4273:
52:
4280:
Chevalley showed in 1958 that the reductive groups over any algebraically closed field are classified up to isomorphism by root data. In particular, the semisimple groups over an algebraically closed field are classified up to central isogenies by their Dynkin diagram, and the simple groups
4055:
2752:, that is, they are direct sums of irreducible representations. That is the source of the name "reductive". Note, however, that complete reducibility fails for reductive groups in positive characteristic (apart from tori). In more detail: an affine group scheme
3560:
3758:(with some edges directed or multiple). The set of vertices of the Dynkin diagram is the set of simple roots. In short, the Dynkin diagram describes the angles between the simple roots and their relative lengths, with respect to a Weyl group-invariant
3198:
4727:
7483:
5314:
and unitary representations have been developed for real reductive groups in this generality. The main differences between this definition and the definition of a reductive algebraic group have to do with the fact that an algebraic group
2316:
5044:
4930:
5862:
As discussed above, the classification of split reductive groups is the same over any field. By contrast, the classification of arbitrary reductive groups can be hard, depending on the base field. Some examples among the
4227:
2678:, which is a complex reductive algebraic group. In fact, this construction gives a one-to-one correspondence between compact connected Lie groups and complex reductive groups, up to isomorphism. For a compact Lie group
2664:
4337:. It is remarkable that the classification of reductive groups is independent of the characteristic. For comparison, there are many more simple Lie algebras in positive characteristic than in characteristic zero.
3935:
7231:, the Galois action on its Dynkin diagram, and a Galois-invariant subset of the vertices of the Dynkin diagram. Traditionally, the Tits index is drawn by circling the Galois orbits in the given subset.
1580:
2182:
8209:
8015:
5794:) of the Schur module ∇(λ), but it need not be equal to the Schur module. The dimension and character of the Schur module are given by the Weyl character formula (as in characteristic zero), by
5708:
has a unique highest weight vector up to scalars; the corresponding "highest weight" λ is dominant; and every dominant weight λ is the highest weight of a unique irreducible representation
3488:
3637:
3309:
3251:
3133:
6030:
7128:
reduces the classification of central simple algebras over a field to the case of division algebras. Generalizing these results, Tits showed that a reductive group over a field
7641:
7609:
7577:
6131:
5929:
5134:
1702:
1277:
2449:
2358:
2105:
7211:
6599:-point of a maximal torus), a graph automorphism (corresponding to an automorphism of the Dynkin diagram), and a field automorphism (coming from an automorphism of the field
6597:
6166:
5964:
5173:
3599:
3125:
3093:
3065:
3041:
3017:
2977:
2420:
2211:
2138:
2068:
1774:
1670:
1304:
5802:(λ) are in general unknown, although a large body of theory has been developed to analyze these representations. One important result is that the dimension and character of
497:
472:
435:
4551:
1436:. (Some authors call this property "almost simple".) This differs slightly from the terminology for abstract groups, in that a simple algebraic group may have nontrivial
2558:
2514:
1218:
7395:
1394:
2391:
2219:
3858:
are in one-to-one correspondence with the subsets of the set Δ of simple roots (or equivalently, the subsets of the set of vertices of the Dynkin diagram). Let
2469:
1364:
1238:
1178:
1150:
1130:
1099:
7246:
with the given action. (For a quasi-split group, every Galois orbit in the Dynkin diagram is circled.) Moreover, any other simply connected semisimple group
7124:
says that a nondegenerate quadratic form over a field is determined up to isomorphism by its Witt index together with its anisotropic kernel. Likewise, the
4967:
4787:
8233:
Building on the Hasse principle, the classification of semisimple groups over number fields is well understood. For example, there are exactly three
7392:
is nontrivial, then it is encoded in the Galois action on the Dynkin diagram: the index-2 subgroup of the Galois group that acts as the identity is
1488:
scheme. Every reductive group over a field admits a central isogeny from the product of a torus and some simple groups. For example, over any field
6264:
algebraically closed, and they are understood for some other fields such as number fields, but for arbitrary fields there are many open questions.
4157:
2015:
is algebraically closed, any two (nondegenerate) quadratic forms of the same dimension are isomorphic, and so it is reasonable to call this group
8227:
2566:
7120:, which reduces the problem to the case of anisotropic groups. This reduction generalizes several fundamental theorems in algebra. For example,
4050:{\displaystyle \left\{{\begin{bmatrix}*&*&*&*\\*&*&*&*\\0&0&*&*\\0&0&0&*\end{bmatrix}}\right\}}
799:
5299:*. The problem of classifying the real reductive groups largely reduces to classifying the simple Lie groups. These are classified by their
1585:
It is slightly awkward that the definition of a reductive group over a field involves passage to the algebraic closure. For a perfect field
9349:
4333:
in the 1880s and 1890s. In particular, the dimensions, centers, and other properties of the simple algebraic groups can be read from the
4449:
with a given Dynkin diagram, with simple groups corresponding to the connected diagrams. At the other extreme, a semisimple group is of
7234:
There is a full classification of quasi-split groups in these terms. Namely, for each action of the absolute Galois group of a field
2953:
17:
8295:
5283:
In particular, every connected semisimple Lie group (meaning that its Lie algebra is semisimple) is reductive. Also, the Lie group
4232:
For the orthogonal group or the symplectic group, the projective homogeneous varieties have a similar description as varieties of
2832:, as in the theories of complex semisimple Lie algebras or compact Lie groups. Here is the way roots appear for reductive groups.
6365:
8317:
1720:. These kinds of groups are useful because their classification can be described through combinatorical data called root data.
934:
357:
9227:
9059:
9020:
8933:
8861:
3384:; this is a combinatorial structure which can be completely classified. More generally, the roots of a reductive group form a
1498:
9409:
9323:
9253:
9102:
9028:
8941:
8869:
7132:
is determined up to isomorphism by its Tits index together with its anisotropic kernel, an associated anisotropic semisimple
2143:
8127:
7933:
3875:
3467:
9067:
4325:. This result is essentially identical to the classifications of compact Lie groups or complex semisimple Lie algebras, by
2757:
307:
1461:
3929:
above are the groups of invertible matrices with zero entries below a given set of squares along the diagonal, such as:
9143:
8988:
8902:
7880:
7858:
5455:
of non-compact type. In fact, every symmetric space of non-compact type arises this way. These are central examples in
792:
302:
9286:
9178:
8829:
8739:
3676:
means a positive root that is not a sum of two other positive roots. Write Δ for the set of simple roots. The number
9501:
8972:
7370:
7091:
3311:
as the direct sum of the diagonal matrices and the 1-dimensional subspaces indexed by the off-diagonal positions (
7121:
7010:
3555:{\displaystyle {\mathfrak {b}}={\mathfrak {t}}\oplus \bigoplus _{\alpha \in \Phi ^{+}}{\mathfrak {g}}_{\alpha }.}
9217:
9164:
1157:
8813:
7900:
5839:
1949:
718:
9393:
9278:
9209:
9094:
8339:
5255:) (in the classical topology). It is also standard to assume that the image of the adjoint representation Ad(
5187:
are classified by root data. This statement includes the existence of
Chevalley groups as group schemes over
3193:{\displaystyle {\mathfrak {g}}={\mathfrak {t}}\oplus \bigoplus _{\alpha \in \Phi }{\mathfrak {g}}_{\alpha }.}
785:
8086:
of positive characteristic was proved earlier by Harder (1975): for every simply connected semisimple group
9484:
Schémas en groupes (SGA 3), II: Groupes de type multiplicatif, et structure des schémas en groupes généraux
8969:
Schémas en groupes (SGA 3), II: Groupes de type multiplicatif, et structure des schémas en groupes généraux
7125:
3755:
3604:
3276:
3218:
2675:
1346:
Over fields of characteristic zero another equivalent definition of a reductive group is a connected group
3762:
on the weight lattice. The connected Dynkin diagrams (corresponding to simple groups) are pictured below.
9204:
9119:
8312:
7066:
6400:
6001:
3773:, an important point is that a root α determines not just a 1-dimensional subspace of the Lie algebra of
402:
216:
1327:
9127:
8326:
8285:
7614:
7582:
7550:
7540:
6104:
5902:
5786:
are typically not direct sums of irreducibles. For a dominant weight λ, the irreducible representation
5107:
1675:
1250:
1004:
904:
849:
134:
5719:
There remains the problem of describing the irreducible representation with given highest weight. For
2425:
2334:
2081:
9199:
7189:
6575:
6144:
5942:
5311:
5304:
5151:
4334:
3580:
3106:
3074:
3046:
3022:
2998:
2958:
2749:
2741:
2396:
2187:
2140:
through the diagonal, and from this representation, their unipotent radical is trivial. For example,
2114:
2046:
1750:
1648:
1282:
5723:
of characteristic zero, there are essentially complete answers. For a dominant weight λ, define the
4722:{\displaystyle q(x_{1},\ldots ,x_{2n+1})=x_{1}x_{2}+x_{3}x_{4}+\cdots +x_{2n-1}x_{2n}+x_{2n+1}^{2};}
1638:
1319:
8765:
8300:
8264:
5445:
4938:
4104:
1366:
admitting a faithful semisimple representation which remains semisimple over its algebraic closure
1245:
922:. Reductive groups over an arbitrary field are harder to classify, but for many fields such as the
868:
600:
334:
211:
99:
8230:, saying that a central simple algebra over a number field is determined by its local invariants.
3474:Φ ⊂ Φ, with the property that Φ is the disjoint union of Φ and −Φ. Explicitly, the Lie algebra of
480:
455:
418:
8330:
7826:
6846:
6526:
5563:, the analog of a metric with nonpositive curvature. The dimension of the affine building is the
5335:
3427:. The Weyl group is in fact a finite group generated by reflections. For example, for the group
2519:
9478:
9048:
9010:
8964:
8923:
8290:
7478:{\displaystyle \operatorname {Gal} (k_{s}/k({\sqrt {d}}))\subset \operatorname {Gal} (k_{s}/k)}
6036:
5823:
5764:
5756:
5520:
4364:
2945:
1606:
1026:
1020:
1008:
1000:
919:
821:
750:
540:
2483:
1187:
9431:
7148:
5422:
1437:
968:
624:
2311:{\displaystyle (a_{1},a_{2})\mapsto {\begin{bmatrix}a_{1}&0\\0&a_{2}\end{bmatrix}}.}
1369:
9460:
9419:
9370:
9333:
9296:
9241:
9188:
9153:
9112:
9084:
9077:
9038:
8998:
8951:
8912:
8879:
8839:
8794:
8774:
8749:
8334:
8304:
8268:
7877:.) It follows, for example, that every reductive group over a finite field is quasi-split.
6929:
5827:
2699:
2369:
1834:
853:
837:
564:
552:
170:
104:
8:
9506:
8850:
8321:
8261:
are the finite simple groups constructed from simple algebraic groups over finite fields.
8025:
6953:
6837:
simply connected and quasi-split, the
Whitehead group is trivial, and so the whole group
5791:
5782:
of positive characteristic, the situation is far more subtle, because representations of
5585:
5460:
5456:
4236:
flags with respect to a given quadratic form or symplectic form. For any reductive group
3681:
3463:
2805:
2331:
is not reductive since its unipotent radical is itself. This includes the additive group
916:
864:
852:. Reductive groups include some of the most important groups in mathematics, such as the
825:
139:
34:
9245:
8778:
5854:-Kazhdan-Lusztig polynomials, which are even more complex, but at least are computable.
9448:
9374:
9231:
8798:
7726:. Namely, such forms (up to isomorphism) are in one-to-one correspondence with the set
7644:
6569:
6279:-rank greater than 0 (that is, if it contains a nontrivial split torus), and otherwise
5847:
5536:
5503:
4958:
4375:
4233:
4096:
2454:
1953:
1481:
1349:
1223:
1163:
1135:
1115:
1084:
124:
96:
9310:, Progress in Mathematics, vol. 9 (2nd ed.), Boston, MA: Birkhäuser Boston,
9405:
9319:
9282:
9249:
9195:
9174:
9160:
9139:
9098:
9063:
9024:
8984:
8937:
8898:
8865:
8825:
8802:
8763:(1971), "Éléments unipotents et sous-groupes paraboliques de groupes réductifs. I.",
8735:
8377:
8308:
8276:
8258:
7874:
7850:
7267:
7238:
on a Dynkin diagram, there is a unique simply connected semisimple quasi-split group
7173:
5406:
5378:
5365:)) is a real reductive group that cannot be viewed as an algebraic group. Similarly,
5039:{\displaystyle \operatorname {Aut} (G)\cong \operatorname {Out} (G)\ltimes (G/Z)(k),}
4925:{\displaystyle q(x_{1},\ldots ,x_{2n})=x_{1}x_{2}+x_{3}x_{4}+\cdots +x_{2n-1}x_{2n}.}
4341:
1314:. (This is equivalent to the definition of reductive groups in the introduction when
1307:
912:
529:
372:
266:
9378:
7660:
5341:(2) is connected as an algebraic group over any field, but its group of real points
5183:
and
Grothendieck showed that split reductive group schemes over any nonempty scheme
4103:. Thus the classification of parabolic subgroups amounts to a classification of the
2471:
on the diagonal. This is an example of a non-reductive group which is not unipotent.
695:
9440:
9397:
9385:
9358:
9344:
9340:
9311:
9131:
8976:
8895:
Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs
8809:
8782:
8727:
8272:
8238:
6997:
6522:
6198:
5768:
5736:
5476:
5232:
4438:
4379:
2793:
1919:
1885:
1485:
971:
in various contexts. First, one can study the representations of a reductive group
900:
879:
841:
680:
672:
664:
656:
648:
636:
576:
516:
506:
348:
290:
165:
9482:
9014:
8927:
7912:
4441:
in the classical topology.) Chevalley's classification gives that, over any field
9474:
9456:
9415:
9366:
9329:
9292:
9184:
9149:
9108:
9073:
9034:
9006:
8994:
8960:
8947:
8919:
8908:
8886:
8875:
8835:
8821:
8790:
8745:
8723:
7916:
7116:
In seeking to classify reductive groups which need not be split, one step is the
6969:
6941:
5864:
5609:
5452:
5410:
5180:
5101:
5065:
has a simpler description: it is the automorphism group of the Dynkin diagram of
4326:
3444:
2328:
2108:
1424:) if it is semisimple, nontrivial, and every smooth connected normal subgroup of
1109:
1078:
987:-vector spaces. But also, one can study the complex representations of the group
764:
743:
700:
588:
511:
341:
255:
195:
75:
8515:
SGA 3 (2011), v. 3, Théorème XXV.1.1; Conrad (2014), Theorems 6.1.16 and 6.1.17.
