4887:
2861:
798:
1369:. There is a subtle distinction between weights of group representations and Lie algebra representations, which is that there is a different notion of integrality condition in the two cases; see below. (The integrality condition is more restrictive in the group case, reflecting that not every representation of the Lie algebra comes from a representation of the group.)
5461:
case, the dominant integral elements live in a 60-degree sector. The notion of being dominant is not the same as being higher than zero. Note the grey area in the picture on the right is a 120-degree sector, strictly containing the 60-degree sector corresponding to the dominant integral elements.
1122:
3651:
4089:
1435:
3052:. For computations, it is convenient to choose an inner product that is invariant under the Weyl group, that is, under reflections about the hyperplanes orthogonal to the roots. We may then use this inner product to identify
3389:
684:
1256:
2176:
1766:
851:
3225:
5263:
is expressible as a linear combination of positive roots with non-negative real coefficients. This means, roughly, that "higher" means in the directions of the positive roots. We equivalently say that
1957:
3026:
939:
4704:
5816:
4575:
3946:
3553:
4135:
3325:
3391:
The angle bracket here is not an inner product, as it is not symmetric, and is linear only in the first argument. The angle bracket notation should not be confused with the inner product
514:
375:
5501:
5424:
3086:
2902:
636:/. Thus weights are primarily of interest for abelian Lie algebras, where they reduce to the simple notion of a generalized eigenvalue for space of commuting linear transformations.
2568:
5327:
5235:
5125:
4214:
3977:
3457:
3117:
2988:
2933:
2830:
2793:
2756:
2699:
2662:
2625:
2212:
4318:
4275:
3843:
3779:
2531:
3424:
3260:
2118:
2052:
5844:
5779:
5755:
5715:
5658:
5583:
5016:
4864:
4824:
4780:
4728:
4611:
4534:
4242:
4179:
3894:
3803:
3509:
3481:
3293:
3141:
3050:
2957:
1988:
1790:
1711:
1611:
1541:
1493:
1459:
1367:
1339:
1311:
1186:
1014:
963:
875:
787:
763:
735:
711:
4655:
3565:
5261:
2026:
1567:
1664:
1006:
901:
4969:
4942:
4915:
4507:
4480:
4453:
4426:
4399:
4372:
3870:
3701:
2481:
2415:
153:
5781:-module with a highest weight is necessarily a highest-weight module, but in the infinite-dimensional case, a highest weight module need not be irreducible. For each
5627:
5539:
5374:
5281:
5209:
5165:
5094:
4989:
4155:
2455:
2389:
2092:
1814:
5881:
5633:
3739:
3674:
3165:
2246:
1838:
1517:
1158:
5459:
5047:
4345:
5354:
5301:
5185:
5145:
5074:
3989:
2309:
5607:
5559:
2850:
2719:
2588:
2501:
2435:
2369:
2349:
2329:
2286:
2266:
2072:
1687:
1587:
4994:
We now introduce a partial ordering on the set of weights, which will be used to formulate the theorem of the highest weight describing the representations of
1383:
1278:(a basis of simultaneous eigenvectors) for all the represented elements of the algebra, i.e., to there being simultaneously diagonalizable matrices (see
765:. In this section, we describe the concepts needed to formulate the "theorem of the highest weight" classifying the finite-dimensional representations of
5850:(λ), but this module is infinite dimensional unless λ is dominant integral. It can be shown that each highest weight module with highest weight λ is a
3872:
in any finite-dimensional representation must be an integer. We conclude that, as stated above, the weight of any finite-dimensional representation of
789:. Notably, we will explain the notion of a "dominant integral element." The representations themselves are described in the article linked to above.
3334:
1720:
From the perspective of representation theory, the significance of the roots and root vectors is the following elementary but important result: If
1201:
6220:
6225:
2126:
1723:
808:
3173:
5886:
2864:
Algebraically integral elements (triangular lattice), dominant integral elements (black dots), and fundamental weights for sl(3,C)
1378:
685:
Representation theory of semisimple Lie algebras § Classifying finite-dimensional representations of semisimple Lie algebras
1160:) is a linear functional λ such that the corresponding weight space is nonzero. Nonzero elements of the weight space are called
1846:
2993:
906:
6195:
6169:
6146:
6105:
6069:
573:), then instead of requiring multiplicativity of a character, one requires that it maps any Lie bracket to the corresponding
6031:
together with the general result on complete reducibility of finite-dimensional representations of semisimple Lie algebras
5819:
4660:
6124:
4157:
is then algebraically integral if and only if it is an integral combination of the fundamental weights. The set of all
17:
5784:
4543:
3905:
3521:
4097:
3298:
5920:
5167:
are integral, but this assumption is not necessary to the definition we are about to introduce. We then say that
460:
321:
6041:
5516:
5468:
5391:
1461:, the space over which the representation acts is the Lie algebra itself. Then the nonzero weights are called
6053:
3055:
2871:
2536:
671:
on its Lie algebra by differentiation. (For Lie groups, this is differentiation at the identity element of
5673:(4) every dominant, algebraically integral element is the highest weight of an irreducible representation.
5306:
5214:
4886:
5099:
5050:
4188:
3980:
3951:
3431:
3091:
2962:
2907:
2798:
2761:
2724:
2667:
2630:
2593:
2181:
4283:
4247:
3808:
3744:
2506:
1117:{\displaystyle V_{\lambda }:=\{v\in V:\forall H\in {\mathfrak {h}},\quad (\sigma (H))(v)=\lambda (H)v\}}
249:. This map is also multiplicative, and sends the identity to 1; thus it is an algebra homomorphism from
6045:
5916:
3646:{\displaystyle (\lambda ,H_{\alpha })=2{\frac {(\lambda ,\alpha )}{(\alpha ,\alpha )}}\in \mathbb {Z} }
3459:. The motivation for these definitions is simple: The weights of finite-dimensional representations of
167:
86:
3394:
3233:
2097:
2031:
5825:
5760:
5736:
5696:
5639:
5564:
4997:
4845:
4805:
4761:
4709:
4592:
4515:
4223:
4160:
3875:
3784:
3490:
3462:
3269:
3122:
3031:
2938:
1969:
1771:
1692:
1592:
1522:
1474:
1440:
1348:
1320:
1292:
1167:
944:
856:
768:
744:
716:
692:
4616:
5240:
1993:
1546:
257:
186:
78:
1619:
1164:. That is to say, a weight vector is a simultaneous eigenvector for the action of the elements of
984:
884:
5891:
4947:
4920:
4893:
4867:
4485:
4458:
4431:
4404:
4377:
4350:
3848:
3679:
2460:
2394:
535:
196:
132:
66:
5612:
5524:
5359:
5266:
5194:
5150:
5079:
4974:
4140:
2440:
2374:
2077:
1799:
6215:
4084:{\displaystyle 2{\frac {(\omega _{i},\alpha _{j})}{(\alpha _{j},\alpha _{j})}}=\delta _{i,j}}
3710:
3659:
3150:
2217:
1823:
1502:
1279:
1143:
182:
159:
32:
253:
to the base field. This "generalized eigenvalue" is a prototype for the notion of a weight.
6179:
6079:
5437:
5025:
4323:
397:
276:
155:
90:
54:
40:
5339:
5286:
5170:
5130:
5059:
8:
630:
570:
179:
82:
47:
2291:
5677:
The last point is the most difficult one; the representations may be constructed using
5592:
5544:
2835:
2704:
2573:
2486:
2420:
2354:
2334:
2314:
2271:
2251:
2057:
1672:
1572:
265:
163:
2503:
maps the weight spaces to themselves. In the fundamental representation, with weights
1430:{\displaystyle \mathrm {ad} :{\mathfrak {g}}\to \operatorname {End} ({\mathfrak {g}})}
6191:
6165:
6142:
6135:
6120:
6101:
6083:
6065:
4871:
738:
223:
5670:(3) two irreducible representations with the same highest weight are isomorphic, and
675:, and the algebraic group case is an abstraction using the notion of a derivation.)
6057:
4787:
308:
678:
6175:
6161:
6075:
5851:
4182:
648:
626:
94:
5818:—not necessarily dominant or integral—there exists a unique (up to isomorphism)
5664:(1) every irreducible (finite-dimensional) representation has a highest weight,
6061:
3384:{\displaystyle \langle \lambda ,\alpha \rangle \equiv (\lambda ,H_{\alpha }).}
3295:
In addition to this inner product, it is common for an angle bracket notation
546:
to any simultaneous eigenspace, this corresponds an algebra homomorphism from
6209:
6087:
5730:
3263:
2860:
5667:(2) the highest weight is always a dominant, algebraically integral element,
241:; this functional is defined as the map which associates to each element of
5855:
5678:
261:
235:
189:
1251:{\displaystyle V=\bigoplus _{\lambda \in {\mathfrak {h}}^{*}}V_{\lambda }}
6119:, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,
6117:
Lie Groups, Lie
Algebras, and Representations: An Elementary Introduction
5896:
3328:
1714:
1463:
582:
566:
416:: the eigenvalue on this common eigenspace of each element of the group.
206:
28:
5660:
is by means of a "theorem of the highest weight." The theorem says that
85:. The importance of the concept, however, stems from its application to
4706:. The reason for making this definition is that if a representation of
1275:
574:
440:
110:
106:
1136:(the representation is often referred to in short by the vector space
419:
The notion of multiplicative character can be extended to any algebra
6056:, Readings in Mathematics. Vol. 129. New York: Springer-Verlag.
