32:
8202:
7428:
7810:
3486:. Thus, for semisimple Lie algebras, a classification of irreducible (i.e. simple) representations leads immediately to classification of all representations. For other Lie algebra, which do not have this special property, classifying the irreducible representations may not help much in classifying general representations.
1699:
8105:
3545:
is completely reducible, as we have just noted. In the other direction, the definition of a reductive Lie algebra means that it decomposes as a direct sum of ideals (i.e., invariant subspaces for the adjoint representation) that have no nontrivial sub-ideals. Some of these ideals will be
7309:
7725:
4200:
4082:
3904:
5441:
5138:
6028:
The universal enveloping algebra plays an important role in the representation theory of semisimple Lie algebras, described above. Specifically, the finite-dimensional irreducible representations are constructed as quotients of
1577:
2604:. Many other interesting Lie algebras (and their representations) arise in other parts of quantum physics. Indeed, the history of representation theory is characterized by rich interactions between mathematics and physics.
8197:{\displaystyle \operatorname {Ind} _{\mathfrak {h}}^{\mathfrak {g}}\circ \operatorname {Res} _{\mathfrak {h}}\simeq \operatorname {Res} _{\mathfrak {g}}\circ \operatorname {Ind} _{\mathfrak {h_{1}}}^{\mathfrak {g_{1}}}}
4748:
4450:
acts on the second factor in the tensor product. In the context of representations of the Lie algebra su(2), the tensor product of representations goes under the name "addition of angular momentum." In this context,
4372:
6838:
6165:
2663:
7624:
7246:
7468:
4611:
8100:
8047:
7423:{\displaystyle \operatorname {Hom} _{\mathfrak {g}}(\operatorname {Ind} _{\mathfrak {h}}^{\mathfrak {g}}W,E)\simeq \operatorname {Hom} _{\mathfrak {h}}(W,\operatorname {Res} _{\mathfrak {h}}^{\mathfrak {g}}E)}
2599:
will constitute a representation of the Lie algebra so(3). An understanding of the representation theory of so(3) is of great help in, for example, analyzing
Hamiltonians with rotational symmetry, such as the
2549:
1089:
7180:
4821:
4909:
2003:
1883:
7888:
7849:
2325:
8271:
is a better place for the representation theory in the semisimple case in zero characteristic. For instance, the category O turned out to be of a right size to formulate the celebrated BGG reciprocity.
4995:
1467:
1211:
7922:
6994:
943:
2178:
1942:
3130:
7805:{\displaystyle \operatorname {Ind} _{\mathfrak {h}}^{\mathfrak {g}}\simeq \operatorname {Ind} _{\mathfrak {h'}}^{\mathfrak {g}}\circ \operatorname {Ind} _{\mathfrak {h}}^{\mathfrak {h'}}}
1378:
5605:
5543:
5361:
5260:
6291:
4090:
6391:
3087:
2210:
6458:
3646:
2825:
2289:
2230:
1726:
983:
887:
811:
8290:
One of the most important applications of Lie algebra representations is to the representation theory of real reductive Lie groups. The application is based on the idea that if
7582:
7549:
7121:
7060:
7027:
6928:
6700:
6616:
6352:
6218:
6067:
6023:
5942:
5909:
5848:
5791:
3995:
3675:
3177:
Similarly, many other constructions from module theory in abstract algebra carry over to this setting: submodule, quotient, subquotient, direct sum, Jordan-Hölder series, etc.
2103:
4856:
8367:
8336:
8265:
8234:
7994:
7970:
7946:
7652:
7516:
7492:
7298:
7270:
7204:
7084:
6960:
6895:
6752:
6724:
6667:
6640:
6579:
6482:
6419:
6319:
6099:
5990:
5966:
5876:
5815:
5758:
5726:
5649:
5567:
5465:
5284:
5222:
5198:
5076:
4635:
4559:
3770:
3718:
3584:
3543:
3515:
3468:
3397:
3308:
3225:
3031:
3007:
2947:
2911:
2687:
2349:
2258:
2030:
1818:
1794:
1569:
1498:
1335:
1275:
1242:
1011:
755:
8581:
and the adjoint representation of the Lie algebra on itself is a representation on an algebra (i.e., acts by derivations on the associative algebra structure), then it is a
7689:
6871:
3987:
3947:
2769:
2354:
A partial converse to this statement says that every representation of a finite-dimensional (real or complex) Lie algebra lifts to a unique representation of the associated
4521:
4485:
4448:
4412:
2142:
3778:
5366:
3616:
5052:
5025:
4254:
4227:
6185:
3340:
3257:
2979:
6532:
2795:
5676:
4662:
2875:
1754:
1304:
1113:
8387:
8308:
7717:
5174:
3160:
8407:
6552:
5625:
5505:
5485:
5081:
4929:
4682:
3360:
3277:
2849:
2727:
2707:
2597:
2577:
1031:
851:
831:
775:
689:
669:
1694:{\displaystyle {\textrm {ad}}:{\mathfrak {g}}\to {\mathfrak {gl}}({\mathfrak {g}}),\quad X\mapsto \operatorname {ad} _{X},\quad \operatorname {ad} _{X}(Y)=.}
8745:
Bernstein I.N., Gelfand I.M., Gelfand S.I., "Structure of
Representations that are generated by vectors of highest weight," Functional. Anal. Appl. 5 (1971)
8767:
A. Beilinson and J. Bernstein, "Localisation de g-modules," Comptes Rendus de l'Académie des
Sciences, SĂ©rie I, vol. 292, iss. 1, pp. 15â18, 1981.
3546:
one-dimensional and the rest are simple Lie algebras. Thus, a reductive Lie algebra is a direct sum of a commutative algebra and a semisimple algebra.
702:. Roughly speaking, the representations of Lie algebras are the differentiated form of representations of Lie groups, while the representations of the
8919:
5687:
449:
4687:
9091:
4278:
2358:
Lie group, so that representations of simply-connected Lie groups are in one-to-one correspondence with representations of their Lie algebras.
6757:
6104:
2620:
497:
7587:
7209:
7437:
4564:
4272:
In the physics literature, the tensor product with the identity operator is often suppressed in the notation, with the formula written as
8052:
7999:
502:
2380:
1042:
8608:
492:
487:
7126:
4756:
8623:
8236:
be a finite-dimensional semisimple Lie algebra over a field of characteristic zero. (in the solvable or nilpotent case, one studies
5851:
4861:
8999:
8851:
1953:
1838:
307:
8409:-module structure allows harmonic analysis to be carried out in a way similar to that on connected compact semisimple Lie groups.
2294:
4937:
3483:
1542:
1386:
571:
454:
8968:
1121:
8267:
turns out to be too large especially for homological algebra methods to be useful: it was realized that a smaller subcategory
8950:
8833:
8803:
7854:
7815:
7893:
6965:
892:
602:
2147:
8628:
1902:
9009:
8884:
8866:
5142:
If we work in a basis, then the transpose in the above definition can be interpreted as the ordinary matrix transpose.
9033:
8929:
8902:
8780:
8758:
4265:
3691:
3092:
1347:
8633:
4195:{\displaystyle (\rho _{1}\otimes \rho _{2})(X)=\rho _{1}(X)\otimes \mathrm {I} +\mathrm {I} \otimes \rho _{2}(X)}
2555:
Thus, the span of these three operators forms a Lie algebra, which is isomorphic to the Lie algebra so(3) of the
464:
8817:
8603:
5572:
5510:
5289:
5227:
6234:
8598:
8464:
6357:
3036:
699:
459:
439:
2183:
9064:
5699:
1894:
714:
404:
312:
6424:
4077:{\displaystyle \rho _{1}\otimes \rho _{2}:{\mathfrak {g}}\rightarrow {\mathfrak {gl}}(V_{1}\otimes V_{2})}
3621:
3493:
if the adjoint representation is semisimple. Certainly, every (finite-dimensional) semisimple Lie algebra
2800:
2271:
9023:
2215:
1711:
952:
856:
780:
7554:
7521:
7093:
7032:
6999:
6900:
6672:
6588:
6324:
6190:
6039:
5995:
5914:
5881:
5820:
5763:
8821:
3651:
2050:
1833:
717:, associated with the Lie algebra plays an important role. The universality of this ring says that the
4829:
1889:
at the identities is a Lie algebra homomorphism. In particular, for a finite-dimensional vector space
1547:
The most basic example of a Lie algebra representation is the adjoint representation of a Lie algebra
8348:
8317:
8246:
8215:
7975:
7951:
7927:
7633:
7497:
7473:
7279:
7251:
7185:
7065:
6941:
6876:
6733:
6705:
6648:
6621:
6560:
6463:
6400:
6300:
6080:
5971:
5947:
5857:
5796:
5793:. The universal property of the universal enveloping algebra guarantees that every representation of
5739:
5707:
5630:
5548:
5446:
5265:
5203:
5179:
5057:
4616:
4540:
3751:
3699:
3565:
3524:
3496:
3449:
3378:
3289:
3206:
3191:
A simple but useful tool in studying irreducible representations is Schur's lemma. It has two parts:
3012:
2988:
2928:
2892:
2668:
2579:
is any subspace of the quantum
Hilbert space that is invariant under the angular momentum operators,
2371:
2330:
2239:
2011:
1799:
1775:
1550:
1479:
1316:
1256:
1223:
992:
736:
692:
7661:
6843:
3952:
3912:
3899:{\displaystyle X\cdot (v_{1}\otimes v_{2})=(X\cdot v_{1})\otimes v_{2}+v_{1}\otimes (X\cdot v_{2}).}
2736:
8339:
5436:{\displaystyle \operatorname {Hom} _{\mathfrak {g}}(V,W)=\operatorname {Hom} (V,W)^{\mathfrak {g}}}
1034:
595:
79:
4490:
4454:
4417:
4381:
2108:
8976:
8790:
BĂ€uerle, G.G.A; de Kerf, E.A.; ten Kroode, A.P.E. (1997). A. van
Groesen; E.M. de Jager (eds.).
4261:
3589:
8618:
6727:
5030:
5003:
4257:
4232:
4205:
3471:
2261:
718:
399:
362:
330:
317:
6170:
3313:
3230:
2952:
8285:
3734:
as their underlying vector spaces, then the tensor product of the representations would have
3490:
620:
431:
99:
6487:
2774:
174:
164:
154:
144:
8843:
8792:
8586:
8564:
8560:
8439:
5654:
4640:
2854:
2556:
2033:
1739:
1289:
1098:
722:
691:
satisfying some fixed set of commutation relations, such as the relations satisfied by the
636:
59:
49:
8372:
8293:
5133:{\displaystyle (AB)^{\operatorname {tr} }=B^{\operatorname {tr} }A^{\operatorname {tr} }.}
8:
8574:
8456:
7694:
6726:-module by extending the adjoint representation. But one can also use the left and right
6582:
5153:
4532:
3139:
588:
576:
417:
247:
9004:. Cambridge Studies in Advanced Mathematics. Vol. 113. Cambridge University Press.
8563:
of itself. This is a representation on an algebra: the (anti)derivation property is the
8310:
is a
Hilbert-space representation of, say, a connected real semisimple linear Lie group
3407:(or semisimple) if it is isomorphic to a direct sum of irreducible representations (cf.
9068:
8638:
8392:
6537:
5733:
5610:
5490:
5470:
4914:
4667:
3345:
3262:
2834:
2712:
2692:
2601:
2582:
2562:
1016:
836:
816:
760:
710:
674:
654:
348:
338:
8989:
6033:, and Verma modules are constructed as quotients of the universal enveloping algebra.
9029:
9005:
8964:
8946:
8925:
8898:
8880:
8862:
8829:
8799:
8776:
8754:
3408:
412:
375:
2366:
In quantum theory, one considers "observables" that are self-adjoint operators on a
527:
265:
9019:
8984:
8960:
3186:
2355:
547:
227:
219:
211:
203:
195:
128:
109:
69:
8207:
2370:. The commutation relations among these operators are then an important tool. The
706:
of a Lie group are the integrated form of the representations of its Lie algebra.
8938:
8839:
8582:
8567:
8237:
7276:. It satisfies (and is in fact characterized by) the universal property: for any
1705:
703:
532:
285:
270:
41:
8556:, we simply drop all the gradings and the (−1) to the some power factors.
8538:
8468:
8430:
6074:
552:
370:
275:
6397:
implies that the canonical map is actually injective. Thus, every Lie algebra
537:
9085:
8809:
8794:
Finite and infinite dimensional Lie algebras and their application in physics
8773:
Finite and infinite dimensional Lie algebras and their application in physics
6221:
2367:
1886:
260:
89:
8771:
BĂ€uerle, G.G.A; de Kerf, E.A. (1990). A. van
Groesen; E.M. de Jager (eds.).
6962:
be a finite-dimensional Lie algebra over a field of characteristic zero and
4743:{\displaystyle \rho ^{*}:{\mathfrak {g}}\rightarrow {\mathfrak {gl}}(V^{*})}
9067:(2012). "Beilinson-Bernstein localization over the Harish-Chandra center".
8828:. Graduate Texts in Mathematics. Vol. 129. New York: Springer-Verlag.
8499:
8483:
8281:
6669:
is a module over itself via adjoint representation, the enveloping algebra
6030:
1765:
1342:
644:
640:
557:
542:
343:
325:
255:
9025:
Representation theory of semisimple groups. An overview based on examples.
8879:, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,
8877:
Lie Groups, Lie
Algebras, and Representations: An Elementary Introduction
8613:
8578:
8426:
6394:
4367:{\displaystyle (\rho _{1}\otimes \rho _{2})(X)=\rho _{1}(X)+\rho _{2}(X)}
2914:
1821:
632:
616:
383:
299:
23:
8585:. The analogous observation for Lie superalgebras gives the notion of a
6833:{\displaystyle l_{X}(Y)=XY,X\in {\mathfrak {g}},Y\in U({\mathfrak {g}})}
6160:{\displaystyle T=\oplus _{n=0}^{\infty }\otimes _{1}^{n}{\mathfrak {g}}}
5968:. Thus, there is a one-to-one correspondence between representations of
2658:{\displaystyle \rho :{\mathfrak {g}}\rightarrow \operatorname {End} (V)}
8268:
648:
522:
388:
280:
8897:, Oxford Graduate Texts in Mathematics, Oxford Science Publications,
7619:{\displaystyle \operatorname {Ind} _{\mathfrak {h}}^{\mathfrak {g}}W}
7241:{\displaystyle \operatorname {Ind} _{\mathfrak {h}}^{\mathfrak {g}}W}
1769:
19:
9073:
8240:
of the enveloping algebra; cf. Dixmier for the definitive account.)
