727:
8721:
8718:
8716:
8713:
8711:
8708:
8706:
8703:
8701:
8698:
8696:
8693:
8691:
8688:
8686:
8683:
8681:
8678:
8676:
8673:
8671:
8668:
8666:
8663:
8659:
8656:
8654:
8651:
8650:
8649:
8646:
8644:
8641:
8640:
8638:
8634:
8628:
8625:
8623:
8620:
8619:
8616:
8612:
8605:
8600:
8598:
8593:
8591:
8586:
8585:
8582:
8575:
8569:
8565:
8561:
8557:
8553:
8549:
8548:
8545:
8540:
8538:3-540-64243-9
8534:
8530:
8526:
8522:
8521:
8516:
8512:
8511:
8503:
8500:
8498:
8495:
8493:
8490:
8488:
8485:
8483:
8480:
8479:
8473:
8459:
8456:
8453:
8450:
8447:
8424:
8421:
8415:
8412:
8409:
8403:
8400:
8397:
8394:
8391:
8385:
8382:
8379:
8373:
8370:
8367:
8361:
8358:
8355:
8342:
8341:
8340:
8326:
8321:
8317:
8307:
8305:
8280:
8276:
8273:
8270:
8252:
8234:
8231:
8228:
8224:
8217:
8214:
8211:
8205:
8200:
8196:
8192:
8187:
8183:
8174:
8170:
8166:
8162:
8158:
8154:
8150:
8146:
8142:
8137:
8123:
8120:
8117:
8114:
8111:
8091:
8088:
8085:
8082:
8077:
8074:
8071:
8067:
8063:
8058:
8054:
8025:
8022:
8019:
8015:
8011:
8008:
8005:
8000:
7997:
7994:
7990:
7983:
7975:
7971:
7967:
7964:
7961:
7956:
7952:
7943:
7938:
7935:
7932:
7928:
7924:
7916:
7912:
7908:
7905:
7902:
7897:
7893:
7879:
7878:
7877:
7873:
7863:
7849:
7844:
7840:
7830:
7817:
7814:
7806:
7790:
7787:
7784:
7781:
7758:
7755:
7749:
7743:
7740:
7734:
7725:
7722:
7719:
7709:
7708:
7707:
7690:
7687:
7681:
7675:
7672:
7666:
7657:
7654:
7651:
7641:
7640:
7639:
7618:
7615:
7613:
7605:
7602:
7599:
7596:
7593:
7591:
7583:
7580:
7577:
7574:
7571:
7568:
7565:
7562:
7559:
7557:
7546:
7543:
7540:
7537:
7534:
7531:
7528:
7525:
7516:
7514:
7503:
7500:
7497:
7494:
7491:
7488:
7485:
7465:
7462:
7456:
7447:
7445:
7437:
7425:
7411:
7408:
7402:
7385:
7384:
7383:
7369:
7366:
7363:
7336:
7333:
7331:
7323:
7320:
7317:
7314:
7312:
7301:
7298:
7295:
7286:
7284:
7273:
7270:
7267:
7247:
7244:
7238:
7229:
7227:
7219:
7207:
7193:
7190:
7184:
7167:
7166:
7165:
7151:
7148:
7145:
7119:
7113:
7110:
7096:
7090:
7087:
7084:
7081:
7078:
7072:
7058:
7055:
7049:
7039:
7038:
7037:
7032:Compatibility
7009:
7005:
7001:
6998:
6995:
6990:
6986:
6980:
6972:
6969:
6963:
6961:
6948:
6944:
6937:
6934:
6931:
6928:
6920:
6916:
6909:
6906:
6904:
6894:
6890:
6886:
6883:
6880:
6875:
6871:
6864:
6853:
6852:
6851:
6834:
6831:
6828:
6825:
6819:
6813:
6810:
6804:
6798:
6795:
6789:
6786:
6783:
6777:
6770:
6769:
6768:
6754:
6749:
6745:
6741:
6738:
6735:
6732:
6709:
6706:
6703:
6697:
6691:
6684:
6683:
6682:
6668:
6663:
6659:
6655:
6652:
6649:
6646:
6623:
6620:
6614:
6608:
6601:
6600:
6599:
6585:
6580:
6576:
6572:
6569:
6566:
6563:
6543:
6535:
6525:
6511:
6508:
6505:
6502:
6499:
6496:
6470:
6467:
6464:
6455:
6449:
6446:
6443:
6431:
6418:
6417:
6416:
6402:
6399:
6396:
6376:
6373:
6370:
6367:
6347:
6344:
6341:
6338:
6332:
6329:
6326:
6320:
6291:
6282:
6268:
6265:
6262:
6248:
6239:
6230:
6224:
6214:
6213:
6212:
6198:
6193:
6189:
6183:
6179:
6175:
6170:
6166:
6162:
6157:
6153:
6132:
6124:
6120:
6101:
6098:
6092:
6089:
6086:
6080:
6077:
6071:
6068:
6065:
6053:
6050:
6040:
6039:
6038:
6035:
6018:
6015:
6009:
6006:
6003:
5997:
5994:
5988:
5979:
5973:
5964:
5961:
5948:
5947:
5946:
5932:
5929:
5926:
5923:
5920:
5897:
5894:
5891:
5888:
5885:
5882:
5879:
5876:
5866:
5865:
5864:
5847:
5842:
5829:
5826:
5823:
5820:
5813:
5812:
5811:
5806:Compatibility
5803:
5786:
5783:
5778:
5775:
5762:
5759:
5756:
5751:
5748:
5735:
5729:
5726:
5721:
5718:
5702:
5692:
5691:
5690:
5676:
5673:
5670:
5667:
5647:
5644:
5641:
5638:
5613:
5610:
5597:
5594:
5591:
5586:
5583:
5570:
5567:
5564:
5556:
5553:
5540:
5537:
5531:
5521:
5520:
5519:
5517:
5501:
5498:
5495:
5492:
5489:
5481:
5465:
5462:
5459:
5456:
5453:
5450:
5430:
5407:
5401:
5398:
5395:
5388:
5387:
5386:
5369:
5366:
5360:
5357:
5354:
5347:
5346:
5345:
5337:
5335:
5319:
5279:
5239:
5219:
5199:
5196:
5193:
5190:
5187:
5181:
5178:
5175:
5146:
5143:
5140:
5134:
5131:
5128:
5125:
5115:
5114:
5113:
5096:
5093:
5087:
5084:
5081:
5078:
5075:
5072:
5062:
5061:
5060:
5052:
5050:
5040:
5026:
5023:
5020:
5017:
5014:
5011:
5008:
4981:
4978:
4976:
4968:
4965:
4962:
4959:
4956:
4953:
4951:
4940:
4934:
4931:
4928:
4925:
4919:
4913:
4910:
4907:
4904:
4902:
4891:
4888:
4885:
4882:
4879:
4876:
4873:
4864:
4861:
4847:
4845:
4837:
4825:
4819:
4816:
4791:
4790:
4789:
4772:
4766:
4752:
4749:
4743:
4732:
4726:
4720:
4717:
4699:
4698:
4697:
4683:
4680:
4677:
4672:
4668:
4664:
4661:
4656:
4652:
4648:
4645:
4622:
4616:
4613:
4610:
4603:
4602:
4601:
4587:
4567:
4562:
4558:
4554:
4551:
4548:
4545:
4525:
4519:
4516:
4513:
4493:
4490:
4487:
4467:
4461:
4458:
4455:
4435:
4429:
4426:
4423:
4420:
4392:
4384:
4381:
4378:
4371:
4367:
4364:
4359:
4355:
4349:
4344:
4341:
4338:
4334:
4327:
4322:
4318:
4314:
4304:
4303:
4302:
4299:
4282:
4279:
4276:
4273:
4270:
4247:
4244:
4221:
4216:
4210:
4206:
4200:
4193:
4190:
4187:
4182:
4179:
4176:
4170:
4165:
4161:
4157:
4151:
4147:
4141:
4134:
4131:
4128:
4123:
4120:
4117:
4111:
4106:
4097:
4094:
4091:
4088:
4085:
4079:
4076:
4072:
4068:
4060:
4056:
4052:
4049:
4046:
4041:
4037:
4023:
4022:
4021:
4018:
4016:
3998:
3994:
3973:
3970:
3962:
3958:
3954:
3933:
3927:
3923:
3919:
3916:
3913:
3905:
3902:
3899:
3893:
3889:
3863:
3857:
3853:
3849:
3846:
3843:
3835:
3829:
3825:
3816:
3813:} → {1, ...,
3812:
3808:
3804:
3781:
3772:
3766:
3763:
3760:
3757:
3751:
3748:
3745:
3739:
3736:
3730:
3727:
3724:
3718:
3704:
3698:
3692:
3689:
3686:
3683:
3677:
3671:
3668:
3662:
3656:
3652:
3644:
3631:
3628:
3625:
3622:
3619:
3616:
3613:
3601:
3598:
3595:
3592:
3589:
3586:
3583:
3577:
3574:
3568:
3562:
3559:
3556:
3550:
3547:
3536:
3535:
3534:
3532:
3530:
3526:
3522:
3516:
3500:
3472:
3463:
3457:
3453:
3449:
3446:
3443:
3435:
3432:
3429:
3423:
3419:
3414:
3410:
3406:
3397:
3391:
3387:
3383:
3380:
3377:
3369:
3363:
3359:
3354:
3344:
3341:
3338:
3335:
3332:
3318:
3315:
3311:
3304:
3299:
3296:
3293:
3289:
3285:
3283:
3274:
3268:
3264:
3260:
3257:
3254:
3249:
3246:
3243:
3239:
3234:
3229:
3224:
3218:
3214:
3210:
3207:
3204:
3199:
3195:
3190:
3184:
3179:
3176:
3173:
3169:
3165:
3163:
3150:
3146:
3136:
3133:
3130:
3122:
3118:
3108:
3106:
3096:
3092:
3088:
3085:
3082:
3077:
3073:
3055:
3054:
3053:
3051:
3032:
3027:
3024:
3021:
3017:
3013:
3010:
3005:
3001:
2995:
2990:
2987:
2984:
2980:
2973:
2968:
2964:
2960:
2950:
2949:
2948:
2945:
2932:
2929:
2926:
2923:
2920:
2917:
2914:
2911:
2908:
2905:
2885:
2882:
2879:
2849:
2846:
2843:
2837:
2834:
2831:
2828:
2825:
2822:
2819:
2816:
2813:
2810:
2807:
2804:
2801:
2795:
2792:
2789:
2783:
2781:
2770:
2767:
2764:
2761:
2758:
2755:
2752:
2746:
2740:
2737:
2734:
2731:
2728:
2725:
2722:
2716:
2714:
2706:
2703:
2700:
2683:
2682:
2681:
2661:
2652:
2646:
2634:
2631:
2628:
2625:
2615:
2614:
2613:
2599:
2594:
2590:
2586:
2583:
2580:
2577:
2549:
2533:
2525:
2509:
2506:
2486:
2483:
2440:
2437:
2434:
2431:
2428:
2425:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2395:
2392:
2389:
2383:
2371:
2362:
2357:
2354:
2331:
2330:
2329:
2315:
2312:
2292:
2286:
2283:
2278:
2275:
2239:
2231:
2228:
2215:
2206:
2200:
2191:
2186:
2183:
2163:
2162:
2161:
2147:
2144:
2141:
2138:
2115:
2112:
2109:
2106:
2103:
2100:
2097:
2094:
2088:
2085:
2082:
2076:
2073:
2067:
2054:
2053:
2052:
2038:
2018:
2015:
2012:
2009:
2004:
2000:
1996:
1993:
1990:
1987:
1964:
1961:
1958:
1952:
1949:
1939:
1938:
1937:
1920:
1917:
1914:
1911:
1908:
1905:
1902:
1896:
1893:
1883:
1882:
1881:
1867:
1864:
1861:
1858:
1855:
1852:
1823:
1808:
1788:
1768:
1760:
1744:
1724:
1701:
1695:
1675:
1672:
1649:
1646:
1643:
1640:
1637:
1631:
1628:
1625:
1615:
1614:
1613:
1611:
1601:
1598:
1593:
1588:
1582:
1576:
1570:
1564:
1560:
1556:
1551:
1522:
1501:
1492:
1479:
1476:
1459:
1455:
1452:
1449:
1432:
1428:
1394:
1373:
1365:
1360:
1346:
1326:
1318:
1313:
1311:
1307:
1303:
1299:
1289:
1287:
1283:
1279:
1275:
1271:
1267:
1263:
1259:
1248:
1246:
1228:
1224:
1220:
1217:
1214:
1209:
1205:
1184:
1179:
1175:
1166:
1145:
1141:
1134:
1127:, but rather
1111:
1105:
1097:
1092:
1090:
1086:
1082:
1081:associativity
1078:
1074:
1070:
1069:
1064:
1060:
1059:basis vectors
1056:
1052:
1048:
1044:
1034:
1030:
1026:
1019:
1015:
1009:
1003:
997:
991:
985:
979:
974:
970:
965:
959:
957:
951:
947:
940:
936:
930:
925:
920:
910:
903:
898:
891:
887:
882:
877:
871:
865:
848:
842:
839:
836:
829:
825:
824:
823:
821:
817:
812:
809:
804:
800:
795:
790:
786:
778:
774:
770:
751:
745:
739:
731:
726:
721:
720:
713:
709:
698:
696:
692:
689:
686:
682:
678:
674:
670:
668:
663:
660:
655:
653:
634:
628:
625:
620:
616:
607:
603:
599:
596:
592:
588:
584:
580:
576:
557:
552:
549:
546:
542:
535:
530:
526:
522:
519:
514:
510:
502:
501:
500:
498:
494:
475:
472:
469:
463:
460:
457:
454:
451:
445:
439:
436:
433:
427:
424:
421:
418:
415:
412:
407:
403:
392:
389:
386:
382:
378:
372:
366:
359:
358:
357:
355:
351:
348:
344:
340:
336:
331:
329:
326:
322:
319:
315:
312:
308:
304:
301:
282:
279:
276:
273:
270:
267:
264:
261:
258:
253:
250:
246:
242:
239:
234:
230:
222:
221:
220:
218:
214:
210:
206:
202:
200:
195:
192:
188:
185:
181:
177:
167:
165:
161:
157:
153:
149:
147:
143:
139:
135:
130:
128:
124:
120:
116:
112:
108:
104:
100:
95:
93:
89:
85:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
30:
26:
22:
9745:Hermann Weyl
9549:Vector space
9534:Pseudotensor
9499:Fiber bundle
9452:abstractions
9347:Mixed tensor
9332:Tensor field
9194:
9139:
9066:Order theory
9056:Field theory
9023:
8922:Affine space
8855:Vector space
8710:Order theory
8555:
8543:
8519:
8439:
8308:
8250:
8172:
8169:free algebra
8164:
8160:
8155:denotes the
8152:
8148:
8144:
8138:
8136:trivially).
8045:
7875:
7831:
7804:
7773:
7705:
7638:Recall that
7637:
7355:
7137:
7035:
6849:
6724:
6681:is given by
6638:
6598:is given by
6534:Hopf algebra
6531:
6528:Hopf algebra
6488:
6312:
6122:
6118:
6116:
6036:
6033:
5912:
5862:
5809:
5801:
5630:
5479:
5422:
5384:
5343:
5333:
5161:
5111:
5058:
5046:
5000:
4787:
4637:
4412:
4300:
4236:
4019:
4014:
3960:
3956:
3952:
3814:
3810:
3806:
3805:0) and Sh(0,
3802:
3800:
3528:
3524:
3520:
3492:
3049:
3047:
2946:
2871:
2679:
2455:
2257:
2130:
1979:
1935:
1824:
1758:
1664:
1607:
1596:
1586:
1580:
1574:
1568:
1562:
1552:
1493:
1361:
1314:
1306:Hopf algebra
1295:
1282:Weyl algebra
1265:
1261:
1254:
1164:
1095:
1093:
1091:-linearity.
1088:
1076:
1072:
1067:
1062:
1054:
1050:
1046:
1042:
1040:
1028:
1024:
1017:
1013:
1007:
1001:
995:
989:
983:
977:
963:
960:
949:
945:
938:
934:
928:
918:
915:
896:
889:
885:
875:
869:
863:
819:
813:
807:
799:left adjoint
793:
772:
724:
719:free algebra
717:
711:
707:
704:
694:
690:
684:
680:
672:
666:
661:
656:
651:
605:
601:
597:
594:
586:
582:
578:
574:
572:
496:
492:
490:
353:
349:
346:
338:
334:
332:
327:
325:ground field
320:
317:
313:
306:
302:
299:
297:
216:
215:with itself
212:
204:
198:
197:
193:
186:
180:vector space
175:
173:
170:Construction
151:
150:
146:Hopf algebra
131:
123:Weyl algebra
106:
102:
96:
83:
76:left adjoint
71:
68:free algebra
66:. It is the
59:
47:
43:
39:
35:
31:
29:vector space
24:
18:
9685:Élie Cartan
9633:Spin tensor
9607:Weyl tensor
9565:Mathematics
9529:Multivector
9320:definitions
9218:Engineering
9157:Mathematics
9071:Ring theory
9033:Topic lists
8993:Multivector
8979:Free object
8897:Dot product
8883:Determinant
8869:Linear and
4413:The counit
2524:free object
1592:natural map
769:linear maps
767:extends to
667:commutative
160:associative
148:structure.