801:
Example of the weights of a representation of the Lie algebra sl(3,C)
644:
98:
3428:
We now define two different notions of integrality for elements of
2171:{\displaystyle {\mathfrak {g}}={\mathfrak {su}}_{\mathbb {C} }(2)}
1761:{\displaystyle \sigma :{\mathfrak {g}}\to \operatorname {End} (V)}
846:{\displaystyle \sigma :{\mathfrak {g}}\to \operatorname {End} (V)}
6160:, Graduate Texts in Mathematics, vol. 21, Berlin, New York:
3220:{\displaystyle H_{\alpha }=2{\frac {\alpha }{(\alpha ,\alpha )}}}
3983:. That is, the fundamental weights are defined by the condition
3144:
1140:
over which elements of the Lie algebra act rather than the map
679:
Weights in the representation theory of semisimple Lie algebras
5882:
Classifying finite-dimensional representations of Lie algebras
5634:
classifying the finite-dimensional irreducible representations
585:
this simply means that this map must vanish on Lie brackets:
1345:
is then simply a weight of the associated representation of
797:
4881:
412:, if such exists, determines a multiplicative character on
5923:, without needing to assume that they are diagonalizable.
2028:
is either the zero vector or a weight vector with weight
1952:{\displaystyle \sigma (H)(\sigma (X)(v))=(\sigma (X)(v))}
3676:. The motivation for this condition is that the coroot
3021:{\displaystyle \lambda :{\mathfrak {h}}\to \mathbb {C} }
934:{\displaystyle \lambda :{\mathfrak {h}}\to \mathbb {C} }
5434:
integer combination of the fundamental weights. In the
3511:, the weights of finite-dimensional representations of
6137:
Introduction to Lie
Algebras and Representation Theory
6098:
Representations and
Invariants of the Classical Groups
5887:
Representation theory of a connected compact Lie group
5689:
A representation (not necessarily finite dimensional)
5465:
The set of all λ (not necessarily integral) such that
3948:
are defined by the property that they form a basis of
6027:
This follows from (the proof of) Proposition 6.13 in
5828:
5787:
5763:
5739:
5699:
5642:
5615:
5595:
5567:
5547:
5527:
5471:
5440:
5394:
5362:
5342:
5309:
5289:
5269:
5243:
5217:
5197:
5173:
5153:
5133:
5102:
5082:
5062:
5028:
5000:
4977:
4971:, the shaded region is the set of points higher than
4950:
4923:
4896:
4848:
4808:
4764:
4712:
4663:
4619:
4595:
4546:
4518:
4509:. The weight lattice is then the triangular lattice.
4488:
4461:
4434:
4407:
4380:
4353:
4326:
4286:
4250:
4226:
4191:
4163:
4143:
4100:
3992:
3954:
3908:
3878:
3851:
3811:
3787:
3747:
3713:
3682:
3662:
3568:
3524:
3493:
3465:
3434:
3397:
3337:
3301:
3272:
3236:
3176:
3153:
3125:
3094:
3058:
3034:
2996:
2965:
2941:
2910:
2874:
2838:
2801:
2764:
2727:
2707:
2670:
2633:
2596:
2576:
2539:
2509:
2489:
2463:
2443:
2423:
2397:
2377:
2357:
2337:
2317:
2294:
2274:
2254:
2220:
2184:
2129:
2100:
2080:
2060:
2034:
1996:
1972:
1849:
1826:
1802:
1774:
1726:
1695:
1675:
1622:
1595:
1575:
1549:
1525:
1505:
1477:
1471:, and the weight vectors, which are thus elements of
1443:
1386:
1351:
1323:
1295:
1204:
1170:
1146:
1017:
987:
947:
909:
887:
859:
811:
771:
747:
719:
695:
463:
324:
135:
408:, each simultaneous eigenspace for every element of
5915:In fact, given a set of commuting matrices over an
6134:
5838:
5810:
5773:
5749:
5709:
5652:
5621:
5601:
5577:
5553:
5533:
5495:
5453:
5418:
5368:
5348:
5321:
5295:
5275:
5255:
5229:
5203:
5179:
5159:
5139:
5119:
5088:
5068:
5041:
5010:
4983:
4963:
4936:
4909:
4858:
4818:
4774:
4722:
4699:{\displaystyle (\lambda ,t)\in 2\pi i\mathbb {Z} }
4698:
4649:
4605:
4569:
4528:
4501:
4474:
4447:
4420:
4393:
4366:
4339:
4312:
4269:
4236:
4208:
4173:
4149:
4129:
4083:
3971:
3940:
3888:
3864:
3837:
3797:
3773:
3733:
3695:
3668:
3645:
3547:
3503:
3483:satisfy the first integrality condition, while if
3475:
3451:
3418:
3383:
3319:
3287:
3254:
3219:
3159:
3135:
3111:
3080:
3044:
3020:
2982:
2951:
2927:
2896:
2844:
2824:
2787:
2750:
2713:
2693:
2656:
2619:
2582:
2562:
2525:
2495:
2475:
2449:
2429:
2409:
2383:
2363:
2343:
2323:
2311:respectively. The Cartan subalgebra is spanned by
2303:
2280:
2260:
2240:
2206:
2170:
2112:
2086:
2066:
2046:
2020:
1982:
1951:
1832:
1808:
1784:
1760:
1705:
1681:
1658:
1605:
1581:
1561:
1535:
1511:
1487:
1453:
1429:
1361:
1333:
1305:
1250:
1180:
1152:
1116:
1000:
957:
933:
895:
869:
845:
781:
757:
729:
705:
508:
369:
147:
4734:, then the weights of the representation will be
1188:, with the corresponding eigenvalues given by λ.
120:
6207:
5846:-module with highest weight λ, which is denoted
5510:
4280:The figure shows the example of the Lie algebra
5811:{\displaystyle \lambda \in {\mathfrak {h}}^{*}}
5507:associated to the given set of positive roots.
4570:{\displaystyle \lambda \in {\mathfrak {h}}_{0}}
3941:{\displaystyle \omega _{1},\ldots ,\omega _{n}}
3548:{\displaystyle \lambda \in {\mathfrak {h}}_{0}}
256:The notion is closely related to the idea of a
6095:
4130:{\displaystyle \alpha _{1},\ldots \alpha _{n}}
5127:. We are mainly interested in the case where
3979:dual to the set of coroots associated to the
3320:{\displaystyle \langle \cdot ,\cdot \rangle }
1372:
1313:, every finite-dimensional representation of
792:
6040:
3350:
3338:
3314:
3302:
2351:classifies the weight spaces. The action of
1111:
1031:
4455:and should project orthogonally to half of
6155:
6132:
4830:is not simply connected, then the lattice
3515:satisfy the second integrality condition.
509:{\displaystyle \chi (ab)=\chi (a)\chi (b)}
370:{\displaystyle \chi (gh)=\chi (g)\chi (h)}
5729:that is annihilated by the action of all
5430:. Equivalently, λ is dominant if it is a
4692:
4347:root system. There are two simple roots,
4303:
3828:
3764:
3639:
3014:
2153:
1274:this corresponds to there being a common
927:
889:
6096:Goodman, Roe; Wallach, Nolan R. (1998),
5684:
5496:{\displaystyle (\lambda ,\gamma )\geq 0}
5419:{\displaystyle (\lambda ,\gamma )\geq 0}
4885:
4882:Partial ordering on the space of weights
2859:
796:
713:be a complex semisimple Lie algebra and
105:is a generalization of the notion of an
3081:{\displaystyle {\mathfrak {h}}_{0}^{*}}
2897:{\displaystyle {\mathfrak {h}}_{0}^{*}}
1195:is the direct sum of its weight spaces
14:
6208:
5872:highest weight module is irreducible.
5721:if it is generated by a weight vector
5022:is the set of roots; we now fix a set
2563:{\displaystyle V_{\pm {\frac {1}{2}}}}
881:over a field of characteristic 0, say
629:and hence descends to a weight on the
158:over the same field, each of which is
6221:Representation theory of Lie algebras
6185:
6050:Representation theory. A first course
5865:in the definition of a Verma module.
