4057:
4475:
3618:
is Weyl-invariant. The contours of the degree 3 Weyl-invariant polynomial (for a particular choice of standard representation where one of the reflections is across the x-axis) are shown below. These two polynomials generate the polynomial algebra, and are the fundamental invariants for
1625:
1756:
2725:
for a more general statement). Therefore, the center of the universal enveloping algebra of a simple Lie algebra is isomorphic to a polynomial algebra. The degrees of the generators of the algebra are the degrees of the fundamental invariants given in the following table.
1900:
5239:
4225:
2327:
5371:
1510:
1634:
4666:
4153:
1791:
3845:
3371:
2124:
2001:
1478:
2409:
1338:
2434:
Further, it is a necessary condition for the existence of a non-zero homomorphism of some highest weight modules (a homomorphism of such modules preserves central character). A simple consequence is that for
1952:
3731:
5132:
4945:
5488:
2061:
5415:
795:
3616:
1393:
1015:
393:
3891:
3417:
1230:
960:
97:
3275:
2547:
5279:
3653:. In the isomorphism, these correspond to a degree 2 polynomial on the CSA. Since the Weyl group acts by reflections on the CSA, they are isometries, so the degree 2 invariant polynomial is
914:
846:
1055:
3091:
5299:
5127:
4892:
4729:
3146:
2695:
2234:
1270:
200:
5596:
3198:
1786:
5539:
4854:
4470:{\displaystyle (h_{1}\mapsto -h_{1},h_{2}\mapsto h_{2}),(h_{1}\mapsto h_{1},h_{2}\mapsto -h_{2}),(h_{1}\mapsto h_{2},h_{2}\mapsto h_{1}),(h_{1}\mapsto -h_{2},h_{2}\mapsto h_{1})}
3027:
1146:
428:
2181:
1426:
706:
236:
133:
743:
3524:
2354:
1081:
5096:
5060:
3775:
3231:
2651:
2627:
2579:
2148:
616:
456:
346:
318:
263:
160:
4045:
5005:
3301:
2467:
1107:
643:
513:
4220:
3967:
3457:
5802:
2262:
5616:
5032:
1505:
486:
4758:
2487:
873:
670:
564:
540:
2254:
1166:
4529:
4502:
4180:
3938:
3644:
3484:
2507:
584:
5958:
4784:
2427:, so that only the quadratic Casimir operator is required; there is a corresponding streamlined treatment proof of the character formula in the second edition of
4965:
3996:
3911:
3751:
2719:
2599:
286:
5292:
The isomorphism in this case is an isomorphism between the Feigin–Frenkel center and the W-algebra constructed associated to the
Langlands dual Lie algebra by
1628:
2722:
4534:
4077:
3526:
which acts on the CSA in the standard representation. Since the Weyl group acts by reflections, they are isometries and so the degree 2 polynomial
1620:{\displaystyle n^{+}=\bigoplus _{\alpha \in \Phi _{+}}{\mathfrak {g}}_{\alpha },n^{-}=\bigoplus _{\alpha \in \Phi _{-}}{\mathfrak {g}}_{\alpha }}
2006:
5939:
5906:
5884:
712:
acts on the modules by scalar multiplication (this follows from the fact that the modules are generated by a highest weight vector). So, for
3783:
3309:
2071:
1957:
1434:
396:
2359:
1751:{\displaystyle U({\mathfrak {g}})=U({\mathfrak {h}})\oplus (U({\mathfrak {g}}){\mathfrak {n}}^{+}+{\mathfrak {n}}^{-}U({\mathfrak {g}})).}
1287:
1905:
3656:
3303:, by negation, so the invariant of the Weyl group is the square of the generator of the Cartan subalgebra, which is also of degree 2.
5853:
800:
1895:{\displaystyle z\in U({\mathfrak {h}})\oplus (U({\mathfrak {g}}){\mathfrak {n}}^{+}\cap {\mathfrak {n}}^{-}U({\mathfrak {g}})).}
4897:
5420:
5376:
748:
3529:
1343:
977:
355:
5963:
5787:
3850:
3376:
1189:
1184:
919:
673:
100:
56:
3236:
2512:
1180:
20:
5244:
2420:
878:
5740:
Feigin, Boris; Frenkel, Edward; Reshetikhin, Nikolai (3 Apr 1994). "Gaudin Model, Bethe Ansatz and
Critical Level".
5234:{\displaystyle {\mathfrak {Z}}({\hat {\mathfrak {g}}}):=\{S\in V_{\text{cri}}({\mathfrak {g}})|{\mathfrak {g}}S=0\}}
5898:
1020:
5695:
Molev, Alexander (19 January 2021). "On Segal–Sugawara vectors and
Casimir elements for classical Lie algebras".
5800:
Harish-Chandra (1951), "On some applications of the universal enveloping algebra of a semisimple Lie algebra",
3097:
of the Lie group and vice versa. In another direction, fundamental invariants are related to cohomology of the
3032:
2440:
1627:
as the corresponding positive nilpotent subalgebra and negative nilpotent subalgebra respectively, due to the
5632:
5101:
4866:
4671:
3104:
2660:
2186:
1235:
165:
5544:
3151:
4800:
4668:
Despite this being a two-dimensional vector space, this contributes only one new fundamental invariant as
1761:
321:
5493:
4826:
2986:
1120:
402:
4796:
2153:
1398:
678:
543:
208:
136:
105:
4731:
lies in the space. In this case, there is no unique choice of quartic invariant as any polynomial with
715:
3489:
2332:
1060:
5068:
5041:
3756:
3212:
2632:
2608:
2560:
2129:
1110:
597:
437:
327:
299:
244:
141:
51:
4001:
5627:
2743:
2322:{\displaystyle {\tilde {\gamma }}=\tau \circ \gamma :{\mathfrak {Z}}\rightarrow S({\mathfrak {h}})}
5676:
Borel, Armand (Apr 1954). "Sur la cohomologie des espaces homogenes des groupes de Lie compacts".
4974:
3649:
For all the Lie algebras in the classification, there is a fundamental invariant of degree 2, the
3284:
2445:
1086:
621:
491:
5872:
5821:
5767:
5749:
5722:
5704:
5366:{\displaystyle {\mathfrak {Z}}({\hat {\mathfrak {g}}})\cong {\mathcal {W}}(^{L}{\mathfrak {g}}).}
4968:
4804:
4185:
3943:
3422:
5601:
5010:
1483:
1272:(the elements of the symmetric algebra of the Cartan subalgebra fixed by the Weyl group) is an
464:
5935:
5927:
5902:
5880:
5849:
5281:
is the affine vertex algebra at the critical level. Elements of this center are also known as
4734:
3278:
3098:
2472:
851:
648:
549:
518:
431:
349:
239:
203:
2423:
for finite-dimensional irreducible representations. The proof has been further simplified by
2239:
1284:
More explicitly, the isomorphism can be constructed as the composition of two maps, one from
1151:
5923:
5811:
5759:
5714:
2654:
709:
5916:
5863:
5833:
4507:
4480:
4158:
3916:
3622:
3462:
5912:
5859:
5845:
5829:
5035:
4531:, and invariance under exchange of coordinates means any invariant quartic can be written
2980:
2698:
2492:
569:
5417:
as a polynomial algebra in a finite number of countably infinite families of generators,
4763:
4950:
4820:
4812:
3972:
3896:
3736:
2735:
2704:
2584:
271:
35:
3281:
of degree 2, and the Weyl group acts on the Cartan subalgebra, which is isomorphic to
5952:
5726:
3419:
is isomorphic to a polynomial algebra of Weyl-invariant polynomials in two variables
1169:
459:
5771:
4222:, and is generated (non-minimally) by four reflections, which act on coordinates as
3847:, the Cartan subalgebra is one-dimensional, and the Harish-Chandra isomorphism says
3093:. Due to this, the degrees of the fundamental invariants can be calculated from the
5063:
4808:
4056:
3094:
2436:
2003:, and is a homomorphism of algebras. This is related to the central characters by
3893:
is isomorphic to the algebra of Weyl-invariant polynomials in a single variable
2356:
is not actually Weyl-invariant, but it can be proven that the twisted character
1273:
47:
43:
27:
5718:
4815:
showing that an algebra known as the Feigin–Frenkel center is isomorphic to a
3969:. The subalgebra of Weyl-invariant polynomials in the full polynomial algebra
2424:
1114:
266:
4661:{\displaystyle f_{4}(h_{1},h_{2})=ah_{1}^{4}+bh_{1}^{2}h_{2}^{2}+ah_{2}^{4}.}
4148:{\displaystyle {\mathfrak {g}}=B_{2}={\mathfrak {so}}(5)={\mathfrak {sp}}(4)}
5293:
4816:
2976:
5618:
is the (negative of) the natural derivative operator on the loop algebra.
3277:, then the center of the universal enveloping algebra is generated by the
2983:
of a Lie group. In particular, if the fundamental invariants have degrees
16:
Isomorphism of commutative rings constructed in the theory of Lie algebras
5844:. Graduate Texts in Mathematics. Vol. 9 (Second revised ed.).