7463:{\displaystyle \operatorname {Ind} _{\mathfrak {h}}^{\mathfrak {g}}}
4606:{\displaystyle \rho :{\mathfrak {g}}\rightarrow {\mathfrak {gl}}(V)}
8095:{\displaystyle {\mathfrak {h}}_{1}={\mathfrak {h}}/{\mathfrak {n}}}
8042:{\displaystyle {\mathfrak {g}}_{1}={\mathfrak {g}}/{\mathfrak {n}}}
5681:
3427:
is an invariant subspace, then there is another invariant subspace
3419:
is completely reducible if and only if every invariant subspace of
721:
of representations of a Lie algebra is the same as the category of
9028:, Princeton Landmarks in Mathematics, Princeton University Press,
8389:
allows algebraic especially homological methods to be applied and
2544:{\displaystyle =i\hbar L_{z},\;\;=i\hbar L_{x},\;\;=i\hbar L_{y},}
2268:. Applying the preceding, one gets the Lie algebra representation
1084:{\displaystyle \rho \colon {\mathfrak {g}}\to {\mathfrak {gl}}(V)}
8959:
479:
7175:{\displaystyle U({\mathfrak {g}})\otimes _{U({\mathfrak {h}})}W}
8798:. Studies in mathematical physics. Vol. 7. North-Holland.
8775:. Studies in mathematical physics. Vol. 1. North-Holland.
4816:{\displaystyle \rho ^{*}(X)=-(\rho (X))^{\operatorname {tr} },}
4664:
be the dual space, that is, the space of linear functionals on
709:
In the study of representations of a Lie algebra, a particular
4904:{\displaystyle A^{\operatorname {tr} }:V^{*}\rightarrow V^{*}}
8243:
The category of (possibly infinite-dimensional) modules over
1998:{\displaystyle d\phi :{\mathfrak {g}}\to {\mathfrak {gl}}(V)}
1878:{\displaystyle d_{e}\phi :{\mathfrak {g}}\to {\mathfrak {h}}}
3992:
In the language of homomorphisms, this means that we define
2320:{\displaystyle d_{e}\operatorname {Ad} =\operatorname {ad} }
4990:{\displaystyle (A^{\operatorname {tr} }\phi )(v)=\phi (Av)}
1462:{\displaystyle \cdot v=X\cdot (Y\cdot v)-Y\cdot (X\cdot v)}
651:. In the language of physics, one looks for a vector space
31:
8861:, Graduate Texts in Mathematics, vol. 267, Springer,
4487:
might, for example, be the orbital angular momentum while
1731:
6930:. The right regular representation is defined similarly.
2612:
1206:{\displaystyle \rho ()=\rho (X)\rho (Y)-\rho (Y)\rho (X)}
889:
into a Lie algebra with bracket given by the commutator:
8924:, Graduate Texts in Mathematics, vol. 9, Springer,
8789:
8314:, then it has two natural actions: the complexification
1508:. This is related to the previous definition by setting
7883:{\displaystyle {\mathfrak {h}}\subset {\mathfrak {h}}'}
7844:{\displaystyle {\mathfrak {h'}}\subset {\mathfrak {g}}}
1736:
A Lie algebra representation also arises in nature. If
8921:
Introduction to Lie
Algebras and Representation Theory
8912:
D-modules, perverse sheaves, and representation theory
8624:
Representation theory of a connected compact Lie group
7917:{\displaystyle {\mathfrak {h}}\subset {\mathfrak {g}}}
6989:{\displaystyle {\mathfrak {h}}\subset {\mathfrak {g}}}
938:{\displaystyle =\rho \circ \sigma -\sigma \circ \rho }
8852:
8395:
8375:
8351:
8320:
8296:
8249:
8218:
8208:
Infinite-dimensional representations and "category O"
8108:
8055:
8002:
7978:
7954:
7930:
7896:
7857:
7818:
7728:
7697:
7664:
7636:
7590:
7557:
7524:
7500:
7476:
7440:
7312:
7282:
7254:
7212:
7188:
7129:
7096:
7068:
7035:
7002:
6968:
6944:
6903:
6879:
6846:
6760:
6736:
6708:
6675:
6651:
6624:
6591:
6563:
6540:
6490:
6466:
6427:
6403:
6360:
6327:
6303:
6237:
6193:
6173:
6107:
6083:
6042:
5998:
5974:
5950:
5917:
5884:
5860:
5823:
5799:
5766:
5742:
5710:
5657:
5633:
5613:
5575:
5551:
5513:
5493:
5473:
5449:
5369:
5292:
5268:
5230:
5206:
5182:
5156:
5084:
5060:
5033:
5006:
4940:
4917:
4864:
4832:
4759:
4690:
4670:
4643:
4619:
4567:
4543:
4493:
4457:
4420:
4384:
4281:
4235:
4208:
4093:
3998:
3955:
3915:
3781:
3754:
3702:
3685:
3654:
3624:
3592:
3568:
3527:
3499:
3452:
3381:
3348:
3316:
3292:
3265:
3233:
3209:
3142:
3095:
3039:
3015:
2991:
2955:
2931:
2895:
2857:
2837:
2803:
2777:
2739:
2715:
2695:
2671:
2623:
2585:
2565:
2383:
2333:
2297:
2274:
2242:
2218:
2186:
2150:
2111:
2053:
2014:
1956:
1905:
1841:
1802:
1778:
1742:
1714:
1580:
1553:
1482:
1389:
1350:
1319:
1292:
1259:
1226:
1124:
1101:
1045:
1019:
995:
955:
895:
859:
839:
819:
783:
763:
739:
677:
657:
647:) in such a way that the Lie bracket is given by the
8910:
Ryoshi Hotta, Kiyoshi Takeuchi, Toshiyuki Tanisaki,
5545:
that are invariant under the just-defined action of
2173:{\displaystyle \operatorname {GL} ({\mathfrak {g}})}
4414:acts on the first factor in the tensor product and
3748:as the underlying vector space, with the action of
1937:{\displaystyle \phi :G\to \operatorname {GL} (V)\,}
8895:Lie Groups - An Introduction Through Linear Groups
8791:
8401:
8381:
8361:
8330:
8302:
8259:
8228:
8196:
8094:
8041:
7988:
7964:
7940:
7916:
7882:
7843:
7804:
7711:
7683:
7646:
7618:
7576:
7543:
7510:
7486:
7462:
7422:
7292:
7264:
7240:
7198:
7174:
7115:
7078:
7054:
7021:
6988:
6954:
6922:
6889:
6865:
6832:
6746:
6718:
6694:
6661:
6634:
6610:
6573:
6546:
6526:
6476:
6452:
6413:
6385:
6346:
6313:
6285:
6212:
6179:
6159:
6093:
6061:
6017:
5984:
5960:
5936:
5903:
5870:
5842:
5809:
5785:
5752:
5720:
5670:
5643:
5619:
5599:
5561:
5537:
5499:
5479:
5459:
5435:
5355:
5278:
5254:
5216:
5192:
5168:
5132:
5070:
5046:
5019:
4989:
4923:
4903:
4850:
4815:
4742:
4676:
4656:
4629:
4605:
4553:
4515:
4479:
4442:
4406:
4366:
4248:
4221:
4194:
4076:
3981:
3941:
3898:
3764:
3712:
3669:
3648:. The set of all invariant elements is denoted by
3640:
3610:
3578:
3537:
3509:
3462:
3391:
3354:
3334:
3302:
3271:
3251:
3219:
3154:
3124:
3081:
3025:
3001:
2973:
2941:
2905:
2869:
2843:
2819:
2789:
2763:
2721:
2701:
2681:
2657:
2591:
2571:
2543:
2343:
2319:
2283:
2252:
2224:
2204:
2172:
2136:
2097:
2024:
1997:
1936:
1877:
1812:
1788:
1748:
1720:
1693:
1563:
1492:
1461:
1372:
1329:
1298:
1269:
1236:
1205:
1107:
1083:
1025:
1005:
977:
937:
881:
845:
825:
805:
769:
749:
683:
663:
8559:A Lie (super)algebra is an algebra and it has an
2374:, for example, satisfy the commutation relations
9083:
6228:by the ideal generated by elements of the form
5688:Representation theory of semisimple Lie algebras
5682:Representation theory of semisimple Lie algebras
5145:
3696:If we have two representations of a Lie algebra
2881:is also used for an irreducible representation.
1279:. (Many authors abuse terminology and refer to
450:Representation theory of semisimple Lie algebras
8412:
7890:. The induction commutes with restriction: let
5627:to be the base field, we recover the action of
3310:-module over an algebraically closed field and
9001:An Introduction to Lie Groups and Lie Algebras
8770:
8753:, Amsterdam, New York, Oxford: North-Holland,
9062:
7630:is simple (resp. absolutely simple). Here, a
6618:is the symmetric algebra of the vector space
3125:{\displaystyle X\in {\mathfrak {g}},\,v\in V}
1115:should be a linear map and it should satisfy
596:
8816:
6421:can be embedded into an associative algebra
6354:obtained by restricting the quotient map of
2864:
2858:
1373:{\displaystyle {\mathfrak {g}}\times V\to V}
8969:"Representation of Jordan and Lie Algebras"
5736:called the universal enveloping algebra of
3772:uniquely determined by the assumption that
833:, that is, the space of all linear maps of
698:The notion is closely related to that of a
671:together with a collection of operators on
7626:is simple (resp. absolutely simple), then
3423:has an invariant complement. (That is, if
2489:
2488:
2436:
2435:
1536:
603:
589:
488:Particle physics and representation theory
30:
9072:
8988:
8917:
7470:is an exact functor from the category of
6933:
6167:and the multiplication on it is given by
5600:{\displaystyle \operatorname {Hom} (V,W)}
5538:{\displaystyle \operatorname {Hom} (V,W)}
5356:{\displaystyle (X\cdot f)(v)=Xf(v)-f(Xv)}
5255:{\displaystyle \operatorname {Hom} (V,W)}
3112:
2827:. A nonzero representation is said to be
1933:
8997:
8937:
8892:
8730:
6286:{\displaystyle -(X\otimes Y-Y\otimes X)}
3474:over a field of characteristic zero and
3366:
8748:
8694:
8609:Weyl's theorem on complete reducibility
6386:{\displaystyle T\to U({\mathfrak {g}})}
4526:
3082:{\displaystyle f(X\cdot v)=X\cdot f(v)}
1732:Infinitesimal Lie group representations
1543:Adjoint representation of a Lie algebra
455:Representations of classical Lie groups
9084:
5693:
4684:. Then we can define a representation
4202:. This is called the Kronecker sum of
3680:
2613:Invariant subspaces and irreducibility
2205:{\displaystyle \operatorname {Ad} (g)}
1947:determines a Lie algebra homomorphism
9092:Representation theory of Lie algebras
9046:
9018:
8826:Representation theory. A first course
3375:be a representation of a Lie algebra
3362:is a scalar multiple of the identity.
2361:
813:denote the space of endomorphisms of
8874:
8856:
8718:
8706:
8682:
8670:
8658:
8425:on an algebra is a (not necessarily
6453:{\displaystyle A=U({\mathfrak {g}})}
5000:The minus sign in the definition of
4911:is defined as the "composition with
3641:{\displaystyle x\in {\mathfrak {g}}}
3484:Weyl's complete reducibility theorem
3166:. Such maps are also referred to as
2831:if the only invariant subspaces are
2820:{\displaystyle X\in {\mathfrak {g}}}
2284:{\displaystyle d\operatorname {Ad} }
2040:), i.e. the endomorphism algebra of
728:
308:Lie groupâLie algebra correspondence
8354:
8323:
8252:
8221:
8186:
8182:
8172:
8168:
8152:
8137:
8122:
8115:
8087:
8075:
8059:
8034:
8022:
8006:
7981:
7957:
7933:
7909:
7899:
7871:
7860:
7836:
7822:
7792:
7784:
7769:
7758:
7742:
7735:
7639:
7604:
7597:
7566:
7533:
7518:-modules. These uses the fact that
7503:
7479:
7454:
7447:
7405:
7398:
7374:
7344:
7337:
7319:
7285:
7257:
7226:
7219:
7191:
7159:
7138:
7105:
7071:
7044:
7011:
6981:
6971:
6947:
6912:
6882:
6822:
6800:
6754:-module; namely, with the notation
6739:
6711:
6684:
6654:
6627:
6600:
6566:
6469:
6442:
6406:
6375:
6336:
6306:
6297:There is a natural linear map from
6202:
6152:
6086:
6051:
6007:
5977:
5953:
5926:
5893:
5863:
5832:
5802:
5775:
5745:
5713:
5636:
5554:
5452:
5427:
5376:
5271:
5209:
5185:
5063:
4719:
4716:
4706:
4622:
4589:
4586:
4576:
4546:
4040:
4037:
4027:
3757:
3705:
3661:
3633:
3571:
3530:
3502:
3455:
3384:
3295:
3212:
3104:
3018:
2994:
2934:
2898:
2812:
2674:
2632:
2336:
2245:
2225:{\displaystyle \operatorname {Ad} }
2162:
2017:
1981:
1978:
1968:
1870:
1860:
1805:
1781:
1721:{\displaystyle \operatorname {ad} }
1616:
1606:
1603:
1593:
1556:
1485:
1353:
1322:
1262:
1248:, together with the representation
1229:
1067:
1064:
1054:
998:
978:{\displaystyle {\mathfrak {gl}}(V)}
961:
958:
882:{\displaystyle {\mathfrak {gl}}(V)}
865:
862:
806:{\displaystyle {\mathfrak {gl}}(V)}
789:
786:
742:
13:
9056:
9049:Lie Groups Beyond and Introduction
7691:is simple for any field extension
7577:{\displaystyle U({\mathfrak {h}})}
7544:{\displaystyle U({\mathfrak {g}})}
7116:{\displaystyle U({\mathfrak {g}})}
7055:{\displaystyle U({\mathfrak {g}})}
7022:{\displaystyle U({\mathfrak {h}})}
6923:{\displaystyle U({\mathfrak {g}})}
6695:{\displaystyle U({\mathfrak {g}})}
6611:{\displaystyle U({\mathfrak {g}})}
6460:in such a way that the bracket on
6347:{\displaystyle U({\mathfrak {g}})}
6213:{\displaystyle U({\mathfrak {g}})}
6130:
6062:{\displaystyle U({\mathfrak {g}})}
6018:{\displaystyle U({\mathfrak {g}})}
5937:{\displaystyle U({\mathfrak {g}})}
5911:, so that every representation of
5904:{\displaystyle U({\mathfrak {g}})}
5843:{\displaystyle U({\mathfrak {g}})}
5817:gives rise to a representation of
5786:{\displaystyle U({\mathfrak {g}})}
5678:given in the previous subsection.