21:mathematics
9760:Categories
9514:Linear map
9382:Operations
9049:Glossaries
9003:Polynomial
8983:Free group
8908:Linear map
8765:Inequality
8552:Serge Lang
8508:References
8502:Fock space
7706:and that
3531:)-shuffles
3493:where the
2499:, because
971:from the
828:linear map
730:functorial
356:= 0,1,2,…
343:direct sum
207:to be the
50:), is the
34:, denoted
9653:EM tensor
9489:Dimension
9440:Transpose
8775:Operation
8457:∈
8422:⊠
8413:⊗
8398:⊠
8383:⊗
8374:⊠
8359:⊗
8350:Δ
8193:⋅
8151:), where
8115:⊗
8089:∈
8012:⊗
8009:⋯
8006:⊗
7984:⊠
7968:⊗
7965:⋯
7962:⊗
7929:∑
7909:⊗
7906:⋯
7903:⊗
7887:Δ
7785:∈
7744:η
7726:ϵ
7723:∘
7720:η
7676:η
7658:ϵ
7655:∘
7652:η
7597:−
7581:⊗
7569:⊗
7563:−
7544:⊠
7532:⊠
7526:−
7520:∇
7501:⊠
7489:⊠
7466:⊠
7457:∘
7454:∇
7429:Δ
7426:∘
7412:⊠
7403:∘
7400:∇
7367:∈
7321:⊗
7299:⊠
7290:∇
7271:⊠
7248:⊠
7239:∘
7236:∇
7211:Δ
7208:∘
7194:⊠
7185:∘
7182:∇
7149:∈
7123:Δ
7120:∘
7111:⊠
7097:∘
7094:∇
7088:ϵ
7085:∘
7082:η
7076:Δ
7073:∘
7059:⊠
7050:∘
7047:∇
7002:⊗
6999:⋯
6996:⊗
6970:−
6935:⊗
6932:⋯
6929:⊗
6887:⊗
6884:⋯
6881:⊗
6832:⊗
6811:⊗
6787:⊗
6742:∈
6736:⊗
6707:−
6650:∈
6567:∈
6506:∈
6468:⊗
6459:Δ
6447:⊠
6435:∇
6432:∘
6429:Δ
6371:⊂
6345:⊠
6330:⊠
6321:τ
6295:Δ
6292:⊠
6289:Δ
6283:∘
6269:⊠
6266:τ
6263:⊠
6249:∘
6243:∇
6240:⊠
6237:∇
6228:∇
6225:∘
6222:Δ
6163:⊗
6117:whenever
6090:⊗
6081:ϵ
6069:⊠
6057:∇
6054:∘
6051:ϵ
6016:≅
6007:⊠
5983:Δ
5965:η
5962:∘
5959:Δ
5930:≅
5924:⊠
5898:η
5895:≅
5892:η
5889:⊠
5886:η
5880:η
5877:∘
5874:Δ
5827:η
5824:∘
5821:ϵ
5787:η
5784:⋅
5760:η
5757:⊗
5730:η
5727:⊠
5703:∘
5700:∇
5674:⊠
5642:⊠
5598:⋅
5595:η
5571:⊗
5568:η
5541:⊠
5538:η
5532:∘
5529:∇
5493:∈
5454:⊗
5431:⊗
5405:↦
5396:η
5364:→
5355:η
5320:⊠
5300:∇
5280:⊗
5260:∇
5240:⊗
5220:⊠
5194:⊗
5179:⊠
5170:∇
5162:That is,
5144:⊗
5138:↦
5132:⊠
5123:∇
5091:→
5082:⊠
5070:∇
5049:bialgebra
5043:Bialgebra
5021:≅
5015:⊠
4979:≅
4966:⊠
4935:ϵ
4932:⊠
4914:ϵ
4911:⊠
4889:⊠
4877:⊠
4865:ϵ