3940:
acting as reflection, with non-trivial element acting on polynomials by
5825:
5763:
5754:
5934:, Progress in Mathematics, vol. 140, Springer, pp. 246–258,
5928:"V. Finite Dimensional Representations §5. Harish-Chandra Isomorphism"
3840:{\displaystyle {\mathfrak {g}}=A_{1}={\mathfrak {sl}}(2,\mathbb {C} )}
3366:{\displaystyle {\mathfrak {g}}=A_{2}={\mathfrak {sl}}(3,\mathbb {C} )}
2119:{\displaystyle \tau :S({\mathfrak {h}})\rightarrow S({\mathfrak {h}})}
2979:
is equal to its rank. Fundamental invariants are also related to the
1996:{\displaystyle \gamma :{\mathfrak {Z}}\rightarrow S({\mathfrak {h}})}
1473:{\displaystyle \gamma :{\mathfrak {Z}}\rightarrow S({\mathfrak {h}})}
5816:
2419:
The theorem has been used to obtain a simple Lie algebraic proof of
2404:{\displaystyle {\tilde {\chi }}_{\lambda }=\chi _{\lambda -\delta }}
5709:
1333:{\displaystyle {\mathfrak {Z}}={\mathcal {Z}}(U({\mathfrak {g}}))}
3148:
is isomorphic to a polynomial algebra on generators with degrees
5877:
Representations of semisimple Lie algebras in the BGG category O
2329:
is the isomorphism. The reason this twist is introduced is that
1947:{\displaystyle U({\mathfrak {g}})\rightarrow U({\mathfrak {h}})}
3726:{\displaystyle f_{2}(\mathbf {h} )=h_{1}^{2}+\cdots +h_{r}^{2}}
4894:
is not exactly the center of the universal enveloping algebra
5335:
4903:
3856:
3382:
1303:
1195:
925:
760:
62:
3029:, then the generators of the cohomology ring have degrees
5868:(Contains an improved proof of Weyl's character formula.)
4940:{\displaystyle {\mathcal {Z}}(U({\hat {\mathfrak {g}}}))}
5483:{\displaystyle \partial ^{n}S_{i},i=1,\cdots ,l,n\geq 0}
2056:{\displaystyle \chi _{\lambda }(x)=\gamma (x)(\lambda )}
5410:{\displaystyle {\mathfrak {Z}}({\hat {\mathfrak {g}}})}
790:{\displaystyle x\in {\mathcal {Z}}(U({\mathfrak {g}}))}
5842:
Introduction to Lie algebras and representation theory
3611:{\displaystyle f_{2}(h_{1},h_{2})=h_{1}^{2}+h_{2}^{2}}
3459:(since the Cartan subalgebra is two-dimensional). For
1388:{\displaystyle U({\mathfrak {h}})=S({\mathfrak {h}}),}
5604:
5547:
5496:
5423:
5379:
5302:
5247:
5135:
5104:
5071:
5044:
5013:
4977:
4953:
4900:
4869:
4829:
4766:
4737:
4674:
4537:
4510:
4483:
4228:
4188:
4161:
4080:
4066:, corresponding to the degree 3 fundamental invariant
4004:
3998:
is therefore only the even polynomials, generated by
3975:
3946:
3919:
3899:
3853:
3786:
3759:
3739:
3659:
3625:
3532:
3492:
3465:
3425:
3379:
3312:
3287:
3239:
3215:
3154:
3107:
3035:
2989:
2707:
2663:
2635:
2611:
2587:
2563:
2515:
2495:
2475:
2448:
2362:
2335:
2265:
2242:
2189:
2156:
2132:
2074:
2009:
1960:
1908:
1794:
1764:
1637:
1513:
1486:
1437:
1401:
1346:
1290:
1238:
1192:
1154:
1123:
1089:
1063:
1023:
980:
922:
881:
854:
803:
751:
718:
681:
651:
624:
600:
572:
552:
521:
494:
467:
440:
405:
358:
330:
302:
274:
247:
211:
168:
144:
108:
59:
1010:{\displaystyle \lambda ,\mu \in {\mathfrak {h}}^{*}}
388:{\displaystyle \lambda ,\mu \in {\mathfrak {h}}^{*}}
5895:
Cohomological induction and unitary representations
5610:
5590:
5533:
5482:
5409:
5365:
5273:
5233:
5121:
5090:
5054:
5026:
4999:
4959:
4939:
4886:
4848:
4778:
4752:
4723:
4660:
4523:
4496:
4469:
4214:
4174:
4147:
4039:
3990:
3961:
3932:
3905:
3886:{\displaystyle {\mathcal {Z}}(U({\mathfrak {g}}))}
3885:
3839:
3769:
3745:
3725:
3638:
3610:
3518:
3478:
3451:
3412:{\displaystyle {\mathcal {Z}}(U({\mathfrak {g}}))}
3411:
3365:
3295:
3269:
3225:
3192:
3140:
3085:
3021:
2713:
2689:
2645:
2621:
2605:, that is, the dimension of any Cartan subalgebra
2593:
2573:
2541:
2501:
2481:
2461:
2403:
2348:
2321:
2248:
2228:
2175:
2142:
2118:
2055:
1995:
1946:
1894:
1780:
1750:
1619:
1499:
1472:
1420:
1387:
1332:
1264:
1225:{\displaystyle {\mathcal {Z}}(U({\mathfrak {g}}))}
1224:
1160:
1140:
1101:
1075:
1049:
1009:
955:{\displaystyle {\mathcal {Z}}(U({\mathfrak {g}}))}
954:
908:
867:
840:
789:
737:
700:
664:
637:
610:
578:
558:
534:
507:
480:
450:
422:
387:
340:
312:
280:
257:
230:
194:
154:
127:
92:{\displaystyle {\mathcal {Z}}(U({\mathfrak {g}}))}
91:
46:of commutative rings constructed in the theory of
5803:Transactions of the American Mathematical Society
3270:{\displaystyle {\mathfrak {sl}}(2,\mathbb {R} )}
2542:{\displaystyle V_{\lambda }\rightarrow V_{\mu }}
1179:Another closely related formulation is that the
5897:, Princeton Mathematical Series, vol. 45,
5274:{\displaystyle V_{\text{cri}}({\mathfrak {g}})}
909:{\displaystyle \chi _{\lambda },\,\chi _{\mu }}
2975:The number of the fundamental invariants of a
841:{\displaystyle x\cdot v:=\chi _{\lambda }(x)v}
39:
4477:. Any invariant quartic must be even in both
1050:{\displaystyle \chi _{\lambda }=\chi _{\mu }}
8:
5228:
5166:
3777:, also known as the rank of the Lie algebra.
5893:Knapp, Anthony W.; Vogan, David A. (1995),
5815:
5753:
5708:
5663:
5651:
5603:
5552:
5546:
5501:
5495:
5438:
5428:
5422:
5393:
5391:
5390:
5381:
5380:
5378:
5351:
5350:
5344:
5334:
5333:
5316:
5314:
5313:
5304:
5303:
5301:
5262:
5261:
5252:
5246:
5204:
5203:
5198:
5189:
5188:
5179:
5149:
5147:
5146:
5137:
5136:
5134:
5108:
5106:
5105:
5103:
5073:
5072:
5070:
5046:
5045:
5043:
5018:
5012:
4991:
4976:
4952:
4920:
4918:
4917:
4902:
4901:
4899:
4873:
4871:
4870:
4868:
4840:
4839:
4833:
4828:
4765:
4736:
4715:
4705:
4692:
4679:
4673:
4649:
4644:
4628:
4623:
4613:
4608:
4592:
4587:
4568:
4555:
4542:
4536:
4515:
4509:
4488:
4482:
4458:
4445:
4432:
4416:
4397:
4384:
4371:
4358:
4339:
4323:
4310:
4297:
4278:
4265:
4252:
4236:
4227:
4206:
4193:
4187:
4166:
4160:
4127:
4126:
4105:
4104:
4095:
4082:
4081:
4079:
4031:
4009:
4003:
3974:
3945:
3924:
3918:
3898:
3871:
3870:
3855:
3854:
3852:
3830:
3829:
3811:
3810:
3801:
3788:
3787:
3785:
3761:
3760:
3758:
3738:
3717:
3712:
3693:
3688:
3673:
3664:
3658:
3630:
3624:
3602:
3597:
3584:
3579:
3563:
3550:
3537:
3531:
3510:
3497:
3491:
3470:
3464:
3443:
3430:
3424:
3397:
3396:
3381:
3380:
3378:
3356:
3355:
3337:
3336:
3327:
3314:
3313:
3311:
3289:
3288:
3286:
3260:
3259:
3241:
3240:
3238:
3217:
3216:
3214:
3184:
3162:
3153:
3131:
3130:
3112:
3106:
3086:{\displaystyle 2d_{1}-1,\cdots ,2d_{r}-1}
3071:
3043:
3034:
3013:
2994:
2988:
2706:
2681:
2671:
2670:
2662:
2637:
2636:
2634:
2613:
2612:
2610:
2586:
2565:
2564:
2562:
2533:
2520:
2514:
2494:
2489:, there exist only finitely many weights