4166:
4158:
3686:Tensor products of representations
14:
9103:
8990:10.1090/s0002-9947-1949-0029366-6
8859:Quantum Theory for Mathematicians
8459:and in addition, the elements of
7062:from the right and thus, for any
6730:to make the enveloping algebra a
4266:Tensor product of representations
3692:Tensor product of representations
3670:{\displaystyle V^{\mathfrak {g}}}
3279:is either zero or an isomorphism.
2607:
2525:
2472:
2419:
2144:at the identity is an element of
2098:{\displaystyle c_{g}(x)=gxg^{-1}}
8629:Whitehead's lemma (Lie algebras)
5054:is actually a representation of
4851:{\displaystyle A:V\rightarrow V}
3180:
2884:
2327:, the adjoint representation of
9039:(elementary treatment for SL(2,
8914:; translated by Kiyoshi Takeuch
8554:representation of a Lie algebra
8362:{\displaystyle {\mathfrak {g}}}
8331:{\displaystyle {\mathfrak {g}}}
8275:
8260:{\displaystyle {\mathfrak {g}}}
8229:{\displaystyle {\mathfrak {g}}}
7989:{\displaystyle {\mathfrak {h}}}
7965:{\displaystyle {\mathfrak {g}}}
7941:{\displaystyle {\mathfrak {n}}}
7647:{\displaystyle {\mathfrak {g}}}
7511:{\displaystyle {\mathfrak {g}}}
7487:{\displaystyle {\mathfrak {h}}}
7293:{\displaystyle {\mathfrak {g}}}
7265:{\displaystyle {\mathfrak {g}}}
7199:{\displaystyle {\mathfrak {g}}}
7079:{\displaystyle {\mathfrak {h}}}
6955:{\displaystyle {\mathfrak {g}}}
6890:{\displaystyle {\mathfrak {g}}}
6747:{\displaystyle {\mathfrak {g}}}
6719:{\displaystyle {\mathfrak {g}}}
6662:{\displaystyle {\mathfrak {g}}}
6635:{\displaystyle {\mathfrak {g}}}
6574:{\displaystyle {\mathfrak {g}}}
6477:{\displaystyle {\mathfrak {g}}}
6414:{\displaystyle {\mathfrak {g}}}
6314:{\displaystyle {\mathfrak {g}}}
6094:{\displaystyle {\mathfrak {g}}}
5985:{\displaystyle {\mathfrak {g}}}
5961:{\displaystyle {\mathfrak {g}}}
5871:{\displaystyle {\mathfrak {g}}}
5810:{\displaystyle {\mathfrak {g}}}
5753:{\displaystyle {\mathfrak {g}}}
5721:{\displaystyle {\mathfrak {g}}}
5644:{\displaystyle {\mathfrak {g}}}
5562:{\displaystyle {\mathfrak {g}}}
5460:{\displaystyle {\mathfrak {g}}}
5279:{\displaystyle {\mathfrak {g}}}
5217:{\displaystyle {\mathfrak {g}}}
5193:{\displaystyle {\mathfrak {g}}}
5071:{\displaystyle {\mathfrak {g}}}
4630:{\displaystyle {\mathfrak {g}}}
4554:{\displaystyle {\mathfrak {g}}}
3765:{\displaystyle {\mathfrak {g}}}
3713:{\displaystyle {\mathfrak {g}}}
3579:{\displaystyle {\mathfrak {g}}}
3538:{\displaystyle {\mathfrak {g}}}
3510:{\displaystyle {\mathfrak {g}}}
3463:{\displaystyle {\mathfrak {g}}}
3392:{\displaystyle {\mathfrak {g}}}
3303:{\displaystyle {\mathfrak {g}}}
3220:{\displaystyle {\mathfrak {g}}}
3026:{\displaystyle {\mathfrak {g}}}
3002:{\displaystyle {\mathfrak {g}}}
2942:{\displaystyle {\mathfrak {g}}}
2906:{\displaystyle {\mathfrak {g}}}
2682:{\displaystyle {\mathfrak {g}}}
2344:{\displaystyle {\mathfrak {g}}}
2253:{\displaystyle {\mathfrak {g}}}
2025:{\displaystyle {\mathfrak {g}}}
1813:{\displaystyle {\mathfrak {h}}}
1789:{\displaystyle {\mathfrak {g}}}
1728:is a Lie algebra homomorphism.
1647:
1627:
1564:{\displaystyle {\mathfrak {g}}}
1493:{\displaystyle {\mathfrak {g}}}
1330:{\displaystyle {\mathfrak {g}}}
1283:itself as the representation).
1270:{\displaystyle {\mathfrak {g}}}
1237:{\displaystyle {\mathfrak {g}}}
1006:{\displaystyle {\mathfrak {g}}}
750:{\displaystyle {\mathfrak {g}}}
629:representation of a Lie algebra
8724:
8712:
8700:
8688:
8676:
8664:
8652:
8604:Weight (representation theory)
8417:If we have a Lie superalgebra
7684:{\displaystyle V\otimes _{k}F}
7571:
7561:
7538:
7528:
7417:
7383:
7362:
7328:
7164:
7154:
7143:
7133:
7110:
7100:
7049:
7039:
7016:
7006:
6917:
6907:
6866:{\displaystyle X\mapsto l_{X}}
6850:
6827:
6817:
6777:
6771:
6689:
6679:
6605:
6595:
6503:
6491:
6447:
6437:
6380:
6370:
6364:
6341:
6331:
6280:
6256:
6250:
6238:
6207:
6197:
6056:
6046:
6012:
6002:
5931:
5921:
5898:
5888:
5837:
5827:
5780:
5770:
5732:, one can associate a certain
5594:
5582:
5532:
5520:
5422:
5409:
5397:
5385:
5350:
5341:
5332:
5326:
5314:
5308:
5305:
5293:
5249:
5237:
5095:
5085:
4984:
4975:
4966:
4960:
4957:
4941:
4888:
4842:
4801:
4797:
4791:
4785:
4776:
4770:
4737:
4724:
4711:
4600:
4594:
4581:
4523:is the spin angular momentum.
4510:
4504:
4474:
4468:
4437:
4431:
4401:
4395:
4361:
4355:
4339:
4333:
4317:
4311:
4308:
4282:
4189:
4183:
4151:
4145:
4129:
4123:
4120:
4094:
4071:
4045:
4032:
3982:{\displaystyle v_{2}\in V_{2}}
3942:{\displaystyle v_{1}\in V_{1}}
3890:
3871:
3839:
3820:
3814:
3788:
3326:
3243:
3076:
3070:
3055:
3043:
2965:
2764:{\displaystyle \rho (X)w\in W}
2749:
2743:
2652:
2646:
2637:
2516:
2490:
2463:
2437:
2410:
2384:
2199:
2193:
2167:
2157:
2128:
2070:
2064:
1992:
1986:
1973:
1930:
1924:
1915:
1865:
1685:
1673:
1667:
1661:
1631:
1621:
1611:
1598:
1456:
1444:
1432:
1420:
1402:
1390:
1364:
1313:One can equivalently define a
1200:
1194:
1188:
1182:
1173:
1167:
1161:
1155:
1146:
1143:
1131:
1128:
1078:
1072:
1059:
972:
966:
908:
896:
876:
870:
800:
794:
503:Galilean group representations
498:Poincaré group representations
1:
9051:(second ed.), Birkhauser
8739:
8599:Representation of a Lie group
8573:If a vector space is both an
8445:which is a representation of
7722:The induction is transitive:
5146:Representation on linear maps
4258:Matrix addition#Kronecker_sum
3549:
2212:one obtains a representation
725:over its enveloping algebra.
700:representation of a Lie group
493:Lorentz group representations
460:Theorem of the highest weight
8413:Representation on an algebra
7551:is a free right module over
7494:-modules to the category of
6873:defines a representation of
5700:Universal enveloping algebra
4516:{\displaystyle \rho _{2}(X)}
4480:{\displaystyle \rho _{1}(X)}
4443:{\displaystyle \rho _{2}(x)}
4407:{\displaystyle \rho _{1}(x)}
4378:where it is understood that
4262:Kronecker product#Properties
3489:A Lie algebra is said to be
3478:is finite-dimensional, then
3415:is finite-dimensional, then
2949:-modules. Then a linear map
2137:{\displaystyle c_{g}:G\to G}
1895:representation of Lie groups
1095:Explicitly, this means that
715:universal enveloping algebra
7:
8634:KazhdanâLusztig conjectures
8592:
8421:, then a representation of
5507:are simply the elements of
5467:-module homomorphisms from
5078:, in light of the identity
4264:, and more specifically in
2105:. Then the differential of
1531:
10:
9108:
9047:Knapp, Anthony W. (2002),
8279:
5697:
4530:
3689:
3611:{\displaystyle x\cdot v=0}
3184:
2851:itself and the zero space
2372:angular momentum operators
2032:to the Lie algebra of the
1540:
1337:-module as a vector space
777:be a vector space. We let
693:angular momentum operators
625:Lie algebra representation
445:Lie algebra representation
8918:Humphreys, James (1972),
8641:- analog of Schur's lemma
6393:to degree one piece. The
5047:{\displaystyle \rho ^{*}}
5027:is needed to ensure that
5020:{\displaystyle \rho ^{*}}
4858:, the transpose operator
4249:{\displaystyle \rho _{2}}
4222:{\displaystyle \rho _{1}}
2689:, we say that a subspace
1704:Indeed, by virtue of the
757:be a Lie algebra and let
8645:
8340:maximal compact subgroup
7658:is absolutely simple if
7090:, one can form the left
6180:{\displaystyle \otimes }
3470:is a finite-dimensional
3342:is a homomorphism, then
3335:{\displaystyle f:V\to V}
3259:is a homomorphism, then
3252:{\displaystyle f:V\to W}
2974:{\displaystyle f:V\to W}
1035:Lie algebra homomorphism
440:Lie group representation
8977:Trans. Amer. Math. Soc.
8893:Rossmann, Wulf (2002),
8875:Hall, Brian C. (2015),
8857:Hall, Brian C. (2013),
8552:Now, for the case of a
7851:and any Lie subalgebra
7812:for any Lie subalgebra
6101:. Thus, by definition,
4826:where for any operator
4613:be a representation of
3482:is semisimple; this is
2617:Given a representation
2291:. It can be shown that
1832:respectively, then the
1537:Adjoint representations
465:BorelâWeilâBott theorem
8619:Weyl character formula
8561:adjoint representation
8478:More specifically, if
8403:
8383:
8363:
8332:
8304:
8261:
8230:
8198:
8096:
8043:
7990:
7966:
7942:
7918:
7884:
7845:
7806:
7713:
7685:
7648:
7620:
7578:
7545:
7512:
7488:
7464:
7424:
7294:
7266:
7242:
7200:
7176:
7117:
7080:
7056:
7023:
6990:
6956:
6934:Induced representation
6924:
6891:
6867:
6834:
6748:
6728:regular representation
6720:
6696:
6663:
6636:
6612:
6575:
6548:
6528:
6527:{\displaystyle =XY-YX}
6478:
6454:
6415:
6387:
6348:
6315:
6287:
6214:
6181:
6161:
6095:
6063:
6019:
5986:
5962:
5938:
5905:
5872:
5844:
5811:
5787:
5754:
5722:
5672:
5645:
5621:
5601:
5563:
5539:
5501:
5481:
5461:
5443:; that is to say, the
5437:
5357:
5280:
5256:
5218:
5194:
5170:
5134:
5072:
5048:
5021:
4991:
4925:
4905:
4852:
4817:
4744:
4678:
4658:
4631:
4607:
4555:
4517:
4481:
4444:
4408:
4368:
4250:
4223:
4196:
4078:
3983:
3943:
3900:
3766:
3714:
3671:
3642:
3612:
3580:
3539:
3511:
3472:semisimple Lie algebra
3464:
3393:
3356:
3336:
3304:
3273:
3253:
3221:
3156:
3126:
3083:
3027:
3003:
2975:
2943:
2907:
2871:
2845:
2821:
2791:
2790:{\displaystyle w\in W}
2765:
2723:
2703:
2683:
2659:
2593:
2573:
2545:
2345:
2321:
2285:
2262:adjoint representation
2254:
2226:
2206:
2174:
2138:
2099:
2026:
1999:
1938:
1879:
1814:
1790:
1750:
1722:
1695:
1565:
1494:
1463:
1374:
1331:
1300:
1271:
1238:
1207:
1109:
1085:
1027:
1007:
979:
939:
883:
847:
827:
807:
771:
751:
685:
665:
631:is a way of writing a
363:Semisimple Lie algebra
318:Adjoint representation
8998:Kirillov, A. (2008).