7359:, and so its automorphism group is of order 2, switching the two "legs" of the D
5369:(2) is simply connected as an algebraic group over any field, but the Lie group
5287:
is reductive in this sense, since it can be viewed as the identity component of
3890:). As a result, there are exactly 2 conjugacy classes of parabolic subgroups in
9303:
9262:
9170:
8890:
6063:
5831:
5815:
5507:
5425:, that is, the product of a semisimple Lie algebra and an abelian Lie algebra.
5394:
5300:
4426:
3751:
3459:
3399:
2918:
2828:
The classification of reductive algebraic groups is in terms of the associated
1930:
1106:
950:
908:
771:
707:
397:
377:
314:
279:
200:
190:
175:
160:
114:
91:
9315:
8731:
5632:. In particular, this parametrization is independent of the characteristic of
4330:
9495:
8281:
7908:
7652:
6388:
4281:
correspond to the connected diagrams. Thus there are simple groups of types A
4088:
3759:
2711:
1929:) is the subgroup of the general linear group that preserves a nondegenerate
1908:
1062:
1034:
833:
690:
612:
446:
319:
185:
7903:(which has cohomological dimension 2). More generally, for any number field
5850:
conjectured the irreducible characters of a reductive group in terms of the
4222:{\displaystyle 0\subset S_{a_{1}}\subset \cdots \subset S_{a_{i}}\subset V.}
2884:(as an algebraic group) is a direct sum of 1-dimensional representations. A
9426:
9016:
Schémas en groupes (SGA 3), III: Structure des schémas en groupes réductifs
8846:
8760:
8756:
8715:
8080:
6708:
6059:
5795:
5625:
5078:
2659:{\displaystyle B_{n}/(R_{u}(B_{n}))\cong \prod _{i=1}^{n}\mathbb {G} _{m}.}
1912:
1038:
958:
930:
545:
244:
233:
180:
155:
150:
109:
80:
43:
9135:
8929:
Schémas en groupes (SGA 3), I: Propriétés générales des schémas en groupes
1220:. (Some authors do not require reductive groups to be connected.) A group
9124:
Algebraic Groups: The Theory of Group
Schemes of Finite Type over a Field
8722:, Graduate Texts in Mathematics, vol. 126 (2nd ed.), New York:
7672:
6626:) is close to being simple, under mild assumptions. Namely, suppose that
6560:, Steinberg also determined the automorphism group of the abstract group
6545:(the root subgroups), with relations determined by the Dynkin diagram of
6252:
essentially includes the problem of classifying all quadratic forms over
4355:
had been constructed earlier, at least in the form of the abstract group
4272:
2829:
2364:
1849:
923:
813:
9401:
7086:-invariant measure). For example, a discrete subgroup Γ is a lattice if
2768:
if its finite-dimensional representations are completely reducible. For
903:
showed that the classification of reductive groups is the same over any
9452:
9362:
9266:
9088:
8980:
8786:
7255:
7117:
7111:
4536:
4520:
3803:
3403:
3390:
3385:
2043:, although they all have the same base change to the algebraic closure
1011:. The structure theory of reductive groups is used in all these areas.
845:
712:
440:
6853:-simple groups the Whitehead group is trivial. In all known examples,
5212:
3898:. Explicitly, the parabolic subgroup corresponding to a given subset
3127:
together with 1-dimensional subspaces indexed by the set Φ of roots:
533:
9444:
6508:
1999:) can always be defined as the maximal smooth connected subgroup of
1597:
is reductive if and only if every smooth connected unipotent normal
9236:
6718:
The exceptions for fields of order 2 or 3 are well understood. For
5798:. The dimensions and characters of the irreducible representations
4484:
For example, the simply connected split simple groups over a field
3826:
and the positive root subgroups. In fact, a split semisimple group
1440:(although the center must be finite). For example, for any integer
70:
5636:. In more detail, fix a split maximal torus and a Borel subgroup,
4127:, parametrizing sequences of linear subspaces of given dimensions
1605:
is trivial. For an arbitrary field, the latter property defines a
8971:. Lecture Notes in Mathematics. Vol. 152. Berlin; New York:
6712:
6707:) by its center is simple (as an abstract group). The proof uses
4488:
corresponding to the "classical" Dynkin diagrams are as follows:
3043:
corresponding to each root is 1-dimensional, and the subspace of
2934:
412:
326:
7643:
on itself by left translation. A torsor can also be viewed as a
5421:) is not a real reductive group, even though its Lie algebra is
4405:
reductive groups is the same over any field. A semisimple group
3798:. The root subgroup is the unique copy of the additive group in
7829:, which are invariants taking values in Galois cohomology with
7519:
5556:
5331:) is not connected, and likewise for simply connected groups.
2892:
means an isomorphism class of 1-dimensional representations of
1132:
is trivial. More generally, a connected linear algebraic group
907:. In particular, the simple algebraic groups are classified by
51:
8121:, the Hasse principle holds in a weaker form: the natural map
7738:)). For example, (nondegenerate) quadratic forms of dimension
6248:
As a result, the problem of classifying reductive groups over
5612:, which are defined as the intersection of the weight lattice
2669:
6572:, a diagonal automorphism (meaning conjugation by a suitable
6379:
of characteristic zero (such as the real numbers), the group
5763:(λ) is isomorphic to the Schur module ∇(λ). Furthermore, the
5704:). Chevalley showed that every irreducible representation of
5191:, and it says that every split reductive group over a scheme
5144:, the corresponding geometric fiber means the base change of
4949:
is isomorphic to the automorphism group of the root datum of
4398:
over a field of positive characteristic were completely new.
6399:
is reductive and anisotropic. Example: the orthogonal group
4445:, there is a unique simply connected split semisimple group
2776:
is linearly reductive if and only if the identity component
2111:. They are examples of reductive groups since they embed in
828:. One definition is that a connected linear algebraic group
7369:
on the Dynkin diagram is trivial if and only if the signed
5323:
may be connected as an algebraic group while the Lie group
5195:
is isomorphic to the base change of a
Chevalley group from
4453:
if its center is trivial. The split semisimple groups over
4267:
7853:'s "Conjecture I": for a connected linear algebraic group
7164:) acts (continuously) on the "absolute" Dynkin diagram of
7105:
5771:(and in particular the dimension) of this representation.
5349:) has two connected components. The identity component of
4111:(with smooth stabilizer group; that is no restriction for
3577:), then this is the obvious decomposition of the subspace
8542:
Jantzen (2003), Proposition II.4.5 and
Corollary II.5.11.
7873:) = 1. (The case of a finite field was known earlier, as
6928:) is compact in the classical topology. Since it is also
5834:'s conjecture in that case). Their character formula for
4429:, being simply connected in this sense is equivalent to
3462:
containing a given maximal torus, and they are permuted
1575:{\displaystyle GL(n)\cong (G_{m}\times SL(n))/\mu _{n}.}
2177:{\displaystyle \mathbb {G} _{m}\times \mathbb {G} _{m}}
937:
says that most finite simple groups arise as the group
8639:
Tits (1964), Main
Theorem; Gille (2009), Introduction.
8204:{\displaystyle H^{1}(k,G)\to \prod _{v}H^{1}(k_{v},G)}
8010:{\displaystyle H^{1}(k,G)\to \prod _{v}H^{1}(k_{v},G)}
7883:
predicts that for a simply connected semisimple group
6952:) contains infinitely many normal subgroups of finite
6729:, Tits's simplicity theorem remains valid except when
6537:). It is generated by copies of the additive group of
5591:
3948:
3569:
is the Borel subgroup of upper-triangular matrices in
2260:
8130:
7936:
7617:
7585:
7553:
7398:
7321:
be a nondegenerate quadratic form of even dimension 2
7192:
6578:
6147:
6107:
6004:
5982:(the maximum dimension of an isotropic subspace over
5945:
5905:
5154:
5110:
4970:
4790:
4554:
4160:
3938:
3810:
and which has the given Lie algebra. The whole group
3607:
3583:
3491:
3279:
3221:
3136:
3109:
3077:
3049:
3025:
3001:
2961:
2569:
2522:
2486:
2457:
2428:
2399:
2372:
2337:
2222:
2190:
2146:
2117:
2084:
2049:
1833:
under multiplication. Another reductive group is the
1753:
1678:
1651:
1501:
1372:
1352:
1285:
1253:
1226:
1190:
1166:
1138:
1118:
1087:
483:
458:
421:
9005:
8959:
8918:
8488:
Chevalley (2005); Springer (1998), 9.6.2 and 10.1.1.
7810:)). These problems motivate the systematic study of
7365:
diagram. The action of the absolute Galois group of
6391:
in the classical topology (based on the topology of
5409:) are real reductive groups. On the other hand, the
5247:) whose kernel is finite and whose image is open in
4417:
if every central isogeny from a semisimple group to
2987:
means a nonzero weight that occurs in the action of
9488:
Revised and annotated edition of the 1970 original.
9043:
Revised and annotated edition of the 1970 original.
8956:
Revised and annotated edition of the 1970 original.
8818:
Classification des groupes algébriques semi-simples
8113:In the slightly different case of an adjoint group
7887:over a field of cohomological dimension at most 2,
6845:) is simple modulo its center. More generally, the
5842:, which are combinatorially complex. For any prime
8203:
8066:) is trivial for every nonarchimidean local field
8009:
7635:
7603:
7571:
7512:
7477:
7205:
6591:
6160:
6125:
6024:
5958:
5923:
5167:
5128:
5038:
4924:
4721:
4221:
4049:
3704:is semisimple). For example, the simple roots for
3631:
3593:
3554:
3470:). A choice of Borel subgroup determines a set of
3303:
3245:
3192:
3119:
3087:
3059:
3035:
3011:
2971:
2658:
2552:
2508:
2463:
2443:
2414:
2385:
2352:
2310:
2205:
2176:
2132:
2099:
2062:
1768:
1744:A fundamental example of a reductive group is the
1696:
1664:
1574:
1388:
1358:
1298:
1271:
1232:
1212:
1172:
1144:
1124:
1093:
491:
466:
429:
9345:"Regular elements of semisimple algebraic groups"
9159:
8431:Demazure & Gabriel (1970), Théorème IV.3.3.6.
6959:
6509:Structure of semisimple groups as abstract groups
4527:+1) associated to a quadratic form of dimension 2
4457:with given Dynkin diagram are exactly the groups
3818:and the root subgroups, while the Borel subgroup
3750:Root systems are classified by the corresponding
2480:Note that the normality of the unipotent radical
9493:
9473:
9429:(1964), "Algebraic and abstract simple groups",
5608:(as an algebraic group) are parametrized by the
5057:. For a split semisimple simply connected group
2744:zero, all finite-dimensional representations of
1589:, that can be avoided: a linear algebraic group
9215:
8885:
5531:plays the role of the symmetric space. Namely,
4766:) associated to a quadratic form of dimension 2
1180:is trivial. This normal subgroup is called the
999:is a finite field, or the infinite-dimensional
7333:≥ 5. (These restrictions can be avoided.) Let
6552:For a simply connected split semisimple group
6513:For a simply connected split semisimple group
5061:over a field, the outer automorphism group of
3335:for the standard basis for the weight lattice
3273:. Then the root-space decomposition expresses
1057:. Equivalently, a linear algebraic group over
8702:Platonov & Rapinchuk (1994), Theorem 6.4.
8693:Platonov & Rapinchuk (1994), section 6.8.
8684:Platonov & Rapinchuk (1994), Theorem 6.6.
8666:Platonov & Rapinchuk (1994), section 9.1.
8603:Platonov & Rapinchuk (1994), Theorem 3.1.
7698:Torsors arise whenever one seeks to classify
7266:is the group associated to an element of the
6497:| ≤ 1, and it is quasi-split if and only if |
6453:. A split reductive group is quasi-split. If
5857:
5747:associated to λ; this is a representation of
5692:acts on that line through its quotient group
4386:. By contrast, the Chevalley groups of type F
2475:
1907:) that preserves a nondegenerate alternating
1341:
1152:over an algebraically closed field is called
1101:over an algebraically closed field is called
1072:
964:, or as minor variants of that construction.
933:, the classification is well understood. The
793:
8244:, corresponding to the three real forms of E
7094:says, in particular: for a simple Lie group
6302:contains a copy of the multiplicative group
6019:
6005:
5223:such that there is a linear algebraic group
3680:of simple roots is equal to the rank of the
2702:, with respect to the classical topology on
979:as an algebraic group, which are actions of
9392:, University Lecture Series, vol. 66,
8560:Riche & Williamson (2018), section 1.8.
8041:is the corresponding local field (possibly
6884:) can be far from simple. For example, let
6691:) is nontrivial, and even Zariski dense in
6568:). Every automorphism is the product of an
6426:), and so it is anisotropic if and only if
6356:perfect, it is also equivalent to say that
6291:, the following conditions are equivalent:
5656:with a smooth connected unipotent subgroup
2670:Other characterizations of reductive groups
1987:) is in fact connected but not smooth over
9216:Riche, Simon; Williamson, Geordie (2018),
9194:
7919:: for a simply connected semisimple group
7915:and Vladimir Chernousov (1989) proved the
7814:-torsors, especially for reductive groups
7762:)), and central simple algebras of degree
7098:of real rank at least 2, every lattice in
6979:can be extended to an affine group scheme
6754:, or non-split (that is, unitary) of type
6081:means the square root of the dimension of
5696:, by some element λ of the weight lattice
5072:
3917:. For example, the parabolic subgroups of
3862:be the order of Δ, the semisimple rank of
3830:is generated by the root subgroups alone.
1399:
800:
786:
9384:
9339:
9235:
8808:
8755:
8357:SGA 3 (2011), v. 3, Définition XIX.1.6.1.
7695:), in the language of Galois cohomology.
6794:, in order to understand the whole group
6699:is infinite.) Then the quotient group of
5759:says that the irreducible representation
5397:. By definition, all finite coverings of
4079:-subgroup such that the quotient variety
2643:
2431:
2340:
2164:
2149:
2087:
2027:, different quadratic forms of dimension
1867:
1612:
1244:is called semisimple or reductive if the
485:
460:
423:
9302:
9271:Automorphic Forms, Representations, and
9261:
8594:Borel & Tits (1971), Corollaire 3.8.