2474:
2453:
2447:
2428:
2389:
2376:
2365:
2364:
2361:
2340:
2334:
2310:
2309:
2294:
2293:
2267:
2266:
2264:
2241:
2188:
2164:
2163:
2155:
2134:
2133:
2131:
2107:
2106:
2088:
2087:
2073:
2014:
2008:
1984:
1983:
1968:
1967:
1959:
1935:
1934:
1916:
1915:
1907:
1877:
1876:
1864:
1858:
1857:
1847:
1841:
1840:
1830:
1829:
1808:
1807:
1793:
1772:
1771:
1763:
1733:
1732:
1720:
1714:
1713:
1703:
1697:
1696:
1686:
1685:
1664:
1663:
1645:
1644:
1636:
1611:
1605:
1604:
1595:
1584:
1571:
1558:
1552:
1551:
1542:
1531:
1518:
1512:
1491:
1485:
1461:
1460:
1445:
1444:
1436:
1409:
1408:
1400:
1373:
1372:
1354:
1353:
1345:
1318:
1317:
1302:
1301:
1292:
1291:
1289:
1256:
1246:
1245:
1237:
1210:
1209:
1194:
1193:
1191:
1153:
1132:
1126:
1125:
1122:
1088:
1062:
1041:
1028:
1022:
1001:
995:
994:
979:
940:
939:
924:
923:
921:
900:
895:
886:
880:
859:
853:
820:
802:
775:
774:
759:
758:
750:
729:
717:
689:
688:
680:
656:
650:
629:
623:
602:
601:
599:
571:
551:
526:
520:
499:
493:
472:
466:
442:
441:
439:
414:
408:
407:
404:
379:
373:
372:
357:
332:
331:
329:
304:
303:
301:
273:
249:
248:
246:
219:
218:
210:
186:
176:
175:
167:
146:
145:
143:
116:
115:
107:
77:
76:
61:
60:
58:
2728:
5788:Notes on the Harish-Chandra isomorphism
5644:
5122:{\displaystyle {\hat {\mathfrak {g}}}}
5062:which are annihilated by the positive
4887:{\displaystyle {\hat {\mathfrak {g}}}}
4724:{\displaystyle f_{2}(h_{1},h_{2})^{2}}
3373:, the Harish-Chandra isomorphism says
3141:{\displaystyle H^{*}(BG,\mathbb {R} )}
2690:{\displaystyle S({\mathfrak {h}})^{W}}
2229:{\displaystyle \tau (h)=h-\delta (h)1}
1265:{\displaystyle S({\mathfrak {h}})^{W}}
195:{\displaystyle S({\mathfrak {h}})^{W}}
5959:Representation theory of Lie algebras
5591:{\displaystyle d_{i}+1,i=1,\cdots ,l}
4791:Generalization to affine Lie algebras
3193:{\displaystyle 2d_{1},\cdots ,2d_{r}}
7:
1781:{\displaystyle z\in {\mathfrak {Z}}}
5534:{\displaystyle S_{i},i=1,\cdots ,l}
5394:
5382:
5352:
5317:
5305:
5263:
5205:
5190:
5150:
5138:
5109:
5074:
5047:
4921:
4874:
4849:{\displaystyle ^{L}{\mathfrak {g}}}
4841:
4131:
4128:
4109:
4106:
4083:
3872:
3815:
3812:
3789:
3762:
3398:
3341:
3338:
3315:
3245:
3242:
3218:
3022:{\displaystyle d_{1},\cdots ,d_{r}}
2672:
2638:
2614:
2566:
2509:for which a non-zero homomorphism
2311:
2295:
2165:
2135:
2108:
2089:
1985:
1969:
1936:
1917:
1878:
1859:
1842:
1831:
1809:
1773:
1734:
1715:
1698:
1687:
1665:
1646:
1606:
1553:
1462:
1446:
1410:
1374:
1355:
1319:
1293:
1247:
1211:
1141:{\displaystyle {\mathfrak {h}}^{*}}
1127:
996:
970:Statement of Harish-Chandra theorem
941:
776:
690:
603:
443:
423:{\displaystyle {\mathfrak {h}}^{*}}
409:
374:
333:
305:
250:
220:
177:
147:
117:
78:
5605:
5425:
2748:Degrees of fundamental invariants
2176:{\displaystyle U({\mathfrak {h}})}
1902:The restriction of the projection
1592:
1539:
1488:
1421:{\displaystyle S({\mathfrak {h}})}
701:{\displaystyle U({\mathfrak {g}})}
469:
231:{\displaystyle S({\mathfrak {h}})}
128:{\displaystyle U({\mathfrak {g}})}
14:
5932:Lie Groups Beyond an Introduction
1480:. For a choice of positive roots
738:{\displaystyle v\in V_{\lambda }}
4055:
3674:
3519:{\displaystyle S_{3}\cong D_{6}}
2349:{\displaystyle \chi _{\lambda }}
1076:{\displaystyle \lambda +\delta }
5697:Letters in Mathematical Physics
5678:American Journal of Mathematics
5373:There is also a description of
5091:{\displaystyle {\mathfrak {g}}}
5055:{\displaystyle {\mathfrak {g}}}
4803:. There is a generalization to
3770:{\displaystyle {\mathfrak {h}}}
3226:{\displaystyle {\mathfrak {g}}}
2723:Chevalley–Shephard–Todd theorem
2646:{\displaystyle {\mathfrak {g}}}
2622:{\displaystyle {\mathfrak {h}}}
2574:{\displaystyle {\mathfrak {g}}}
2143:{\displaystyle {\mathfrak {h}}}
611:{\displaystyle {\mathfrak {g}}}
451:{\displaystyle {\mathfrak {h}}}
341:{\displaystyle {\mathfrak {h}}}
313:{\displaystyle {\mathfrak {g}}}
258:{\displaystyle {\mathfrak {h}}}
155:{\displaystyle {\mathfrak {g}}}
5404:
5398:
5387:
5357:
5341:
5327:
5321:
5310:
5268:
5258:
5216:
5210:
5199:
5195:
5185:
5160:
5154:
5143:
5113:
5085:
5079:
4934:
4931:
4925:
4914:
4908:
4878:
4712:
4685:
4574:
4548:
4464:
4451:
4422:
4409:
4403:
4390:
4364:
4351:
4345:
4329:
4303:
4290:
4284:
4271:
4242:
4229:
4142:
4136:
4120:
4114:
4040:{\displaystyle f_{2}(h)=h^{2}}
4021:
4015:
3985:
3979:
3950:
3880:
3877:
3867:
3861:
3834:
3820:
3678:
3670:
3569:
3543:
3406:
3403:
3393:
3387:
3360:
3346:
3264:
3250:
3135:
3118:
2678:
2667:
2526:
2370:
2316:
2306:
2300:
2272:
2220:
2214:
2199:
2193:
2170:
2160:
2113:
2103:
2097:
2094:
2084:
2050:
2044:
2041:
2035:
2026:
2020:
1990:
1980:
1974:
1941:
1931:
1925:
1922:
1912:
1886:
1883:
1873:
1836:
1826:
1820:
1814:
1804:
1742:
1739:
1729:
1692:
1682:
1676:
1670:
1660:
1651:
1641:
1629:Poincaré–Birkhoff–Witt theorem
1467:
1457:
1451:
1415:
1405:
1379:
1369:
1360:
1350:
1327:
1324:
1314:
1308:
1253:
1242:
1219:
1216:
1206:
1200:
949:
946:
936:
930:
832:
826:
784:
781:
771:
765:
695:
685:
225:
215:
183:
172:
122:
112:
86:
83:
73:
67:
1:
2935:2, 8, 12, 14, 18, 20, 24, 30
265:that are invariant under the
5840:Humphreys, James E. (1978).
5000:{\displaystyle k=-h^{\vee }}
4182:, acting on two coordinates
3753:is the dimension of the CSA
3296:{\displaystyle \mathbb {R} }
2462:{\displaystyle V_{\lambda }}
1185:universal enveloping algebra
1102:{\displaystyle \mu +\delta }
674:universal enveloping algebra
638:{\displaystyle V_{\lambda }}
508:{\displaystyle V_{\lambda }}
101:universal enveloping algebra
4795:The above result holds for
4215:{\displaystyle h_{1},h_{2}}
3962:{\displaystyle h\mapsto -h}
3452:{\displaystyle h_{1},h_{2}}
1181:Harish-Chandra homomorphism
672:are representations of the
458:) and assume that a set of
50:. The isomorphism maps the
21:Harish-Chandra homomorphism
5980:
5899:Princeton University Press
5719:10.1007/s11005-020-01344-3
5294:Drinfeld–Sokolov reduction
4062:Weyl-invariant cubic for A
2581:a simple Lie algebra, let
1788:is central, then in fact
1431:The first is a projection
32:Harish-Chandra isomorphism
18:
5611:{\displaystyle \partial }
5027:{\displaystyle h^{\vee }}
4863:of an affine Lie algebra
2441:generalized Verma modules
1631:there is a decomposition
1500:{\displaystyle \Phi _{+}}
1172:, sometimes known as the
481:{\displaystyle \Phi _{+}}
4786:not both zero) suffices.