853:be a representation of a Lie algebra
651:, then a multiplicative character θ:
65:, or equivalently, a one-dimensional
6114:
6028:
6015:
6003:
5991:
5979:
5967:
5955:
5943:
5336:ordering; it can easily happen that
5322:{\displaystyle \lambda \preceq \mu }
5230:{\displaystyle \mu \succeq \lambda }
3331:, with the angle bracket defined as
1341:. A weight of the representation of
214:. Each of these common eigenvectors
6226:Representation theory of Lie groups
5861:(λ). This is just a restatement of
5831:
5797:
5766:
5742:
5702:
5645:
5570:
5120:{\displaystyle {\mathfrak {h}}_{0}}
5106:
5003:
4851:
4811:
4767:
4715:
4598:
4556:
4521:
4259:
4229:
4209:{\displaystyle {\mathfrak {h}}_{0}}
4195:
4166:
3972:{\displaystyle {\mathfrak {h}}_{0}}
3958:
3881:
3790:
3534:
3496:
3468:
3452:{\displaystyle {\mathfrak {h}}_{0}}
3438:
3128:
3112:{\displaystyle {\mathfrak {h}}_{0}}
3098:
3062:
3037:
3005:
2990:is the space of linear functionals
2983:{\displaystyle {\mathfrak {h}}^{*}}
2969:
2944:
2928:{\displaystyle {\mathfrak {h}}^{*}}
2914:
2878:
2855:
2825:{\displaystyle V_{-{\frac {1}{2}}}}
2788:{\displaystyle V_{+{\frac {1}{2}}}}
2751:{\displaystyle V_{-{\frac {1}{2}}}}
2694:{\displaystyle V_{+{\frac {1}{2}}}}
2657:{\displaystyle V_{-{\frac {1}{2}}}}
2620:{\displaystyle V_{+{\frac {1}{2}}}}
2207:{\displaystyle {\mathfrak {su}}(2)}
2190:
2187:
2146:
2143:
2132:
1975:
1777:
1735:
1698:
1598:
1528:
1480:
1446:
1419:
1400:
1354:
1326:
1298:
1225:
1173:
1057:
950:
918:
862:
820:
774:
750:
722:
698:
24:
5994:Corollary 13.8 and Corollary 13.20
5379:
4746:-integral weights is a sublattice
4536:is the Lie algebra of a Lie group
4313:{\displaystyle sl(3,\mathbb {C} )}
4270:{\displaystyle P({\mathfrak {g}})}
3838:{\displaystyle sl(2,\mathbb {C} )}
3774:{\displaystyle sl(2,\mathbb {C} )}
2852:maps each weight space to itself.
2526:{\displaystyle \pm {\frac {1}{2}}}
2248:span the algebra and have weights
2094:into the weight space with weight
2074:maps the weight space with weight
1713:. The collection of roots forms a
1499:. Explicitly, a linear functional
1391:
1388:
1046:
245:its eigenvalue on the eigenvector
77:. It is the algebra analogue of a
25:
6237:
6188:Lie Groups Beyond an Introduction
5356:is neither higher nor lower than
4512:Suppose now that the Lie algebra
4137:are the simple roots. An element
234: ) generated by the set of
4730:arises from a representation of
4401:. The first fundamental weight,
3419:{\displaystyle (\cdot ,\cdot ).}
3255:{\displaystyle (\alpha ,\beta )}
3143:. With this identification, the
2113:{\displaystyle \lambda +\alpha }
2047:{\displaystyle \lambda +\alpha }
5921:simultaneously triangularizable
5839:{\displaystyle {\mathfrak {g}}}
5774:{\displaystyle {\mathfrak {g}}}
5750:{\displaystyle {\mathfrak {g}}}
5710:{\displaystyle {\mathfrak {g}}}
5653:{\displaystyle {\mathfrak {g}}}
5578:{\displaystyle {\mathfrak {g}}}
5011:{\displaystyle {\mathfrak {g}}}
4859:{\displaystyle {\mathfrak {g}}}
4819:{\displaystyle {\mathfrak {g}}}
4775:{\displaystyle {\mathfrak {g}}}
4723:{\displaystyle {\mathfrak {g}}}
4606:{\displaystyle {\mathfrak {h}}}
4529:{\displaystyle {\mathfrak {g}}}
4237:{\displaystyle {\mathfrak {g}}}
4174:{\displaystyle {\mathfrak {g}}}
3889:{\displaystyle {\mathfrak {g}}}
3798:{\displaystyle {\mathfrak {g}}}
3504:{\displaystyle {\mathfrak {g}}}
3476:{\displaystyle {\mathfrak {g}}}
3288:{\displaystyle \alpha ,\beta .}
3136:{\displaystyle {\mathfrak {h}}}
3045:{\displaystyle {\mathfrak {h}}}
2952:{\displaystyle {\mathfrak {g}}}
2214:complexified, the root vectors
1983:{\displaystyle {\mathfrak {h}}}
1796:is a weight vector with weight
1785:{\displaystyle {\mathfrak {g}}}
1706:{\displaystyle {\mathfrak {h}}}
1606:{\displaystyle {\mathfrak {g}}}
1536:{\displaystyle {\mathfrak {h}}}
1488:{\displaystyle {\mathfrak {g}}}
1467:, the weight spaces are called
1454:{\displaystyle {\mathfrak {g}}}
1362:{\displaystyle {\mathfrak {g}}}
1334:{\displaystyle {\mathfrak {g}}}
1306:{\displaystyle {\mathfrak {g}}}
1181:{\displaystyle {\mathfrak {h}}}
1065:
958:{\displaystyle {\mathfrak {h}}}
870:{\displaystyle {\mathfrak {g}}}
782:{\displaystyle {\mathfrak {g}}}
758:{\displaystyle {\mathfrak {g}}}
730:{\displaystyle {\mathfrak {h}}}
706:{\displaystyle {\mathfrak {g}}}
87:representations of Lie algebras
6100:, Cambridge University Press,
6021:
6009:
5997:
5985:
5973:
5961:
5949:
5937:
5909:
5517:Theorem on the highest weights
5484:
5472:
5407:
5395:
4676:
4664:
4650:{\displaystyle \exp(t)=1\in G}
4632:
4626:
4307:
4293:
4264:
4254:
4056:
4030:
4025:
3999:
3832:
3818:
3768:
3754:
3629:
3617:
3612:
3600:
3588:
3569:
3410:
3398:
3375:
3356:
3249:
3237:
3211:
3199:
3010:
2457:to the weight space of weight
2437:maps a weight space of weight
2391:to the weight space of weight
2371:maps a weight space of weight
2201:
2195:
2165:
2159:
2015:
2009:
2006:
2000:
1946:
1943:
1937:
1934:
1928:
1922:
1919:
1916:
1910:
1907:
1895:
1892:
1886:
1883:
1877:
1874:
1868:
1862:
1859:
1853:
1755:
1749:
1740:
1650:
1644:
1635:
1623:
1424:
1414:
1405:
1105:
1099:
1090:
1084:
1081:
1078:
1072:
1066:
923:
840:
834:
825:
621:. Any weight on a Lie algebra
503:
497:
491:
485:
476:
467:
364:
358:
352:
346:
337:
328:
121:Motivation and general concept
13:
1:
6156:Humphreys, James E. (1972b),
6133:Humphreys, James E. (1972a),
6054:Graduate Texts in Mathematics
5930:
5511:Theorem of the highest weight
5256:{\displaystyle \mu -\lambda }
3327:to be used in discussions of
2021:{\displaystyle \sigma (X)(v)}
1562:{\displaystyle \alpha \neq 0}
554:assigning to each element of
6190:(2nd ed.), Birkhäuser,
3805:. By elementary results for
3487:is a group with Lie algebra
1659:{\displaystyle =\alpha (H)X}
1317:induces a representation of
1001:{\displaystyle V_{\lambda }}
896:{\displaystyle \mathbb {C} }
174:. Equivalently, for any set
7:
5875:
4964:{\displaystyle \alpha _{3}}
4937:{\displaystyle \alpha _{2}}
4910:{\displaystyle \alpha _{1}}
4742:semisimple, the set of all
4502:{\displaystyle \omega _{2}}
4475:{\displaystyle \gamma _{1}}
4448:{\displaystyle \gamma _{2}}
4421:{\displaystyle \omega _{1}}
4394:{\displaystyle \gamma _{2}}
4367:{\displaystyle \gamma _{1}}
4320:, whose root system is the
3896:is algebraically integral.