8404:
8384:
8369:-module structure of
8364:
8333:
8305:
8286:Harish-Chandra module
8262:
8231:
8199:
8097:
8044:
7991:
7972:that is contained in
7967:
7943:
7919:
7885:
7846:
7807:
7714:
7686:
7649:
7621:
7579:
7546:
7513:
7489:
7465:
7425:
7295:
7267:
7243:
7201:
7177:
7118:
7081:
7057:
7024:
6991:
6957:
6925:
6892:
6868:
6835:
6749:
6721:
6697:
6664:
6637:
6613:
6576:
6549:
6529:
6479:
6455:
6416:
6388:
6349:
6316:
6288:
6215:
6182:
6162:
6096:
6064:
6020:
5987:
5963:
5944:can be restricted to
5939:
5906:
5873:
5845:
5812:
5788:
5755:
5723:
5673:
5671:{\displaystyle V^{*}}
5646:
5622:
5602:
5564:
5540:
5502:
5482:
5462:
5438:
5358:
5281:
5257:
5219:
5195:
5171:
5135:
5073:
5049:
5022:
4992:
4926:
4906:
4853:
4818:
4745:
4679:
4659:
4657:{\displaystyle V^{*}}
4632:
4608:
4561:be a Lie algebra and
4556:
4518:
4482:
4445:
4409:
4369:
4251:
4224:
4197:
4079:
3984:
3944:
3901:
3767:
3715:
3672:
3643:
3613:
3581:
3540:
3512:
3465:
3435:is the direct sum of
3394:
3367:Complete reducibility
3357:
3337:
3305:
3274:
3254:
3222:
3157:
3127:
3084:
3028:
3004:
2976:
2944:
2908:
2872:
2870:{\displaystyle \{0\}}
2846:
2822:
2792:
2766:
2724:
2704:
2684:
2660:
2594:
2574:
2546:
2346:
2322:
2286:
2255:
2227:
2207:
2175:
2139:
2100:
2027:
2000:
1939:
1880:
1815:
1791:
1768:of (real or complex)
1751:
1749:{\displaystyle \phi }
1723:
1696:
1566:
1495:
1464:
1375:
1332:
1301:
1299:{\displaystyle \rho }
1272:
1239:
1208:
1110:
1108:{\displaystyle \rho }
1086:
1028:
1008:
980:
940:
884:
848:
828:
808:
772:
752:
686:
666:
621:representation theory
432:Representation theory
8749:Dixmier, J. (1977),
8587:Poisson superalgebra
8393:
8382:{\displaystyle \pi }
8373:
8349:
8318:
8303:{\displaystyle \pi }
8294:
8247:
8216:
8106:
8053:
8000:
7976:
7952:
7928:
7894:
7855:
7816:
7726:
7695:
7662:
7634:
7588:
7584:. In particular, if
7555:
7522:
7498:
7474:
7438:
7310:
7280:
7252:
7210:
7186:
7127:
7094:
7066:
7033:
7000:
6966:
6942:
6901:
6877:
6844:
6758:
6734:
6706:
6673:
6649:
6622:
6589:
6561:
6538:
6488:
6464:
6425:
6401:
6358:
6325:
6301:
6235:
6191:
6171:
6105:
6081:
6077:of the vector space
6040:
6036:The construction of
5996:
5972:
5948:
5915:
5882:
5858:
5821:
5797:
5764:
5740:
5708:
5704:To each Lie algebra
5655:
5631:
5611:
5573:
5549:
5511:
5491:
5471:
5447:
5367:
5290:
5266:
5228:
5224:a Lie algebra. Then
5204:
5180:
5154:
5082:
5058:
5031:
5004:
4938:
4915:
4862:
4830:
4757:
4688:
4668:
4641:
4617:
4565:
4541:
4527:Dual representations
4491:
4455:
4418:
4382:
4279:
4233:
4206:
4091:
3996:
3953:
3913:
3779:
3752:
3700:
3652:
3622:
3590:
3566:
3525:
3517:is reductive, since
3497:
3450:
3405:completely reducible
3379:
3346:
3314:
3290:
3263:
3231:
3207:
3140:
3093:
3037:
3033:-equivariant; i.e.,
3013:
2989:
2953:
2929:
2893:
2855:
2835:
2801:
2775:
2737:
2713:
2693:
2669:
2621:
2583:
2563:
2557:rotation group SO(3)
2381:
2331:
2295:
2272:
2240:
2236:on the vector space
2216:
2184:
2148:
2109:
2051:
2034:general linear group
2012:
1954:
1903:
1839:
1800:
1776:
1740:
1712:
1578:
1551:
1480:
1387:
1348:
1317:
1310:if it is injective.
1290:
1257:
1224:
1122:
1099:
1043:
1017:
993:
953:
893:
857:
837:
817:
781:
761:
737:
675:
655:
8751:Enveloping Algebras
8575:associative algebra
8457:graded vector space
8193:
8127:
7801:
7774:
7747:
7712:{\displaystyle F/k}
7609:
7459:
7410:
7349:
7272:-module induced by
7231:
7206:-module denoted by
6149:
6134:
6069:is as follows. Let
5694:Enveloping algebras
5286:-module by setting
5169:{\displaystyle V,W}
4533:Dual representation
3681:Basic constructions
3155:{\displaystyle V,W}
1286:The representation
1244:. The vector space
853:to itself. We make
577:Table of Lie groups
418:Compact Lie algebra
8399:
8379:
8359:
8338:and the connected
8328:
8300:
8257:
8226:
8194:
8161:
8109:
8092:
8039:
7986:
7962:
7938:
7924:be subalgebra and
7914:
7880:
7841:
7802:
7778:
7751:
7729:
7709:
7681:
7644:
7616:
7591:
7574:
7541:
7508:
7484:
7460:
7441:
7420:
7392:
7331:
7290:
7262:
7238:
7213:
7196:
7172:
7113:
7076:
7052:
7019:
6986:
6952:
6920:
6887:
6863:
6830:
6744:
6716:
6692:
6659:
6632:
6608:
6571:
6544:
6524:
6474:
6450:
6411:
6383:
6344:
6311:
6283:
6210:
6177:
6157:
6135:
6114:
6091:
6059:
6015:
5982:
5958:
5934:
5901:
5868:
5850:. Conversely, the
5840:
5807:
5783:
5750:
5718:
5668:
5641:
5617:
5597:
5559:
5535:
5497:
5477:
5457:
5433:
5353:
5276:
5252:
5214:
5190:
5166:
5130:
5068:
5044:
5017:
4987:
4921:
4901:
4848:
4813:
4740:
4674:
4654:
4627:
4603:
4551:
4513:
4477:
4440:
4404:
4364:
4246:
4219:
4192:
4074:
3979:
3939:
3896:
3762:
3710:
3667:
3638:
3608:
3576:
3535:
3521:representation of
3507:
3460:
3389:
3352:
3332:
3300:
3286:is an irreducible
3269:
3249:
3217:
3152:
3122:
3079:
3023:
3009:-modules if it is
2999:
2971:
2939:
2903:
2867:
2841:
2817:
2787:
2761:
2719:
2699:
2679:
2655:
2589:
2569:
2541:
2362:In quantum physics
2341:
2317:
2281:
2250:
2222:
2202:
2170:
2134:
2095:
2022:
1995:
1934:
1875:
1810:
1786:
1746:
1718:
1691:
1561:
1490:
1459:
1370:
1327:
1296:
1267:
1234:
1203:
1105:
1081:
1023:
1003:
975:
935:
879:
843:
823:
803:
767:
747:
681:
661:
349:Affine Lie algebra
339:Simple Lie algebra
80:Special orthogonal
9020:Knapp, Anthony W.
8965:Philip M. Whitman
8952:978-0-486-63832-4
8835:978-0-387-97495-8
8805:978-0-444-82836-1
8402:{\displaystyle K}
6547:{\displaystyle A}
5620:{\displaystyle W}
5500:{\displaystyle W}
5480:{\displaystyle V}
5363:. In particular,
4924:{\displaystyle A}
4677:{\displaystyle V}
3409:semisimple module
3355:{\displaystyle f}
3272:{\displaystyle f}
3168:intertwining maps
2844:{\displaystyle V}
2722:{\displaystyle V}
2702:{\displaystyle W}
2665:of a Lie algebra
2592:{\displaystyle V}
2572:{\displaystyle V}
2180:. Denoting it by
2047:For example, let
1585:
1026:{\displaystyle V}
846:{\displaystyle V}
826:{\displaystyle V}
770:{\displaystyle V}
729:Formal definition
684:{\displaystyle V}
664:{\displaystyle V}
613:
612:
413:Split Lie algebra
376:Cartan subalgebra
238:
237:
129:Simple Lie groups
9099:
9078:
9076:
9063:Ben-Zvi, David;
9052:
9038:
9015:
8994:
8992:
8973:
8961:Garrett Birkhoff
8956:
8939:Jacobson, Nathan
8934:
8907:
8889:
8871:
8847:
8813:
8797:
8786:
8763:
8733:
8728:
8722:
8716:
8710:
8704:
8698:
8692:
8686:
8680:
8674:
8668:
8662:
8656:
8408:
8406:
8405:
8400:
8388:
8386:
8385:
8380:
8368:
8366:
8365:
8360:
8358:
8357:
8337:
8335:
8334:
8329:
8327:
8326:
8309:
8307:
8306:
8301:
8266:
8264:
8263:
8258:
8256:
8255:
8238:primitive ideals
8235:
8233:
8232:
8227:
8225:
8224:
8203:
8201:
8200:
8195:
8192:
8191:
8190:
8189:
8178:
8177:
8176:
8175:
8157:
8156:
8155:
8142:
8141:
8140:
8126:
8125:
8119:
8118:
8101:
8099:
8098:
8093:
8091:
8090:
8084:
8079:
8078:
8069:
8068:
8063:
8062:
8048:
8046:
8045:
8040:
8038:
8037:
8031:
8026:
8025:
8016:
8015:
8010:
8009:
7995:
7993:
7992:
7987:
7985:
7984:
7971:
7969:
7968:
7963:
7961:
7960:
7947:
7945:
7944:
7939:
7937:
7936:
7923:
7921:
7920:
7915:
7913:
7912:
7903:
7902:
7889:
7887:
7886:
7881:
7879:
7875:
7874:
7864:
7863:
7850:
7848:
7847:
7842:
7840:
7839:
7830:
7829:
7828:
7811:
7809:
7808:
7803:
7800:
7799:
7798:
7788:
7787:
7773:
7772:
7766:
7765:
7764:
7746:
7745:
7739:
7738:
7718:
7716:
7715:
7710:
7705:
7690:
7688:
7687:
7682:
7677:
7676:
7653:
7651:
7650:
7645:
7643:
7642:
7625:
7623:
7622:
7617:
7608:
7607:
7601:
7600:
7583:
7581:
7580:
7575:
7570:
7569:
7550:
7548:
7547:
7542:
7537:
7536:
7517:
7515:
7514:
7509:
7507:
7506:
7493:
7491:
7490:
7485:
7483:
7482:
7469:
7467:
7466:
7461:
7458:
7457:
7451:
7450:
7429:
7427:
7426:
7421:
7409:
7408:
7402:
7401:
7379:
7378:
7377:
7348:
7347:
7341:
7340:
7324:
7323:
7322:
7299:
7297:
7296:
7291:
7289:
7288:
7271:
7269:
7268:
7263:
7261:
7260:
7247:
7245:
7244:
7239:
7230:
7229:
7223:
7222:
7205:
7203:
7202:
7197:
7195:
7194:
7181:
7179:
7178:
7173:
7168:
7167:
7163:
7162:
7142:
7141:
7122:
7120:
7119:
7114:
7109:
7108:
7085:
7083:
7082:
7077:
7075:
7074:
7061:
7059:
7058:
7053:
7048:
7047:
7028:
7026:
7025:
7020:
7015:
7014:
6995:
6993:
6992:
6987:
6985:
6984:
6975:
6974:
6961:
6959:
6958:
6953:
6951:
6950:
6929:
6927:
6926:
6921:
6916:
6915:
6896:
6894:
6893:
6888:
6886:
6885:
6872:
6870:
6869:
6864:
6862:
6861:
6839:
6837:
6836:
6831:
6826:
6825:
6804:
6803:
6770:
6769:
6753:
6751:
6750:
6745:
6743:
6742:
6725:
6723:
6722:
6717:
6715:
6714:
6701:
6699:
6698:
6693:
6688:
6687:
6668:
6666:
6665:
6660:
6658:
6657:
6641:
6639:
6638:
6633:
6631:
6630:
6617:
6615:
6614:
6609:
6604:
6603:
6580:
6578:
6577:
6572:
6570:
6569:
6553:
6551:
6550:
6545:
6533:
6531:
6530:
6525:
6483:
6481:
6480:
6475:
6473:
6472:
6459:
6457:
6456:
6451:
6446:
6445:
6420:
6418:
6417:
6412:
6410:
6409:
6392:
6390:
6389:
6384:
6379:
6378:
6353:
6351:
6350:
6345:
6340:
6339:
6320:
6318:
6317:
6312:
6310:
6309:
6292:
6290:
6289:
6284:
6219:
6217:
6216:
6211:
6206:
6205:
6186:
6184:
6183:
6178:
6166:
6164:
6163:
6158:
6156:
6155:
6148:
6143:
6133:
6128:
6100:
6098:
6097:
6092:
6090:
6089:
6068:
6066:
6065:
6060:
6055:
6054:
6024:
6022:
6021:
6016:
6011:
6010:
5991:
5989:
5988:
5983:
5981:
5980:
5967:
5965:
5964:
5959:
5957:
5956:
5943:
5941:
5940:
5935:
5930:
5929:
5910:
5908:
5907:
5902:
5897:
5896:
5877:
5875:
5874:
5869:
5867:
5866:
5849:
5847:
5846:
5841:
5836:
5835:
5816:
5814:
5813:
5808:
5806:
5805:
5792:
5790:
5789:
5784:
5779:
5778:
5759:
5757:
5756:
5751:
5749:
5748:
5727:
5725:
5724:
5719:
5717:
5716:
5677:
5675:
5674:
5669:
5667:
5666:
5650:
5648:
5647:
5642:
5640:
5639:
5626:
5624:
5623:
5618:
5606:
5604:
5603:
5598:
5568:
5566:
5565:
5560:
5558:
5557:
5544:
5542:
5541:
5536:
5506:
5504:
5503:
5498:
5486:
5484:
5483:
5478:
5466:
5464:
5463:
5458:
5456:
5455:
5442:
5440:
5439:
5434:
5432:
5431:
5430:
5381:
5380:
5379:
5362:
5360:
5359:
5354:
5285:
5283:
5282:
5277:
5275:
5274:
5261:
5259:
5258:
5253:
5223:
5221:
5220:
5215:
5213:
5212:
5199:
5197:
5196:
5191:
5189:
5188:
5175:
5173:
5172:
5167:
5139:
5137:
5136:
5131:
5126:
5125:
5116:
5115:
5103:
5102:
5077:
5075:
5074:
5069:
5067:
5066:
5053:
5051:
5050:
5045:
5043:
5042:
5026:
5024:
5023:
5018:
5016:
5015:
4996:
4994:
4993:
4988:
4953:
4952:
4930:
4928:
4927:
4922:
4910:
4908:
4907:
4902:
4900:
4899:
4887:
4886:
4874:
4873:
4857:
4855:
4854:
4849:
4822:
4820:
4819:
4814:
4809:
4808:
4769:
4768:
4749:
4747:
4746:
4741:
4736:
4735:
4723:
4722:
4710:
4709:
4700:
4699:
4683:
4681:
4680:
4675:
4663:
4661:
4660:
4655:
4653:
4652:
4636:
4634:
4633:
4628:
4626:
4625:
4612:
4610:
4609:
4604:
4593:
4592:
4580:
4579:
4560:
4558:
4557:
4552:
4550:
4549:
4522:
4520:
4519:
4514:
4503:
4502:
4486:
4484:
4483:
4478:
4467:
4466:
4449:
4447:
4446:
4441:
4430:
4429:
4413:
4411:
4410:
4405:
4394:
4393:
4373:
4371:
4370:
4365:
4354:
4353:
4332:
4331:
4307:
4306:
4294:
4293:
4255:
4253:
4252:
4247:
4245:
4244:
4228:
4226:
4225:
4220:
4218:
4217:
4201:
4199:
4198:
4193:
4182:
4181:
4169:
4161:
4144:
4143:
4119:
4118:
4106:
4105:
4084:by the formula
4083:
4081:
4080:
4075:
4070:
4069:
4057:
4056:
4044:
4043:
4031:
4030:
4021:
4020:
4008:
4007:
3988:
3986:
3985:
3980:
3978:
3977:
3965:
3964:
3948:
3946:
3945:
3940:
3938:
3937:
3925:
3924:
3905:
3903:
3902:
3897:
3889:
3888:
3867:
3866:
3854:
3853:
3838:
3837:
3813:
3812:
3800:
3799:
3771:
3769:
3768:
3763:
3761:
3760:
3719:
3717:
3716:
3711:
3709:
3708:
3676:
3674:
3673:
3668:
3666:
3665:
3664:
3647:
3645:
3644:
3639:
3637:
3636:
3617:
3615:
3614:
3609:
3585:
3583:
3582:
3577:
3575:
3574:
3544:
3542:
3541:
3536:
3534:
3533:
3516:
3514:
3513:
3508:
3506:
3505:
3469:
3467:
3466:
3461:
3459:
3458:
3398:
3396:
3395:
3390:
3388:
3387:
3361:
3359:
3358:
3353:
3341:
3339:
3338:
3333:
3309:
3307:
3306:
3301:
3299:
3298:
3278:
3276:
3275:
3270:
3258:
3256:
3255:
3250:
3226:
3224:
3223:
3218:
3216:
3215:
3203:are irreducible
3161:
3159:
3158:
3153:
3131:
3129:
3128:
3123:
3108:
3107:
3088:
3086:
3085:
3080:
3032:
3030:
3029:
3024:
3022:
3021:
3008:
3006:
3005:
3000:
2998:
2997:
2980:
2978:
2977:
2972:
2948:
2946:
2945:
2940:
2938:
2937:
2912:
2910:
2909:
2904:
2902:
2901:
2876:
2874:
2873:
2868:
2850:
2848:
2847:
2842:
2826:
2824:
2823:
2818:
2816:
2815:
2796:
2794:
2793:
2788:
2770:
2768:
2767:
2762:
2728:
2726:
2725:
2720:
2708:
2706:
2705:
2700:
2688:
2686:
2685:
2680:
2678:
2677:
2664:
2662:
2661:
2656:
2636:
2635:
2598:
2596:
2595:
2590:
2578:
2576:
2575:
2570:
2550:
2548:
2547:
2542:
2537:
2536:
2515:
2514:
2502:
2501:
2484:
2483:
2462:
2461:
2449:
2448:
2431:
2430:
2409:
2408:
2396:
2395:
2356:simply connected
2350:
2348:
2347:
2342:
2340:
2339:
2326:
2324:
2323:
2318:
2307:
2306:
2290:
2288:
2287:
2282:
2259:
2257:
2256:
2251:
2249:
2248:
2231:
2229:
2228:
2223:
2211:
2209:
2208:
2203:
2179:
2177:
2176:
2171:
2166:
2165:
2143:
2141:
2140:
2135:
2121:
2120:
2104:
2102:
2101:
2096:
2094:
2093:
2063:
2062:
2031:
2029:
2028:
2023:
2021:
2020:
2004:
2002:
2001:
1996:
1985:
1984:
1972:
1971:
1943:
1941:
1940:
1935:
1884:
1882:
1881:
1876:
1874:
1873:
1864:
1863:
1851:
1850:
1819:
1817:
1816:
1811:
1809:
1808:
1795:
1793:
1792:
1787:
1785:
1784:
1755:
1753:
1752:
1747:
1727:
1725:
1724:
1719:
1700:
1698:
1697:
1692:
1657:
1656:
1643:
1642:
1620:
1619:
1610:
1609:
1597:
1596:
1587:
1586:
1583:
1570:
1568:
1567:
1562:
1560:
1559:
1499:
1497:
1496:
1491:
1489:
1488:
1468:
1466:
1465:
1460:
1379:
1377:
1376:
1371:
1357:
1356:
1341:together with a
1336:
1334:
1333:
1328:
1326:
1325:
1305:
1303:
1302:
1297:
1276:
1274:
1273:
1268:
1266:
1265:
1243:
1241:
1240:
1235:
1233:
1232:
1212:
1210:
1209:
1204:
1114:
1112:
1111:
1106:
1090:
1088:
1087:
1082:
1071:
1070:
1058:
1057:
1032:
1030:
1029:
1024:
1012:
1010:
1009:
1004:
1002:
1001:
984:
982:
981:
976:
965:
964:
944:
942:
941:
936:
888:
886:
885:
880:
869:
868:
852:
850:
849:
844:
832:
830:
829:
824:
812:
810:
809:
804:
793:
792:
776:
774:
773:
768:
756:
754:
753:
748:
746:
745:
690:
688:
687:
682:
670:
668:
667:
662:
605:
598:
591:
548:Claude Chevalley
405:Complexification
248:Other Lie groups
134:
133:
42:Classical groups
34:
16:
15:
9107:
9106:
9102:
9101:
9100:
9098:
9097:
9096:
9082:
9081:
9059:
9057:Further reading
9036:
9012:
8971:
8953:
8932:
8905:
8887:
8869:
8836:
8806:
8783:
8761:
8742:
8737:
8736:
8729:
8725:
8717:
8713:
8705:
8701:
8697:, Theorem 1.6.3
8693:
8689:
8681:
8677:
8669:
8665:
8657:
8653:
8648:
8639:Quillen's lemma
8595:
8583:Poisson algebra
8568:Jacobi identity
8469:antiderivations
8455:
8436:
8415:
8394:
8391:
8390:
8374:
8371:
8370:
8353:
8352:
8350:
8347:
8346:
8322:
8321:
8319:
8316:
8315:
8295:
8292:
8291:
8288:
8280:Main articles:
8278:
8251:
8250:
8248:
8245:
8244:
8220:
8219:
8217:
8214:
8213:
8210:
8185:
8181:
8180:
8179:
8171:
8167:
8166:
8165:
8151:
8150:
8146:
8136:
8135:
8131:
8121:
8120:
8114:
8113:
8107:
8104:
8103:
8086:
8085:
8080:
8074:
8073:
8064:
8058:
8057:
8056:
8054:
8051:
8050:
8033:
8032:
8027:
8021:
8020:
8011:
8005:
8004:
8003:
8001:
7998:
7997:
7980:
7979:
7977:
7974:
7973:
7956:
7955:
7953:
7950:
7949:
7932:
7931:
7929:
7926:
7925:
7908:
7907:
7898:
7897:
7895:
7892:
7891:
7870:
7869:
7868:
7859:
7858:
7856:
7853:
7852:
7835:
7834:
7821:
7820:
7819:
7817:
7814:
7813:
7791:
7790:
7789:
7783:
7782:
7768:
7767:
7757:
7756:
7755:
7741:
7740:
7734:
7733:
7727:
7724:
7723:
7701:
7696:
7693:
7692:
7672:
7668:
7663:
7660:
7659:
7638:
7637:
7635:
7632:
7631:
7603:
7602:
7596:
7595:
7589:
7586:
7585:
7565:
7564:
7556:
7553:
7552:
7532:
7531:
7523:
7520:
7519:
7502:
7501:
7499:
7496:
7495:
7478:
7477:
7475:
7472:
7471:
7453:
7452:
7446:
7445:
7439:
7436:
7435:
7404:
7403:
7397:
7396:
7373:
7372:
7368:
7343:
7342:
7336:
7335:
7318:
7317:
7313:
7311:
7308:
7307:
7284:
7283:
7281:
7278:
7277:
7256:
7255:
7253:
7250:
7249:
7248:and called the
7225:
7224:
7218:
7217:
7211:
7208:
7207:
7190:
7189:
7187:
7184:
7183:
7158:
7157:
7150:
7146:
7137:
7136:
7128:
7125:
7124:
7104:
7103:
7095:
7092:
7091:
7070:
7069:
7067:
7064:
7063:
7043:
7042:
7034:
7031:
7030:
7010:
7009:
7001:
6998:
6997:
6980:
6979:
6970:
6969:
6967:
6964:
6963:
6946:
6945:
6943:
6940:
6939:
6936:
6911:
6910:
6902:
6899:
6898:
6881:
6880:
6878:
6875:
6874:
6857:
6853:
6845:
6842:
6841:
6821:
6820:
6799:
6798:
6765:
6761:
6759:
6756:
6755:
6738:
6737:
6735:
6732:
6731:
6710:
6709:
6707:
6704:
6703:
6683:
6682:
6674:
6671:
6670:
6653:
6652:
6650:
6647:
6646:
6626:
6625:
6623:
6620:
6619:
6599:
6598:
6590:
6587:
6586:
6565:
6564:
6562:
6559:
6558:
6539:
6536:
6535:
6489:
6486:
6485:
6468:
6467:
6465:
6462:
6461:
6441:
6440:
6426:
6423:
6422:
6405:
6404:
6402:
6399:
6398:
6374:
6373:
6359:
6356:
6355:
6335:
6334:
6326:
6323:
6322:
6305:
6304:
6302:
6299:
6298:
6236:
6233:
6232:
6201:
6200:
6192:
6189:
6188:
6172:
6169:
6168:
6151:
6150:
6144:
6139:
6129:
6118:
6106:
6103:
6102:
6085:
6084:
6082:
6079:
6078:
6050:
6049:
6041:
6038:
6037:
6006:
6005:
5997:
5994:
5993:
5976:
5975:
5973:
5970:
5969:
5952:
5951:
5949:
5946:
5945:
5925:
5924:
5916:
5913:
5912:
5892:
5891:
5883:
5880:
5879:
5862:
5861:
5859:
5856:
5855:
5831:
5830:
5822:
5819:
5818:
5801:
5800:
5798:
5795:
5794:
5774:
5773:
5765:
5762:
5761:
5744:
5743:
5741:
5738:
5737:
5712:
5711:
5709:
5706:
5705:
5702:
5696:
5684:
5662:
5658:
5656:
5653:
5652:
5635:
5634:
5632:
5629:
5628:
5612:
5609:
5608:
5574:
5571:
5570:
5553:
5552:
5550:
5547:
5546:
5512:
5509:
5508:
5492:
5489:
5488:
5472:
5469:
5468:
5451:
5450:
5448:
5445:
5444:
5426:
5425:
5421:
5375:
5374:
5370:
5368:
5365:
5364:
5291:
5288:
5287:
5270:
5269:
5267:
5264:
5263:
5229:
5226:
5225:
5208:
5207:
5205:
5202:
5201:
5184:
5183:
5181:
5178:
5177:
5155:
5152:
5151:
5148:
5121:
5117:
5111:
5107:
5098:
5094:
5083:
5080:
5079:
5062:
5061:
5059:
5056:
5055:
5038:
5034:
5032:
5029:
5028:
5011:
5007:
5005:
5002:
5001:
4948:
4944:
4939:
4936:
4935:
4916:
4913:
4912:
4895:
4891:
4882:
4878:
4869:
4865:
4863:
4860:
4859:
4831:
4828:
4827:
4804:
4800:
4764:
4760:
4758:
4755:
4754:
4750:by the formula
4731:
4727:
4715:
4714:
4705:
4704:
4695:
4691:
4689:
4686:
4685:
4669:
4666:
4665:
4648:
4644:
4642:
4639:
4638:
4621:
4620:
4618:
4615:
4614:
4585:
4584:
4575:
4574:
4566:
4563:
4562:
4545:
4544:
4542:
4539:
4538:
4535:
4529:
4498:
4494:
4492:
4489:
4488:
4462:
4458:
4456:
4453:
4452:
4425:
4421:
4419:
4416:
4415:
4389:
4385:
4383:
4380:
4379:
4349:
4345:
4327:
4323:
4302:
4298:
4289:
4285:
4280:
4277:
4276:
4240:
4236:
4234:
4231:
4230:
4213:
4209:
4207:
4204:
4203:
4177:
4173:
4165:
4157:
4139:
4135:
4114:
4110:
4101:
4097:
4092:
4089:
4088:
4065:
4061:
4052:
4048:
4036:
4035:
4026:
4025:
4016:
4012:
4003:
3999:
3997:
3994:
3993:
3973:
3969:
3960:
3956:
3954:
3951:
3950:
3933:
3929:
3920:
3916:
3914:
3911:
3910:
3884:
3880:
3862:
3858:
3849:
3845:
3833:
3829:
3808:
3804:
3795:
3791:
3780:
3777:
3776:
3756:
3755:
3753:
3750:
3749:
3747:
3740:
3733:
3726:
3704:
3703:
3701:
3698:
3697:
3694:
3688:
3683:
3660:
3659:
3655:
3653:
3650:
3649:
3632:
3631:
3623:
3620:
3619:
3591:
3588:
3587:
3570:
3569:
3567:
3564:
3563:
3552:
3529:
3528:
3526:
3523:
3522:
3501:
3500:
3498:
3495:
3494:
3454:
3453:
3451:
3448:
3447:
3383:
3382:
3380:
3377:
3376:
3369:
3347:
3344:
3343:
3315:
3312:
3311:
3294:
3293:
3291:
3288:
3287:
3264:
3261:
3260:
3232:
3229:
3228:
3211:
3210:
3208:
3205:
3204:
3189:
3183:
3162:are said to be
3141:
3138:
3137:
3103:
3102:
3094:
3091:
3090:
3038:
3035:
3034:
3017:
3016:
3014:
3011:
3010:
2993:
2992:
2990:
2987:
2986:
2954:
2951:
2950:
2933:
2932:
2930:
2927:
2926:
2897:
2896:
2894:
2891:
2890:
2887:
2856:
2853:
2852:
2836:
2833:
2832:
2811:
2810:
2802:
2799:
2798:
2776:
2773:
2772:
2738:
2735:
2734:
2714:
2711:
2710:
2694:
2691:
2690:
2673:
2672:
2670:
2667:
2666:
2631:
2630:
2622:
2619:
2618:
2615:
2610:
2584:
2581:
2580:
2564:
2561:
2560:
2532:
2528:
2510:
2506:
2497:
2493:
2479:
2475:
2457:
2453:
2444:
2440:
2426:
2422:
2404:
2400:
2391:
2387:
2382:
2379:
2378:
2364:
2335:
2334:
2332:
2329:
2328:
2302:
2298:
2296:
2293:
2292:
2273:
2270:
2269:
2244:
2243:
2241:
2238:
2237:
2217:
2214:
2213:
2185:
2182:
2181:
2161:
2160:
2149:
2146:
2145:
2116:
2112:
2110:
2107:
2106:
2086:
2082:
2058:
2054:
2052:
2049:
2048:
2016:
2015:
2013:
2010:
2009:
1977:
1976:
1967:
1966:
1955:
1952:
1951:
1904:
1901:
1900:
1869:
1868:
1859:
1858:
1846:
1842:
1840:
1837:
1836:
1804:
1803:
1801:
1798:
1797:
1780:
1779:
1777:
1774:
1773:
1741:
1738:
1737:
1734:
1713:
1710:
1709:
1706:Jacobi identity
1652:
1648:
1638:
1634:
1615:
1614:
1602:
1601:
1592:
1591:
1582:
1581:
1579:
1576:
1575:
1555:
1554:
1552:
1549:
1548:
1545:
1539:
1534:
1484:
1483:
1481:
1478:
1477:
1388:
1385:
1384:
1352:
1351:
1349:
1346:
1345:
1321:
1320:
1318:
1315:
1314:
1291:
1288:
1287:
1261:
1260:
1258:
1255:
1254:
1228:
1227:
1225:
1222:
1221:
1123:
1120:
1119:
1100:
1097:
1096:
1063:
1062:
1053:
1052:
1044:
1041:
1040:
1018:
1015:
1014:
997:
996:
994:
991:
990:
957:
956:
954:
951:
950:
894:
891:
890:
861:
860:
858:
855:
854:
838:
835:
834:
818:
815:
814:
785:
784:
782:
779:
778:
762:
759:
758:
741:
740:
738:
735:
734:
731:
704:universal cover
676:
673:
672:
656:
653:
652:
609:
564:
563:
562:
533:Wilhelm Killing
517:
509:
508:
507:
482:
471:
470:
469:
434:
424:
423:
422:
409:
393:
371:Dynkin diagrams
365:
355:
354:
353:
335:
313:Exponential map
302:
292:
291:
290:
271:Conformal group
250:
240:
239:
231:
223:
215:
207:
199:
180:
170:
160:
150:
131:
121:
120:
119:
100:Special unitary
44:
12:
11:
5:
9105:
9095:
9094:
9080:
9079:
9058:
9055:
9054:
9053:
9044:
9034:
9016:
9011:978-0521889698
9010:
8995:
8957:
8951:
8935:
8930:
8915:
8908:
8903:
8890:
8886:978-3319134666
8885:
8872:
8868:978-1461471158
8867:
8854:
8850:D. Gaitsgory,
8848:
8834:
8814:
8804:
8787:
8781:
8768:
8765:
8759:
8746:
8741:
8738:
8735:
8734:
8723:
8711:
8699:
8687:
8675:
8663:
8650:
8649:
8647:
8644:
8643:
8642:
8636:
8631:
8626:
8621:
8616:
8611:
8606:
8601:
8594:
8591:
8550:
8549:
8531:
8530:
8453:
8434:
8414:
8411:
8398:
8378:
8356:
8325:
8299:
8277:
8274:
8254:
8223:
8209:
8206:
8188:
8184:
8174:
8170:
8164:
8160:
8154:
8149:
8145:
8139:
8134:
8130:
8124:
8117:
8112:
8089:
8083:
8077:
8072:
8067:
8061:
8036:
8030:
8024:
8019:
8014:
8008:
7983:
7959:
7935:
7911:
7906:
7901:
7878:
7873:
7867:
7862:
7838:
7833:
7827:
7824:
7797:
7794:
7786:
7781:
7777:
7771:
7763:
7760:
7754:
7750:
7744:
7737:
7732:
7708:
7704:
7700:
7680:
7675:
7671:
7667:
7641:
7615:
7612:
7606:
7599:
7594:
7573:
7568:
7563:
7560:
7540:
7535:
7530:
7527:
7505:
7481:
7456:
7449:
7444:
7432:
7431:
7419:
7416:
7413:
7407:
7400:
7395:
7391:
7388:
7385:
7382:
7376:
7371:
7367:
7364:
7361:
7358:
7355:
7352:
7346:
7339:
7334:
7330:
7327:
7321:
7316:
7287:
7259:
7237:
7234:
7228:
7221:
7216:
7193:
7171:
7166:
7161:
7156:
7153:
7149:
7145:
7140:
7135:
7132:
7112:
7107:
7102:
7099:
7073:
7051:
7046:
7041:
7038:
7018:
7013:
7008:
7005:
6996:a subalgebra.
6983:
6978:
6973:
6949:
6935:
6932:
6919:
6914:
6909:
6906:
6884:
6860:
6856:
6852:
6849:
6840:, the mapping
6829:
6824:
6819:
6816:
6813:
6810:
6807:
6802:
6797:
6794:
6791:
6788:
6785:
6782:
6779:
6776:
6773:
6768:
6764:
6741:
6713:
6691:
6686:
6681:
6678:
6656:
6629:
6607:
6602:
6597:
6594:
6568:
6543:
6523:
6520:
6517:
6514:
6511:
6508:
6505:
6502:
6499:
6496:
6493:
6471:
6449:
6444:
6439:
6436:
6433:
6430:
6408:
6382:
6377:
6372:
6369:
6366:
6363:
6343:
6338:
6333:
6330:
6308:
6295:
6294:
6282:
6279:
6276:
6273:
6270:
6267:
6264:
6261:
6258:
6255:
6252:
6249:
6246:
6243:
6240:
6209:
6204:
6199:
6196:
6176:
6154:
6147:
6142:
6138:
6132:
6127:
6124:
6121:
6117:
6113:
6110:
6088:
6075:tensor algebra
6058:
6053:
6048:
6045:
6014:
6009:
6004:
6001:
5979:
5955:
5933:
5928:
5923:
5920:
5900:
5895:
5890:
5887:
5865:
5854:tells us that
5839:
5834:
5829:
5826:
5804:
5782:
5777:
5772:
5769:
5747:
5715:
5698:Main article:
5695:
5692:
5683:
5680:
5665:
5661:
5638:
5616:
5596:
5593:
5590:
5587:
5584:
5581:
5578:
5556:
5534:
5531:
5528:
5525:
5522:
5519:
5516:
5496:
5476:
5454:
5429:
5424:
5420:
5417:
5414:
5411:
5408:
5405:
5402:
5399:
5396:
5393:
5390:
5387:
5384:
5378:
5373:
5352:
5349:
5346:
5343:
5340:
5337:
5334:
5331:
5328:
5325:
5322:
5319:
5316:
5313:
5310:
5307:
5304:
5301:
5298:
5295:
5273:
5251:
5248:
5245:
5242:
5239:
5236:
5233:
5211:
5187:
5165:
5162:
5159:
5147:
5144:
5129:
5124:
5120:
5114:
5110:
5106:
5101:
5097:
5093:
5090:
5087:
5065:
5041:
5037:
5014:
5010:
4998:
4997:
4986:
4983:
4980:
4977:
4974:
4971:
4968:
4965:
4962:
4959:
4956:
4951:
4947:
4943:
4920:
4898:
4894:
4890:
4885:
4881:
4877:
4872:
4868:
4847:
4844:
4841:
4838:
4835:
4824:
4823:
4812:
4807:
4803:
4799:
4796:
4793:
4790:
4787:
4784:
4781:
4778:
4775:
4772:
4767:
4763:
4739:
4734:
4730:
4726:
4721:
4718:
4713:
4708:
4703:
4698:
4694:
4673:
4651:
4647:
4624:
4602:
4599:
4596:
4591:
4588:
4583:
4578:
4573:
4570:
4548:
4531:Main article:
4528:
4525:
4512:
4509:
4506:
4501:
4497:
4476:
4473:
4470:
4465:
4461:
4439:
4436:
4433:
4428:
4424:
4403:
4400:
4397:
4392:
4388:
4376:
4375:
4363:
4360:
4357:
4352:
4348:
4344:
4341:
4338:
4335:
4330:
4326:
4322:
4319:
4316:
4313:
4310:
4305:
4301:
4297:
4292:
4288:
4284:
4270:
4269:
4243:
4239:
4216:
4212:
4191:
4188:
4185:
4180:
4176:
4172:
4168:
4164:
4160:
4156:
4153:
4150:
4147:
4142:
4138:
4134:
4131:
4128:
4125:
4122:
4117:
4113:
4109:
4104:
4100:
4096:
4073:
4068:
4064:
4060:
4055:
4051:
4047:
4042:
4039:
4034:
4029:
4024:
4019:
4015:
4011:
4006:
4002:
3976:
3972:
3968:
3963:
3959:
3936:
3932:
3928:
3923:
3919:
3907:
3906:
3895:
3892:
3887:
3883:
3879:
3876:
3873:
3870:
3865:
3861:
3857:
3852:
3848:
3844:
3841:
3836:
3832:
3828:
3825:
3822:
3819:
3816:
3811:
3807:
3803:
3798:
3794:
3790:
3787:
3784:
3759:
3745:
3738:
3731:
3724:
3707:
3690:Main article:
3687:
3684:
3682:
3679:
3663:
3658:
3635:
3630:
3627:
3607:
3604:
3601:
3598:
3595:
3586:-invariant if
3573:
3562:is said to be
3551:
3548:
3532:
3504:
3457:
3403:is said to be
3386:
3368:
3365:
3364:
3363:
3351:
3331:
3328:
3325:
3322:
3319:
3297:
3280:
3268:
3248:
3245:
3242:
3239:
3236:
3214:
3185:Main article:
3182:
3179:
3151:
3148:
3145:
3136:is bijective,
3121:
3118:
3115:
3111:
3106:
3101:
3098:
3078:
3075:
3072:
3069:
3066:
3063:
3060:
3057:
3054:
3051:
3048:
3045:
3042:
3020:
2996:
2970:
2967:
2964:
2961:
2958:
2936:
2900:
2886:
2883:
2866:
2863:
2860:
2840:
2814:
2809:
2806:
2786:
2783:
2780:
2760:
2757:
2754:
2751:
2748:
2745:
2742:
2718:
2698:
2676:
2654:
2651:
2648:
2645:
2642:
2639:
2634:
2629:
2626:
2614:
2611:
2609:
2608:Basic concepts
2606:
2588:
2568:
2553:
2552:
2540:
2535:
2531:
2527:
2524:
2521:
2518:
2513:
2509:
2505:
2500:
2496:
2492:
2487:
2482:
2478:
2474:
2471:
2468:
2465:
2460:
2456:
2452:
2447:
2443:
2439:
2434:
2429:
2425:
2421:
2418:
2415:
2412:
2407:
2403:
2399:
2394:
2390:
2386:
2363:
2360:
2338:
2316:
2313:
2310:
2305:
2301:
2280:
2277:
2260:. This is the
2247:
2221:
2201:
2198:
2195:
2192:
2189:
2169:
2164:
2159:
2156:
2153:
2133:
2130:
2127:
2124:
2119:
2115:
2092:
2089:
2085:
2081:
2078:
2075:
2072:
2069:
2066:
2061:
2057:
2019:
2006:
2005:
1994:
1991:
1988:
1983:
1980:
1975:
1970:
1965:
1962:
1959:
1945:
1944:
1932:
1929:
1926:
1923:
1920:
1917:
1914:
1911:
1908:
1887:tangent spaces
1872:
1867:
1862:
1857:
1854:
1849:
1845:
1807:
1783:
1745:
1733:
1730:
1717:
1702:
1701:
1690:
1687:
1684:
1681:
1678:
1675:
1672:
1669:
1666:
1663:
1660:
1655:
1651:
1646:
1641:
1637:
1633:
1630:
1626:
1623:
1618:
1613:
1608:
1605:
1600:
1595:
1590:
1558:
1541:Main article:
1538:
1535:
1533:
1530:
1487:
1470:
1469:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1401:
1398:
1395:
1392:
1369:
1366:
1363:
1360:
1355:
1324:
1306:is said to be
1295:
1264:
1252:, is called a
1231:
1214:
1213:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1127:
1104:
1093:
1092:
1080:
1077:
1074:
1069:
1066:
1061:
1056:
1051:
1048:
1022:
1000:
987:representation
974:
971:
968:
963:
960:
934:
931:
928:
925:
922:
919:
916:
913:
910:
907:
904:
901:
898:
878:
875:
872:
867:
864:
842:
822:
802:
799:
796:
791:
788:
766:
744:
730:
727:
680:
660:
611:
610:
608:
607:
600:
593:
585:
582:
581:
580:
579:
574:
566:
565:
561:
560:
555:
553:Harish-Chandra
550:
545:
540:
535:
530:
528:Henri Poincaré
525:
519:
518:
515:
514:
511:
510:
506:
505:
500:
495:
490:
484:
483:
478:Lie groups in
477:
476:
473:
472:
468:
467:
462:
457:
452:
447:
442:
436:
435:
430:
429:
426:
425:
421:
420:
415:
410:
408:
407:
402:
396:
394:
392:
391:
386:
380:
378:
373:
367:
366:
361:
360:
357:
356:
352:
351:
346:
341:
336:
334:
333:
328:
322:
320:
315:
310:
304:
303:
298:
297:
294:
293:
289:
288:
283:
278:
276:Diffeomorphism
273:
268:
263:
258:
252:
251:
246:
245:
242:
241:
236:
235:
234:
233:
229:
225:
221:
217:
213:
209:
205:
201:
197:
190:
189:
185:
184:
183:
182:
176:
172:
166:
162:
156:
152:
146:
139:
138:
132:
127:
126:
123:
122:
118:
117:
107:
97:
87:
77:
67:
60:Special linear
57:
50:General linear
46:
45:
40:
39:
36:
35:
27:
26:
9:
6:
4:
3:
2:
9104:
9093:
9090:
9089:
9087:
9075:
9070:
9066:
9065:Nadler, David
9061:
9060:
9050:
9045:
9042:
9037:
9035:0-691-09089-0
9031:
9027:
9026:
9021:
9017:
9013:
9007:
9003:
9002:
8996:
8991:
8986:
8982:
8979:
8978:
8970:
8966:
8962:
8958:
8954:
8948:
8944:
8940:
8936:
8933:
8931:9781461263982
8927:
8923:
8922:
8916:
8913:
8909:
8906:
8904:0-19-859683-9
8900:
8896:
8891:
8888:
8882:
8878:
8873:
8870:
8864:
8860:
8855:
8853:
8849:
8845:
8841:
8837:
8831:
8827:
8823:
8819:
8815:
8811:
8810:ScienceDirect
8807:
8801:
8796:
8795:
8788:
8784:
8782:0-444-88776-8
8778:
8774:
8769:
8766:
8762:
8760:0-444-11077-1
8756:
8752:
8747:
8744:
8743:
8732:
8731:Jacobson 1962
8727:
8720:
8715:
8708:
8703:
8696:
8691:
8684:
8679:
8672:
8667:
8660:
8655:
8651:
8640:
8637:
8635:
8632:
8630:
8627:
8625:
8622:
8620:
8617:
8615:
8612:
8610:
8607:
8605:
8602:
8600:
8597:
8596:
8590:
8588:
8584:
8580:
8576:
8571:
8569:
8566:
8562:
8557:
8555:
8547:
8544:
8543:
8542:
8540:
8536:
8528:
8524:
8520:
8516:
8512:
8509:
8508:
8507:
8505:
8501:
8500:pure elements
8497:
8493:
8489:
8485:
8481:
8476:
8474:
8470:
8466:
8462:
8458:
8452:
8448:
8444:
8441:
8438:
8433:
8428:
8424:
8420:
8410:
8396:
8376:
8344:
8341:
8313:
8297:
8287:
8283:
8273:
8270:
8241:
8239:
8205:
8162:
8158:
8147:
8143:
8132:
8128:
8110:
8081:
8070:
8065:
8028:
8017:
8012:
7904:
7876:
7865:
7831:
7825:
7795:
7779:
7775:
7761:
7752:
7748:
7730:
7720:
7706:
7702:
7698:
7678:
7673:
7669:
7665:
7657:
7629:
7613:
7610:
7592:
7558:
7525:
7442:
7434:Furthermore,
7414:
7411:
7393:
7389:
7386:
7380:
7369:
7365:
7359:
7356:
7353:
7350:
7332:
7325:
7314:
7306:
7305:
7304:
7303:
7275:
7235:
7232:
7214:
7169:
7151:
7147:
7130:
7097:
7089:
7036:
7003:
6976:
6931:
6904:
6858:
6854:
6847:
6814:
6811:
6808:
6805:
6795:
6792:
6789:
6786:
6783:
6780:
6774:
6766:
6762:
6729:
6676:
6643:
6592:
6584:
6555:
6541:
6521:
6518:
6515:
6512:
6509:
6506:
6500:
6497:
6494:
6434:
6431:
6428:
6396:
6367:
6361:
6328:
6277:
6274:
6271:
6268:
6265:
6262:
6259:
6253:
6247:
6244:
6241:
6231:
6230:
6229:
6227:
6223:
6222:quotient ring
6194:
6174:
6145:
6140:
6136:
6125:
6122:
6119:
6115:
6111:
6108:
6076:
6072:
6043:
6034:
6032:
6031:Verma modules
6026:
5999:
5992:and those of
5918:
5885:
5853:
5824:
5767:
5735:
5731:
5728:over a field
5701:
5691:
5689:
5679:
5663:
5659:
5614:
5607:. If we take
5591:
5588:
5585:
5579:
5576:
5529:
5526:
5523:
5517:
5514:
5494:
5474:
5418:
5415:
5412:
5406:
5403:
5400:
5394:
5391:
5388:
5382:
5371:
5347:
5344:
5338:
5335:
5329:
5323:
5320:
5317:
5311:
5302:
5299:
5296:
5246:
5243:
5240:
5234:
5231:
5163:
5160:
5157:
5143:
5140:
5127:
5122:
5118:
5112:
5108:
5104:
5099:
5091:
5088:
5039:
5035:
5012:
5008:
4981:
4978:
4972:
4969:
4963:
4954:
4949:
4945:
4934:
4933:
4932:
4918:
4896:
4892:
4883:
4879:
4875:
4870:
4866:
4845:
4839:
4836:
4833:
4810:
4805:
4794:
4788:
4782:
4779:
4773:
4765:
4761:
4753:
4752:
4751:
4732:
4728:
4701:
4696:
4692:
4671:
4649:
4645:
4597:
4571:
4568:
4534:
4524:
4507:
4499:
4495:
4471:
4463:
4459:
4434:
4426:
4422:
4398:
4390:
4386:
4358:
4350:
4346:
4342:
4336:
4328:
4324:
4320:
4314:
4303:
4299:
4295:
4290:
4286:
4275:
4274:
4273:
4267:
4263:
4259:
4256:, defined in
4241:
4237:
4214:
4210:
4186:
4178:
4174:
4170:
4162:
4154:
4148:
4140:
4136:
4132:
4126:
4115:
4111:
4107:
4102:
4098:
4087:
4086:
4085:
4066:
4062:
4058:
4053:
4049:
4022:
4017:
4013:
4009:
4004:
4000:
3990:
3974:
3970:
3966:
3961:
3957:
3934:
3930:
3926:
3921:
3917:
3893:
3885:
3881:
3877:
3874:
3868:
3863:
3859:
3855:
3850:
3846:
3842:
3834:
3830:
3826:
3823:
3817:
3809:
3805:
3801:
3796:
3792:
3785:
3782:
3775:
3774:
3773:
3744:
3737:
3730:
3723:
3693:
3678:
3656:
3628:
3625:
3605:
3602:
3599:
3596:
3593:
3561:
3557:
3547:
3520:
3492:
3487:
3485:
3481:
3477:
3473:
3444:
3442:
3438:
3434:
3430:
3426:
3422:
3418:
3414:
3410:
3406:
3402:
3374:
3349:
3329:
3323:
3320:
3317:
3285:
3281:
3266:
3246:
3240:
3237:
3234:
3227:-modules and
3202:
3198:
3194:
3193:
3192:
3188:
3187:Schur's lemma
3181:Schur's lemma
3178:
3175:
3173:
3169:
3165:
3149:
3146:
3143:
3135:
3119:
3116:
3113:
3109:
3099:
3096:
3073:
3067:
3064:
3061:
3058:
3052:
3049:
3046:
3040:
2984:
2968:
2962:
2959:
2956:
2924:
2920:
2916:
2885:Homomorphisms
2882:
2880:
2879:simple module
2861:
2838:
2830:
2807:
2804:
2784:
2781:
2778:
2758:
2755:
2752:
2746:
2740:
2732:
2716:
2696:
2649:
2643:
2640:
2627:
2624:
2605:
2603:
2602:hydrogen atom
2586:
2566:
2558:
2538:
2533:
2529:
2522:
2519:
2511:
2507:
2503:
2498:
2494:
2485:
2480:
2476:
2469:
2466:
2458:
2454:
2450:
2445:
2441:
2432:
2427:
2423:
2416:
2413:
2405:
2401:
2397:
2392:
2388:
2377:
2376:
2375:
2373:
2369:
2368:Hilbert space
2359:
2357:
2352:
2314:
2311:
2308:
2303:
2299:
2278:
2275:
2267:
2263:
2235:
2219:
2196:
2190:
2187:
2154:
2151:
2131:
2125:
2122:
2117:
2113:
2090:
2087:
2083:
2079:
2076:
2073:
2067:
2059:
2055:
2045:
2043:
2039:
2035:
1989:
1963:
1960:
1957:
1950:
1949:
1948:
1927:
1921:
1918:
1912:
1909:
1906:
1899:
1898:
1897:
1896:
1892:
1888:
1855:
1852:
1847:
1843:
1835:
1831:
1827:
1823:
1771:
1767:
1763:
1759:
1743:
1729:
1715:
1707:
1688:
1682:
1679:
1676:
1670:
1664:
1658:
1653:
1649:
1644:
1639:
1635:
1628:
1624:
1588:
1574:
1573:
1572:
1544:
1529:
1527:
1523:
1519:
1515:
1511:
1507:
1503:
1475:
1453:
1450:
1447:
1441:
1438:
1435:
1429:
1426:
1423:
1417:
1414:
1411:
1408:
1405:
1399:
1396:
1393:
1383:
1382:
1381:
1367:
1361:
1358:
1344:
1340:
1311:
1309:
1293:
1284:
1282:
1278:
1251:
1247:
1219:
1197:
1191:
1185:
1179:
1176:
1170:
1164:
1158:
1152:
1149:
1140:
1137:
1134:
1125:
1118:
1117:
1116:
1102:
1075:
1049:
1046:
1039:
1038:
1037:
1036:
1020:
988:
969:
948:
932:
929:
926:
923:
920:
917:
914:
911:
905:
902:
899:
873:
840:
820:
797:
764:
726:
724:
720:
716:
713:, called the
712:
707:
705:
701:
696:
694:
678:
658:
650:
646:
642:
641:endomorphisms
638:
634:
630:
626:
622:
618:
606:
601:
599:
594:
592:
587:
586:
584:
583:
578:
575:
573:
570:
569:
568:
567:
559:
556:
554:
551:
549:
546:
544:
541:
539:
536:
534:
531:
529:
526:
524:
521:
520:
513:
512:
504:
501:
499:
496:
494:
491:
489:
486:
485:
481:
475:
474:
466:
463:
461:
458:
456:
453:
451:
448:
446:
443:
441:
438:
437:
433:
428:
427:
419:
416:
414:
411:
406:
403:
401:
398:
397:
395:
390:
387:
385:
382:
381:
379:
377:
374:
372:
369:
368:
364:
359:
358:
350:
347:
345:
342:
340:
337:
332:
329:
327:
324:
323:
321:
319:
316:
314:
311:
309:
306:
305:
301:
296:
295:
287:
284:
282:
279:
277:
274:
272:
269:
267:
264:
262:
259:
257:
254:
253:
249:
244:
243:
232:
226:
224:
218:
216:
210:
208:
202:
200:
194:
193:
192:
191:
187:
186:
181:
179:
173:
171:
169:
163:
161:
159:
153:
151:
149:
143:
142:
141:
140:
136:
135:
130:
125:
124:
115:
111:
108:
105:
101:
98:
95:
91:
88:
85:
81:
78:
75:
71:
68:
65:
61:
58:
55:
51:
48:
47:
43:
38:
37:
33:
29:
28:
25:
21:
18:
17:
9048:
9040:
9024:
9000:
8980:
8975:
8943:Lie algebras
8942:
8920:
8911:
8894:
8876:
8858:
8825:
8808:– via
8793:
8772:
8750:
8726:
8714:
8702:
8695:Dixmier 1977
8690:
8685:Theorem 4.29
8678:
8673:Section 17.3
8666:
8654:
8572:
8558:
8553:
8551:
8545:
8534:
8532:
8526:
8522:
8521:+ (−1)
8518:
8514:
8510:
8503:
8495:
8491:
8487:
8484:pure element
8479:
8477:
8472:
8460:
8450:
8446:
8442:
8431:
8422:
8418:
8416:
8342:
8311:
8289:
8282:(g,K)-module
8276:(g,K)-module
8242:
8211:
7948:an ideal of
7721:
7655:
7627:
7433:
7301:
7273:
7087:
6937:
6644:
6556:
6484:is given by
6296:
6225:
6070:
6035:
6027:
5878:sits inside
5760:and denoted
5729:
5703:
5685:
5149:
5141:
4999:
4931:" operator:
4825:
4536:
4377:
4271:
3991:
3908:
3742:
3735:
3728:
3721:
3695:
3559:
3555:
3553:
3518:
3488:
3479:
3475:
3445:
3440:
3436:
3432:
3428:
3424:
3420:
3416:
3412:
3404:
3400:
3372:
3370:
3283:
3200:
3196:
3190:
3176:
3171:
3167:
3163:
3133:
2983:homomorphism
2982:
2922:
2918:
2888:
2878:
2828:
2730:
2616:
2554:
2365:
2353:
2265:
2233:
2046:
2041:
2037:
2007:
1946:
1890:
1834:differential
1829:
1825:
1822:Lie algebras
1766:homomorphism
1761:
1757:
1735:
1703:
1546:
1525:
1521:
1517:
1513:
1509:
1505:
1501:
1473:
1471:
1343:bilinear map
1338:
1312:
1307:
1285:
1280:
1253:
1249:
1245:
1217:
1215:
1094:
986:
946:
732:
708:
697:
645:vector space
635:as a set of
628:
624:
617:mathematical
614:
558:Armand Borel
543:Hermann Weyl
444:
344:Loop algebra
326:Killing form
300:Lie algebras
177:
167:
157:
147:
113:
103:
93:
83:
73:
63:
53:
24:Lie algebras
9074:1209.0188v1
8983:: 116â136.