4753:{\displaystyle b\neq 2a}
2918:2, 6, 8, 10, 12, 14, 18
2482:{\displaystyle \lambda }
2421:Weyl's character formula
2150:viewed as a subspace of
868:{\displaystyle V_{\mu }}
665:{\displaystyle V_{\mu }}
559:{\displaystyle \lambda }
535:{\displaystyle V_{\mu }}
292:Introduction and setting
19:Not to be confused with
5633:Infinitesimal character
4801:semisimple Lie algebras
2249:{\displaystyle \delta }
1183:from the center of the
1168:is the half-sum of the
1161:{\displaystyle \delta }
916:are homomorphisms from
395:be two elements of the
5612:
5592:
5535:
5484:
5411:
5367:
5287:Segal–Sugawara vectors
5275:
5235:
5123:
5092:
5056:
5028:
5001:
4961:
4941:
4888:
4850:
4780:
4754:
4725:
4662:
4525:
4498:
4471:
4216:
4176:
4149:
4041:
3992:
3963:
3934:
3907:
3887:
3841:
3771:
3747:
3727:
3640:
3612:
3520:
3480:
3453:
3413:
3367:
3297:
3271:
3227:
3194:
3142:
3101:. The cohomology ring
3087:
3023:
2715:
2691:
2647:
2623:
2595:
2575:
2553:Fundamental invariants
2543:
2503:
2483:
2463:
2405:
2350:
2323:
2250:
2230:
2177:
2144:
2120:
2065:The second map is the
2057:
1997:
1948:
1896:
1782:
1752:
1621:
1501:
1474:
1422:
1389:
1334:
1266:
1226:
1162:
1142:
1103:
1077:
1051:
1011:
956:
910:
875:, where the functions
869:
842:
791:
739:
702:
666:
639:
612:
580:
560:
544:highest weight modules
536:
509:
482:
452:
424:
389:
342:
322:semisimple Lie algebra
314:
282:
259:
232:
196:
156:
129:
93:
5613:
5593:
5536:
5485:
5412:
5368:
5276:
5236:
5124:
5093:
5057:
5029:
5002:
4969:affine vertex algebra
4962:
4942:
4889:
4861:Feigin–Frenkel center
4851:
4781:
4755:
4726:
4663:
4526:
4524:{\displaystyle h_{2}}
4499:
4497:{\displaystyle h_{1}}
4472:
4217:
4177:
4175:{\displaystyle D_{8}}
4150:
4042:
3993:
3964:
3935:
3933:{\displaystyle S_{2}}
3908:
3888:
3842:
3772:
3748:
3728:
3641:
3639:{\displaystyle A_{2}}
3613:
3521:
3481:
3479:{\displaystyle A_{2}}
3454:
3414:
3368:
3298:
3272:
3228:
3195:
3143:
3088:
3024:
2884: − 2
2716:
2692:
2648:
2624:
2596:
2576:
2544:
2504:
2484:
2464:
2431:, pp. 143–144).
2406:
2351:
2324:
2251:
2231:
2178:
2145:
2121:
2058:
1998:
1949:
1897:
1783:
1753:
1622:
1502:
1475:
1423:
1390:
1335:
1267:
1227:
1163:
1143:
1104:
1078:
1052:
1012:
957:
911:
870:
843:
792:
740:
703:
667:
640:
613:
581:
561:
546:with highest weights
537:
510:
488:have been fixed. Let
483:
453:
425:
390:
343:
315:
283:
260:
233:
197:
157:
137:reductive Lie algebra
130:
94:
5602:
5545:
5494:
5421:
5377:
5300:
5245:
5133:
5102:
5069:
5042:
5011:
4975:
4951:
4947:. They are elements
4898:
4867:
4827:
4799:, and in particular
4764:
4735:
4672:
4535:
4508:
4481:
4226:
4186:
4159:
4155:, the Weyl group is
4078:
4002:
3973:
3944:
3917:
3913:. The Weyl group is
3897:
3851:
3784:
3757:
3737:
3657:
3623:
3530:
3490:
3486:, the Weyl group is
3463:
3423:
3377:
3310:
3285:
3237:
3213:
3152:
3105:
3033:
2987:
2874: − 2
2867: − 2
2816: − 1
2705:
2661:
2633:
2609:
2585:
2561:
2513:
2502:{\displaystyle \mu }
2493:
2473:
2469:with highest weight
2446:
2360:
2333:
2263:
2240:
2187:
2154:
2130:
2072:
2007:
1958:
1906:
1792:
1762:
1635:
1511:
1484:
1435:
1399:
1344:
1288:
1280:Explicit isomorphism
1236:
1190:
1152:
1121:
1087:
1061:
1021:
978:
920:
879:
852:
801:
749:
716:
679:
649:
622:
598:
579:{\displaystyle \mu }
570:
550:
519:
492:
465:
438:
403:
356:
328:
300:
272:
245:
209:
166:
142:
106:
57:
5964:Theorems in algebra
5879:, AMS, p. 26,
5873:Humphreys, James E.
5666:, pp. 135–141.
5628:Translation functor
5036:dual Coxeter number
4805:affine Lie algebras
4779:{\displaystyle a,b}
4654:
4633:
4618:
4597:
3722:
3698:
3607:
3589:
3233:is the Lie algebra
2744:Dual Coxeter number
2697:is isomorphic to a
5783:External resources
5764:10.1007/BF02099300
5742:Commun. Math. Phys
5608:
5588:
5531:
5480:
5407:
5363:
5271:
5231:
5119:
5088:
5052:
5024:
4997:
4971:at critical level
4957:
4937:
4884:
4846:
4819:associated to the
4776:
4750:
4721:
4658:
4640:
4619:
4604:
4583:
4521:
4494:
4467:
4212:
4172:
4145:
4037:
3988:
3959:
3930:
3903:
3883:
3837:
3767:
3743:
3723:
3708:
3684:
3636:
3608:
3593:
3575:
3516:
3476:
3449:
3409:
3363:
3293:
3267:
3223:
3190:
3138:
3083:
3019:
2901:2, 5, 6, 8, 9, 12
2711:
2699:polynomial algebra
2687:
2643:
2619:
2591:
2571:
2539:
2499:
2479:
2459:
2401:
2346:
2319:
2246:
2226:
2173:
2140:
2116:
2053:
1993:
1944:
1892:
1778:
1748:
1617:
1602:
1549:
1497:
1470:
1418:
1385:
1330:
1262:
1222:
1158:
1138:
1099:
1073:
1047:
1007:
964:central characters
962:to scalars called
952:
906:
865:
848:and similarly for
838:
787:
735:
698:
662:
635:
608:
590:Central characters
576:
556:
532:
505:
478:
448:
420:
385:
338:
310:
278:
255:
228:
192:
152:
125:
89:
5941:978-1-4757-2453-0
5924:Knapp, Anthony W.
5908:978-0-691-03756-1
5886:978-0-8218-4678-0
5401:
5324:
5255:
5182:
5157:
5116:
4960:{\displaystyle S}
4928:
4881:
3991:{\displaystyle K}
3906:{\displaystyle h}
3746:{\displaystyle r}
3651:quadratic Casimir
3279:Casimir invariant
3099:classifying space
2973:
2972:
2880:; 2, 4, 6, ..., 2
2714:{\displaystyle r}
2594:{\displaystyle r}
2373:
2275:
2256:the Weyl vector.
1954:to the centre is
1580:
1527:
1395:and another from
1017:, the characters
350:Cartan subalgebra
281:{\displaystyle W}
240:Cartan subalgebra
204:symmetric algebra
5971:
5944:
5919:
5889:
5867:
5836:
5819:
5776:
5775:
5757:
5737:
5731:
5730:
5712:
5692:
5686:
5685:
5673:
5667:
5661:
5655:
5649:
5617:
5615:
5614:
5609:
5597:
5595:
5594:
5589:
5557:
5556:
5540:
5538:
5537:
5532:
5506:
5505:
5489:
5487:
5486:
5481:
5443:
5442:
5433:
5432:
5416:
5414:
5413:
5408:
5403:
5402:
5397:
5392:
5386:
5385:
5372:
5370:
5369:
5364:
5356:
5355:
5349:
5348:
5339:
5338:
5326:
5325:
5320:
5315:
5309:
5308:
5283:singular vectors
5280:
5278:
5277:
5272:
5267:
5266:
5257:
5256:
5253:
5240:
5238:
5237:
5232:
5209:
5208:
5202:
5194:
5193:
5184:
5183:
5180:
5159:
5158:
5153:
5148:
5142:
5141:
5128:
5126:
5125:
5120:
5118:
5117:
5112:
5107:
5097:
5095:
5094:
5089:
5078:
5077:
5061:
5059:
5058:
5053:
5051:
5050:
5033:
5031:
5030:
5025:
5023:
5022:
5006:
5004:
5003:
4998:
4996:
4995:
4966:
4964:
4963:
4958:
4946:
4944:
4943:
4938:
4930:
4929:
4924:
4919:
4907:
4906:
4893:
4891:
4890:
4885:
4883:
4882:
4877:
4872:
4855:
4853:
4852:
4847:
4845:
4844:
4838:
4837:
4785:
4783:
4782:
4777:
4759:
4757:
4756:
4751:
4730:
4728:
4727:
4722:
4720:
4719:
4710:
4709:
4697:
4696:
4684:
4683:
4667:
4665:
4664:
4659:
4653:
4648:
4632:
4627:
4617:
4612:
4596:
4591:
4573:
4572:
4560:
4559:
4547:
4546:
4530:
4528:
4527:
4522:
4520:
4519:
4503:
4501:
4500:
4495:
4493:
4492:
4476:
4474:
4473:
4468:
4463:
4462:
4450:
4449:
4437:
4436:
4421:
4420:
4402:
4401:
4389:
4388:
4376:
4375:
4363:
4362:
4344:
4343:
4328:
4327:
4315:
4314:
4302:
4301:
4283:
4282:
4270:
4269:
4257:
4256:
4241:
4240:
4221:
4219:
4218:
4213:
4211:
4210:
4198:
4197:
4181:
4179:
4178:
4173:
4171:
4170:
4154:
4152:
4151:
4146:
4135:
4134:
4113:
4112:
4100:
4099:
4087:
4086:
4059:
4046:
4044:
4043:
4038:
4036:
4035:
4014:
4013:
3997:
3995:
3994:
3989:
3968:
3966:
3965:
3960:
3939:
3937:
3936:
3931:
3929:
3928:
3912:
3910:
3909:
3904:
3892:
3890:
3889:
3884:
3876:
3875:
3860:
3859:
3846:
3844:
3843:
3838:
3833:
3819:
3818:
3806:
3805:
3793:
3792:
3776:
3774:
3773:
3768:
3766:
3765:
3752:
3750:
3749:
3744:
3732:
3730:
3729:
3724:
3721:
3716:
3697:
3692:
3677:
3669:
3668:
3645:
3643:
3642:
3637:
3635:
3634:
3617:
3615:
3614:
3609:
3606:
3601:
3588:
3583:
3568:
3567:
3555:
3554:
3542:
3541:
3525:
3523:
3522:
3517:
3515:
3514:
3502:
3501:
3485:
3483:
3482:
3477:
3475:
3474:
3458:
3456:
3455:
3450:
3448:
3447:
3435:
3434:
3418:
3416:
3415:
3410:
3402:
3401:
3386:
3385:
3372:
3370:
3369:
3364:
3359:
3345:
3344:
3332:
3331:
3319:
3318:
3302:
3300:
3299:
3294:
3292:
3276:
3274:
3273:
3268:
3263:
3249:
3248:
3232:
3230:
3229:
3224:
3222:
3221:
3199:
3197:
3196:
3191:
3189:
3188:
3167:
3166:
3147:
3145:
3144:
3139:
3134:
3117:
3116:
3092:
3090:
3089:
3084:
3076:
3075:
3048:
3047:
3028:
3026:
3025:
3020:
3018:
3017:
2999:
2998:
2729:
2720:
2718:
2717:
2712:
2696:
2694:
2693:
2688:
2686:
2685:
2676:
2675:
2655:H. S. M. Coxeter
2652:
2650:
2649:
2644:
2642:
2641:
2628:
2626:
2625:
2620:
2618:
2617:
2600:
2598:
2597:
2592:
2580:
2578:
2577:
2572:
2570:
2569:
2548:
2546:
2545:
2540:
2538:
2537:
2525:
2524:
2508:
2506:
2505:
2500:
2488:
2486:
2485:
2480:
2468:
2466:
2465:
2460:
2458:
2457:
2410:
2408:
2407:
2402:
2400:
2399:
2381:
2380:
2375:
2374:
2366:
2355:
2353:
2352:
2347:
2345:
2344:
2328:
2326:
2325:
2320:
2315:
2314:
2299:
2298:
2277:
2276:
2268:
2255:
2253:
2252:
2247:
2235:
2233:
2232:
2227:
2182:
2180:
2179:
2174:
2169:
2168:
2149:
2147:
2146:
2141:
2139:
2138:
2125:
2123:
2122:
2117:
2112:
2111:
2093:
2092:
2062:
2060:
2059:
2054:
2019:
2018:
2002:
2000:
1999:
1994:
1989:
1988:
1973:
1972:
1953:
1951:
1950:
1945:
1940:
1939:
1921:
1920:
1901:
1899:
1898:
1893:
1882:
1881:
1869:
1868:
1863:
1862:
1852:
1851:
1846:
1845:
1835:
1834:
1813:
1812:
1787:
1785:
1784:
1779:
1777:
1776:
1757:
1755:
1754:
1749:
1738:
1737:
1725:
1724:
1719:
1718:
1708:
1707:
1702:
1701:
1691:
1690:
1669:
1668:
1650:
1649:
1626:
1624:
1623:
1618:
1616:
1615:
1610:
1609:
1601:
1600:
1599:
1576:
1575:
1563:
1562:
1557:
1556:
1548:
1547:
1546:
1523:
1522:
1506:
1504:
1503:
1498:
1496:
1495:
1479:
1477:
1476:
1471:
1466:
1465:
1450:
1449:
1427:
1425:
1424:
1419:
1414:
1413:
1394:
1392:
1391:
1386:
1378:
1377:
1359:
1358:
1339:
1337:
1336:
1331:
1323:
1322:
1307:
1306:
1297:
1296:
1271:
1269:
1268:
1263:
1261:
1260:
1251:
1250:
1231:
1229:
1228:
1223:
1215:
1214:
1199:
1198:
1167:
1165:
1164:
1159:
1147:
1145:
1144:
1139:
1137:
1136:
1131:
1130:
1109:are on the same
1108:
1106:
1105:
1100:
1082:
1080:
1079:
1074:
1056:
1054:
1053:
1048:
1046:
1045:
1033:
1032:
1016:
1014:
1013:
1008:
1006:
1005:
1000:
999:
961:
959:
958:
953:
945:
944:
929:
928:
915:
913:
912:
907:
905:
904:
891:
890:
874:
872:
871:
866:
864:
863:
847:
845:
844:
839:
825:
824:
796:
794:
793:
788:
780:
779:
764:
763:
744:
742:
741:
736:
734:
733:
707:
705:
704:
699:
694:
693:
671:
669:
668:
663:
661:
660:
644:
642:
641:
636:
634:
633:
617:
615:
614:
609:
607:
606:
585:
583:
582:
577:
565:
563:
562:
557:
541:
539:
538:
533:
531:
530:
514:
512:
511:
506:
504:
503:
487:
485:
484:
479:
477:
476:
457:
455:
454:
449:
447:
446:
429:
427:
426:
421:
419:
418:
413:
412:
394:
392:
391:
386:
384:
383:
378:
377:
347:
345:
344:
339:
337:
336:
319:
317:
316:
311:
309:
308:
287:
285:
284:
279:
264:
262:
261:
256:
254:
253:
237:
235:
234:
229:
224:
223:
201:
199:
198:
193:
191:
190:
181:
180:
162:to the elements
161:
159:
158:
153:
151:
150:
134:
132:
131:
126:
121:
120:
98:
96:
95:
90:
82:
81:
66:
65:
34:, introduced by
5979:
5978:
5974:
5973:
5972:
5970:
5969:
5968:
5949:
5948:
5947:
5942:
5922:
5909:
5892:
5887:
5871:
5856:
5846:Springer-Verlag
5839:
5817:10.2307/1990524
5799:
5795:
5785:
5780:
5779:
5739:
5738:
5734:
5694:
5693:
5689:
5675:
5674:
5670:
5662:
5658:
5650:
5646:
5641:
5624:
5600:
5599:
5548:
5543:
5542:
5497:
5492:
5491:
5434:
5424:
5419:
5418:
5375:
5374:
5340:
5298:
5297:
5248:
5243:
5242:
5175:
5131:
5130:
5100:
5099:
5067:
5066:
5040:
5039:
5014:
5009:
5008:
4987:
4973:
4972:
4949:
4948:
4896:
4895:
4865:
4864:
4830:
4825:
4824:
4793:
4762:
4761:
4733:
4732:
4711:
4701:
4688:
4675:
4670:
4669:
4564:
4551:
4538:
4533:
4532:
4511:
4506:
4505:
4484:
4479:
4478:
4454:
4441:
4428:
4412:
4393:
4380:
4367:
4354:
4335:
4319:
4306:
4293:
4274:
4261:
4248:
4232:
4224:
4223:
4202:
4189:
4184:
4183:
4162:
4157:
4156:
4091:
4076:
4075:
4071:
4070:
4069:
4068:
4067:
4065:
4060:
4027:
4005:
4000:
3999:
3971:
3970:
3942:
3941:
3920:
3915:
3914:
3895:
3894:
3849:
3848:
3797:
3782:
3781:
3755:
3754:
3735:
3734:
3660:
3655:
3654:
3626:
3621:
3620:
3559:
3546:
3533:
3528:
3527:
3506:
3493:
3488:
3487:
3466:
3461:
3460:
3439:
3426:
3421:
3420:
3375:
3374:
3323:
3308:
3307:
3283:
3282:
3235:
3234:
3211:
3210:
3206:
3180:
3158:
3150:
3149:
3108:
3103:
3102:
3067:
3039:
3031:
3030:
3009:
2990:
2985:
2984:
2981:cohomology ring
2960:
2943:
2926:
2909:
2892:
2860:
2847:2, 4, 6, ..., 2
2832:
2819:2, 4, 6, ..., 2
2803:
2793: + 1
2774:
2721:variables (see
2703:
2702:
2677:
2659:
2658:
2631:
2630:
2607:
2606:
2583:
2582:
2559:
2558:
2555:
2529:
2516:
2511:
2510:
2491:
2490:
2471:
2470:
2449:
2444:
2443:
2429:Humphreys (1978
2417:
2385:
2363:
2358:
2357:
2336:
2331:
2330:
2261:
2260:
2238:
2237:
2185:
2184:
2152:
2151:
2128:
2127:
2070:
2069:
2010:
2005:
2004:
1956:
1955:
1904:
1903:
1856:
1839:
1790:
1789:
1760:
1759:
1712:
1695:
1633:
1632:
1603:
1591:
1567:
1550:
1538:
1514:
1509:
1508:
1487:
1482:
1481:
1433:
1432:
1397:
1396:
1342:
1341:
1286:
1285:
1282:
1252:
1234:
1233:
1188:
1187:
1150:
1149:
1124:
1119:
1118:
1085:
1084:
1059:
1058:
1057:if and only if
1037:
1024:
1019:
1018:
993:
976:
975:
972:
918:
917:
896:
882:
877:
876:
855:
850:
849:
816:
799:
798:
747:
746:
725:
714:
713:
677:
676:
652:
647:
646:
625:
620:
619:
596:
595:
592:
568:
567:
548:
547:
522:
517:
516:
495:
490:
489:
468:
463:
462:
436:
435:
406:
401:
400:
371:
354:
353:
326:
325:
298:
297:
294:
270:
269:
243:
242:
207:
206:
182:
164:
163:
140:
139:
104:
103:
55:
54:
24:
17:
12:
11:
5:
5977:
5975:
5967:
5966:
5961:
5951:
5950:
5946:
5945:
5940:
5920:
5907:
5890:
5885:
5869:
5854:
5837:
5796:
5794:
5791:
5784:
5781:
5778:
5777:
5755:hep-th/9402022
5732:
5687:
5668:
5664:Humphreys 1978
5656:
5654:, p. 130.