3865:{\displaystyle H_{\alpha }}
3703:can be identified with the
3696:{\displaystyle H_{\alpha }}
1820:is a root vector with root
1569:and there exists a nonzero
569:(which is generally not an
166:, it is always possible to
10:
6242:
6186:Knapp, Anthony W. (2002),
5946:Theorem 7.19 and Eq. (7.9)
5917:algebraically closed field
5514:
4890:If the positive roots are
4428:, should be orthogonal to
2935:generated by the roots of
2476:{\displaystyle \lambda -1}
2410:{\displaystyle \lambda +1}
1373:Action of the root vectors
1289:is group with Lie algebra
941:be a linear functional on
793:Weight of a representation
682:
168:simultaneously diagonalize
103:weight of a representation
6062:10.1007/978-1-4612-0979-9
5589:if every other weight of
5384:An integral element λ is
148:{\displaystyle n\times n}
5902:
5622:{\displaystyle \lambda }
5534:{\displaystyle \lambda }
5505:fundamental Weyl chamber
5369:{\displaystyle \lambda }
5276:{\displaystyle \lambda }
5204:{\displaystyle \lambda }
5160:{\displaystyle \lambda }
5089:{\displaystyle \lambda }
4984:{\displaystyle \lambda }
4150:{\displaystyle \lambda }
2904:be the real subspace of
2450:{\displaystyle \lambda }
2384:{\displaystyle \lambda }
2087:{\displaystyle \lambda }
1809:{\displaystyle \lambda }
258:multiplicative character
109:, and the corresponding
79:multiplicative character
6158:Linear Algebraic Groups
6115:Hall, Brian C. (2015),
5892:Highest-weight category
5426:for each positive root
4181:-integral weights is a
3734:{\displaystyle {X,Y,H}}
3669:{\displaystyle \alpha }
3160:{\displaystyle \alpha }
2241:{\displaystyle {H,X,Y}}
1833:{\displaystyle \alpha }
1768:is a representation of
1512:{\displaystyle \alpha }
1153:{\displaystyle \sigma }
170:all of the elements of
162:, and any two of which
5840:
5812:
5775:
5751:
5711:
5654:
5623:
5603:
5579:
5555:
5535:
5497:
5455:
5420:
5370:
5350:
5323:
5297:
5277:
5257:
5231:
5205:
5181:
5161:
5141:
5121:
5090:
5070:
5056:Consider two elements
5043:
5012:
4991:
4985:
4965:
4938:
4911:
4860:
4820:
4776:
4724:
4700:
4651:
4607:
4571:
4530:
4503:
4476:
4449:
4422:
4395:
4368:
4341:
4314:
4271:
4238:
4210:
4175:
4151:
4131:
4085:
3973:
3942:
3890:
3866:
3839:
3799:
3775:
3735:
3707:element in a standard
3697:
3670:
3647:
3557:algebraically integral
3549:
3505:
3477:
3453:
3420:
3385:
3321:
3289:
3256:
3221:
3161:
3137:
3113:
3082:
3046:
3022:
2984:
2953:
2929:
2898:
2865:
2846:
2826:
2789:
2752:
2715:
2695:
2658:
2621:
2584:
2564:
2527:
2497:
2477:
2451:
2431:
2411:
2385:
2365:
2345:
2325:
2305:
2282:
2262:
2242:
2208:
2172:
2114:
2088:
2068:
2054:. Thus, the action of
2048:
2022:
1984:
1953:
1834:
1810:
1786:
1762:
1707:
1683:
1660:
1607:
1583:
1563:
1537:
1513:
1489:
1455:
1431:
1379:adjoint representation
1363:
1335:
1307:
1252:
1182:
1154:
1132:of the representation
1118:
1002:
959:
935:
897:
871:
847:
802:
783:
759:
731:
707:
510:
371:
183:linear transformations
178:of mutually commuting
149:
5863:universality property
5841:
5813:
5776:
5752:
5719:highest-weight module
5712:
5685:Highest-weight module
5655:
5624:
5604:
5580:
5556:
5536:
5498:
5456:
5454:{\displaystyle A_{2}}
5421:
5371:
5351:
5324:
5298:
5278:
5258:
5232:
5206:
5182:
5162:
5142:
5122:
5091:
5071:
5044:
5042:{\displaystyle R^{+}}
5013:
4986:
4966:
4939:
4912:
4889:
4870:is isomorphic to the
4861:
4821:
4777:
4725:
4701:
4652:
4608:
4579:analytically integral
4572:
4531:
4504:
4477:
4450:
4423:
4396:
4369:
4342:
4340:{\displaystyle A_{2}}
4315:
4272:
4239:
4211:
4176:
4152:
4132:
4086:
3974:
3943:
3891:
3867:
3845:, the eigenvalues of
3840:
3800:
3776:
3736:
3698:
3671:
3648:
3550:
3506:
3478:
3454:
3421:
3386:
3322:
3290:
3257:
3222:
3162:
3147:associated to a root
3138:
3114:
3083:
3047:
3023:
2985:
2954:
2930:
2899:
2863:
2847:
2827:
2790:
2753:
2716:
2696:
2659:
2622:
2585:
2565:
2528:
2498:
2478:
2452:
2432:
2412:
2386:
2366:
2346:
2326:
2306:
2283:
2263:
2243:
2209:
2173:
2115:
2089:
2069:
2049:
2023:
1985:
1954:
1835:
1811:
1787:
1763:
1708:
1684:
1661:
1608:
1584:
1564:
1538:
1514:
1490:
1456:
1432:
1364:
1336:
1308:
1280:diagonalizable matrix
1253:
1183:
1155:
1119:
1003:
960:
936:
898:
872:
848:
800:
784:
760:
732:
708:
609:with λ() = 0 for all
511:
372:
150:
101:. In this context, a
33:representation theory
6018:Theorems 9.4 and 9.5
5826:
5785:
5761:
5757:. Every irreducible
5737:
5697:
5640:
5613:
5593:
5565:
5545:
5541:of a representation
5525:
5469:
5438:
5392:
5360:
5349:{\displaystyle \mu }
5340:
5307:
5303:, which we write as
5296:{\displaystyle \mu }
5287:
5267:
5241:
5215:
5211:, which we write as
5195:
5180:{\displaystyle \mu }
5171:
5151:
5140:{\displaystyle \mu }
5131:
5100:
5080:
5069:{\displaystyle \mu }
5060:
5026:
4998:
4975:
4948:
4921:
4894:
4846:
4806:
4762:
4710:
4661:
4617:
4593:
4544:
4516:
4486:
4482:, and similarly for
4459:
4432:
4405:
4378:
4351:
4324:
4284:
4248:
4224:
4189:
4161:
4141:
4098:
3990:
3952:
3906:
3876:
3849:
3809:
3785:
3745:
3711:
3680:
3660:
3566:
3522:
3491:
3463:
3432:
3395:
3335:
3299:
3270:
3234:
3174:
3151:
3123:
3092:
3056:
3032:
3028:, the dual space to
2994:
2963:
2939:
2908:
2872:
2836:
2799:
2762:
2725:
2705:
2668:
2631:
2594:
2574:
2537:
2507:
2487:
2483:, and the action of
2461:
2441:
2421:
2395:
2375:
2355:
2335:
2331:, and the action of
2315:
2292:
2272:
2252:
2218:
2182:
2127:
2098:
2078:
2058:
2032:
1994:
1970:
1847:
1824:
1800:
1772:
1724:
1693:
1673:
1620:
1593:
1573:
1547:
1543:is called a root if
1523:
1503:
1475:
1441:
1384:
1349:
1321:
1293:
1202:
1168:
1144:
1015:
985:
945:
907:
885:
857:
809:
769:
745:
717:
693:
461:
322:
277:multiplicative group
133:
55:algebra homomorphism
4540:. Then we say that
3901:fundamental weights
3077:
2893:
631:abelian Lie algebra
601:is a linear map λ:
571:associative algebra
210:of all elements of
5870:finite-dimensional
5836:
5808:
5771:
5747:
5707:
5650:
5619:
5599:
5575:
5551:
5531:
5493:
5451:
5416:
5366:
5346:
5319:
5293:
5273:
5253:
5227:
5201:
5177:
5157:
5137:
5117:
5086:
5066:
5039:
5008:
4992:
4981:
4961:
4934:
4907:
4856:
4838:) is smaller than
4816:
4772:
4720:
4696:
4647:
4603:
4567:
4526:
4499:
4472:
4445:
4418:
4391:
4364:
4337:
4310:
4267:
4234:
4206:
4171:
4147:
4127:
4081:
3969:
3938:
3886:
3862:
3835:
3795:
3771:
3731:
3693:
3666:
3643:
3545:
3501:
3473:
3449:
3416:
3381:
3317:
3285:
3252:
3217:
3157:
3133:
3109:
3078:
3059:
3042:
3018:
2980:
2949:
2925:
2894:
2875:
2866:
2842:
2822:
2785:
2748:
2711:
2691:
2654:
2617:
2580:
2560:
2533:and weight spaces
2523:
2493:
2473:
2447:
2427:
2417:and the action of
2407:
2381:
2361:
2341:
2321:
2304:{\displaystyle -1}
2301:
2278:
2258:
2238:
2204:
2168:
2110:
2084:
2064:
2044:
2018:
1980:
1949:
1830:
1806:
1782:
1758:
1703:
1679:
1656:
1603:
1579:
1559:
1533:
1509:
1485:
1451:
1427:
1359:
1331:
1303:
1248:
1237:
1178:
1150:
1114:
998:
955:
931:
893:
877:on a vector space
867:
843:
803:
779:
755:
727:
703:
538:on a vector space
506:
400:on a vector space
367:
226:on the subalgebra
187:finite-dimensional
145:
89:and hence also to
18:Fundamental weight
6197:978-0-8176-4259-4
6171:978-0-387-90108-4
6148:978-0-387-90053-7
6107:978-0-521-66348-9
6071:978-0-387-97495-8
5602:{\displaystyle V}
5554:{\displaystyle V}
4872:fundamental group
4060:
3633:
3215:
2845:{\displaystyle H}
2818:
2781:
2744:
2714:{\displaystyle Y}
2687:
2650:
2613:
2583:{\displaystyle X}
2556:
2521:
2496:{\displaystyle H}
2430:{\displaystyle Y}
2364:{\displaystyle X}
2344:{\displaystyle H}
2324:{\displaystyle H}
2281:{\displaystyle 1}
2261:{\displaystyle 0}
2067:{\displaystyle X}
1682:{\displaystyle H}
1582:{\displaystyle X}
1211:
739:Cartan subalgebra
659:induces a weight
593:on a Lie algebra
224:linear functional
16:(Redirected from
6233:
6200:
6182:
6151:
6140:
6129:
6110:
6091:
6032:
6025:
6019:
6013:
6007:
6001:
5995:
5989:
5983:
5982:Proposition 12.5
5977:
5971:
5970:Proposition 8.36
5965:
5959:
5953:
5947:
5941:
5924:
5913:
5845:
5843:
5842:
5837:
5835:
5834:
5817:
5815:
5814:
5809:
5807:
5806:
5801:
5800:
5780:
5778:
5777:
5772:
5770:
5769:
5756:
5754:
5753:
5748:
5746:
5745:
5716:
5714:
5713:
5708:
5706:
5705:
5659:
5657:
5656:
5651:
5649:
5648:
5628:
5626:
5625:
5620:
5608:
5606:
5605:
5600:
5584:
5582:
5581:
5576:
5574:
5573:
5560:
5558:
5557:
5552:
5540:
5538:
5537:
5532:
5503:is known as the
5502:
5500:
5499:
5494:
5460:
5458:
5457:
5452:
5450:
5449:
5425:
5423:
5422:
5417:
5375:
5373:
5372:
5367:
5355:
5353:
5352:
5347:
5328:
5326:
5325:
5320:
5302:
5300:
5299:
5294:
5283:is "lower" than
5282:
5280:
5279:
5274:
5262:
5260:
5259:
5254:
5236:
5234:
5233:
5228:
5210:
5208:
5207:
5202:
5186:
5184:
5183:
5178:
5166:
5164:
5163:
5158:
5146:
5144:
5143:
5138:
5126:
5124:
5123:
5118:
5116:
5115:
5110:
5109:
5095:
5093:
5092:
5087:
5075:
5073:
5072:
5067:
5048:
5046:
5045:
5040:
5038:
5037:
5017:
5015:
5014:
5009:
5007:
5006:
4990:
4988:
4987:
4982:
4970:
4968:
4967:
4962:
4960:
4959:
4943:
4941:
4940:
4935:
4933:
4932:
4916:
4914:
4913:
4908:
4906:
4905:
4865:
4863:
4862:
4857:
4855:
4854:
4825:
4823:
4822:
4817:
4815:
4814:
4788:simply connected
4781:
4779:
4778:
4773:
4771:
4770:
4729:
4727:
4726:
4721:
4719:
4718:
4705:
4703:
4702:
4697:
4695:
4656:
4654:
4653:
4648:
4612:
4610:
4609:
4604:
4602:
4601:
4576:
4574:
4573:
4568:
4566:
4565:
4560:
4559:
4535:
4533:
4532:
4527:
4525:
4524:
4508:
4506:
4505:
4500:
4498:
4497:
4481:
4479:
4478:
4473:
4471:
4470:
4454:
4452:
4451:
4446:
4444:
4443:
4427:
4425:
4424:
4419:
4417:
4416:
4400:
4398:
4397:
4392:
4390:
4389:
4373:
4371:
4370:
4365:
4363:
4362:
4346:
4344:
4343:
4338:
4336:
4335:
4319:
4317:
4316:
4311:
4306:
4276:
4274:
4273:
4268:
4263:
4262:
4243:
4241:
4240:
4235:
4233:
4232:
4215:
4213:
4212:
4207:
4205:
4204:
4199:
4198:
4180:
4178:
4177:
4172:
4170:
4169:
4156:
4154:
4153:
4148:
4136:
4134:
4133:
4128:
4126:
4125:
4110:
4109:
4090:
4088:
4087:
4082:
4080:
4079:
4061:
4059:
4055:
4054:
4042:
4041:
4028:
4024:
4023:
4011:
4010:
3997:
3978:
3976:
3975:
3970:
3968:
3967:
3962:
3961:
3947:
3945:
3944:
3939:
3937:
3936:
3918:
3917:
3895:
3893:
3892:
3887:
3885:
3884:
3871:
3869:
3868:
3863:
3861:
3860:
3844:
3842:
3841:
3836:
3831:
3804:
3802:
3801:
3796:
3794:
3793:
3780:
3778:
3777:
3772:
3767:
3740:
3738:
3737:
3732:
3730:
3702:
3700:
3699:
3694:
3692:
3691:
3675:
3673:
3672:
3667:
3652:
3650:
3649:
3644:
3642:
3634:
3632:
3615:
3598:
3587:
3586:
3554:
3552:
3551:
3546:
3544:
3543:
3538:
3537:
3510:
3508:
3507:
3502:
3500:
3499:
3482:
3480:
3479:
3474:
3472:
3471:
3458:
3456:
3455:
3450:
3448:
3447:
3442:
3441:
3425:
3423:
3422:
3417:
3390:
3388:
3387:
3382:
3374:
3373:
3326:
3324:
3323:
3318:
3294:
3292:
3291:
3286:
3261:
3259:
3258:
3253:
3226:
3224:
3223:
3218:
3216:
3214:
3194:
3186:
3185:
3166:
3164:
3163:
3158:
3142:
3140:
3139:
3134:
3132:
3131:
3118:
3116:
3115:
3110:
3108:
3107:
3102:
3101:
3088:with a subspace
3087:
3085:
3084:
3079:
3076:
3071:
3066:
3065:
3051:
3049:
3048:
3043:
3041:
3040:
3027:
3025:
3024:
3019:
3017:
3009:
3008:
2989:
2987:
2986:
2981:
2979:
2978:
2973:
2972:
2958:
2956:
2955:
2950:
2948:
2947:
2934:
2932:
2931:
2926:
2924:
2923:
2918:
2917:
2903:
2901:
2900:
2895:
2892:
2887:
2882:
2881:
2856:Integral element
2851:
2849:
2848:
2843:
2831:
2829:
2828:
2823:
2821:
2820:
2819:
2811:
2794:
2792:
2791:
2786:
2784:
2783:
2782:
2774:
2757:
2755:
2754:
2749:
2747:
2746:
2745:
2737:
2720:
2718:
2717:
2712:
2700:
2698:
2697:
2692:
2690:
2689:
2688:
2680:
2663:
2661:
2660:
2655:
2653:
2652:
2651:
2643:
2626:
2624:
2623:
2618:
2616:
2615:
2614:
2606:
2589:
2587:
2586:
2581:
2569:
2567:
2566:
2561:
2559:
2558:
2557:
2549:
2532:
2530:
2529:
2524:
2522:
2514:
2502:
2500:
2499:
2494:
2482:
2480:
2479:
2474:
2456:
2454:
2453:
2448:
2436:
2434:
2433:
2428:
2416:
2414:
2413:
2408:
2390:
2388:
2387:
2382:
2370:
2368:
2367:
2362:
2350:
2348:
2347:
2342:
2330:
2328:
2327:
2322:
2310:
2308:
2307:
2302:
2287:
2285:
2284:
2279:
2267:
2265:
2264:
2259:
2247:
2245:
2244:
2239:
2237:
2213:
2211:
2210:
2205:
2194:
2193:
2177:
2175:
2174:
2169:
2158:
2157:
2156:
2150:
2149:
2136:
2135:
2123:For example, if
2119:
2117:
2116:
2111:
2093:
2091:
2090:
2085:
2073:
2071:
2070:
2065:
2053:
2051:
2050:
2045:
2027:
2025:
2024:
2019:
1989:
1987:
1986:
1981:
1979:
1978:
1958:
1956:
1955:
1950:
1839:
1837:
1836:
1831:
1815:
1813:
1812:
1807:
1791:
1789:
1788:
1783:
1781:
1780:
1767:
1765:
1764:
1759:
1739:
1738:
1712:
1710:
1709:
1704:
1702:
1701:
1688:
1686:
1685:
1680:
1665:
1663:
1662:
1657:
1612:
1610:
1609:
1604:
1602:
1601:
1588:
1586:
1585:
1580:
1568:
1566:
1565:
1560:
1542:
1540:
1539:
1534:
1532:
1531:
1518:
1516:
1515:
1510:
1494:
1492:
1491:
1486:
1484:
1483:
1460:
1458:
1457:
1452:
1450:
1449:
1436:
1434:
1433:
1428:
1423:
1422:
1404:
1403:
1394:
1368:
1366:
1365:
1360:
1358:
1357:
1340:
1338:
1337:
1332:
1330:
1329:
1312:
1310:
1309:
1304:
1302:
1301:
1271:
1270:
1257:
1255:
1254:
1249:
1247:
1246:
1236:
1235:
1234:
1229:
1228:
1187:
1185:
1184:
1179:
1177:
1176:
1159:
1157:
1156:
1151:
1123:
1121:
1120:
1115:
1061:
1060:
1027:
1026:
1007:
1005:
1004:
999:
997:
996:
981:is the subspace
971:
970:
964:
962:
961:
956:
954:
953:
940:
938:
937:
932:
930:
922:
921:
902:
900:
899:
894:
892:
876:
874:
873:
868:
866:
865:
852:
850:
849:
844:
824:
823:
788:
786:
785:
780:
778:
777:
764:
762:
761:
756:
754:
753:
736:
734:
733:
728:
726:
725:
712:
710:
709:
704:
702:
701:
625:vanishes on the
558:its eigenvalue.