8448:
8446:
8395:Conrad (2014), after Proposition 5.1.17.
7082:/Γ has finite volume (with respect to a
6987:, and this determines an abstract group
6658:-points of copies of the additive group
6646:) be the subgroup of the abstract group
5790:(λ) is the unique simple submodule (the
5206:
4271:
4268:Classification of split reductive groups
3814:is generated (as an algebraic group) by
3777:, but also a copy of the additive group
3478:is the direct sum of the Lie algebra of
3269:be the subgroup of diagonal matrices in
2839:be a split reductive group over a field
2674:Every compact connected Lie group has a
9083:
8572:
8570:
8568:
8566:
8497:Milne (2017), Theorems 23.25 and 23.55.
7849:). In this direction, Steinberg proved
7106:The Galois action on the Dynkin diagram
6371:For a connected linear algebraic group
3380:The roots of a semisimple group form a
2710:). For example, the inclusion from the
2031:can yield non-isomorphic simple groups
14:
9494:
8845:
6335:contains a copy of the additive group
5806:(λ) are known when the characteristic
5104:and affine, and every geometric fiber
4953:. Moreover, the automorphism group of
4401:More generally, the classification of
3833:
935:classification of finite simple groups
358:Classification of finite simple groups
9118:
9046:
9013:(2011) . Gille, P.; Polo, P. (eds.).
8926:(2011) . Gille, P.; Polo, P. (eds.).
8714:
8443:
8296:Weil's conjecture on Tamagawa numbers
7899:) = 1. The conjecture is known for a
7821:When possible, one hopes to classify
7182:(which is also the Dynkin diagram of
6968:be a linear algebraic group over the
6449:if it contains a Borel subgroup over
5998:has the maximum possible Witt index,
5604:, the irreducible representations of
5428:For a connected real reductive group
3788:with the given Lie algebra, called a
3632:{\displaystyle {{\mathfrak {g}}l}(n)}
3304:{\displaystyle {{\mathfrak {g}}l}(n)}
3246:{\displaystyle {{\mathfrak {g}}l}(n)}
2800:is linearly reductive if and only if
1629:if it contains a split maximal torus
9425:
9350:Publications Mathématiques de l'IHÉS
8563:
7314:, as discussed in the next section.
6888:be a division algebra with center a
6256:or all central simple algebras over
6180:is isomorphic to the matrix algebra
3850:that contain a given Borel subgroup
3846:, the smooth connected subgroups of
2422:has a non-trivial unipotent radical
1480:of reductive groups is a surjective
9090:Representations of Algebraic Groups
8228:Albert–Brauer–Hasse–Noether theorem
8079:matter. The analogous result for a
7021:). (Arithmeticity of a subgroup of
6920:-simple group. As mentioned above,
6904:is finite and greater than 1. Then
6321:contains a parabolic subgroup over
6025:{\displaystyle \lfloor n/2\rfloor }
5871:Every nondegenerate quadratic form
5592:Representations of reductive groups
5502:that is complete with respect to a
5235:) is reductive, and a homomorphism
3611:
3586:
3538:
3504:
3494:
3283:
3225:
3176:
3149:
3139:
3112:
3080:
3052:
3028:
3004:
2964:
1963:) is reductive, in fact simple for
24:
9166:Algebraic Groups and Number Theory
9056:Séminaire Bourbaki. Vol. 2007/2008
7790:-forms of a given algebraic group
7029:) is independent of the choice of
6461:, then any two Borel subgroups of
4477:-subgroup scheme of the center of
4469:is the simply connected group and
4143:contained in a fixed vector space
3925:) that contain the Borel subgroup
3524:
3168:
2451:of upper-triangular matrices with
2107:and products of it are called the
1860:) is a simple algebraic group for
1609:, which is somewhat more general.
1279:is semisimple or reductive, where
1003:of a real reductive group, or the
25:
9518:
9467:
8375:
8075:, and so only the real places of
7636:{\displaystyle G_{\overline {k}}}
7604:{\displaystyle G_{\overline {k}}}
7572:{\displaystyle X_{\overline {k}}}
7349:. The absolute Dynkin diagram of
7168:, that is, the Dynkin diagram of
6473:). Example: the orthogonal group
6465:are conjugate by some element of
6126:{\displaystyle G_{\overline {k}}}
5924:{\displaystyle G_{\overline {k}}}
5879:determines a reductive group G =
5731:-vector space of sections of the
5231:whose identity component (in the
5129:{\displaystyle G_{\overline {k}}}
3906:together with the root subgroups
3406:of a maximal torus by the torus,
2896:, or equivalently a homomorphism
2690:into the complex reductive group
1884:An important simple group is the
1697:{\displaystyle G_{\overline {k}}}
1460:is simple, and its center is the
1272:{\displaystyle G_{\overline {k}}}
8551:Jantzen (2003), section II.8.22.
8479:Borel (1991), Proposition 21.12.
8470:Milne (2017), Proposition 17.53.
8366:Milne (2017), Proposition 21.60.
8237:-forms of the exceptional group
6896:. Suppose that the dimension of
5830:, and Wolfgang Soergel (proving
5215:rather than algebraic groups, a
4105:projective homogeneous varieties
3601:of upper-triangular matrices in
3099:. Therefore, the Lie algebra of
2516:implies that the quotient group
2444:{\displaystyle \mathbb {U} _{n}}
2353:{\displaystyle \mathbb {G} _{a}}
2100:{\displaystyle \mathbb {G} _{m}}
1848:, the subgroup of matrices with
1704:). It is equivalent to say that
1156:if the largest smooth connected
50:
8696:
8687:
8678:
8669:
8660:
8651:
8642:
8633:
8624:
8615:
8606:
8597:
8588:
8579:
8554:
8545:
8536:
8527:
8518:
8509:
8500:
8491:
8482:
8473:
8464:
8455:
7513:Torsors and the Hasse principle
7206:{\displaystyle {\overline {k}}}
7074:means a discrete subgroup Γ of
7045:) is an arithmetic subgroup of
6944:(but not finite). As a result,
6772:, the theorem holds except for
6592:{\displaystyle {\overline {k}}}
6267:A reductive group over a field
6161:{\displaystyle {\overline {k}}}
5959:{\displaystyle {\overline {k}}}
5571:. For example, the building of
5303:; or one can just refer to the
5179:.) Extending Chevalley's work,
5168:{\displaystyle {\overline {k}}}
3902:of Δ is the group generated by
3594:{\displaystyle {\mathfrak {b}}}
3120:{\displaystyle {\mathfrak {t}}}
3088:{\displaystyle {\mathfrak {t}}}
3060:{\displaystyle {\mathfrak {g}}}
3036:{\displaystyle {\mathfrak {g}}}
3012:{\displaystyle {\mathfrak {g}}}
2972:{\displaystyle {\mathfrak {g}}}
2929:) isomorphic to the product of
2415:{\displaystyle {\text{GL}}_{n}}
2321:
2206:{\displaystyle {\text{GL}}_{2}}
2133:{\displaystyle {\text{GL}}_{n}}
2063:{\displaystyle {\overline {k}}}
1936:on a vector space over a field
1769:{\displaystyle {\text{GL}}_{n}}
1665:{\displaystyle {\overline {k}}}
1299:{\displaystyle {\overline {k}}}
9481:, Gille, P.; Polo, P. (eds.),
9228:Société Mathématique de France
9060:Société Mathématique de France
9021:Société Mathématique de France
8934:Société Mathématique de France
8862:Société Mathématique de France
8675:Steinberg (1965), Theorem 1.9.
8612:Borel (1991), Theorem 20.9(i).
8506:Milne (2017), Corollary 23.47.
8461:Milne (2017), Corollary 21.12.
8434:
8425:
8422:Milne (2017), Corollary 22.43.
8416:
8407:
8398:
8389:
8369:
8360:
8351:
8198:
8179:
8156:
8153:
8141:
8004:
7985:
7962:
7959:
7947:
7901:totally imaginary number field
7794:(sometimes called "twists" of
7722:over the algebraic closure of
7501:is quasi-split if and only if
7472:
7451:
7439:
7436:
7426:
7405:
7329:of characteristic not 2, with
7217:consists of the root datum of
7092:Margulis arithmeticity theorem
6960:Lattices and arithmetic groups
6260:. These problems are easy for
6069:* (as an algebraic group over
5459:of manifolds with nonpositive
5030:
5024:
5021:
5007:
5001:
4995:
4983:
4977:
4829:
4794:
4599:
4558:
3626:
3620:
3482:and the positive root spaces:
3298:
3292:
3240:
3234:
2614:
2611:
2598:
2585:
2547:
2541:
2503:
2497:
2252:
2249:
2223:
1825:-rational points is the group
1728:
1551:
1548:
1542:
1520:
1514:
1508:
1207:
1201:
1014:
719:Infinite dimensional Lie group
13:
1:
9394:American Mathematical Society
9279:American Mathematical Society
9226:, Astérisque, vol. 397,
9095:American Mathematical Society
9058:, Astérisque, vol. 326,
8858:Autour des schémas en groupes
8708:
8630:Steinberg (2016), Theorem 30.
8524:Springer (1979), section 5.1.
8340:Radical of an algebraic group
8049:). Moreover, the pointed set
6638:has at least 4 elements. Let
6634:, and suppose that the field
6618:says that the abstract group
6046:determines a reductive group
5652:is the semidirect product of
4276:The connected Dynkin diagrams
4115:of characteristic zero). For
3696:(which is simply the rank of
3466:by the Weyl group (acting by
3347:, the roots are the elements
2733:) is a homotopy equivalence.
967:Reductive groups have a rich
9390:Lectures on Chevalley Groups
9163:; Rapinchuk, Andrei (1994),
9049:"Le problème de Kneser–Tits"
8621:Steinberg (2016), Theorem 8.
8452:Milne (2017), Theorem 21.11.
8440:Milne (2017), Theorem 12.12.
8413:Milne (2017), Theorem 22.42.
7702:of a given algebraic object
7627:
7595:
7563:
7497:, the maximum possible, and
7254:with the given action is an
7198:
7122:Witt's decomposition theorem
6584:
6153:
6117:
5951:
5939:) over an algebraic closure
5915:
5774:For a split reductive group
5755:of characteristic zero, the
5596:For a split reductive group
5160:
5120:
3838:For a split reductive group
3765:For a split reductive group
2847:be a split maximal torus in
2748:(as an algebraic group) are
2055:
1688:
1657:
1291:
1263:
1053:, for some positive integer
953:of a simple algebraic group
492:{\displaystyle \mathbb {Z} }
467:{\displaystyle \mathbb {Z} }
430:{\displaystyle \mathbb {Z} }
7:
9205:Encyclopedia of Mathematics
8820:, Collected Works, Vol. 3,
8657:Gille (2009), Théorème 6.1.
8585:Borel (1991), section 23.2.
8576:Borel (1991), section 23.4.
8533:Milne (2017), Theorem 22.2.
8313:geometric Langlands program
8251:
7718:which become isomorphic to
7523:for an affine group scheme
6228:) − 1. So the simple group
5840:Kazhdan–Lusztig polynomials
5491:(2) is hyperbolic 3-space.
5381:isomorphic to the integers
5276:)) (which is automatic for
5136:is reductive. (For a point
4941:of a split reductive group
3253:is the vector space of all
3071:is exactly the Lie algebra
2909:. The weights form a group
1723:
1005:automorphic representations
895:semisimple algebraic groups
850:irreducible representations
217:List of group theory topics
10:
9523:
9128:Cambridge University Press
8327:Geometric invariant theory
7675:of isomorphism classes of
7186:over an algebraic closure
7109:
7033:-structure.) For example,
6675:. (By the assumption that
5858:Non-split reductive groups
5312:admissible representations
5307:(up to finite coverings).
4778:, which can be written as:
3443:)), the Weyl group is the
3388:, a slight variation. The
2880:. Every representation of
2560:is reductive. For example,
2553:{\displaystyle G/R_{u}(G)}
2476:Associated reductive group
1712:that is maximal among all
1342:With representation theory
1105:if every smooth connected
1073:With the unipotent radical
1018:
905:algebraically closed field
9316:10.1007/978-0-8176-4840-4
8851:"Reductive group schemes"
8732:10.1007/978-1-4612-0941-6
8648:Tits (1964), section 1.2.
7258:of the quasi-split group
6849:asks for which isotropic
6616:Tits's simplicity theorem
6489:is split if and only if |
6283:. For a semisimple group
6240:is a matrix algebra over
5684:maps the line spanned by
5305:list of simple Lie groups
4367:. For example, the group
4335:list of simple Lie groups
3878:to a subgroup containing
3639:. The positive roots are
2921:of representations, with
1829:* of nonzero elements of
1444:at least 2 and any field
1404:A linear algebraic group
836:is reductive if it has a
18:Reductive algebraic group
9219:Tilting Modules and the
9047:Gille, Philippe (2009),
8766:Inventiones Mathematicae
8345:
8301:Langlands classification
8265:Generalized flag variety
7857:over a perfect field of
7827:cohomological invariants
7505:has Witt index at least
7489:is split if and only if
7126:Artin–Wedderburn theorem
7102:is an arithmetic group.
6802:), one can consider the
6610:-simple algebraic group
6541:indexed by the roots of
5814:is much bigger than the
5446:maximal compact subgroup
5432:, the quotient manifold
5175:of the residue field of
5148:to an algebraic closure
4939:outer automorphism group
4421:is an isomorphism. (For
3458:There are finitely many
2823:
2772:of characteristic zero,
2509:{\displaystyle R_{u}(G)}
1975:of characteristic 2 and
1240:over an arbitrary field
1213:{\displaystyle R_{u}(G)}
869:special orthogonal group
335:Elementary abelian group
212:Glossary of group theory
9502:Linear algebraic groups
9308:Linear Algebraic Groups
8720:Linear Algebraic Groups
8379:Linear Algebraic Groups
8226:), this amounts to the
7859:cohomological dimension
7531:means an affine scheme
7078:such that the manifold
6298:is isotropic (that is,
5986:). So the simple group
5676:to be a nonzero vector
5336:projective linear group
5073:Reductive group schemes
4762:: the spin group Spin(2
4741:: the symplectic group
2073:
2023:). For a general field
1788:, for a natural number
1432:is trivial or equal to
1400:Simple reductive groups
1081:linear algebraic group
1001:unitary representations
920:semisimple Lie algebras
891:Simple algebraic groups
8860:, vol. 1, Paris:
8404:Borel (1991), 18.2(i).