5652:Humphreys 1978
5643:
5642:
5640:
5637:
5636:
5635:
5630:
5623:
5620:
5607:
5587:
5584:
5581:
5578:
5575:
5572:
5569:
5566:
5563:
5560:
5555:
5551:
5530:
5527:
5524:
5521:
5518:
5515:
5512:
5509:
5504:
5500:
5479:
5476:
5473:
5470:
5467:
5464:
5461:
5458:
5455:
5452:
5449:
5446:
5441:
5437:
5431:
5427:
5406:
5400:
5396:
5389:
5384:
5362:
5359:
5354:
5347:
5343:
5337:
5332:
5329:
5323:
5319:
5312:
5307:
5270:
5265:
5260:
5251:
5230:
5227:
5224:
5221:
5218:
5215:
5212:
5207:
5201:
5197:
5192:
5187:
5178:
5174:
5171:
5168:
5165:
5162:
5156:
5152:
5145:
5140:
5115:
5111:
5087:
5084:
5081:
5076:
5049:
5021:
5017:
4994:
4990:
4986:
4983:
4980:
4967:of the vacuum
4956:
4936:
4933:
4927:
4923:
4916:
4913:
4910:
4905:
4880:
4876:
4843:
4836:
4832:
4821:Langlands dual
4792:
4789:
4788:
4787:
4775:
4772:
4769:
4749:
4746:
4743:
4740:
4718:
4714:
4708:
4704:
4700:
4695:
4691:
4687:
4682:
4678:
4657:
4652:
4647:
4643:
4639:
4636:
4631:
4626:
4622:
4616:
4611:
4607:
4603:
4600:
4595:
4590:
4586:
4582:
4579:
4576:
4571:
4567:
4563:
4558:
4554:
4550:
4545:
4541:
4518:
4514:
4491:
4487:
4466:
4461:
4457:
4453:
4448:
4444:
4440:
4435:
4431:
4427:
4424:
4419:
4415:
4411:
4408:
4405:
4400:
4396:
4392:
4387:
4383:
4379:
4374:
4370:
4366:
4361:
4357:
4353:
4350:
4347:
4342:
4338:
4334:
4331:
4326:
4322:
4318:
4313:
4309:
4305:
4300:
4296:
4292:
4289:
4286:
4281:
4277:
4273:
4268:
4264:
4260:
4255:
4251:
4247:
4244:
4239:
4235:
4231:
4209:
4205:
4201:
4196:
4192:
4169:
4165:
4144:
4141:
4138:
4133:
4130:
4125:
4122:
4119:
4116:
4111:
4108:
4103:
4098:
4094:
4090:
4085:
4063:
4061:
4054:
4053:
4052:
4051:
4050:
4049:
4048:
4034:
4030:
4026:
4023:
4020:
4017:
4012:
4008:
3987:
3984:
3981:
3978:
3958:
3955:
3952:
3949:
3927:
3923:
3902:
3882:
3879:
3874:
3869:
3866:
3863:
3858:
3836:
3832:
3828:
3825:
3822:
3817:
3814:
3809:
3804:
3800:
3796:
3791:
3778:
3764:
3742:
3720:
3715:
3711:
3707:
3704:
3701:
3696:
3691:
3687:
3683:
3680:
3676:
3672:
3667:
3663:
3647:
3633:
3629:
3605:
3600:
3596:
3592:
3587:
3582:
3578:
3574:
3571:
3566:
3562:
3558:
3553:
3549:
3545:
3540:
3536:
3513:
3509:
3505:
3500:
3496:
3473:
3469:
3446:
3442:
3438:
3433:
3429:
3408:
3405:
3400:
3395:
3392:
3389:
3384:
3362:
3358:
3354:
3351:
3348:
3343:
3340:
3335:
3330:
3326:
3322:
3317:
3304:
3291:
3266:
3262:
3258:
3255:
3252:
3247:
3244:
3220:
3205:
3202:
3187:
3183:
3179:
3176:
3173:
3170:
3165:
3161:
3157:
3137:
3133:
3129:
3126:
3123:
3120:
3115:
3111:
3082:
3079:
3074:
3070:
3066:
3063:
3060:
3057:
3054:
3051:
3046:
3042:
3038:
3016:
3012:
3008:
3005:
3002:
2997:
2993:
2971:
2970:
2967:
2964:
2961:
2958:
2954:
2953:
2950:
2947:
2944:
2941:
2937:
2936:
2933:
2930:
2927:
2924:
2920:
2919:
2916:
2913:
2910:
2907:
2903:
2902:
2899:
2896:
2893:
2890:
2886:
2885:
2875:
2868:
2861:
2856:
2852:
2851:
2845:
2844: + 1
2839:
2833:
2828:
2824:
2823:
2817:
2810:
2804:
2799:
2795:
2794:
2789:2, 3, 4, ...,
2787:
2786: + 1
2781:
2780: + 1
2775:
2770:
2766:
2765:
2762:
2759:
2756:
2750:
2749:
2746:
2741:
2736:Coxeter number
2733:
2710:
2684:
2680:
2674:
2669:
2666:
2657:observed that
2640:
2616:
2590:
2568:
2554:
2551:
2536:
2532:
2528:
2523:
2519:
2498:
2478:
2456:
2452:
2416:
2413:
2398:
2395:
2392:
2388:
2384:
2379:
2372:
2369:
2343:
2339:
2318:
2313:
2308:
2305:
2302:
2297:
2292:
2289:
2286:
2283:
2280:
2274:
2271:
2245:
2225:
2222:
2219:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2195:
2192:
2183:it is defined
2172:
2167:
2162:
2159:
2137:
2115:
2110:
2105:
2102:
2099:
2096:
2091:
2086:
2083:
2080:
2077:
2052:
2049:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2017:
2013:
1992:
1987:
1982:
1979:
1976:
1971:
1966:
1963:
1943:
1938:
1933:
1930:
1927:
1924:
1919:
1914:
1911:
1891:
1888:
1885:
1880:
1875:
1872:
1867:
1861:
1855:
1850:
1844:
1838:
1833:
1828:
1825:
1822:
1819:
1816:
1811:
1806:
1803:
1800:
1797:
1775:
1770:
1767:
1747:
1744:
1741:
1736:
1731:
1728:
1723:
1717:
1711:
1706:
1700:
1694:
1689:
1684:
1681:
1678:
1675:
1672:
1667:
1662:
1659:
1656:
1653:
1648:
1643:
1640:
1614:
1608:
1598:
1594:
1590:
1587:
1583:
1579:
1574:
1570:
1566:
1561:
1555:
1545:
1541:
1537:
1534:
1530:
1526:
1521:
1517:
1494:
1490:
1469:
1464:
1459:
1456:
1453:
1448:
1443:
1440:
1417:
1412:
1407:
1404:
1384:
1381:
1376:
1371:
1368:
1365:
1362:
1357:
1352:
1349:
1329:
1326:
1321:
1316:
1313:
1310:
1305:
1300:
1295:
1281:
1278:
1259:
1255:
1249:
1244:
1241:
1221:
1218:
1213:
1208:
1205:
1202:
1197:
1170:positive roots
1157:
1135:
1129:
1098:
1095:
1092:
1072:
1069:
1066:
1044:
1040:
1036:
1031:
1027:
1004:
998:
992:
989:
986:
983:
971:
968:
951:
948:
943:
938:
935:
932:
927:
903:
899:
894:
889:
885:
862:
858:
837:
834:
831:
828:
823:
819:
815:
812:
809:
806:
786:
783:
778:
773:
770:
767:
762:
757:
754:
732:
728:
724:
721:
697:
692:
687:
684:
659:
655:
632:
628:
605:
591:
588:
586:respectively.
575:
555:
529:
525:
502:
498:
475:
471:
460:positive roots
445:
417:
411:
382:
376:
370:
367:
364:
361:
335:
307:
293:
290:
277:
252:
227:
222:
217:
214:
189:
185:
179:
174:
171:
149:
124:
119:
114:
111:
88:
85:
80:
75:
72:
69:
64:
36:Harish-Chandra
15:
13:
10:
9:
6:
4:
3:
2:
5976:
5965:
5962:
5960:
5957:
5956:
5954:
5943:
5937:
5933:
5929:
5925:
5921:
5918:
5914:
5910:
5904:
5900:
5896:
5891:
5888:
5882:
5878:
5874:
5870:
5865:
5861:
5857:
5855:0-387-90053-5
5851:
5847:
5843:
5838:
5835:
5831:
5827:
5823:
5818:
5813:
5809:
5805:
5804:
5798:
5797:
5792:
5790:
5789:
5782:
5773:
5769:
5765:
5761:
5756:
5751:
5747:
5743:
5736:
5733:
5728:
5724:
5720:
5716:
5711:
5706:
5702:
5698:
5691:
5688:
5684:(2): 273–342.