4862:⊠
4829:Δ
4826:∘
4820:ϵ
4817:⊠
4770:Δ
4767:∘
4753:⊠
4750:ϵ
4730:Δ
4727:∘
4721:ϵ
4718:⊠
4684:⋯
4681:⊕
4665:⊕
4649:∈
4620:↦
4611:ϵ
4588:⊗
4549:∈
4523:↦
4514:ϵ
4491:∈
4465:↦
4456:ϵ
4433:→
4421:ϵ
4382:−
4368:⊠
4335:⨁
4331:→
4312:Δ
4277:…
4191:∉
4171:∏
4162:⊠
4132:∈
4112:∏
4092:…
4080:⊆
4073:∑
4053:⊗
4050:⋯
4047:⊗
4031:Δ
4015:preserved
3928:σ
3920:⊗
3917:⋯
3914:⊗
3894:σ
3858:σ
3850:⊗
3847:⋯
3844:⊗
3830:σ
3767:σ
3761:⋯
3740:σ
3719:σ
3714:and
3693:σ
3687:⋯
3672:σ
3657:σ
3645:σ
3639:∣
3620:…
3608:→
3590:…
3575:σ
3551:
3501:ω
3458:σ
3450:⊗
3447:⋯
3444:⊗
3424:σ
3411:⊠
3392:σ
3384:⊗
3381:⋯
3378:⊗
3364:σ
3342:−
3319:∈
3316:σ
3312:∑
3290:∑
3261:⊗
3258:⋯
3255:⊗
3230:ω
3211:⊗
3208:⋯
3205:⊗
3170:∑
3140:Δ
3137:⊗
3134:⋯
3131:⊗
3112:Δ
3089:⊗
3086:⋯
3083:⊗
3067:Δ
3025:−
3014:⊠
2981:⨁
2977:→
2958:Δ
2921:⋅
2909:⊗
2883:⊗
2847:⊗
2838:⊠
2826:⊠
2814:⊠
2802:⊠
2793:⊗
2768:⊠
2756:⊠
2747:⊗
2738:⊠
2726:⊠
2704:⊗
2695:Δ
2656:Δ
2653:⊗
2641:Δ
2638:↦
2632:⊗
2623:Δ
2587:∈
2581:⊗
2558:Δ
2548:generator
2464:Δ
2438:⊠
2432:⊠
2420:⊠
2414:⊠
2402:⊠
2396:⊠
2375:Δ
2372:∘
2366:Δ
2363:⊠
2290:↦
2243:Δ
2240:∘
2216:⊠
2213:Δ
2204:Δ
2201:∘
2195:Δ
2192:⊠
2142:∈
2113:⊠
2101:⊠
2086:⊠
2062:Δ
2013:⊂
1991:∈
1962:⊠
1956:↦
1947:Δ
1918:⊠
1906:⊠
1900:↦
1891:Δ
1862:⊂
1856:∈
1833:Δ
1809:⊠
1789:⊗
1769:⊗
1745:⊗
1725:⊠
1644:⊠
1635:→
1623:Δ
1610:coproduct
1604:Coproduct
1523:⊗
1502:∧
1460:⊗
1433:⊗
1395:⊗
1374:⊗
1347:⊗
1327:∧
1302:bialgebra
1298:coalgebra
1292:Coalgebra
1251:Quotients
1245:covectors
1218:…
1180:∗
1146:∗
846:→
775:from the
743:↦
728:, and is
553:ℓ
539:→
531:ℓ
523:⊗
473:⋯
470:⊕
461:⊗
455:⊗
446:⊕
437:⊗
428:⊕
422:⊕
398:∞
383:⨁
341:) as the
298:That is,
277:⊗
274:⋯
271:⊗
265:⊗
251:⊗
164:coproduct
142:bialgebra
134:coalgebra
9766:Algebras
9519:Manifold
9504:Geodesic
9262:Notation
9089:Category
8799:Variable
8789:Relation
8779:Addition
8755:Function
8741:Equation
8690:K-theory
8554:(2002),
8517:(1989).