8291:Real form (Lie theory)
8286:Deligne–Lusztig theory
8205:
8011:
7637:
7605:
7573:
7479:
7207:
7139:For a reductive group
7001:means any subgroup of
6593:
6529:of the abstract group
6368:element other than 1.
6162:
6127:
6037:central simple algebra
6026:
5960:
5925:
5838:large is based on the
5765:Weyl character formula
5624:with a convex cone (a
5494:For a reductive group
5259:) is contained in Int(
5169:
5130:
5040:
4926:
4723:
4542:, for example the form
4277:
4240:with a Borel subgroup
4223:
4051:
3633:
3595:
3556:
3305:
3247:
3194:
3121:
3089:
3061:
3037:
3013:
2973:
2952:by conjugation on its
2946:adjoint representation
2736:For a reductive group
2682:with complexification
2667:
2660:
2640:
2554:
2510:
2465:
2445:
2416:
2387:
2354:
2319:
2312:
2207:
2178:
2134:
2101:
2064:
1979:odd, the group scheme
1940:. The algebraic group
1808:(1), and so its group
1784:matrices over a field
1770:
1698:
1672:is a maximal torus in
1666:
1613:Split-reductive groups
1607:pseudo-reductive group
1576:
1390:
1389:{\displaystyle k^{al}}
1360:
1300:
1273:
1234:
1214:
1174:
1146:
1126:
1095:
1027:linear algebraic group
1021:Linear algebraic group
1009:adelic algebraic group
911:, as in the theory of
822:linear algebraic group
751:Linear algebraic group
493:
468:
431:
27:Concept in mathematics
9432:Annals of Mathematics
9136:10.1017/9781316711736
9085:Jantzen, Jens Carsten
8206:
8110:has no real places).
8012:
7881:Serre's Conjecture II
7638:
7606:
7574:
7480:
7213:). The Tits index of
7208:
7149:absolute Galois group
6876:, the abstract group
6594:
6556:over a perfect field
6163:
6128:
6058:), the kernel of the
6027:
5961:
5926:
5739:on the flag manifold
5716:, up to isomorphism.
5662:highest weight vector
5207:Real reductive groups
5170:
5131:
5041:
4927:
4724:
4275:
4224:
4067:of a reductive group
4052:
3634:
3596:
3557:
3394:of a reductive group
3306:
3248:
3195:
3122:
3090:
3062:
3038:
3014:
2974:
2686:, the inclusion from
2661:
2620:
2562:
2555:
2511:
2466:
2446:
2417:
2388:
2386:{\displaystyle B_{n}}
2355:
2313:
2215:
2208:
2179:
2135:
2102:
2065:
1792:. In particular, the
1771:
1699:
1667:
1645:whose base change to
1577:
1484:with kernel a finite
1391:
1361:
1301:
1274:
1235:
1215:
1175:
1147:
1127:
1096:
969:representation theory
893:and (more generally)
494:
469:
432:
8331:Luna's slice theorem
8305:Langlands dual group
8269:Bruhat decomposition
8128:
8117:over a number field
8106:) is trivial (since
7934:
7798:) are classified by
7651:with respect to the
7615:
7583:
7551:
7396:
7337:be the simple group
7190:
6930:totally disconnected
6916:) is an anisotropic
6576:
6457:is quasi-split over
6145:
6105:
6089:-vector space. Here
6002:
5943:
5903:
5664:in a representation
5357:) (sometimes called
5217:real reductive group
5152:
5108:
4968:
4788:
4552:
4425:semisimple over the
4158:
3936:
3754:, which is a finite
3605:
3581:
3489:
3277:
3219:
3134:
3107:
3075:
3047:
3023:
2999:
2959:
2750:completely reducible
2700:homotopy equivalence
2567:
2520:
2484:
2455:
2426:
2397:
2370:
2335:
2220:
2188:
2144:
2115:
2082:
2047:
1950:connected components
1835:special linear group
1794:multiplicative group
1751:
1746:general linear group
1708:is a split torus in
1676:
1649:
1499:
1370:
1350:
1328:multiplicative group
1283:
1251:
1224:
1188:
1164:
1136:
1116:
1085:
854:general linear group
481:
456:
419:
9246:2015arXiv151208296R
8864:, pp. 93–444,
8779:1971InMat..12...95B
8385:. pp. 381–394.
8322:essential dimension
8214:is injective. For
8020:is bijective. Here
7833:coefficient groups
7611:with the action of
7302:associated to some
7090:/Γ is compact. The
6868:For an anisotropic
6847:Kneser–Tits problem
6375:over a local field
5461:sectional curvature
5457:Riemannian geometry
5310:Useful theories of
4715:
3882:by some element of
3834:Parabolic subgroups
3682:commutator subgroup
3464:simply transitively
3215:), its Lie algebra
2816:has order prime to
2806:multiplicative type
1991:. The simple group
1473:th roots of unity.
1160:normal subgroup of
865:invertible matrices
125:Group homomorphisms
35:Algebraic structure
9363:10.1007/bf02684397
9304:Springer, Tonny A.
9267:"Reductive groups"
9263:Springer, Tonny A.
9161:Platonov, Vladimir
9062:, pp. 39–81,
8981:10.1007/BFb0059005
8787:10.1007/BF01404653
8259:groups of Lie type
8201:
8168:
8007:
7974:
7770:are classified by
7746:are classified by
7710:, meaning objects
7645:principal G-bundle
7633:
7601:
7569:
7475:
7388:*) is trivial. If
7294:. In other words,
7203:
6679:is isotropic over
6630:is isotropic over
6589:
6570:inner automorphism
6437:A reductive group
6418:has real rank min(
6158:
6123:
6101:at least 2, since
6022:
5956:
5921:
5899:at least 3, since
5848:Geordie Williamson
5846:, Simon Riche and
5757:Borel–Weil theorem
5688:into itself. Then
5539:with an action of
5537:simplicial complex
5504:discrete valuation
5211:In the context of
5165:
5126:
5036:
4959:semidirect product
4922:
4719:
4692:
4376:automorphism group
4342:exceptional groups
4278:
4219:
4095:, or equivalently
4062:parabolic subgroup
4047:
4037:
3868:parabolic subgroup
3629:
3591:
3552:
3534:
3301:
3243:
3203:For example, when
3190:
3172:
3117:
3085:
3057:
3033:
3019:. The subspace of
3009:
2969:
2855:is isomorphic to (
2788:of characteristic
2784:is reductive. For
2766:linearly reductive
2656:
2550:
2506:
2461:
2441:
2412:
2383:
2350:
2308:
2299:
2203:
2174:
2130:
2097:
2060:
1954:identity component
1899:, the subgroup of
1766:
1694:
1662:
1617:A reductive group
1572:
1386:
1356:
1296:
1269:
1230:
1210:
1170:
1142:
1122:
1091:
1065:group scheme over
913:compact Lie groups
840:that has a finite
601:Special orthogonal
489:
464:
427:
308:Lagrange's theorem
9411:978-1-4704-3105-1
9402:10.1090/ulect/066
9386:Steinberg, Robert
9341:Steinberg, Robert
9325:978-0-8176-4021-7
9281:, pp. 3–27,
9255:978-2-85629-880-0
9200:"Reductive group"
9104:978-0-8218-3527-2
9030:978-2-85629-324-9
8943:978-2-85629-323-2
8897:, Paris: Masson,
8871:978-2-85629-794-0
8810:Chevalley, Claude
8335:Haboush's theorem
8309:Langlands program
8277:Schubert calculus
8159:
7965:
7630:
7598:
7579:is isomorphic to
7566:
7434:
7290:is the center of
7268:Galois cohomology
7201:
7174:separable closure
6733:is split of type
6587:
6525:gave an explicit
6395:) if and only if
6156:
6133:is isomorphic to
6120:
5954:
5931:is isomorphic to
5918:
5407:metaplectic group
5393:) has nontrivial
5379:fundamental group
5334:For example, the
5163:
5123:
5053:is the center of
4123:), these are the
4060:By definition, a
3512:
3157:
2948:is the action of
2464:{\displaystyle 1}
2404:
2195:
2122:
2058:
1971:at least 3. (For
1758:
1691:
1660:
1359:{\displaystyle G}
1318:is perfect.) Any
1308:algebraic closure
1294:
1266:
1233:{\displaystyle G}
1182:unipotent radical
1173:{\displaystyle G}
1145:{\displaystyle G}
1125:{\displaystyle G}
1094:{\displaystyle G}
810:
809:
385:
384:
267:Alternating group
224:
223:
16:(Redirected from
9514:
9487:
9479:Grothendieck, A.
9463:
9422:
9381:
9336:
9299:
9258:
9239:
9223:-Canonical Basis
9212:
9191:
9156:
9115:
9093:(2nd ed.),
9080:
9069:978-285629-269-3
9053:
9042:
9011:Grothendieck, A.
9002:
8965:Grothendieck, A.
8955:
8924:Grothendieck, A.
8915:
8887:Demazure, Michel
8882:
8855:
8842:
8805:
8752:
8703:
8700:
8694:
8691:
8685:
8682:
8676:
8673:
8667:
8664:
8658:
8655:
8649:
8646:
8640:
8637:
8631:
8628:
8622:
8619:
8613:
8610:
8604:
8601:
8595:
8592:
8586:
8583:
8577:
8574:
8561:
8558:
8552:
8549:
8543:
8540:
8534:
8531:
8525:
8522:
8516:
8513:
8507:
8504:
8498:
8495:
8489:
8486:
8480:
8477:
8471:
8468:
8462:
8459:
8453:
8450:
8441:
8438:
8432:
8429:
8423:
8420:
8414:
8411:
8405:
8402:
8396:
8393:
8387:
8386:
8384:
8373:
8367:
8364:
8358:
8355:
8273:Schubert variety
8210:
8208:
8207:
8202:
8191:
8190:
8178:
8177:
8167:
8140:
8139:
8016:
8014:
8013:
8008:
7997:
7996:
7984:
7983:
7973:
7946:
7945:
7642:
7640:
7639:
7634:
7632:
7631:
7623:
7610:
7608:
7607:
7602:
7600:
7599:
7591:
7578:
7576:
7575:
7570:
7568:
7567:
7559:
7484:
7482:
7481:
7476:
7468:
7463:
7462:
7435:
7430:
7422:
7417:
7416:
7298:is the twist of
7212:
7210:
7209:
7204:
7202:
7194:
7060:For a Lie group
6998:arithmetic group
6970:rational numbers
6711:'s machinery of
6598:
6596:
6595:
6590:
6588:
6580:
6523:Robert Steinberg
6199:division algebra
6167:
6165:
6164:
6159:
6157:
6149:
6132:
6130:
6129:
6124:
6122:
6121:
6113:
6031:
6029:
6028:
6023:
6015:
5974:is equal to the
5965:
5963:
5962:
5957:
5955:
5947:
5930:
5928:
5927:
5922:
5920:
5919:
5911:
5865:classical groups
5824:Henning Andersen
5610:dominant weights
5477:hyperbolic plane
5233:Zariski topology
5174:
5172:
5171:
5166:
5164:
5156:
5135:
5133:
5132:
5127:
5125:
5124:
5116:
5092:if the morphism
5045:
5043:
5042:
5037:
5017:
4931:
4929:
4928:
4923:
4918:
4917:
4905:
4904:
4877:
4876:
4867:
4866:
4854:
4853:
4844:
4843:
4828:
4827:
4806:
4805:
4774:with Witt index
4728:
4726:
4725:
4720:
4714:
4709:
4688:
4687:
4675:
4674:
4647:
4646:
4637:
4636:
4624:
4623:
4614:
4613:
4598:
4597:
4570:
4569:
4439:simply connected
4415:simply connected
4380:octonion algebra
4228:
4226:
4225:
4220:
4209:
4208:
4207:
4206:
4183:
4182:
4181:
4180:
4056:
4054:
4053:
4048:
4046:
4042:
4041:
3822:is generated by
3638:
3636:
3635:
3630:
3619:
3615:
3614:
3600:
3598:
3597:
3592:
3590:
3589:
3565:For example, if
3561:
3559:
3558:
3553:
3548:
3547:
3542:
3541:
3533:
3532:
3531:
3508:
3507:
3498:
3497:
3310:
3308:
3307:
3302:
3291:
3287:
3286:
3252:
3250:
3249:
3244:
3233:
3229:
3228:
3199:
3197:
3196:
3191:
3186:
3185:
3180:
3179:
3171:
3153:
3152:
3143:
3142:
3126:
3124:
3123:
3118:
3116:
3115:
3103:decomposes into
3094:
3092:
3091:
3086:
3084:
3083:
3066:
3064:
3063:
3058:
3056:
3055:
3042:
3040:
3039:
3034:
3032:
3031:
3018:
3016:
3015:
3010:
3008:
3007:
2978:
2976:
2975:
2970:
2968:
2967:
2794:Masayoshi Nagata
2792:>0, however,
2740:over a field of
2676:complexification
2665:
2663:
2662:
2657:
2652:
2651:
2646:
2639:
2634:
2610:
2609:
2597:
2596:
2584:
2579:
2578:
2559:
2557:
2556:
2551:
2540:
2539:
2530:
2515:
2513:
2512:
2507:
2496:
2495:
2470:
2468:
2467:
2462:
2450:
2448:
2447:
2442:
2440:
2439:
2434:
2421:
2419:
2418:
2413:
2411:
2410:
2405:
2402:
2392:
2390:
2389:
2384:
2382:
2381:
2359:
2357:
2356:
2351:
2349:
2348:
2343:
2317:
2315:
2314:
2309:
2304:
2303:
2296:
2295:
2272:
2271:
2248:
2247:
2235:
2234:
2212:
2210:
2209:
2204:
2202:
2201:
2196:
2193:
2183:
2181:
2180:
2175:
2173:
2172:
2167:
2158:
2157:
2152:
2139:
2137:
2136:
2131:
2129:
2128:
2123:
2120:
2106:
2104:
2103:
2098:
2096:
2095:
2090:
2069:
2067:
2066:
2061:
2059:
2051:
1920:orthogonal group
1918:. Likewise, the
1886:symplectic group
1775:
1773:
1772:
1767:
1765:
1764:
1759:
1756:
1703:
1701:
1700:
1695:
1693:
1692:
1684:
1671:
1669:
1668:
1663:
1661:
1653:
1581:
1579:
1578:
1573:
1568:
1567:
1558:
1532:
1531:
1486:central subgroup
1395:
1393:
1392:
1387:
1385:
1384:
1365:
1363:
1362:
1357:
1338:, is reductive.