5683:
5679:
5672:
5669:
5665:
5660:
5657:
5653:
5648:
5645:
5638:
5634:
5631:
5629:
5626:
5625:
5621:
5619:
5585:
5582:
5579:
5576:
5573:
5570:
5567:
5564:
5561:
5558:
5553:
5549:
5541:have degrees
5528:
5525:
5522:
5519:
5516:
5513:
5510:
5507:
5502:
5498:
5477:
5474:
5471:
5468:
5465:
5462:
5459:
5456:
5453:
5450:
5447:
5444:
5439:
5435:
5429:
5360:
5345:
5330:
5295:
5290:
5288:
5284:
5249:
5225:
5222:
5219:
5213:
5176:
5172:
5169:
5163:
5082:
5065:
5037:
5019:
5015:
4992:
4988:
4984:
4981:
4978:
4970:
4954:
4911:
4862:
4857:
4834:
4831:
4822:
4818:
4814:
4810:
4806:
4802:
4798:
4790:
4773:
4770:
4767:
4747:
4744:
4741:
4738:
4716:
4706:
4702:
4698:
4693:
4689:
4680:
4676:
4655:
4650:
4645:
4641:
4637:
4634:
4629:
4624:
4620:
4614:
4609:
4605:
4601:
4598:
4593:
4588:
4584:
4580:
4577:
4569:
4565:
4561:
4556:
4552:
4543:
4539:
4516:
4512:
4489:
4485:
4459:
4455:
4446:
4442:
4438:
4433:
4429:
4425:
4417:
4413:
4406:
4398:
4394:
4385:
4381:
4377:
4372:
4368:
4359:
4355:
4348:
4340:
4336:
4332:
4324:
4320:
4316:
4311:
4307:
4298:
4294:
4287:
4279:
4275:
4266:
4262:
4258:
4253:
4249:
4245:
4237:
4233:
4207:
4203:
4199:
4194:
4190:
4167:
4163:
4139:
4123:
4117:
4101:
4096:
4092:
4088:
4073:
4072:
4058:
4032:
4028:
4024:
4018:
4010:
4006:
3982:
3976:
3956:
3953:
3947:
3925:
3921:
3900:
3864:
3826:
3823:
3807:
3802:
3798:
3794:
3779:
3740:
3718:
3713:
3709:
3705:
3702:
3699:
3694:
3689:
3685:
3681:
3665:
3661:
3652:
3648:
3631:
3627:
3603:
3598:
3594:
3590:
3585:
3580:
3576:
3572:
3564:
3560:
3556:
3551:
3547:
3538:
3534:
3511:
3507:
3503:
3498:
3494:
3471:
3467:
3444:
3440:
3436:
3431:
3427:
3390:
3352:
3349:
3333:
3328:
3324:
3320:
3305:
3280:
3256:
3253:
3208:
3207:
3203:
3201:
3185:
3181:
3177:
3174:
3171:
3168:
3163:
3159:
3155:
3127:
3124:
3121:
3113:
3109:
3100:
3096:
3095:Betti numbers
3080:
3077:
3072:
3068:
3064:
3061:
3058:
3055:
3052:
3049:
3044:
3040:
3036:
3014:
3010:
3006:
3003:
3000:
2995:
2991:
2982:
2978:
2968:
2965:
2962:
2956:
2955:
2951:
2948:
2945:
2939:
2938:
2934:
2931:
2928:
2922:
2921:
2917:
2914:
2911:
2905:
2904:
2900:
2897:
2894:
2888:
2887:
2883:
2879:
2876:
2873:
2869:
2866:
2862:
2859:
2854:
2853:
2850:
2846:
2843:
2840:
2838:
2834:
2831:
2826:
2825:
2822:
2818:
2815:
2811:
2809:
2805:
2802:
2797:
2796:
2792:
2788:
2785:
2782:
2779:
2776:
2773:
2768:
2767:
2763:
2760:
2757:
2755:
2752:
2751:
2747:
2745:
2742:
2740:
2737:
2734:
2731:
2730:
2727:
2724:
2708:
2700:
2682:
2664:
2656:
2604:
2588:
2552:
2550:
2534:
2530:
2521:
2517:
2496:
2476:
2454:
2450:
2442:
2438:
2437:Verma modules
2432:
2430:
2426:
2422:
2414:
2412:
2396:
2393:
2390:
2386:
2382:
2377:
2367:
2341:
2337:
2303:
2290:
2287:
2284:
2281:
2278:
2269:
2257:
2243:
2223:
2217:
2211:
2208:
2205:
2202:
2196:
2190:
2157:
2100:
2081:
2078:
2075:
2068:
2063:
2047:
2038:
2032:
2029:
2023:
2015:
2011:
1977:
1964:
1961:
1928:
1909:
1889:
1870:
1865:
1853:
1848:
1823:
1817:
1801:
1798:
1795:
1768:
1765:
1745:
1726:
1721:
1709:
1704:
1679:
1673:
1657:
1654:
1638:
1630:
1612:
1596:
1588:
1585:
1581:
1577:
1572:
1568:
1564:
1559:
1543:
1535:
1532:
1528:
1524:
1519:
1515:
1492:
1454:
1441:
1438:
1429:
1402:
1382:
1366:
1363:
1347:
1311:
1298:
1279:
1277:
1275:
1257:
1239:
1203:
1186:
1182:
1177:
1175:
1171:
1155:
1133:
1116:
1112:
1096:
1093:
1090:
1070:
1067:
1064:
1042:
1038:
1034:
1029:
1025:
1002:
990:
987:
984:
981:
969:
967:
965:
933:
901:
897:
892:
887:
883:
860:
856:
835:
829:
821:
817:
813:
810:
807:
804:
768:
755:
752:
730:
726:
722:
719:
711:
682:
675:
657:
653:
630:
626:
589:
587:
573:
553:
545:
527:
523:
500:
496:
473:
461:
433:
415:
398:
380:
368:
365:
362:
359:
351:
323:
291:
289:
275:
268:
241:
212:
205:
187:
169:
138:
109:
102:
70:
53:
49:
45:
41:
37:
33:
29:
22:
5931:
5894:
5876:
5841:
5810:(1): 28–96,
5807:
5801:
5786:
5745:
5741:
5735:
5700:
5696:
5690:
5681:
5677:
5671:
5659:
5647:
5291:
5286:
5282:
5129:, that is,
5064:loop algebra
4860:
4858:
4823:Lie algebra
4794:
3650:
2974:
2952:2, 6, 8, 12
2881:
2877:
2871:
2864:
2857:
2848:
2841:
2836:
2829:
2820:
2813:
2807:
2800:
2790:
2783:
2777:
2771:
2753:
2738:
2602:
2556:
2433:
2418:
2415:Applications
2258:
2066:
2064:
1507:, defining
1430:
1283:
1178:
1173:
973:
963:
593:
397:weight space
295:
48:Lie algebras
31:
25:
2732:Lie algebra
1428:to itself.
1274:isomorphism
1174:Weyl vector
44:isomorphism
28:mathematics
5953:Categories
5793:References
5710:2008.05256
2425:Victor Kac
1115:Weyl group
267:Weyl group
5926:(2013) ,
5748:: 27–62.
5727:254795180
5606:∂
5580:⋯
5523:⋯
5475:≥
5460:⋯
5426:∂
5399:^
5331:≅
5322:^
5173:∈
5155:^
5114:^
5020:∨
4993:∨
4985:−
4926:^
4879:^
4817:W-algebra
4807:shown by
4797:reductive
4742:≠
4452:↦
4426:−
4423:↦
4391:↦
4365:↦
4333:−
4330:↦
4304:↦
4272:↦
4246:−
4243:↦
3954:−
3951:↦
3703:⋯
3504:≅
3172:⋯
3114:∗
3078:−
3059:⋯
3050:−
3004:⋯
2977:Lie group
2535:μ
2527:→
2522:λ
2497:μ
2477:λ
2455:λ
2397:δ
2394:−
2391:λ
2387:χ
2378:λ
2371:~
2368:χ
2342:λ
2338:χ
2301:→
2288:γ
2285:∘
2282:τ
2273:~
2270:γ
2244:δ
2212:δ
2209:−
2191:τ
2098:→
2076:τ
2067:twist map
2048:λ
2033:γ
2016:λ
2012:χ
1975:→
1962:γ
1926:→
1866:−
1854:∩
1818:⊕
1799:∈
1769:∈
1722:−
1674:⊕
1613:α
1597:−
1593:Φ
1589:∈
1586:α
1582:⨁
1573:−
1560:α
1540:Φ
1536:∈
1533:α
1529:⨁
1489:Φ
1452:→
1439:γ
1156:δ
1134:∗
1097:δ
1091:μ
1071:δ
1065:λ
1043:μ
1039:χ
1030:λ
1026:χ
1003:∗
991:∈
988:μ
982:λ
902:μ
898:χ
888:λ
884:χ
861:μ
822:λ
818:χ
808:⋅
756:∈
731:λ
723:∈
658:μ
631:λ
618:-modules
574:μ
554:λ
528:μ
501:λ
470:Φ
416:∗
381:∗
369:∈
366:μ
360:λ
42:), is an
5875:(2008),
5772:17099900
5622:See also
5490:, where
5098:part of
5007:, where
3204:Examples
2549:exists.