531:. If an algebra
515:
513:
512:
507:
376:
374:
373:
368:
309:identity element
154:
152:
151:
146:
21:
6241:
6240:
6236:
6235:
6234:
6232:
6231:
6230:
6206:
6205:
6204:
6198:
6172:
6162:Springer-Verlag
6149:
6127:
6108:
6072:
6042:Fulton, William
6036:
6035:
6026:
6022:
6014:
6010:
6006:Definition 8.39
6002:
5998:
5990:
5986:
5978:
5974:
5966:
5962:
5958:Proposition 9.2
5954:
5950:
5942:
5938:
5933:
5928:
5927:
5914:
5910:
5905:
5878:
5830:
5829:
5827:
5824:
5823:
5822:highest-weight
5802:
5796:
5795:
5794:
5786:
5783:
5782:
5765:
5764:
5762:
5759:
5758:
5741:
5740:
5738:
5735:
5734:
5701:
5700:
5698:
5695:
5694:
5687:
5644:
5643:
5641:
5638:
5637:
5614:
5611:
5610:
5594:
5591:
5590:
5569:
5568:
5566:
5563:
5562:
5546:
5543:
5542:
5526:
5523:
5522:
5519:
5513:
5470:
5467:
5466:
5445:
5441:
5439:
5436:
5435:
5393:
5390:
5389:
5382:
5380:Dominant weight
5361:
5358:
5357:
5341:
5338:
5337:
5332:This is only a
5308:
5305:
5304:
5288:
5285:
5284:
5268:
5265:
5264:
5242:
5239:
5238:
5216:
5213:
5212:
5196:
5193:
5192:
5172:
5169:
5168:
5152:
5149:
5148:
5132:
5129:
5128:
5111:
5105:
5104:
5103:
5101:
5098:
5097:
5081:
5078:
5077:
5061:
5058:
5057:
5033:
5029:
5027:
5024:
5023:
5002:
5001:
4999:
4996:
4995:
4976:
4973:
4972:
4955:
4951:
4949:
4946:
4945:
4928:
4924:
4922:
4919:
4918:
4901:
4897:
4895:
4892:
4891:
4884:
4850:
4849:
4847:
4844:
4843:
4810:
4809:
4807:
4804:
4803:
4766:
4765:
4763:
4760:
4759:
4738:-integral. For
4714:
4713:
4711:
4708:
4707:
4691:
4662:
4659:
4658:
4618:
4615:
4614:
4597:
4596:
4594:
4591:
4590:
4561:
4555:
4554:
4553:
4545:
4542:
4541:
4520:
4519:
4517:
4514:
4513:
4493:
4489:
4487:
4484:
4483:
4466:
4462:
4460:
4457:
4456:
4439:
4435:
4433:
4430:
4429:
4412:
4408:
4406:
4403:
4402:
4385:
4381:
4379:
4376:
4375:
4358:
4354:
4352:
4349:
4348:
4331:
4327:
4325:
4322:
4321:
4302:
4285:
4282:
4281:
4258:
4257:
4249:
4246:
4245:
4228:
4227:
4225:
4222:
4221:
4200:
4194:
4193:
4192:
4190:
4187:
4186:
4165:
4164:
4162:
4159:
4158:
4142:
4139:
4138:
4121:
4117:
4105:
4101:
4099:
4096:
4095:
4069:
4065:
4050:
4046:
4037:
4033:
4029:
4019:
4015:
4006:
4002:
3998:
3996:
3991:
3988:
3987:
3963:
3957:
3956:
3955:
3953:
3950:
3949:
3932:
3928:
3913:
3909:
3907:
3904:
3903:
3880:
3879:
3877:
3874:
3873:
3856:
3852:
3850:
3847:
3846:
3827:
3810:
3807:
3806:
3789:
3788:
3786:
3783:
3782:
3781:-subalgebra of
3763:
3746:
3743:
3742:
3714:
3712:
3709:
3708:
3687:
3683:
3681:
3678:
3677:
3661:
3658:
3657:
3638:
3616:
3599:
3597:
3582:
3578:
3567:
3564:
3563:
3539:
3533:
3532:
3531:
3523:
3520:
3519:
3495:
3494:
3492:
3489:
3488:
3467:
3466:
3464:
3461:
3460:
3443:
3437:
3436:
3435:
3433:
3430:
3429:
3396:
3393:
3392:
3369:
3365:
3336:
3333:
3332:
3300:
3297:
3296:
3271:
3268:
3267:
3235:
3232:
3231:
3198:
3193:
3181:
3177:
3175:
3172:
3171:
3152:
3149:
3148:
3127:
3126:
3124:
3121:
3120:
3103:
3097:
3096:
3095:
3093:
3090:
3089:
3072:
3067:
3061:
3060:
3057:
3054:
3053:
3036:
3035:
3033:
3030:
3029:
3013:
3004:
3003:
2995:
2992:
2991:
2974:
2968:
2967:
2966:
2964:
2961:
2960:
2943:
2942:
2940:
2937:
2936:
2919:
2913:
2912:
2911:
2909:
2906:
2905:
2888:
2883:
2877:
2876:
2873:
2870:
2869:
2858:
2837:
2834:
2833:
2810:
2806:
2802:
2800:
2797:
2796:
2773:
2769:
2765:
2763:
2760:
2759:
2736:
2732:
2728:
2726:
2723:
2722:
2706:
2703:
2702:
2679:
2675:
2671:
2669:
2666:
2665:
2642:
2638:
2634:
2632:
2629:
2628:
2605:
2601:
2597:
2595:
2592:
2591:
2575:
2572:
2571:
2548:
2544:
2540:
2538:
2535:
2534:
2513:
2508:
2505:
2504:
2488:
2485:
2484:
2462:
2459:
2458:
2442:
2439:
2438:
2422:
2419:
2418:
2396:
2393:
2392:
2376:
2373:
2372:
2356:
2353:
2352:
2336:
2333:
2332:
2316:
2313:
2312:
2293:
2290:
2289:
2273:
2270:
2269:
2253:
2250:
2249:
2221:
2219:
2216:
2215:
2186:
2185:
2183:
2180:
2179:
2152:
2151:
2142:
2141:
2140:
2131:
2130:
2128:
2125:
2124:
2099:
2096:
2095:
2079:
2076:
2075:
2059:
2056:
2055:
2033:
2030:
2029:
1995:
1992:
1991:
1974:
1973:
1971:
1968:
1967:
1848:
1845:
1844:
1825:
1822:
1821:
1801:
1798:
1797:
1776:
1775:
1773:
1770:
1769:
1734:
1733:
1725:
1722:
1721:
1697:
1696:
1694:
1691:
1690:
1674:
1671:
1670:
1621:
1618:
1617:
1597:
1596:
1594:
1591:
1590:
1574:
1571:
1570:
1548:
1545:
1544:
1527:
1526:
1524:
1521:
1520:
1504:
1501:
1500:
1479:
1478:
1476:
1473:
1472:
1445:
1444:
1442:
1439:
1438:
1418:
1417:
1399:
1398:
1387:
1385:
1382:
1381:
1375:
1353:
1352:
1350:
1347:
1346:
1325:
1324:
1322:
1319:
1318:
1297:
1296:
1294:
1291:
1290:
1268:
1267:
1242:
1238:
1230:
1224:
1223:
1222:
1215:
1203:
1200:
1199:
1172:
1171:
1169:
1166:
1165:
1145:
1142:
1141:
1056:
1055:
1022:
1018:
1016:
1013:
1012:
992:
988:
986:
983:
982:
968:
967:
949:
948:
946:
943:
942:
926:
917:
916:
908:
905:
904:
888:
886:
883:
882:
861:
860:
858:
855:
854:
819:
818:
810:
807:
806:
795:
773:
772:
770:
767:
766:
749:
748:
746:
743:
742:
721:
720:
718:
715:
714:
697:
696:
694:
691:
690:
687:
681:
649:algebraic group
627:derived algebra
462:
459:
458:
427:, by replacing
323:
320:
319:
195:there exists a
134:
131:
130:
123:
91:representations
23:
22:
15:
12:
11:
5:
6239:
6229:
6228:
6223:
6218:
6203:
6202:
6196:
6183:
6170:
6153:
6147:
6141:, Birkhäuser,
6130:
6126:978-3319134666
6125:
6112:
6106:
6093:
6070:
6037:
6034:
6033:
6020:
6008:
5996:
5984:
5972:
5960:
5948:
5935:
5934:
5932:
5929:
5926:
5925:
5907:
5906:
5904:
5901:
5900:
5899:
5894:
5889:
5884:
5877:
5874:
5833:
5805:
5799:
5793:
5790:
5768:
5744:
5704:
5686:
5683:
5675:
5674:
5671:
5668:
5665:
5647:
5618:
5609:is lower than
5598:
5587:highest weight
5572:
5550:
5530:
5515:Main article:
5512:
5509:
5492:
5489:
5486:
5483:
5480:
5477:
5474:
5448:
5444:
5415:
5412:
5409:
5406:
5403:
5400:
5397:
5381:
5378:
5365:
5345:
5318:
5315:
5312:
5292:
5272:
5252:
5249:
5246:
5226:
5223:
5220:
5200:
5176:
5156:
5136:
5114:
5108:
5085:
5065:
5051:positive roots
5036:
5032:
5018:. Recall that
5005:
4980:
4958:
4954:
4931:
4927:
4904:
4900:
4883:
4880:
4853:
4813:
4769:
4717:
4694:
4690:
4687:
4684:
4681:
4678:
4675:
4672:
4669:
4666:
4646:
4643:
4640:
4637:
4634:
4631:
4628:
4625:
4622:
4600:
4585:) if for each
4564:
4558:
4552:
4549:
4523:
4496:
4492:
4469:
4465:
4442:
4438:
4415:
4411:
4388:
4384:
4361:
4357:
4334:
4330:
4309:
4305:
4301:
4298:
4295:
4292:
4289:
4266:
4261:
4256:
4253:
4231:
4218:weight lattice
4203:
4197:
4168:
4146:
4124:
4120:
4116:
4113:
4108:
4104:
4092:
4091:
4078:
4075:
4072:
4068:
4064:
4058:
4053:
4049:
4045:
4040:
4036:
4032:
4027:
4022:
4018:
4014:
4009:
4005:
4001:
3995:
3966:
3960:
3935:
3931:
3927:
3924:
3921:
3916:
3912:
3883:
3859:
3855:
3834:
3830:
3826:
3823:
3820:
3817:
3814:
3792:
3770:
3766:
3762:
3759:
3756:
3753:
3750:
3729:
3726:
3723:
3720:
3717:
3690:
3686:
3665:
3656:for all roots
3654:
3653:
3641:
3637:
3631:
3628:
3625:
3622:
3619:
3614:
3611:
3608:
3605:
3602:
3596:
3593:
3590:
3585:
3581:
3577:
3574:
3571:
3542:
3536:
3530:
3527:
3498:
3470:
3446:
3440:
3415:
3412:
3409:
3406:
3403:
3400:
3380:
3377:
3372:
3368:
3364:
3361:
3358:
3355:
3352:
3349:
3346:
3343:
3340:
3316:
3313:
3310:
3307:
3304:
3284:
3281:
3278:
3275:
3251:
3248:
3245:
3242:
3239:
3228:
3227:
3213:
3210:
3207:
3204:
3201:
3197:
3192:
3189:
3184:
3180:
3156:
3130:
3106:
3100:
3075:
3070:
3064:
3039:
3016:
3012:
3007:
3002:
2999:
2977:
2971:
2946:
2922:
2916:
2891:
2886:
2880:
2857:
2854:
2841:
2817:
2814:
2809:
2805:
2780:
2777:
2772:
2768:
2743:
2740:
2735:
2731:
2710:
2686:
2683:
2678:
2674:
2649:
2646:
2641:
2637:
2612:
2609:
2604:
2600:
2579:
2555:
2552:
2547:
2543:
2520:
2517:
2512:
2492:
2472:
2469:
2466:
2446:
2426:
2406:
2403:
2400:
2380:
2360:
2340:
2320:
2300:
2297:
2277:
2257:
2236:
2233:
2230:
2227:
2224:
2203:
2200:
2197:
2192:
2189:
2167:
2164:
2161:
2155:
2148:
2145:
2139:
2134:
2109:
2106:
2103:
2083:
2063:
2043:
2040:
2037:
2017:
2014:
2011:
2008:
2005:
2002:
1999:
1977:
1960:
1959:
1948:
1945:
1942:
1939:
1936:
1933:
1930:
1927:
1924:
1921:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1873:
1870:
1867:
1864:
1861:
1858:
1855:
1852:
1829:
1805:
1779:
1757:
1754:
1751:
1748:
1745:
1742:
1737:
1732:
1729:
1700:
1678:
1667:
1666:
1655:
1652:
1649:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1600:
1578:
1558:
1555:
1552:
1530:
1508:
1482:
1448:
1426:
1421:
1416:
1413:
1410:
1407:
1402:
1397:
1393:
1390:
1374:
1371:
1356:
1328:
1300:
1259:
1258:
1245:
1241:
1233:
1227:
1221:
1218:
1214:
1210:
1207:
1175:
1162:weight vectors
1149:
1126:
1125:
1113:
1110:
1107:
1104:
1101:
1098:
1095:
1092:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1064:
1059:
1054:
1051:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
1025:
1021:
995:
991:
952:
929:
925:
920:
915:
912:
891:
864:
842:
839:
836:
833:
830:
827:
822:
817:
814:
794:
791:
776:
752:
724:
700:
680:
677:
517:
516:
505:
502:
499:
496:
493:
490:
487:
484:
481:
478:
475:
472:
469:
466:
391:
390:
366:
363:
360:
357:
354:
351:
348:
345:
342:
339:
336:
333:
330:
327:
203:consisting of
160:diagonalizable
144:
141:
138:
122:
119:
67:representation
9:
6:
4:
3:
2:
6238:
6227:
6224:
6222:
6219:
6217:
6214:
6213:
6211:
6199:
6193:
6189:
6184:
6181:
6177:
6173:
6167:
6163:
6159:
6154:
6150:
6144:
6139:
6138:
6131:
6128:
6122:
6118:
6113:
6109:
6103:
6099:
6094:
6089:
6085:
6081:
6077:
6073:
6067:
6063:
6059:
6055:
6051:
6047:
6043:
6039:
6038:
6030:
6024:
6017:
6012:
6005:
6000:
5993:
5988:
5981:
5976:
5969:
5964:
5957:
5952:
5945:
5940:
5936:
5922:
5918:
5912:
5908:
5898:
5895:
5893:
5890:
5888:
5885:
5883:
5880:
5879:
5873:
5871:
5866:
5864:
5860:
5857:
5853:
5849:
5821:
5803:
5791:
5788:
5732:
5731:positive root
5728:
5724:
5720:
5692:
5682:
5680:
5679:Verma modules
5672:
5669:
5666:
5663:
5662:
5661:
5635:
5630:
5616:
5596:
5588:
5548:
5528:
5518:
5508:
5506:
5490:
5487:
5481:
5478:
5475:
5463:
5446:
5442:
5433:
5429:
5413:
5410:
5404:
5401:
5398:
5387:
5377:
5363:
5343:
5335:
5330:
5316:
5313:
5310:
5290:
5270:
5250:
5247:
5244:
5224:
5221:
5218:
5198:
5190:
5174:
5154:
5134:
5112:
5083:
5063:
5054:
5052:
5034:
5030:
5021:
4978:
4956:
4952:
4929:
4925:
4902:
4898:
4888:
4879:
4877:
4873:
4869:
4841:
4837:
4833:
4829:
4801:
4797:
4793:
4789:
4785:
4757:
4753:
4749:
4745:
4741:
4737:
4733:
4688:
4685:
4682:
4679:
4673:
4670:
4667:
4644:
4641:
4638:
4635:
4629:
4623:
4620:
4588:
4584:
4580:
4562:
4550:
4547:
4539:
4510:
4494:
4490:
4467:
4463:
4440:
4436:
4413:
4409:
4386:
4382:
4359:
4355:
4332:
4328:
4299:
4296:
4290:
4287:
4278:
4251:
4244:, denoted by
4219:
4201:
4184:
4144:
4122:
4118:
4114:
4111:
4106:
4102:
4076:
4073:
4070:
4066:
4062:
4051:
4047:
4043:
4038:
4034:
4020:
4016:
4012:
4007:
4003:
3993:
3986:
3985:
3984:
3982:
3964:
3933:
3929:
3925:
3922:
3919:
3914:
3910:
3902:
3897:
3857:
3853:
3824:
3821:
3815:
3812:
3760:
3757:
3751:
3748:
3741:basis for an
3727:
3724:
3721:
3718:
3715:
3706:
3688:
3684:
3663:
3635:
3626:
3623:
3620:
3609:
3606:
3603:
3594:
3591:
3583:
3579:
3575:
3572:
3562:
3561:
3560:
3558:
3540:
3528:
3525:
3516:
3514:
3486:
3444:
3426:
3413:
3407:
3404:
3401:
3378:
3370:
3366:
3362:
3359:
3353:
3347:
3344:
3341:
3330:
3311:
3308:
3305:
3282:
3279:
3276:
3273:
3265:
3264:inner product
3246:
3243:
3240:
3208:
3205:
3202:
3195:
3190:
3187:
3182:
3178:
3170:
3169:
3168:
3167:is given as
3154:
3146:
3104:
3073:
3068:
3000:
2997:
2975:
2920:
2889:
2884:
2862:
2853:
2839:
2815:
2812:
2807:
2803:
2778:
2775:
2770:
2766:
2741:
2738:
2733:
2729:
2708:
2684:
2681:
2676:
2672:
2647:
2644:
2639:
2635:
2610:
2607:
2602:
2598:
2577:
2553:
2550:
2545:
2541:
2518:
2515:
2510:
2490:
2470:
2467:
2464:
2444:
2424:
2404:
2401:
2398:
2378:
2358:
2338:
2318:
2298:
2295:
2275:
2255:
2234:
2231:
2228:
2225:
2222:
2198:
2162:
2137:
2121:
2107:
2104:
2101:
2081:
2061:
2041:
2038:
2035:
2012:
2003:
1997:
1965:
1940:
1931:
1925:
1913:
1904:
1901:
1898:
1889:
1880:
1871:
1865:
1856:
1850:
1843:
1842:
1841:
1827:
1819:
1803:
1795:
1752:
1746:
1743:
1730:
1727:
1718:
1716:
1676:
1653:
1647:
1641:
1638:
1632:
1629:
1626:
1616:
1615:
1614:
1576:
1556:
1553:
1550:
1506:
1498:
1495:, are called
1470:
1466:
1465:
1411:
1408:
1395:
1380:
1370:
1344:
1316:
1288:
1283:
1281:
1277:
1273:
1269:weight module
1264:
1243:
1239:
1231:
1219:
1216:
1212:
1208:
1205:
1198:
1197:
1196:
1194:
1189:
1163:
1147:
1139:
1135:
1131:
1108:
1102:
1096:
1093:
1087:
1075:
1069:
1062:
1052:
1049:
1043:
1040:
1037:
1034:
1028:
1023:
1019:
1011:
1010:
1009:
993:
989:
980:
976:
972:
913:
910:
880:
837:
831:
828:
815:
812:
799:
790:
740:
686:
676:
674:
670:
666:
662:
658:
654:
650:
646:
642:
637:
635:
632:
628:
624:
620:
616:
612:
608:
604:
600:
597:over a field
596:
592:
588:
584:
580:
576:
572:
568:
564:
559:
557:
553:
549:
545:
541:
537:
534:
530:
526:
522:
500:
494:
488:
482:
479:
473:
470:
464:
457:
456:
455:
453:
449:
445:
442:
438:
434:
430:
426:
422:
417:
415:
411:
407:
403:
399:
396:
388:
384:
380:
361:
355:
349:
343:
340:
334:
331:
325:
318:
317:
316:
314:
310:
306:
303:) = 1 (where
302:
298:
294:
290:
286:
282:
278:
274:
271:from a group
270:
267:
264:, which is a
263:
259:
254:
252:
248:
244:
240:
237:
236:endomorphisms
233:
229:
225:
221:
217:
213:
209:
208:
205:simultaneous
202:
198:
194:
191:
188:
184:
181:
177:
173:
169:
165:
161:
157:
142:
139:
136:
128:
118:
116:
112:
108:
104:
100:
96:
92:
88:
84:
80:
76:
72:
68:
64:
60:
56:
52:
49:
45:
42:
38:
34:
30:
19:
6216:Lie algebras
6187:
6157:
6136:
6116:
6097:
6049:
6023:
6011:
5999:
5987:
5975:
5963:
5951:
5939:
5911:
5869:
5867:
5862:
5858:
5856:Verma module
5847:
5726:
5722:
5718:
5690:
5688:
5676:
5631:
5586:
5585:is called a
5520:
5504:
5464:
5432:non-negative
5431:
5427:
5385:
5383:
5333:
5331:
5188:
5055:
5019:
4993:
4875:
4866:) and their
4839:
4835:
4831:
4827:
4799:
4795:
4791:
4783:
4755:
4751:
4747:
4743:
4739:
4735:
4731:
4586:
4582:
4578:
4537:
4511:
4279:
4217:
4093:
3981:simple roots
3900:
3898:
3704:
3655:
3556:
3517:
3512:
3484:
3427:
3329:root systems
3262:denotes the
3229:
2867:
2758:to zero and
2627:to zero and
2122:
1963:
1961:
1817:
1793:
1719:
1668:
1497:root vectors
1496:
1468:
1462:
1376:
1342:
1314:
1286:
1284:
1266:
1265:is called a
1262:
1260:
1192:
1190:
1161:
1137:
1133:
1129:
1127:
978:
977:with weight
974:
969:weight space
966:
878:
804:
688:
672:
668:
664:
660:
656:
652:
640:
638:
633:
622:
618:
614:
610:
606:
602:
598:
594:
590:
586:
578:
577:; but since
562:
560:
555:
551:
547:
543:
539:
532:
528:
524:
520:
518:
451:
447:
443:
436:
432:
428:
424:
420:
418:
413:
409:
405:
401:
394:
392:
386:
382:
378:
312:
304:
300:
296:
292:
288:
284:
280:
272:
268:
266:homomorphism
262:group theory
255:
250:
246:
242:
238:
231:
227:
219:
215:
211:
207:eigenvectors
204:
200:
192:
190:vector space
175:
171:
126:
125:Given a set
124:
115:weight space
114:
113:is called a
102:
74:
70:
62:
58:
50:
43:
36:
29:mathematical
26:
6046:Harris, Joe
5919:, they are
5897:Root system
5632:The theory
4216:called the
3518:An element
3266:of vectors
1990:. That is,
1715:root system
1469:root spaces
965:. Then the
583:commutative
567:Lie algebra
393:Indeed, if
279:of a field
6210:Categories
5931:References
5733:spaces in
5717:is called
4613:such that
4583:G-integral
1613:such that
1276:eigenbasis
903:, and let
683:See also:
589:() = 0. A
575:commutator
441:linear map
295:satisfies
222:defines a
180:semisimple
111:eigenspace
107:eigenvalue
99:Lie groups
6088:246650103
6029:Hall 2015
6016:Hall 2015
6004:Hall 2015
5992:Hall 2015
5980:Hall 2015
5968:Hall 2015
5956:Hall 2015
5944:Hall 2015
5804:∗
5792:∈
5789:λ
5617:λ
5529:λ
5521:A weight
5488:≥
5482:γ
5476:λ
5411:≥
5405:γ
5399:λ
5364:λ
5344:μ
5317:μ
5314:⪯
5311:λ
5291:μ
5271:λ
5251:λ
5248:−
5245:μ
5225:λ
5222:⪰
5219:μ
5199:λ
5175:μ
5155:λ
5135:μ
5084:λ
5064:μ
4979:λ
4953:α
4926:α
4899:α
4686:π
4680:∈
4668:λ
4642:∈
4624:
4551:∈
4548:λ
4491:ω
4464:γ
4437:γ
4410:ω
4383:γ
4356:γ
4145:λ
4119:α
4115:…
4103:α
4067:δ
4048:α
4035:α
4017:α
4004:ω
3930:ω
3923:…
3911:ω
3858:α
3689:α
3664:α
3636:∈
3627:α
3621:α
3610:α
3604:λ
3584:α
3573:λ
3529:∈
3526:λ
3408:⋅
3402:⋅
3371:α
3360:λ
3354:≡
3351:⟩
3348:α
3342:λ
3339:⟨
3315:⟩
3312:⋅
3306:⋅
3303:⟨
3280:β
3274:α
3247:β
3241:α
3209:α
3203:α
3196:α
3183:α
3155:α
3074:∗
3011:→
2998:λ
2976:∗
2921:∗
2890:∗
2808:−
2734:−
2640:−
2546:±
2511:±
2468:−
2465:λ
2445:λ
2399:λ
2379:λ
2296:−
2108:α
2102:λ
2082:λ
2042:α
2036:λ
1998:σ
1926:σ
1905:α
1899:λ
1866:σ
1851:σ
1828:α
1804:λ
1747:
1741:→
1728:σ
1642:α
1554:≠
1551:α
1507:α
1412:
1406:→
1244:λ
1232:∗
1220:∈
1217:λ
1213:⨁
1148:σ
1097:λ
1070:σ
1053:∈
1047:∀
1038:∈
1024:λ
1008:given by
994:λ
924:→
911:λ
832:
826:→
813:σ
645:Lie group
495:χ
483:χ
465:χ
356:χ
344:χ
326:χ
140:×
95:algebraic
31:field of
6048:(1991).
5876:See also
5852:quotient
5386:dominant
4868:quotient
4657:we have
2959:, where
2701:, while
1962:for all
1669:for all
1377:For the
519:for all
377:for all
156:matrices
6180:0396773
6080:1153249
5854:of the
5334:partial
4790:, then
4183:lattice
4094:where
1840:, then
307:is the
283:. Thus
275:to the
230:of End(
164:commute
46:over a
41:algebra
27:In the
6194:
6178:
6168:
6145:
6123:
6104:
6086:
6078:
6068:
5868:Every
5820:simple
5189:higher
4944:, and
4826:). If
4782:). If
3230:where
3145:coroot
2832:, and
2288:, and
1130:weight
663:= dθ:
647:or an
591:weight
454:with:
315:) and
53:is an
39:of an
37:weight
5903:Notes
5237:, if
5191:than
2721:maps
2590:maps
2178:, or
1464:roots
1261:then
643:is a
565:is a
542:over
439:by a
423:over
404:over
197:basis
185:of a
83:group
81:of a
73:over
57:from
48:field
6192:ISBN
6166:ISBN
6143:ISBN
6121:ISBN
6102:ISBN
6084:OCLC
6066:ISBN
5147:and
5096:of
5076:and
4798:) =
4754:) ⊂
4374:and
4220:for
3899:The
2868:Let
1816:and
1437:of
805:Let
689:Let
536:acts
398:acts
97:and
35:, a
6058:doi
5693:of
5636:of
5561:of
5388:if
5187:is
5049:of
4874:of
4786:is
4621:exp
4589:in
4577:is
4185:in
3559:if
3555:is
3119:of
2795:to
2664:to
1966:in
1744:End
1689:in
1589:in
1519:on
1409:End
1285:If
1282:).
1191:If
973:of
829:End
741:of
639:If
617:in
581:is
561:If
550:to
527:in
385:in
311:of
260:in
199:of
129:of
93:of
69:of
61:to
6212::
6176:MR
6174:,
6164:,
6082:.
6076:MR
6074:.
6064:.
6052:.
6044:;
5725:∈
5681:.
5629:.
5376:.
5329:.
5053:.
4917:,
4878:.
4277:.
2570:,
2268:,
2120:.
1792:,
1717:.
1128:A
1029::=
737:a
667:→
655:→
613:,
605:→
523:,
450:→
446::
435:→
431::
381:,
291:→
287::
218:∈
117:.
6201:.
6152:.
6111:.
6092:.
6090:.
6060::
5859:M
5848:L
5832:g
5798:h
5767:g
5743:g
5727:V
5723:v
5703:g
5691:V
5646:g
5597:V
5571:g
5549:V
5491:0
5485:)
5479:,
5473:(
5447:2
5443:A
5428:γ
5414:0
5408:)
5402:,
5396:(
5113:0
5107:h
5035:+
5031:R
5020:R
5004:g
4957:3
4930:2
4903:1
4876:G
4852:g
4842:(
4840:P
4836:G
4834:(
4832:P
4828:G
4812:g
4802:(
4800:P
4796:G
4794:(
4792:P
4784:G
4768:g
4758:(
4756:P
4752:G
4750:(
4748:P
4744:G
4740:G
4736:G
4732:G
4716:g
4693:Z
4689:i
4683:2
4677:)
4674:t
4671:,
4665:(
4645:G
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