1305:
1303:
1302:
1297:
1295:
1287:
1278:
1276:
1275:
1270:
1268:
1267:
1259:
1239:
1237:
1236:
1231:
1219:
1217:
1216:
1211:
1200:
1199:
1179:
1177:
1176:
1171:
1151:
1149:
1148:
1143:
1131:
1129:
1128:
1123:
1100:
1098:
1097:
1092:
1033:is defined as a
901:Claude Chevalley
880:symplectic group
802:
795:
788:
744:Algebraic groups
517:Hyperbolic group
507:Arithmetic group
498:
496:
495:
490:
488:
473:
471:
470:
465:
463:
436:
434:
433:
428:
426:
349:Schur multiplier
303:Cauchy's theorem
291:Quaternion group
239:
238:
65:
64:
54:
41:
30:
29:
21:
9522:
9521:
9517:
9516:
9515:
9513:
9512:
9511:
9492:
9491:
9470:
9445:10.2307/1970394
9412:
9326:
9289:
9277:, vol. 1,
9256:
9181:
9146:
9105:
9070:
9051:
9031:
8991:
8973:Springer-Verlag
8944:
8905:
8891:Gabriel, Pierre
8872:
8853:
8832:
8822:Springer Nature
8742:
8724:Springer Nature
8711:
8706:
8701:
8697:
8692:
8688:
8683:
8679:
8674:
8670:
8665:
8661:
8656:
8652:
8647:
8643:
8638:
8634:
8629:
8625:
8620:
8616:
8611:
8607:
8602:
8598:
8593:
8589:
8584:
8580:
8575:
8564:
8559:
8555:
8550:
8546:
8541:
8537:
8532:
8528:
8523:
8519:
8514:
8510:
8505:
8501:
8496:
8492:
8487:
8483:
8478:
8474:
8469:
8465:
8460:
8456:
8451:
8444:
8439:
8435:
8430:
8426:
8421:
8417:
8412:
8408:
8403:
8399:
8394:
8390:
8382:
8374:
8370:
8365:
8361:
8356:
8352:
8348:
8254:
8247:
8242:
8186:
8182:
8173:
8169:
8163:
8135:
8131:
8129:
8126:
8125:
8074:
8061:
8040:
7992:
7988:
7979:
7975:
7969:
7941:
7937:
7935:
7932:
7931:
7917:Hasse principle
7825:-torsors using
7667:is smooth over
7622:
7618:
7616:
7613:
7612:
7590:
7586:
7584:
7581:
7580:
7558:
7554:
7552:
7549:
7548:
7515:
7493:has Witt index
7464:
7458:
7454:
7429:
7418:
7412:
7408:
7397:
7394:
7393:
7364:
7358:
7262:, meaning that
7230:
7229:
7193:
7191:
7188:
7187:
7181:
7159:
7114:
7108:
6962:
6942:profinite group
6804:Whitehead group
6782:
6771:
6760:
6753:
6746:
6739:
6728:
6666:
6654:) generated by
6579:
6577:
6574:
6573:
6511:
6343:
6310:
6236:if and only if
6192:
6148:
6146:
6143:
6142:
6112:
6108:
6106:
6103:
6102:
6011:
6003:
6000:
5999:
5994:if and only if
5946:
5944:
5941:
5940:
5910:
5906:
5904:
5901:
5900:
5860:
5594:
5583:
5521:affine building
5517:
5463:. For example,
5453:symmetric space
5411:universal cover
5405:) (such as the
5395:covering spaces
5267:
5219:is a Lie group
5209:
5181:Michel Demazure
5155:
5153:
5150:
5149:
5115:
5111:
5109:
5106:
5105:
5075:
5013:
4969:
4966:
4965:
4910:
4906:
4891:
4887:
4872:
4868:
4862:
4858:
4849:
4845:
4839:
4835:
4820:
4816:
4801:
4797:
4789:
4786:
4785:
4761:
4740:
4710:
4696:
4680:
4676:
4661:
4657:
4642:
4638:
4632:
4628:
4619:
4615:
4609:
4605:
4584:
4580:
4565:
4561:
4553:
4550:
4549:
4518:
4497:
4427:complex numbers
4397:
4393:
4389:
4373:
4354:
4350:
4327:Wilhelm Killing
4324:
4320:
4316:
4312:
4308:
4304:
4298:
4292:
4286:
4270:
4202:
4198:
4197:
4193:
4176:
4172:
4171:
4167:
4159:
4156:
4155:
4142:
4133:
4036:
4035:
4030:
4025:
4020:
4014:
4013:
4008:
4003:
3998:
3992:
3991:
3986:
3981:
3976:
3970:
3969:
3964:
3959:
3954:
3944:
3943:
3939:
3937:
3934:
3933:
3912:
3836:
3797:
3783:
3738:
3728:
3690:semisimple rank
3656:
3647:
3610:
3609:
3608:
3606:
3603:
3602:
3585:
3584:
3582:
3579:
3578:
3543:
3537:
3536:
3535:
3527:
3523:
3516:
3503:
3502:
3493:
3492:
3490:
3487:
3486:
3460:Borel subgroups
3454:
3445:symmetric group
3418:
3364:
3355:
3334:
3325:
3282:
3281:
3280:
3278:
3275:
3274:
3224:
3223:
3222:
3220:
3217:
3216:
3181:
3175:
3174:
3173:
3161:
3148:
3147:
3138:
3137:
3135:
3132:
3131:
3111:
3110:
3108:
3105:
3104:
3079:
3078:
3076:
3073:
3072:
3051:
3050:
3048:
3045:
3044:
3027:
3026:
3024:
3021:
3020:
3003:
3002:
3000:
2997:
2996:
2963:
2962:
2960:
2957:
2956:
2908:
2863:
2826:
2672:
2647:
2642:
2641:
2635:
2624:
2605:
2601:
2592:
2588:
2580:
2574:
2570:
2568:
2565:
2564:
2535:
2531:
2526:
2521:
2518:
2517:
2491:
2487:
2485:
2482:
2481:
2478:
2456:
2453:
2452:
2435:
2430:
2429:
2427:
2424:
2423:
2406:
2401:
2400:
2398:
2395:
2394:
2377:
2373:
2371:
2368:
2367:
2344:
2339:
2338:
2336:
2333:
2332:
2329:unipotent group
2324:
2298:
2297:
2291:
2287:
2285:
2279:
2278:
2273:
2267:
2263:
2256:
2255:
2243:
2239:
2230:
2226:
2221:
2218:
2217:
2197:
2192:
2191:
2189:
2186:
2185:
2168:
2163:
2162:
2153:
2148:
2147:
2145:
2142:
2141:
2124:
2119:
2118:
2116:
2113:
2112:
2091:
2086:
2085:
2083:
2080:
2079:
2076:
2050:
2048:
2045:
2044:
1895:) over a field
1882:
1844:) over a field
1816:
1803:
1760:
1755:
1754:
1752:
1749:
1748:
1742:
1740:
1734:
1726:
1683:
1679:
1677:
1674:
1673:
1652:
1650:
1647:
1646:
1615:
1563:
1559:
1554:
1527:
1523:
1500:
1497:
1496:
1478:central isogeny
1467:
1402:
1377:
1373:
1371:
1368:
1367:
1351:
1348:
1347:
1344:
1337:
1286:
1284:
1281:
1280:
1258:
1254:
1252:
1249:
1248:
1225:
1222:
1221:
1195:
1191:
1189:
1186:
1185:
1184:and is denoted
1165:
1162:
1161:
1137:
1134:
1133:
1117:
1114:
1113:
1110:normal subgroup
1086:
1083:
1082:
1075:
1039:subgroup scheme
1023:
1017:
951:rational points
909:Dynkin diagrams
897:are reductive.
818:reductive group
806:
777:
776:
765:Abelian variety
758:Reductive group
746:
736:
735:
734:
733:
684:
676:
668:
660:
652:
625:Special unitary
536:
522:
521:
503:
502:
484:
482:
479:
478:
459:
457:
454:
453:
422:
420:
417:
416:
408:
407:
398:Discrete groups
387:
386:
342:Frobenius group
287:
274:
263:
256:Symmetric group
252:
236:
226:
225:
76:Normal subgroup
62:
42:
33:
28:
23:
22:
15:
12:
11:
5:
9520:
9510:
9509:
9504:
9490:
9489:
9469:
9468:External links
9466:
9465:
9464:
9439:(2): 313–329,
9423:
9410:
9382:
9337:
9324:
9300:
9287:
9259:
9254:
9213:
9192:
9179:
9171:Academic Press
9157:
9145:978-1107167483
9144:
9116:
9103:
9081:
9068:
9044:
9029:
9003:
8990:978-3540051800
8989:
8957:
8942:
8916:
8904:978-2225616662
8903:
8883:
8870:
8843:
8830:
8806:
8753:
8740:
8710:
8707:
8705:
8704:
8695:
8686:
8677:
8668:
8659:
8650:
8641:
8632:
8623:
8614:
8605:
8596:
8587:
8578:
8562:
8553:
8544:
8535:
8526:
8517:
8508:
8499:
8490:
8481:
8472:
8463:
8454:
8442:
8433:
8424:
8415:
8406:
8397:
8388:
8368:
8359:
8349:
8347:
8344:
8343:
8342:
8337:
8324:
8315:
8298:
8293:
8288:
8279:
8262:
8253:
8250:
8245:
8240:
8212:
8211:
8200:
8197:
8194:
8189:
8185:
8181:
8176:
8172:
8166:
8162:
8158:
8155:
8152:
8149:
8146:
8143:
8138:
8134:
8070:
8057:
8036:
8024:runs over all
8018:
8017:
8006:
8003:
8000:
7995:
7991:
7987:
7982:
7978:
7972:
7968:
7964:
7961:
7958:
7955:
7952:
7949:
7944:
7940:
7875:Lang's theorem
7679:-torsors over
7661:étale topology
7629:
7626:
7621:
7597:
7594:
7589:
7565:
7562:
7557:
7514:
7511:
7474:
7471:
7467:
7461:
7457:
7453:
7450:
7447:
7444:
7441:
7438:
7433:
7428:
7425:
7421:
7415:
7411:
7407:
7404:
7401:
7360:
7354:
7225:
7221:
7200:
7197:
7179:
7155:
7110:Main article:
7107:
7104:
6961:
6958:
6872:-simple group
6865:) is abelian.