1148:, where
974:For any
708:and its
5917:1330919
5864:0499562
5834:0044515
5826:1990524
5034:is the
4813:Frenkel
2601:be its
1113:of the
430:is the
399:(where
202:of the
99:of the
38: (
5938:
5915:
5905:
5883:
5862:
5852:
5832:
5824:
5770:
5725:
5241:where
4809:Feigin
3733:where
710:center
52:center
30:, the
5822:JSTOR
5768:S2CID
5750:arXiv
5723:S2CID
5705:arXiv
5703:(8).
5639:Notes
4760:(and
2969:2, 6
2259:Then
2236:with
2126:. On
1111:orbit
320:be a
238:of a
135:of a
5936:ISBN
5903:ISBN
5881:ISBN
5850:ISBN
5598:and
5038:for
4859:The
4811:and
4504:and
4074:For
3780:For
3306:For
2603:rank
2557:For
2411:is.
1083:and
745:and
645:and
594:The
566:and
515:and
432:dual
352:and
348:its
296:Let
40:1951
5812:doi
5760:doi
5746:166
5715:doi
5701:111
5285:or
5254:cri
5181:cri
3209:If
2701:in
2629:of
2439:or
1758:If
1340:to
1232:to
1117:of
542:be
434:of
26:In
5955::
5930:,
5913:MR
5911:,
5901:,
5860:MR
5858:.
5848:.
5830:MR
5828:,
5820:,
5808:70
5806:,
5766:.
5758:.
5744:.
5721:.
5713:.
5699:.
5682:76
5680:.
5296::
5289:.
5164::=
4856:.
3200:.
2946:12
2932:30
2929:30
2915:18
2912:18
2898:12
2895:12
2764:1
2653:.
1276:.
1176:.
966:.
814::=
797:,
324:,
288:.
5866:.
5814::
5774:.
5762::
5752::
5729:.
5717::
5707::
5586:l
5583:,
5577:,
5574:1
5571:=
5568:i
5565:,
5562:1
5559:+
5554:i
5550:d
5529:l
5526:,
5520:,
5517:1
5514:=
5511:i
5508:,
5503:i
5499:S
5478:0
5472:n
5469:,
5466:l
5463:,
5457:,
5454:1
5451:=
5448:i
5445:,
5440:i
5436:S
5430:n
5405:)
5395:g
5388:(
5383:Z
5361:.
5358:)
5353:g
5346:L
5342:(
5336:W
5328:)
5318:g
5311:(
5306:Z
5269:)
5264:g
5259:(
5250:V
5229:}
5226:0
5223:=
5220:S
5217:]
5214:t
5211:[
5206:g
5200:|
5196:)
5191:g
5186:(
5177:V
5170:S
5167:{
5161:)
5151:g
5144:(
5139:Z
5110:g
5086:]
5083:t
5080:[
5075:g
5048:g
5016:h
4989:h
4982:=
4979:k
4955:S
4935:)
4932:)
4922:g
4915:(
4912:U
4909:(
4904:Z
4875:g
4842:g
4835:L
4774:b
4771:,
4768:a
4748:a
4745:2
4739:b
4717:2
4713:)
4707:2
4703:h
4699:,
4694:1
4690:h
4686:(
4681:2
4677:f
4656:.
4651:4
4646:2
4642:h
4638:a
4635:+
4630:2
4625:2
4621:h
4615:2
4610:1
4606:h
4602:b
4599:+
4594:4
4589:1
4585:h
4581:a
4578:=
4575:)
4570:2
4566:h
4562:,
4557:1
4553:h
4549:(
4544:4
4540:f
4517:2
4513:h
4490:1
4486:h
4465:)
4460:1
4456:h
4447:2
4443:h
4439:,
4434:2
4430:h
4418:1
4414:h
4410:(
4407:,
4404:)
4399:1
4395:h
4386:2
4382:h
4378:,
4373:2
4369:h
4360:1
4356:h
4352:(
4349:,
4346:)
4341:2
4337:h
4325:2
4321:h
4317:,
4312:1
4308:h
4299:1
4295:h
4291:(
4288:,
4285:)
4280:2
4276:h
4267:2
4263:h
4259:,
4254:1
4250:h
4238:1
4234:h
4230:(
4208:2
4204:h
4200:,
4195:1
4191:h
4168:8
4164:D
4143:)
4140:4
4137:(
4132:p
4129:s
4124:=
4121:)
4118:5
4115:(
4110:o
4107:s
4102:=
4097:2
4093:B
4089:=
4084:g
4064:2
4047:.
4033:2
4029:h
4025:=
4022:)
4019:h
4016:(
4011:2
4007:f
3986:]
3983:h
3980:[
3977:K
3957:h
3948:h
3926:2
3922:S
3901:h
3881:)
3878:)
3873:g
3868:(
3865:U
3862:(
3857:Z
3835:)
3831:C
3827:,
3824:2
3821:(
3816:l
3813:s
3808:=
3803:1
3799:A
3795:=
3790:g
3763:h
3741:r
3719:2
3714:r
3710:h
3706:+
3700:+
3695:2
3690:1
3686:h
3682:=
3679:)
3675:h
3671:(
3666:2
3662:f
3646:.
3632:2
3628:A
3604:2
3599:2
3595:h
3591:+
3586:2
3581:1
3577:h
3573:=
3570:)
3565:2
3561:h
3557:,
3552:1
3548:h
3544:(
3539:2
3535:f
3512:6
3508:D
3499:3
3495:S
3472:2
3468:A
3445:2
3441:h
3437:,
3432:1
3428:h
3407:)
3404:)
3399:g
3394:(
3391:U
3388:(
3383:Z
3361:)
3357:C
3353:,
3350:3
3347:(
3342:l
3339:s
3334:=
3329:2
3325:A
3321:=
3316:g
3290:R
3265:)
3261:R
3257:,
3254:2
3251:(
3246:l
3243:s
3219:g
3186:r
3182:d
3178:2
3175:,
3169:,
3164:1
3160:d
3156:2
3136:)
3132:R
3128:,
3125:G
3122:B
3119:(
3110:H
3081:1
3073:r
3069:d
3065:2
3062:,
3056:,
3053:1
3045:1
3041:d
3037:2
3015:r
3011:d
3007:,
3001:,
2996:1
2992:d
2966:4
2963:6
2959:2
2957:G
2949:9
2942:4
2940:F
2925:8
2923:E
2908:7
2906:E
2891:6
2889:E
2882:n
2878:n
2872:n
2870:2
2865:n
2863:2
2858:n
2855:D
2849:n
2842:n
2837:n
2835:2
2830:n
2827:C
2821:n
2814:n
2812:2
2808:n
2806:2
2801:n
2798:B
2791:n
2784:n
2778:n
2772:n
2769:A
2761:0
2758:0
2754:R
2739:h
2709:r
2683:W
2679:)
2673:h
2668:(
2665:S
2639:g
2615:h
2589:r
2567:g
2531:V
2518:V
2451:V
2383:=
2317:)
2312:h
2307:(
2304:S
2296:Z
2291::
2279:=
2224:1
2221:)
2218:h
2215:(
2206:h
2203:=
2200:)
2197:h
2194:(
2171:)
2166:h
2161:(
2158:U
2136:h
2114:)
2109:h
2104:(
2101:S
2095:)
2090:h
2085:(
2082:S
2079::
2051:)
2045:(
2042:)
2039:x
2036:(
2030:=
2027:)
2024:x
2021:(
1991:)
1986:h
1981:(
1978:S
1970:Z
1965::
1942:)
1937:h
1932:(
1929:U
1923:)
1918:g
1913:(
1910:U
1890:.
1887:)
1884:)
1879:g
1874:(
1871:U
1860:n
1849:+
1843:n
1837:)
1832:g
1827:(
1824:U
1821:(
1815:)
1810:h
1805:(
1802:U
1796:z
1774:Z
1766:z
1746:.
1743:)
1740:)
1735:g
1730:(
1727:U
1716:n
1710:+
1705:+
1699:n
1693:)
1688:g
1683:(
1680:U
1677:(
1671:)
1666:h
1661:(
1658:U
1655:=
1652:)
1647:g
1642:(
1639:U
1607:g
1578:=
1569:n
1565:,
1554:g
1544:+
1525:=
1520:+
1516:n
1493:+
1468:)
1463:h
1458:(
1455:S
1447:Z
1442::
1416:)
1411:h
1406:(
1403:S
1383:,
1380:)
1375:h
1370:(
1367:S
1364:=
1361:)
1356:h
1351:(
1348:U
1328:)
1325:)
1320:g
1315:(
1312:U
1309:(
1304:Z
1299:=
1294:Z
1258:W
1254:)
1248:h
1243:(
1240:S
1220:)
1217:)
1212:g
1207:(
1204:U
1201:(
1196:Z
1128:h
1094:+
1068:+
1035:=
997:h
985:,
950:)
947:)
942:g
937:(
934:U
931:(
926:Z
893:,
857:V
836:v
833:)
830:x
827:(
811:v
805:x
785:)
782:)
777:g
772:(
769:U
766:(
761:Z
753:x
727:V
720:v
696:)
691:g
686:(
683:U
654:V
627:V
604:g
524:V
497:V
474:+
444:h
410:h
375:h
363:,
334:h
306:g
276:W
251:h
226:)
221:h
216:(
213:S
188:W
184:)
178:h
173:(
170:S
148:g
123:)
118:g
113:(
110:U
87:)
84:)
79:g
74:(
71:U
68:(
63:Z
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.