8476:See also
7803:that is
7774:for any
5482:and any
4638:for all
3955:= 0 and
2131:for all
777:category
688:bimodule
9776:Tensors
9616:Physics
9450:Related
9213:Physics
9131:Tensors
9001: (
8991: (
8857: (
8847: (
8837: (
8827: (
8817: (
8627:History
8622:Outline
8611:Algebra
8556:Algebra
6725:and on
5689:, that
1559:functor
1098:is not
969:functor
922:is the
801:to the
773:functor
664:over a
323:is the
219:times:
191:integer
182:over a
78:to the
56:tensors
52:algebra
9544:Vector
9539:Spinor
9524:Matrix
9318:Tensor
9015:, ...)
8985:, ...)
8912:Matrix
8859:Vector
8849:theory
8839:theory
8835:Module
8829:theory
8819:theory
8658:Scheme
8570:
8535:
8249:where
6313:where
4409:Counit
2258:where
1980:where
1665:Here,
1280:, the
1272:, the
1083:, the
659:module
156:unital
121:, the
113:, the
23:, the
9464:Basis
9149:Scope
8953:Trace
8878:Basis
8825:Group
8815:Field
8636:Areas
8251:(i,j)
2546:is a
2522:is a
1594:from
1071:) in
975:over
967:is a
956:up to
932:into
916:Here
883:from
873:over
861:from
675:is a
671:. If
593:with
311:order
184:field
178:be a
92:below
90:(see
42:) or
27:of a
8948:Rank
8928:Norm
8845:Ring
8568:ISBN
8533:ISBN
8440:for
8141:dual
6532:The
6489:For
5340:Unit
4538:for
4506:and
4480:for
3881:and
3764:<
3758:<
3737:<
3690:<
3684:<
3669:<
2526:and
1936:and
1284:and
1087:and
1061:for
999:and
826:Any
669:ring
352:for
174:Let
158:and
152:Note
125:and
8173:can
7807:in
7805:not
6767:by
6556:on
6121:or
5631:on
5334:not
4013:is
1587:Sym
1260:of
1053:in
1041:If
1022:to
926:of
894:to
797:is
779:of
345:of
309:of
211:of
203:of
101:of
94:).
70:on
58:on
54:of
19:In
9762::
8781:,
8747:,
8566:,
8558:,
8531:.
8527:.
8523:.
8306:.
8163:→
7925::=
7382::
7164::
5047:A
4298:.
3959:=
3803:m,
3548:Sh
3527:−
3523:,
3052::
1288:.
1276:,
1033:.
822::
654:.
166:.
129:.
117:,
9123:e
9116:t
9109:v
9011:(
9005:)
8995:)
8981:(
8944:)
8940:(
8934:)
8930:(
8924:)
8920:(
8914:)
8910:(
8899:)
8895:(
8861:)
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2080:(
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2074:=
2071:)
2068:k
2065:(
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2019:V
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2010:V
2005:0
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1953:1
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1921:v
1915:1
1912:+
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1865:T
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1853:v
1705:)
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1676:V
1673:T
1650:V
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1626::
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1581:Λ
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1563:T
1534:m
1531:y
1528:S
1480:.
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1471:m
1468:y
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1453:=
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1027:(
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1020:)
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1002:W
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990:K
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978:K
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948:(
946:T
941:)
939:V
937:(
935:T
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919:i
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897:A
892:)
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888:(
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864:V
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843:V
840::
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752:V
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714:)
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685:R
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673:R
662:M
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638:}
635:0
632:{
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621:k
617:T
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595:T
587:V
585:(
583:T
579:V
577:(
575:T
558:V
550:+
547:k
543:T
536:V
527:T
520:V
515:k
511:T
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495:(
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476:.
467:)
464:V
458:V
452:V
449:(
443:)
440:V
434:V
431:(
425:V
419:K
416:=
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408:k
404:T
393:0
390:=
387:k
379:=
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373:V
370:(
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354:k
350:V
347:T
339:V
337:(
335:T
328:K
321:V
318:T
314:k
307:V
303:V
300:T
283:.
280:V
268:V
262:V
259:=
254:k
247:V
243:=
240:V
235:k
231:T
217:k
213:V
205:V
199:k
194:k
187:K
176:V
107:V
105:(
103:T
84:V
72:V
60:V
48:V
46:(
44:T
40:V
38:(
36:T
32:V
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