6790:-simple group
6780:
6769:
6758:
6751:
6744:
6737:
6726:
6662:
6586:
6583:
6510:
6507:
6350:
6349:
6339:
6330:
6316:
6306:
6246:
6245:
6232:is split over
6184:
6176:(meaning that
6155:
6152:
6119:
6116:
6111:
6064:group of units
6033:
6021:
6018:
6014:
6010:
6007:
5990:is split over
5953:
5950:
5917:
5914:
5909:
5895:has dimension
5859:
5856:
5816:Coxeter number
5593:
5590:
5579:
5555:) preserves a
5513:
5508:p-adic numbers
5301:Satake diagram
5263:
5208:
5205:
5162:
5159:
5122:
5119:
5114:
5084:over a scheme
5074:
5071:
5047:
5046:
5035:
5032:
5029:
5026:
5023:
5020:
5016:
5012:
5009:
5006:
5003:
5000:
4997:
4994:
4991:
4988:
4985:
4982:
4979:
4976:
4973:
4935:
4934:
4933:
4932:
4921:
4916:
4913:
4909:
4903:
4900:
4897:
4894:
4890:
4886:
4883:
4880:
4875:
4871:
4865:
4861:
4857:
4852:
4848:
4842:
4838:
4834:
4831:
4826:
4823:
4819:
4815:
4812:
4809:
4804:
4800:
4796:
4793:
4780:
4779:
4757:
4754:
4736:
4732:
4731:
4730:
4729:
4718:
4713:
4708:
4705:
4702:
4699:
4695:
4691:
4686:
4683:
4679:
4673:
4670:
4667:
4664:
4660:
4656:
4653:
4650:
4645:
4641:
4635:
4631:
4627:
4622:
4618:
4612:
4608:
4604:
4601:
4596:
4593:
4590:
4587:
4583:
4579:
4576:
4573:
4568:
4564:
4560:
4557:
4544:
4543:
4514:
4511:
4493:
4395:
4391:
4387:
4371:
4352:
4348:
4322:
4318:
4314:
4310:
4306:
4300:
4294:
4288:
4282:
4269:
4266:
4252:is called the
4230:
4229:
4218:
4215:
4212:
4205:
4201:
4196:
4192:
4189:
4186:
4179:
4175:
4170:
4166:
4163:
4138:
4131:
4125:flag varieties
4058:
4057:
4045:
4040:
4034:
4031:
4029:
4026:
4024:
4021:
4019:
4016:
4015:
4012:
4009:
4007:
4004:
4002:
3999:
3997:
3994:
3993:
3990:
3987:
3985:
3982:
3980:
3977:
3975:
3972:
3971:
3968:
3965:
3963:
3960:
3958:
3955:
3953:
3950:
3949:
3947:
3942:
3910:
3835:
3832:
3795:
3781:
3752:Dynkin diagram
3733:
3724:
3652:
3643:
3628:
3625:
3622:
3618:
3613:
3588:
3563:
3562:
3551:
3546:
3540:
3530:
3526:
3522:
3519:
3515:
3511:
3506:
3501:
3496:
3472:positive roots
3450:
3414:
3400:quotient group
3360:
3351:
3330:
3323:
3300:
3297:
3294:
3290:
3285:
3261:matrices over
3242:
3239:
3236:
3232:
3227:
3201:
3200:
3189:
3184:
3178:
3170:
3167:
3164:
3160:
3156:
3151:
3146:
3141:
3114:
3082:
3054:
3030:
3006:
2966:
2933:copies of the
2919:tensor product
2904:
2859:
2825:
2822:
2742:characteristic
2671:
2668:
2655:
2650:
2645:
2638:
2633:
2630:
2627:
2623:
2619:
2616:
2613:
2608:
2604:
2600:
2595:
2591:
2587:
2583:
2577:
2573:
2549:
2546:
2543:
2538:
2534:
2529:
2525:
2505:
2502:
2499:
2494:
2490:
2477:
2474:
2473:
2472:
2460:
2438:
2433:
2409:
2380:
2376:
2361:
2347:
2342:
2323:
2320:
2307:
2302:
2294:
2290:
2286:
2284:
2281:
2280:
2277:
2274:
2270:
2266:
2262:
2261:
2259:
2254:
2251:
2246:
2242:
2238:
2233:
2229:
2225:
2200:
2171:
2166:
2161:
2156:
2151:
2127:
2109:algebraic tori
2094:
2089:
2075:
2072:
2057:
2054:
1931:quadratic form
1881:
1866:
1812:
1799:
1776:of invertible
1763:
1741:
1736:
1730:
1727:
1725:
1722:
1690:
1687:
1682:
1659:
1656:
1614:
1611:
1583:
1582:
1571:
1566:
1562:
1557:
1553:
1550:
1547:
1544:
1541:
1538:
1535:
1530:
1526:
1522:
1519:
1516:
1513:
1510:
1507:
1504:
1463:
1462:group scheme μ
1401:
1398:
1383:
1380:
1376:
1355:
1343:
1340:
1333:
1326:, such as the
1293:
1290:
1265:
1262:
1257:
1229:
1209:
1206:
1203:
1198:
1194:
1169:
1141:
1121:
1090:
1074:
1071:
1019:Main article:
1016:
1013:
838:representation
808:
807:
805:
804:
797:
790:
782:
779:
778:
775:
774:
772:Elliptic curve
768:
767:
761:
760:
754:
753:
747:
742:
741:
738:
737:
732:
731:
728:
725:
721:
717:
716:
715:
710:
708:Diffeomorphism
704:
703:
698:
693:
687:
686:
682:
678:
674:
670:
666:
662:
658:
654:
650:
645:
644:
633:
632:
621:
620:
609:
608:
597:
596:
585:
584:
573:
572:
565:Special linear
561:
560:
553:General linear
549:
548:
543:
537:
528:
527:
524:
523:
520:
519:
514:
509:
501:
500:
487:
475:
462:
449:
447:Modular groups
445:
444:
443:
438:
425:
409:
406:
405:
400:
394:
393:
392:
389:
388:
383:
382:
381:
380:
375:
370:
367:
361:
360:
354:
353:
352:
351:
345:
344:
338:
337:
332:
323:
322:
320:Hall's theorem
317:
315:Sylow theorems
311:
310:
305:
297:
296:
295:
294:
288:
283:
280:Dihedral group
276:
275:
270:
264:
259:
253:
248:
237:
232:
231:
228:
227:
222:
221:
220:
219:
214:
206:
205:
204:
203:
198:
193:
188:
183:
178:
173:
171:multiplicative
168:
163:
158:
153:
145:
144:
143:
142:
137:
129:
128:
120:
119:
118:
117:
115:Wreath product
112:
107:
102:
100:direct product
94:
92:Quotient group
86:
85:
84:
83:
78:
73:
63:
60:
59:
56:
55:
47:
46:
26:
9:
6:
4:
3:
2:
9519:
9508:
9505:
9503:
9500:
9499:
9497:
9486:
9485:
9480:
9476:
9472:
9471:
9462:
9458:
9454:
9450:
9446:
9442:
9438:
9434:
9433:
9428:
9427:Tits, Jacques
9424:
9421:
9417:
9413:
9407:
9403:
9399:
9395:
9391:
9387:
9383:
9380:
9376:
9372:
9368:
9364:
9360:
9356:
9352:
9351:
9346:
9342:
9338:
9335:
9331:
9327:
9321:
9317:
9313:
9309:
9305:
9301:
9298:
9294:
9290:
9288:0-8218-3347-2
9284:
9280:
9276:
9272:
9268:
9264:
9260:
9257:
9251:
9247:
9243:
9238:
9233:
9229:
9225:
9224:
9220:
9214:
9211:
9207:
9206:
9201:
9197:
9193:
9190:
9186:
9182:
9180:0-12-558180-7
9176:
9172:
9168:
9167:
9162:
9158:
9155:
9151:
9147:
9141:
9137:
9133:
9129:
9125:
9121:
9117:
9114:
9110:
9106:
9100:
9096:
9092:
9091:
9086:
9082:
9079:
9075:
9071:
9065:
9061:
9057:
9050:
9045:
9040:
9036:
9032:
9026:
9022:
9018:
9017:
9012:
9008:
9004:
9000:
8996:
8992:
8986:
8982:
8978:
8974:
8970:
8966:
8962:
8958:
8953:
8949:
8945:
8939:
8935:
8931:
8930:
8925:
8921:
8917:
8914:
8910:
8906:
8900:
8896:
8892:
8888:
8884:
8881:
8877:
8873:
8867:
8863:
8859:
8852:
8848:
8847:Conrad, Brian
8844:
8841:
8837:
8833:
8831:3-540-23031-9
8827:
8823:
8819:
8815:
8811:
8807:
8804:
8800:
8796:
8792:
8788:
8784:
8780:
8776:
8773:(2): 95–104,
8772:
8768:
8767:
8762:
8761:Tits, Jacques
8758:
8757:Borel, Armand
8754:
8751:
8747:
8743:
8741:0-387-97370-2
8737:
8733:
8729:
8725:
8721:
8717:
8716:Borel, Armand
8713:
8712:
8699:
8690:
8681:
8672:
8663:
8654:
8645:
8636:
8627:
8618:
8609:
8600:
8591:
8582:
8573:
8571:
8569:
8567:
8557:
8548:
8539:
8530:
8521:
8512:
8503:
8494:
8485:
8476:
8467:
8458:
8449:
8447:
8437:
8428:
8419:
8410:
8401:
8392:
8381:
8380:
8372:
8363:
8354:
8350:
8341:
8338:
8336:
8332:
8328:
8325:
8323:
8319:
8318:Special group
8316:
8314:
8310:
8306:
8302:
8299:
8297:
8294:
8292:
8289:
8287:
8283:
8282:Schur algebra
8280:
8278:
8274:
8270:
8266:
8263:
8260:
8256:
8255:
8249:
8243:
8236:
8231:
8229:
8225:
8221:
8217:
8195:
8192:
8187:
8183:
8174:
8170:
8164:
8160:
8150:
8147:
8144:
8136:
8132:
8124:
8123:
8122:
8120:
8116:
8111:
8109:
8105:
8101:
8097:
8093:
8089:
8085:
8082:
8078:
8073:
8069:
8065:
8060:
8056:
8052:
8048:
8044:
8039:
8035:
8031:
8027:
8023:
8001:
7998:
7993:
7989:
7980:
7976:
7970:
7966:
7956:
7953:
7950:
7942:
7938:
7930:
7929:
7928:
7926:
7922:
7918:
7914:
7913:Günter Harder
7910:
7909:Martin Kneser
7906:
7902:
7898:
7894:
7890:
7886:
7882:
7878:
7876:
7872:
7868:
7864:
7860:
7856:
7852:
7848:
7844:
7840:
7836:
7832:
7828:
7824:
7819:
7817:
7813:
7809:
7805:
7801:
7797:
7793:
7789:
7785:
7781:
7777:
7773:
7769:
7765:
7761:
7757:
7753:
7749:
7745:
7741:
7737:
7733:
7729:
7725:
7721:
7717:
7713:
7709:
7706:over a field
7705:
7701:
7696:
7694:
7690:
7686:
7682:
7678:
7674:
7670:
7666:
7662:
7658:
7654:
7653:fppf topology
7650:
7646:
7624:
7619:
7592:
7587:
7560:
7555:
7546:
7542:
7538:
7534:
7530:
7527:over a field
7526:
7522:
7521:
7510:
7508:
7504:
7500:
7496:
7492:
7488:
7469:
7465:
7459:
7455:
7448:
7445:
7442:
7431:
7423:
7419:
7413:
7409:
7402:
7399:
7391:
7387:
7383:
7379:
7375:
7372:
7368:
7363:
7357:
7352:
7348:
7344:
7340:
7336:
7332:
7328:
7325:over a field
7324:
7320:
7317:Example: Let
7315:
7313:
7310:-torsor over
7309:
7305:
7301:
7297:
7293:
7289:
7285:
7281:
7277:
7273:
7269:
7265:
7261:
7257:
7253:
7249:
7245:
7241:
7237:
7232:
7228:
7224:
7220:
7216:
7195:
7185:
7178:
7175:
7171:
7167:
7163:
7158:
7154:
7150:
7146:
7143:over a field
7142:
7137:
7135:
7131:
7127:
7123:
7119:
7113:
7103:
7101:
7097:
7093:
7089:
7085:
7081:
7077:
7073:
7069:
7068:
7063:
7058:
7056:
7052:
7048:
7044:
7040:
7036:
7032:
7028:
7024:
7020:
7016:
7012:
7011:commensurable
7008:
7004:
7000:
6999:
6994:
6990:
6986:
6982:
6978:
6974:
6971:
6967:
6957:
6955:
6951:
6947:
6943:
6939:
6935:
6931:
6927:
6923:
6919:
6915:
6911:
6907:
6903:
6899:
6895:
6891:
6887:
6883:
6879:
6875:
6871:
6866:
6864:
6860:
6856:
6852:
6848:
6844:
6840:
6836:
6832:
6828:
6824:
6820:
6816:
6812:
6808:
6805:
6801:
6797:
6793:
6789:
6784:
6779:
6775:
6768:
6764:
6757:
6750:
6743:
6736:
6732:
6725:
6721:
6716:
6714:
6710:
6706:
6702:
6698:
6694:
6690:
6686:
6682:
6678:
6674:
6671:contained in
6670:
6665:
6661:
6657:
6653:
6649:
6645:
6641:
6637:
6633:
6629:
6625:
6621:
6617:
6613:
6609:
6604:
6602:
6581:
6571:
6567:
6563:
6559:
6555:
6550:
6548:
6544:
6540:
6536:
6532:
6528:
6524:
6520:
6517:over a field
6516:
6506:
6504:
6500:
6496:
6492:
6488:
6484:
6480:
6476:
6472:
6468:
6464:
6460:
6456:
6452:
6448:
6444:
6441:over a field
6440:
6435:
6433:
6429:
6425:
6421:
6417:
6413:
6411:
6407:
6403:
6398:
6394:
6390:
6386:
6382:
6378:
6374:
6369:
6367:
6364:) contains a
6363:
6359:
6355:
6347:
6342:
6338:
6334:
6331:
6328:
6325:not equal to
6324:
6320:
6317:
6314:
6309:
6305:
6301:
6297:
6294:
6293:
6292:
6290:
6287:over a field
6286:
6282:
6278:
6274:
6270:
6265:
6263:
6259:
6255:
6251:
6243:
6239:
6235:
6231:
6227:
6223:
6219:
6215:
6211:
6207:
6203:
6200:
6196:
6191:
6187:
6183:
6179:
6175:
6171:
6150:
6140:
6136:
6114:
6109:
6100:
6096:
6093:is simple if
6092:
6088:
6084:
6080:
6076:
6072:
6068:
6065:
6061:
6057:
6053:
6049:
6045:
6041:
6038:
6034:
6016:
6012:
6008:
5997:
5993:
5989:
5985:
5981:
5977:
5973:
5969:
5948:
5938:
5934:
5912:
5907:
5898:
5894:
5891:is simple if
5890:
5886:
5882:
5878:
5875:over a field
5874:
5870:
5869:
5868:
5866:
5855:
5853:
5849:
5845:
5841:
5837:
5833:
5829:
5825:
5821:
5817:
5813:
5809:
5805:
5801:
5797:
5793:
5789:
5785:
5781:
5778:over a field
5777:
5772:
5770:
5766:
5762:
5758:
5754:
5750:
5746:
5742:
5738:
5735:-equivariant
5734:
5730:
5726:
5722:
5717:
5715:
5711:
5707:
5703:
5699:
5695:
5691:
5687:
5683:
5679:
5675:
5671:
5667:
5663:
5659:
5655:
5651:
5647:
5643:
5639:
5635:
5631:
5627:
5623:
5619:
5615:
5611:
5607:
5603:
5600:over a field
5599:
5589:
5587:
5582:
5578:
5574:
5570:
5566:
5562:
5558:
5554:
5550:
5546:
5542:
5538:
5534:
5530:
5526:
5523:
5522:
5516:
5512:
5509:
5506:(such as the
5505:
5501:
5498:over a field
5497:
5492:
5490:
5486:
5482:
5478:
5474:
5470:
5466:
5462:
5458:
5454:
5450:
5447:
5443:
5439:
5435:
5431:
5426:
5424:
5420:
5416:
5412:
5408:
5404:
5400:
5396:
5392:
5388:
5384:
5380:
5376:
5372:
5368:
5364:
5360:
5356:
5352:
5348:
5344:
5340:
5337:
5332:
5330:
5326:
5322:
5318:
5313:
5308:
5306:
5302:
5298:
5294:
5290:
5286:
5281:
5279:
5275:
5271:
5266:
5262:
5258:
5254:
5250:
5246:
5242:
5238:
5234:
5230:
5226:
5222:
5218:
5214:
5204:
5202:
5198:
5194:
5190:
5186:
5182:
5178:
5157:
5147:
5143:
5139:
5117:
5112:
5103:
5099:
5095:
5091:
5087:
5083:
5080:
5070:
5068:
5064:
5060:
5056:
5052:
5033:
5027:
5018:
5014:
5010:
5004:
4998:
4992:
4989:
4986:
4980:
4974:
4971:
4964:
4963:
4962:
4960:
4956:
4952:
4948:
4945:over a field
4944:
4940:
4919:
4914:
4911:
4907:
4901:
4898:
4895:
4892:
4888:
4884:
4881:
4878:
4873:
4869:
4863:
4859:
4855:
4850:
4846:
4840:
4836:
4832:
4824:
4821:
4817:
4813:
4810:
4807:
4802:
4798:
4791:
4784:
4783:
4782:
4781:
4777:
4773:
4769:
4765:
4760:
4755:
4752:
4748:
4744:
4739:
4734:
4733:
4716:
4711:
4706:
4703:
4700:
4697:
4693:
4689:
4684:
4681:
4677:
4671:
4668:
4665:
4662:
4658:
4654:
4651:
4648:
4643:
4639:
4633:
4629:
4625:
4620:
4616:
4610:
4606:
4602:
4594:
4591:
4588:
4585:
4581:
4577:
4574:
4571:
4566:
4562:
4555:
4548:
4547:
4546:
4545:
4541:
4538:
4534:
4530:
4526:
4522:
4517:
4512:
4509:
4505:
4501:
4496:
4491:
4490:
4489:
4487:
4482:
4480:
4476:
4472:
4468:
4464:
4460:
4456:
4452:
4448:
4444:
4440:
4436:
4432:
4428:
4424:
4420:
4416:
4412:
4409:over a field
4408:
4404:
4399:
4385:
4381:
4377:
4370:
4366:
4365:L. E. Dickson
4362:
4358:
4346:
4343:
4338:
4336:
4332:
4328:
4303:
4297:
4291:
4285:
4274:
4265:
4263:
4259:
4258:flag manifold
4255:
4251:
4247:
4243:
4239:
4235:
4216:
4213:
4210:
4203:
4199:
4194:
4190:
4187:
4184:
4177:
4173:
4168:
4164:
4161:
4154:
4153:
4152:
4150:
4147:of dimension
4146:
4141:
4137:
4130:
4126:
4122:
4118:
4114:
4110:
4106:
4102:
4098:
4094:
4090:
4086:
4082:
4078:
4074:
4071:over a field
4070:
4066:
4063:
4043:
4038:
4032:
4027:
4022:
4017:
4010:
4005:
4000:
3995:
3988:
3983:
3978:
3973:
3966:
3961:
3956:
3951:
3945:
3940:
3932:
3931:
3930:
3928:
3924:
3920:
3916:
3909:
3905:
3901:
3897:
3893:
3889:
3885:
3881:
3877:
3873:
3869:
3865:
3861:
3857:
3853:
3849:
3845:
3842:over a field
3841:
3831:
3829:
3825:
3821:
3817:
3813:
3809:
3805:
3801:
3794:
3791:
3790:root subgroup
3787:
3780:
3776:
3772:
3769:over a field
3768:
3763:
3761:
3760:inner product
3757:
3753:
3748:
3746:
3742:
3736:
3732:
3727:
3723:
3719:
3715:
3711:
3707:
3703:
3699:
3695:
3691:
3688:, called the
3687:
3683:
3679:
3675:
3670:
3668:
3664:
3660:
3655:
3651:
3646:
3642:
3623:
3616:
3576:
3572:
3568:
3549:
3544:
3528:
3520:
3517:
3513:
3509:
3499:
3485:
3484:
3483:
3481:
3477:
3473:
3469:
3465:
3461:
3456:
3453:
3449:
3446:
3442:
3438:
3434:
3430:
3426:
3422:
3417:
3413:
3409:
3405:
3401:
3397:
3393:
3392:
3387:
3383:
3378:
3376:
3372:
3368:
3363:
3359:
3354:
3350:
3346:
3342:
3338:
3333:
3329:
3322:
3318:
3314:
3295:
3288:
3272:
3268:
3264:
3260:
3256:
3237:
3230:
3214:
3210:
3207:is the group
3206:
3187:
3182:
3165:
3162:
3158:
3154:
3144:
3130:
3129:
3128:
3102:
3098:
3070:
2994:
2990:
2986:
2982:
2955:
2951:
2947:
2942:
2940:
2936:
2932:
2928:
2924:
2920:
2916:
2912:
2907:
2903:
2899:
2895:
2891:
2887:
2883:
2879:
2875:
2871:
2867:
2862:
2858:
2854:
2850:
2846:
2842:
2838:
2833:
2831:
2821:
2819:
2815:
2811:
2807:
2803:
2799:
2795:
2791:
2787:
2783:
2779:
2775:
2771:
2767:
2763:
2760:over a field
2759:
2755:
2751:
2747:
2743:
2739:
2734:
2732:
2728:
2724:
2720:
2716:
2713:
2712:unitary group
2709:
2705:
2701:
2697:
2693:
2689:
2685:
2681:
2677:
2666:
2653:
2648:
2636:
2631:
2628:
2625:
2621:
2617:
2606:
2602:
2593:
2589:
2581:
2575:
2571:
2561:
2544:
2536:
2532:
2527:
2523:
2500:
2492:
2488:
2458:
2436:
2407:
2378:
2374:
2366:
2362:
2345:
2330:
2326:
2325:
2318:
2305:
2300:
2292:
2288:
2282:
2275:
2268:
2264:
2257:
2244:
2240:
2236:
2231:
2227:
2214:
2198:
2169:
2159:
2154:
2125:
2110:
2092:
2071:
2052:
2042:
2038:
2034:
2030:
2026:
2022:
2018:
2014:
2010:
2006:
2002:
1998:
1994:
1990:
1986:
1982:
1978:
1974:
1970:
1967:of dimension
1966:
1962:
1958:
1955:
1951:
1947:
1943:
1939:
1935:
1932:
1928:
1924:
1921:
1917:
1914:
1910:
1909:bilinear form
1906:
1902:
1898:
1894:
1890:
1887:
1879:
1875:
1871:
1865:
1863:
1859:
1855:
1851:
1847:
1843:
1839:
1836:
1832:
1828:
1824:
1820:
1815:
1811:
1807:
1804:is the group
1802:
1798:
1795:
1791:
1787:
1783:
1779:
1761:
1747:
1739:
1733:
1721:
1719:
1715:
1711:
1707:
1685:
1680:
1654:
1644:
1640:
1636:
1632:
1628:
1624:
1621:over a field
1620:
1610:
1608:
1604:
1601:-subgroup of
1600:
1596:
1592:
1588:
1569:
1564:
1560:
1555:
1545:
1539:
1536:
1533:
1528:
1524:
1517:
1511:
1505:
1502:
1495:
1494:
1493:
1491:
1487:
1483:
1479:
1474:
1472:
1468:
1466:
1459:
1455:
1451:
1447:
1443:
1439:
1435:
1431:
1427:
1423:
1419:
1415:
1411:
1408:over a field
1407:
1397:
1381:
1378:
1374:
1353:
1339:
1336:
1332:
1329:
1325:
1321:
1317:
1313:
1309:
1288:
1260:
1255:
1247:
1243:
1227:
1204:
1196:
1192:
1183:
1167:
1159:
1155:
1139:
1119:
1111:
1108:
1104:
1088:
1080:
1070:
1068:
1064:
1060:
1056:
1052:
1048:
1044:
1040:
1036:
1032:
1029:over a field
1028:
1022:
1012:
1010:
1006:
1002:
998:
994:
990:
986:
982:
978:
975:over a field
974:
970:
965:
963:
960:
956:
952:
948:
944:
940:
936:
932:
928:
925:
921:
918:
914:
910:
906:
902:
898:
896:
892:
888:
884:
881:
877:
873:
870:
866:
862:
858:
855:
851:
847:
843:
839:
835:
834:perfect field
831:
827:
823:
820:is a type of
819:
815:
803:
798:
796:
791:
789:
784:
783:
781:
780:
773:
770:
769:
766:
763:
762:
759:
756:
755:
752:
749:
748:
745:
740:
739:
729:
726:
723:
722:
720:
714:
711:
709:
706:
705:
702:
699:
697:
694:
692:
689:
688:
685:
679:
677:
671:
669:
663:
661:
655:
653:
647:
646:
642:
638:
635:
634:
630:
626:
623:
622:
618:
614:
611:
610:
606:
602:
599:
598:
594:
590:
587:
586:
582:
578:
575:
574:
570:
566:
563:
562:
558:
554:
551:
550:
547:
544:
542:
539:
538:
535:
531:
526:
525:
518:
515:
513:
510:
508:
505:
504:
476:
451:
450:
448:
442:
439:
414:
411:
410:
404:
401:
399:
396:
395:
391:
390:
379:
376:
374:
371:
368:
365:
364:
363:
362:
359:
356:
355:
350:
347:
346:
343:
340:
339:
336:
333:
331:
329:
325:
324:
321:
318:
316:
313:
312:
309:
306:
304:
301:
300:
299:
298:
292:
289:
286:
281:
278:
277:
273:
268:
265:
262:
257:
254:
251:
246:
243:
242:
241:
240:
235:
234:Finite groups
230:
229:
218:
215:
213:
210:
209:
208:
207:
202:
199:
197:
194:
192:
189:
187:
184:
182:
179:
177:
174:
172:
169:
167:
164:
162:
159:
157:
154:
152:
149:
148:
147:
146:
141:
138:
136:
133:
132:
131:
130:
127:
126:
122:
121:
116:
113:
111:
108:
106:
103:
101:
98:
95:
93:
90:
89:
88:
87:
82:
79:
77:
74:
72:
69:
68:
67:
66:
61:Basic notions
58:
57:
53:
49:
48:
45:
40:
36:
32:
31:
19:
9483:
9475:Demazure, M.
9436:
9430:
9389:
9354:
9348:
9307:
9274:
9270:
9222:
9218:
9203:
9165:
9123:
9120:Milne, J. S.
9089:
9055:
9015:
9007:Demazure, M.
8968:
8961:Demazure, M.
8928:
8920:Demazure, M.
8894:
8857:
8817:
8770:
8764:
8719:
8698:
8689:
8680:
8671:
8662:
8653:
8644:
8635:
8626:
8617:
8608:
8599:
8590:
8581:
8556:
8547:
8538:
8529:
8520:
8511:
8502:
8493:
8484:
8475:
8466:
8457:
8436:
8427:
8418:
8409:
8400:
8391:
8378:
8371:
8362:
8353:
8234:
8232:
8223:
8219:
8215:
8213:
8118:
8114:
8112:
8107:
8103:
8099:
8095:
8091:
8087:
8083:
8081:global field
8076:
8071:
8067:
8063:
8058:
8054:
8050:
8046:
8042:
8037:
8033:
8029:
8021:
8019:
7924:
7920:
7904:
7896:
7892:
7888:
7884:
7879:
7870:
7866:
7862:
7854:
7846:
7842:
7838:
7834:
7830:
7822:
7820:
7815:
7811:
7807:
7803:
7799:
7795:
7791:
7787:
7783:
7779:
7775:
7771:
7767:
7763:
7759:
7755:
7751:
7747:
7743:
7739:
7735:
7731:
7727:
7723:
7719:
7715:
7711:
7707:
7703:
7699:
7697:
7692:
7688:
7684:
7680:
7676:
7668:
7664:
7656:
7648:
7544:
7536:
7532:
7528:
7524:
7518:
7516:
7506:
7502:
7498:
7494:
7490:
7486:
7485:. The group
7389:
7385:
7381:
7377:
7373:
7371:discriminant
7366:
7361:
7355:
7353:is of type D
7350:
7346:
7342:
7338:
7334:
7330:
7326:
7322:
7318:
7316:
7311:
7307:
7303:
7299:
7295:
7291:
7287:
7283:
7279:
7275:
7271:
7263:
7259:
7251:
7247:
7243:
7239:
7235:
7233:
7226:
7222:
7218:
7214:
7183:
7176:
7169:
7165:
7161:
7156:
7152:
7144:
7140:
7138:
7133:
7129:
7115:
7099:
7095:
7087:
7083:
7079:
7075:
7071:
7065:
7061:
7059:
7054:
7050:
7046:
7042:
7038:
7034:
7030:
7026:
7022:
7018:
7014:
7006:
7002:
6996:
6992:
6988:
6984:
6980:
6976:
6972:
6965:
6963:
6949:
6945:
6937:
6933:
6925:
6921:
6917:
6913:
6909:
6905:
6901:
6897:
6893:
6892:-adic field
6889:
6885:
6881:
6877:
6873:
6869:
6867:
6862:
6858:
6854:
6850:
6842:
6838:
6834:
6830:
6826:
6822:
6818:
6814:
6810:
6806:
6803:
6799:
6795:
6791:
6787:
6785:
6777:
6773:
6766:
6762:
6755:
6748:
6741:
6734:
6730:
6723:
6719:
6717:
6709:Jacques Tits
6704:
6700:
6696:
6692:
6688:
6684:
6683:, the group
6680:
6676:
6672:
6668:
6663:
6659:
6655:
6651:
6647:
6643:
6639:
6635:
6631:
6627:
6623:
6619:
6615:
6611:
6607:
6605:
6600:
6565:
6561:
6557:
6553:
6551:
6546:
6542:
6538:
6534:
6530:
6527:presentation
6518:
6514:
6512:
6502:
6498:
6494:
6490:
6486:
6482:
6478:
6474:
6470:
6466:
6462:
6458:
6454:
6450:
6446:
6442:
6438:
6436:
6431:
6427:
6423:
6419:
6415:
6409:
6405:
6401:
6396:
6392:
6384:
6380:
6376:
6372:
6370:
6361:
6357:
6353:
6351:
6345:
6340:
6336:
6332:
6326:
6322:
6318:
6312:
6307:
6303:
6299:
6295:
6288:
6284:
6280:
6276:
6272:
6268:
6266:
6261:
6257:
6253:
6249:
6247:
6241:
6237:
6233:
6229:
6225:
6221:
6217:
6213:
6212:), then the
6209:
6205:
6201:
6194:
6189:
6185:
6181:
6177:
6173:
6169:
6138:
6134:
6098:
6094:
6090:
6086:
6082:
6078:
6074:
6070:
6066:
6060:reduced norm
6055:
6051:
6047:
6043:
6039:
5995:
5991:
5987:
5983:
5979:
5975:
5971:
5967:
5936:
5932:
5896:
5892:
5888:
5884:
5880:
5876:
5872:
5861:
5851:
5843:
5835:
5828:Jens Jantzen
5819:
5811:
5807:
5803:
5799:
5796:George Kempf
5787:
5783:
5779:
5775:
5773:
5760:
5752:
5748:
5744:
5740:
5732:
5728:
5727:∇(λ) as the
5725:Schur module
5724:
5720:
5718:
5713:
5709:
5705:
5701:
5697:
5693:
5689:
5685:
5681:
5677:
5673:
5669:
5665:
5661:
5657:
5653:
5649:
5645:
5641:
5637:
5633:
5629:
5626:Weyl chamber
5621:
5617:
5613:
5605:
5601:
5597:
5595:
5580:
5576:
5572:
5568:
5564:
5560:
5552:
5548:
5544:
5540:
5532:
5528:
5524:
5519:
5514:
5510:
5499:
5495:
5493:
5488:
5484:
5480:
5472:
5468:
5464:
5448:
5441:
5437:
5433:
5429:
5427:
5418:
5414:
5402:
5398:
5390:
5386:
5382:
5374:
5370:
5366:
5362:
5358:
5354:
5350:
5346:
5342:
5338:
5333:
5328:
5324:
5320:
5316:
5309:
5296:
5292:
5288:
5284:
5282:
5280:connected).