8721:Section 9.5
8709:Section 4.3
8661:Theorem 5.6
8614:Root system
8579:Lie algebra
8465:derivations
8427:associative
6395:PBW theorem
5852:PBW theorem
3554:An element
2915:Lie algebra
2877:. The term
2829:irreducible
1571:on itself:
633:Lie algebra
538:Ălie Cartan
384:Root system
188:Exceptional
8822:Harris, J.
8818:Fulton, W.
8740:References
8269:category O
7182:. It is a
6702:becomes a
5262:becomes a
5200:-modules,
3550:Invariants
3431:such that
3164:equivalent
2559:. Then if
1770:Lie groups
1380:such that
649:commutator
523:Sophus Lie
516:Scientists
389:Weyl group
110:Symplectic
70:Orthogonal
20:Lie groups
8945:. Dover.
8941:(1979) .
8719:Hall 2015
8707:Hall 2015
8683:Hall 2015
8671:Hall 2013
8659:Hall 2015
8533:Also, if
8377:π
8298:π
8159:∘
8144:≃
8129:∘
7905:⊂
7866:⊂
7832:⊂
7776:∘
7749:≃
7670:⊗
7611:
7412:
7381:
7366:≃
7351:
7326:
7233:
7148:⊗
6977:⊂
6851:↦
6812:∈
6796:∈
6516:−
6365:→
6275:⊗
6269:−
6263:⊗
6254:−
6175:⊗
6137:⊗
6131:∞
6116:⊕
5664:∗
5580:
5518:
5407:
5383:
5336:−
5300:⋅
5235:
5040:∗
5036:ρ
5013:∗
5009:ρ
4973:ϕ
4955:ϕ
4897:∗
4889:→
4884:∗
4843:→
4789:ρ
4783:−
4766:∗
4762:ρ
4733:∗
4712:→
4697:∗
4693:ρ
4650:∗
4582:→
4569:ρ
4496:ρ
4460:ρ
4423:ρ
4387:ρ
4347:ρ
4325:ρ
4300:ρ
4296:⊗
4287:ρ
4238:ρ
4211:ρ
4175:ρ
4171:⊗
4155:⊗
4137:ρ
4112:ρ
4108:⊗
4099:ρ
4059:⊗
4033:→
4014:ρ
4010:⊗
4001:ρ
3967:∈
3927:∈
3878:⋅
3869:⊗
3843:⊗
3827:⋅
3802:⊗
3786:⋅
3629:∈
3597:⋅
3491:reductive
3327:→
3244:→
3172:morphisms
3117:∈
3100:∈
3065:⋅
3050:⋅
2966:→
2808:∈
2782:∈
2756:∈
2741:ρ
2731:invariant
2644:
2638:→
2625:ρ
2526:ℏ
2473:ℏ
2420:ℏ
2191:
2155:
2129:→
2088:−
1974:→
1961:ϕ
1922:
1916:→
1907:ϕ
1866:→
1853:ϕ
1744:ϕ
1659:
1632:↦
1599:→
1451:⋅
1442:⋅
1436:−
1427:⋅
1418:⋅
1406:⋅
1365:→
1359:×
1294:ρ
1192:ρ
1180:ρ
1177:−
1165:ρ
1153:ρ
1126:ρ
1103:ρ
1060:→
1050::
1047:ρ
985:. Then a
933:ρ
930:∘
927:σ
924:−
921:σ
918:∘
915:ρ
906:σ
900:ρ
619:field of
400:Real form
286:Euclidean
137:Classical
9086:Category
9022:(2001),
8967:(1949).
8824:(1991).
8593:See also
8463:acts as
7877:′
7826:′
7796:′
7762:′
7654:-module
7300:-module
7123:-module
7086:-module
7029:acts on
3909:for all
3618:for all
3089:for any
2771:for all
1820:are the
1532:Examples
1472:for all
1308:faithful
1216:for all
945:for all
719:category
637:matrices
572:Glossary
266:Poincaré
8844:1153249
8541:, then
8440:algebra
8102:. Then
6585:, then
6583:abelian
6220:be the
6073:be the
3720:, with
3399:. Then
1277:-module
723:modules
615:In the
480:physics
261:Lorentz
90:Unitary
9032:
9008:
8949:
8928:
8901:
8883:
8865:
8842:
8832:
8802:
8779:
8757:
8577:and a
8539:unital
8437:graded
8345:. The
7996:. Set
6645:Since
6187:. Let
4637:. Let
3411:). If
2917:. Let
1772:, and
256:Circle
9069:arXiv
8972:(PDF)
8646:Notes
8565:super
8482:is a
8449:as a
6321:into
3519:every
3132:. If
2981:is a
2913:be a
2008:from
1764:is a
1033:is a
643:of a
331:Index
9030:ISBN
9006:ISBN
8947:ISBN
8926:ISBN
8899:ISBN
8881:ISBN
8863:ISBN
8830:ISBN
8800:ISBN
8777:ISBN
8755:ISBN
8498:are
8494:and
8490:and
8284:and
8212:Let
8049:and
6938:Let
5734:ring
5686:See
5150:Let
4537:Let
4260:and
4229:and
3949:and
3727:and
3439:and
3371:Let
2889:Let
2797:and
1893:, a
1828:and
1796:and
1500:and
1218:X, Y
733:Let
711:ring
639:(or
623:, a
281:Loop
22:and
8985:doi
8548:= 0
8537:is
8513:= (
8502:of
8486:of
8471:on
8163:Ind
8148:Res
8133:Res
8111:Ind
7780:Ind
7753:Ind
7731:Ind
7593:Ind
7443:Ind
7394:Res
7370:Hom
7333:Ind
7315:Hom
7215:Ind
6897:on
6581:is
6557:If
6534:in
6224:of
5651:on
5577:Hom
5569:on
5515:Hom
5487:to
5404:Hom
5372:Hom
5232:Hom
5176:be
3558:of
3446:If
3443:.)
3282:If
3195:If
3170:or
2985:of
2925:be
2733:if
2729:is
2709:of
2641:End
2264:of
2232:of
2036:GL(
1885:on
1824:of
1528:).
1504:in
1476:in
1474:X,Y
1220:in
1013:on
989:of
949:in
947:Ï,Ï
627:or
112:Sp(
102:SU(
82:SO(
62:SL(
52:GL(
9088::
9043:))
8981:65
8974:.
8963:;
8840:MR
8838:.
8820:;
8589:.
8570:.
8506:,
8475:.
8429:)
8204:.
7719:.
6642:.
6554:.
6025:.
5690:.
5123:tr
5113:tr
5100:tr
4950:tr
4871:tr
4806:tr
3989:.
3741:â
3677:.
3199:,
3174:.
2921:,
2351:.
2315:ad
2309:Ad
2279:Ad
2220:Ad
2188:Ad
2152:GL
2044:.
1919:GL
1760:â
1756::
1716:ad
1708:,
1650:ad
1636:ad
1584:ad
1524:)(
1516:=
1512:â
695:.
92:U(
72:O(
9077:.
9071::
9041:C
9014:.
8993:.
8987::
8955:.
8846:.
8812:.
8785:.
8764:.
8546:H
8535:A
8529:)
8527:H
8525:(
8523:x
8519:y
8517:)
8515:H
8511:H
8504:A
8496:y
8492:x
8488:L
8480:H
8473:A
8467:/
8461:L
8454:2
8451:Z
8447:L
8443:A
8435:2
8432:Z
8423:L
8419:L
8397:K
8355:g
8343:K
8324:g
8312:G
8253:g
8222:g
8187:1
8183:g
8173:1
8169:h
8153:g
8138:h
8123:g
8116:h
8088:n
8082:/
8076:h
8071:=
8066:1
8060:h
8035:n
8029:/
8023:g
8018:=
8013:1
8007:g
7982:h
7958:g
7934:n
7910:g
7900:h
7872:h
7861:h
7837:g
7823:h
7793:h
7785:h
7770:g
7759:h
7743:g
7736:h
7707:k
7703:/
7699:F
7679:F
7674:k
7666:V
7656:V
7640:g
7628:W
7614:W
7605:g
7598:h
7572:)
7567:h
7562:(
7559:U
7539:)
7534:g
7529:(
7526:U
7504:g
7480:h
7455:g
7448:h
7430:.
7418:)
7415:E
7406:g
7399:h
7390:,
7387:W
7384:(
7375:h
7363:)
7360:E
7357:,
7354:W
7345:g
7338:h
7329:(
7320:g
7302:E
7286:g
7274:W
7258:g
7236:W
7227:g
7220:h
7192:g
7170:W
7165:)
7160:h
7155:(
7152:U
7144:)
7139:g
7134:(
7131:U
7111:)
7106:g
7101:(
7098:U
7088:W
7072:h
7050:)
7045:g
7040:(
7037:U
7017:)
7012:h
7007:(
7004:U
6982:g
6972:h
6948:g
6918:)
6913:g
6908:(
6905:U
6883:g
6859:X
6855:l
6848:X
6828:)
6823:g
6818:(
6815:U
6809:Y
6806:,
6801:g
6793:X
6790:,
6787:Y
6784:X
6781:=
6778:)
6775:Y
6772:(
6767:X
6763:l
6740:g
6712:g
6690:)
6685:g
6680:(
6677:U
6655:g
6628:g
6606:)
6601:g
6596:(
6593:U
6567:g
6542:A
6522:X
6519:Y
6513:Y
6510:X
6507:=
6504:]
6501:Y
6498:,
6495:X
6492:[
6470:g
6448:)
6443:g
6438:(
6435:U
6432:=
6429:A
6407:g
6381:)
6376:g
6371:(
6368:U
6362:T
6342:)
6337:g
6332:(
6329:U
6307:g
6293:.
6281:)
6278:X
6272:Y
6266:Y
6260:X
6257:(
6251:]
6248:Y
6245:,
6242:X
6239:[
6226:T
6208:)
6203:g
6198:(
6195:U
6153:g
6146:n
6141:1
6126:0
6123:=
6120:n
6112:=
6109:T
6087:g
6071:T
6057:)
6052:g
6047:(
6044:U
6013:)
6008:g
6003:(
6000:U
5978:g
5954:g
5932:)
5927:g
5922:(
5919:U
5899:)
5894:g
5889:(
5886:U
5864:g
5838:)
5833:g
5828:(
5825:U
5803:g
5781:)
5776:g
5771:(
5768:U
5746:g
5730:k
5714:g
5660:V
5637:g
5615:W
5595:)
5592:W
5589:,
5586:V
5583:(
5555:g
5533:)
5530:W
5527:,
5524:V
5521:(
5495:W
5475:V
5453:g
5428:g
5423:)
5419:W
5416:,
5413:V
5410:(
5401:=
5398:)
5395:W
5392:,
5389:V
5386:(
5377:g
5351:)
5348:v
5345:X
5342:(
5339:f
5333:)
5330:v
5327:(
5324:f
5321:X
5318:=
5315:)
5312:v
5309:(
5306:)
5303:f
5297:X
5294:(
5272:g
5250:)
5247:W
5244:,
5241:V
5238:(
5210:g
5186:g
5164:W
5161:,
5158:V
5128:.
5119:A
5109:B
5105:=
5096:)
5092:B
5089:A
5086:(
5064:g
4985:)
4982:v
4979:A
4976:(
4970:=
4967:)
4964:v
4961:(
4958:)
4946:A
4942:(
4919:A
4893:V
4880:V
4876::
4867:A
4846:V
4840:V
4837::
4834:A
4811:,
4802:)
4798:)
4795:X
4792:(
4786:(
4780:=
4777:)
4774:X
4771:(
4738:)
4729:V
4725:(
4720:l
4717:g
4707:g
4702::
4672:V
4646:V
4623:g
4601:)
4598:V
4595:(
4590:l
4587:g
4577:g
4572::
4547:g
4511:)
4508:X
4505:(
4500:2
4475:)
4472:X
4469:(
4464:1
4438:)
4435:x
4432:(
4427:2
4402:)
4399:x
4396:(
4391:1
4374:,
4362:)
4359:X
4356:(
4351:2
4343:+
4340:)
4337:X
4334:(
4329:1
4321:=
4318:)
4315:X
4312:(
4309:)
4304:2
4291:1
4283:(
4268:.
4242:2
4215:1
4190:)
4187:X
4184:(
4179:2
4167:I
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1683:Y
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1674:[
1671:=
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1640:X
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1412:=
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1403:]
1400:Y
1397:,
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1391:[
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1162:)
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1150:=
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1138:,
1135:X
1132:[
1129:(
1091:.
1079:)
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116:)
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96:)
94:n
86:)
84:n
76:)
74:n
66:)
64:n
56:)
54:n
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