5277:
5273:
5269:
5264:
5260:
5256:
5252:
5248:
5244:
5240:
5236:
5228:
5224:
5220:
5216:
5210:
5200:
5196:
5192:
5188:
5184:
5176:
5145:
5141:
5137:
5097:
5093:
5089:
5085:
5081:
5079:group scheme
5076:
5066:
5062:
5058:
5054:
5050:
5048:
4957:splits as a
4954:
4950:
4946:
4942:
4936:
4775:
4771:
4767:
4763:
4758:
4750:
4746:
4742:
4737:
4539:
4532:
4528:
4524:
4515:
4507:
4503:
4499:
4494:
4485:
4483:
4478:
4474:
4470:
4466:
4462:
4458:
4454:
4451:adjoint type
4450:
4446:
4442:
4434:
4430:
4422:
4418:
4414:
4410:
4406:
4402:
4400:
4383:
4368:
4360:
4356:
4344:
4339:
4301:
4295:
4289:
4283:
4279:
4261:
4257:
4254:flag variety
4253:
4249:
4245:
4241:
4237:
4231:
4148:
4144:
4139:
4135:
4128:
4124:
4120:
4116:
4112:
4108:
4100:
4092:
4084:
4080:
4076:
4075:is a smooth
4072:
4068:
4064:
4061:
4059:
3926:
3922:
3918:
3914:
3907:
3903:
3899:
3895:
3891:
3887:
3883:
3879:
3871:
3867:
3863:
3859:
3855:
3851:
3847:
3843:
3839:
3837:
3827:
3823:
3819:
3815:
3811:
3807:
3799:
3792:
3789:
3785:
3778:
3774:
3770:
3766:
3764:
3749:
3744:
3740:
3734:
3730:
3725:
3721:
3717:
3713:
3709:
3705:
3701:
3697:
3693:
3689:
3685:
3677:
3673:
3671:
3666:
3662:
3658:
3653:
3649:
3644:
3640:
3574:
3570:
3566:
3564:
3479:
3475:
3471:
3457:
3451:
3447:
3440:
3436:
3432:
3428:
3424:
3420:
3415:
3411:
3407:
3395:
3389:
3381:
3379:
3374:
3370:
3366:
3361:
3357:
3352:
3348:
3344:
3340:
3336:
3331:
3327:
3320:
3316:
3312:
3270:
3266:
3262:
3258:
3254:
3212:
3208:
3204:
3202:
3100:
3096:
3068:
2992:
2988:
2984:
2980:
2949:
2943:
2938:
2930:
2926:
2922:
2914:
2910:
2905:
2901:
2897:
2893:
2889:
2885:
2881:
2877:
2873:
2869:
2865:
2860:
2856:
2852:
2848:
2844:
2840:
2836:
2834:
2827:
2817:
2813:
2809:
2801:
2797:
2796:showed that
2789:
2785:
2781:
2777:
2773:
2769:
2765:
2761:
2753:
2745:
2737:
2735:
2730:
2726:
2722:
2718:
2714:
2707:
2703:
2695:
2691:
2687:
2683:
2679:
2673:
2563:
2479:
2322:Non-examples
2216:
2213:from the map
2077:
2040:
2036:
2032:
2028:
2024:
2020:
2016:
2012:
2008:
2004:
2000:
1996:
1992:
1988:
1984:
1980:
1976:
1972:
1968:
1964:
1960:
1956:
1945:
1941:
1937:
1933:
1926:
1922:
1915:
1913:vector space
1904:
1900:
1896:
1892:
1888:
1883:
1877:
1873:
1869:
1864:at least 2.
1861:
1857:
1853:
1852:1. In fact,
1845:
1841:
1837:
1830:
1826:
1822:
1818:
1813:
1809:
1805:
1800:
1796:
1793:
1789:
1785:
1781:
1777:
1745:
1743:
1737:
1731:
1717:
1713:
1709:
1705:
1642:
1637:(that is, a
1634:
1630:
1626:
1622:
1618:
1616:
1602:
1598:
1594:
1590:
1586:
1584:
1489:
1482:homomorphism
1477:
1475:
1470:
1464:
1457:
1453:
1449:
1448:, the group
1445:
1441:
1433:
1429:
1425:
1421:
1417:
1413:
1409:
1405:
1403:
1345:
1334:
1330:
1323:
1315:
1311:
1241:
1181:
1153:
1102:
1076:
1066:
1061:is a smooth
1058:
1054:
1050:
1046:
1042:
1030:
1024:
996:
992:
988:
984:
980:
976:
972:
966:
961:
959:finite field
954:
946:
942:
938:
931:number field
926:
924:real numbers
899:
894:
890:
886:
882:
875:
871:
860:
856:
829:
817:
811:
757:
640:
628:
616:
604:
592:
580:
568:
556:
327:
284:
271:
260:
249:
245:Cyclic group
123:
110:Free product
81:Group action
44:Group theory
39:Group theory
38:
8814:Cartier, P.
7861:at most 1,
7673:pointed set
6447:quasi-split
6281:anisotropic
6097:has degree
5737:line bundle
5660:. Define a
5475:(2) is the
4331:Élie Cartan
3674:simple root
3468:conjugation
3382:root system
3319:). Writing
2954:Lie algebra
2872:called the
2864:) for some
2830:root system
2758:finite type
2365:Borel group
1850:determinant
1639:split torus
1246:base change
1015:Definitions
878:), and the
814:mathematics
530:Topological
369:alternating
9507:Lie groups
9496:Categories
9275:-functions
9237:1512.08296
9196:V.L. Popov
8709:References
7927:, the map
7786:)). Also,
7683:is called
7547:such that
7256:inner form
7118:Tits index
7112:Tits index
7009:) that is
6445:is called
6275:if it has
6271:is called
6204:of degree
6172:has index
5976:Witt index
5767:gives the
5680:such that
5559:metric on
5295:) ≅
5213:Lie groups
5088:is called
4537:Witt index
4521:spin group
4413:is called
4097:projective
3804:normalized
3404:normalizer
3398:means the
3391:Weyl group
3386:root datum
3373:from 1 to
2843:, and let
2764:is called
2184:embeds in
2078:The group
1952:, and its
1948:) has two
1876:), and Sp(
1625:is called
1412:is called
1103:semisimple
846:direct sum
637:Symplectic
577:Orthogonal
534:Lie groups
441:Free group
166:continuous
105:Direct sum
9388:(2016) ,
9357:: 49–80,
9210:EMS Press
9198:(2001) ,
9087:(2003) ,
8812:(2005) ,
8803:119837998
8718:(1991) ,
8161:∏
8157:→
7967:∏
7963:→
7659:, or the
7628:¯
7596:¯
7564:¯
7449:
7443:⊂
7403:
7286:), where
7199:¯
6585:¯
6434:is zero.
6366:unipotent
6273:isotropic
6216:-rank of
6154:¯
6118:¯
6020:⌋
6006:⌊
5970:-rank of
5952:¯
5916:¯
5769:character
5567:-rank of
5423:reductive
5385:, and so
5161:¯
5121:¯
5090:reductive
5005:⋉
4993:
4987:≅
4975:
4899:−
4882:⋯
4811:…
4669:−
4652:⋯
4575:…
4506:+1) over
4347:of type G
4234:isotropic
4211:⊂
4191:⊂
4188:⋯
4185:⊂
4165:⊂
4033:∗
4011:∗
4006:∗
3989:∗
3984:∗
3979:∗
3974:∗
3967:∗
3962:∗
3957:∗
3952:∗
3913:for α in
3876:conjugate
3802:which is
3545:α
3525:Φ
3521:∈
3518:α
3514:⨁
3510:⊕
3183:α
3169:Φ
3166:∈
3163:α
3159:⨁
3155:⊕
3067:fixed by
2622:∏
2618:≅
2253:↦
2160:×
2056:¯
1716:-tori in
1689:¯
1658:¯
1561:μ
1534:×
1518:≅
1292:¯
1264:¯
1158:unipotent
1154:reductive
1079:connected
844:and is a
701:Conformal
589:Euclidean
196:nilpotent
9379:55638217
9343:(1965),
9306:(1998),
9265:(1979),
9122:(2017),
8967:(1970).
8893:(1970),
8849:(2014),
8252:See also
7539:with an
7136:-group.
6776:of type
6713:BN-pairs
6197:) for a
5887:). Here
4531:+1 over
4465:, where
4437:) being
3866:. Every
3739:for 1 ≤
3657:for 1 ≤
3365:for all
2935:integers
2917:) under
2011:.) When
1724:Examples
1107:solvable
696:Poincaré
541:Solenoid
413:Integers
403:Lattices
378:sporadic
373:Lie type
201:solvable
191:dihedral
176:additive
161:infinite
71:Subgroup
9461:0164968
9453:1970394
9420:3616493
9371:0180554
9334:1642713
9297:0546587
9242:Bibcode
9189:1278263
9154:3729270
9113:2015057
9078:2605318
9039:2867622
8999:0274459
8952:2867621
8913:0302656
8880:3309122
8840:2124841
8816:(ed.),
8795:0294349
8775:Bibcode
8750:1102012
8376:Milne.
7831:abelian
7345:) over
7172:over a
7067:lattice
6975:. Then
6940:) is a
6833:). For
6505:| ≤ 2.
6485:) over
6389:compact
6141:) over
6073:). The
6062:on the
5832:Lusztig
5712:(λ) of
5648:. Then
5584:) is a
5547:), and
5518:), the
5268:) = Ad(
4749:) over
4374:is the
3720:)) are
3402:of the
2868:, with
2698:) is a
2039:) over
2007:) over
1911:on the
1456:) over
1049:) over
1037:closed
995:) when
957:over a
917:complex
832:over a
824:over a
691:Lorentz
613:Unitary
512:Lattice
452:PSL(2,
186:abelian
97:(Semi-)
9459:
9451:
9418:
9408:
9377:
9369:
9332:
9322:
9295:
9285:
9252:
9187:
9177:
9152:
9142:
9111:
9101:
9076:
9066:
9037:
9027:
8997:
8987:
8950:
8940:
8911:
8901:
8878:
8868:
8838:
8828:
8801:
8793:
8748:
8738:
8032:, and
8026:places
7671:. The
7541:action
7520:torsor
7147:, the
6995:). An
6786:For a
6761:. For
6606:For a
6075:degree
6035:Every
5966:. The
5751:. For
5557:CAT(0)
5479:, and
5377:) has
5102:smooth
5049:where
4523:Spin(2
4519:: the
4378:of an
4363:), by
4089:proper
3712:) (or
3435:) (or
3265:. Let
2886:weight
2804:is of
1872:), SO(
1735:and SL
1438:center
1422:simple
1414:simple
1306:is an
1063:affine
1035:smooth
1007:of an
867:, the
842:kernel
546:Circle
477:SL(2,
366:cyclic
330:-group
181:cyclic
156:finite
151:simple
135:kernel
9449:JSTOR
9375:S2CID
9232:arXiv
9052:(PDF)
8854:(PDF)
8799:S2CID
8383:(PDF)
8346:Notes
8090:over
7923:over
7851:Serre
7806:,Aut(
7766:over
7742:over
7734:,Aut(
7714:over
7700:forms
7647:over
7535:over
7509:− 1.
7250:over
7242:over
7013:with
6983:over
6954:index
6900:over
6747:, or
6667:over
6414:over
6387:) is
6344:over
6311:over
6208:over
6168:. If
6085:as a
6042:over
5867:are:
5822:, by
5792:socle
5672:over
5628:) in
5620:) ≅
5535:is a
5451:is a
5444:by a
5319:over
5227:over
4770:over
4535:with
4473:is a
4403:split
4382:over
4351:and E
4134:,...,
4099:over
4091:over
3894:over
3756:graph
3747:− 1.
3661:<
3326:,...,
2851:; so
2824:Roots
2721:) to
1821:) of
1633:over
1627:split
1593:over
1428:over
1322:over
1320:torus
945:) of
929:or a
863:) of
826:field
730:Sp(∞)
727:SU(∞)
140:image
9406:ISBN
9320:ISBN
9283:ISBN
9250:ISBN
9175:ISBN
9140:ISBN
9099:ISBN
9064:ISBN
9025:ISBN
8985:ISBN
8938:ISBN
8899:ISBN
8866:ISBN
8826:ISBN
8736:ISBN
8257:The
7270:set
7151:Gal(
7064:, a
6964:Let
6352:For
6220:is (
5586:tree
4937:The
4340:The
4329:and
4305:, E
4299:, D
4287:, B
4107:for
3343:) ≅
2981:root
2979:. A
2944:The
2888:for
2874:rank
2835:Let
2808:and
2363:The
2327:Any
2074:Tori
1416:(or
816:, a
724:O(∞)
713:Loop
532:and
9441:doi
9398:doi
9359:doi
9312:doi
9132:doi
8977:doi
8783:doi
8728:doi
8220:PGL
8045:or
8028:of
7780:PGL
7663:if
7655:on
7543:of
7446:Gal
7400:Gal
7384:*/(
7380:in
7376:of
7070:in
7057:).
6912:(1,
6695:if
6603:).
6430:or
6077:of
6054:(1,
5978:of
5818:of
5810:of
5668:of
5575:(2,
5527:of
5483:(2,
5467:(2,
5440:of
5417:(2,
5413:of
5401:(2,
5389:(2,
5373:(2,
5361:(2,
5359:PSL
5353:(2,
5351:PGL
5345:(2,
5343:PGL
5339:PGL
5291:(1,
5199:to
5140:in
5100:is
4990:Out
4972:Aut
4394:, E
4390:, E
4321:, G
4317:, F
4313:, E
4309:, E
4293:, C
4260:of
4256:or
4087:is
3874:is
3870:of
3854:of
3806:by
3784:in
3700:if
3692:of
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247:Z
20:)
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