Kac-Moody Lie algebras, spectral sequences, and the Witt formula

Kang, Seok-Jin

doi:10.1090/S0002-9947-1993-1102889-0

Kac-Moody Lie algebras, spectral sequences, and the Witt formula
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Trans. Amer. Math. Soc. 339 (1993), 463-493 Request permission

Abstract:

In this work, we develop a homological theory for the graded Lie algebras, which gives new information on the structure of the Lorentzian Kac-Moody Lie algebras. The technique of the Hochschild-Serre spectral sequences offers a uniform method of studying the higher level root multiplicities and the principally specialized affine characters of Lorentzian Kac-Moody Lie algebras.

References

Georgia Benkart, Seok-Jin Kang, and Kailash C. Misra, Graded Lie algebras of Kac-Moody type, Adv. Math. 97 (1993), no. 2, 154–190. MR 1201842, DOI 10.1006/aima.1993.1005
S. Berman and R. V. Moody, Lie algebra multiplicities, Proc. Amer. Math. Soc. 76 (1979), no. 2, 223–228. MR 537078, DOI 10.1090/S0002-9939-1979-0537078-X
N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
Alex Jay Feingold, Tensor products of certain modules for the generalized Cartan matrix Lie algebra $A_{1}^{(1)}$, Comm. Algebra 9 (1981), no. 12, 1323–1341. MR 618906, DOI 10.1080/00927878108822649
Alex J. Feingold, Some applications of vertex operators to Kac-Moody algebras, Vertex operators in mathematics and physics (Berkeley, Calif., 1983) Math. Sci. Res. Inst. Publ., vol. 3, Springer, New York, 1985, pp. 185–206. MR 781378, DOI 10.1007/978-1-4613-9550-8_{9}
Alex J. Feingold and Igor B. Frenkel, A hyperbolic Kac-Moody algebra and the theory of Siegel modular forms of genus $2$, Math. Ann. 263 (1983), no. 1, 87–144. MR 697333, DOI 10.1007/BF01457086
Alex J. Feingold and James Lepowsky, The Weyl-Kac character formula and power series identities, Adv. in Math. 29 (1978), no. 3, 271–309. MR 509801, DOI 10.1016/0001-8708(78)90020-8
I. B. Frenkel, Representations of Kac-Moody algebras and dual resonance models, Applications of group theory in physics and mathematical physics (Chicago, 1982) Lectures in Appl. Math., vol. 21, Amer. Math. Soc., Providence, RI, 1985, pp. 325–353. MR 789298
I. B. Frenkel and V. G. Kac, Basic representations of affine Lie algebras and dual resonance models, Invent. Math. 62 (1980/81), no. 1, 23–66. MR 595581, DOI 10.1007/BF01391662
D. B. Fuks, Cohomology of infinite-dimensional Lie algebras, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1986. Translated from the Russian by A. B. Sosinskiĭ. MR 874337
Ofer Gabber and Victor G. Kac, On defining relations of certain infinite-dimensional Lie algebras, Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 2, 185–189. MR 621889, DOI 10.1090/S0273-0979-1981-14940-5
Howard Garland and James Lepowsky, Lie algebra homology and the Macdonald-Kac formulas, Invent. Math. 34 (1976), no. 1, 37–76. MR 414645, DOI 10.1007/BF01418970
G. Hochschild and J.-P. Serre, Cohomology of Lie algebras, Ann. of Math. (2) 57 (1953), 591–603. MR 54581, DOI 10.2307/1969740
Nathan Jacobson, Basic algebra. II, W. H. Freeman and Co., San Francisco, Calif., 1980. MR 571884
Nathan Jacobson, Lie algebras, Dover Publications, Inc., New York, 1979. Republication of the 1962 original. MR 559927
V. G. Kac, Simple irreducible graded Lie algebras of finite growth, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 1323–1367 (Russian). MR 0259961
Victor G. Kac, Infinite-dimensional Lie algebras, 2nd ed., Cambridge University Press, Cambridge, 1985. MR 823672
V. G. Kac, R. V. Moody, and M. Wakimoto, On $E_{10}$, Differential geometrical methods in theoretical physics (Como, 1987) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 250, Kluwer Acad. Publ., Dordrecht, 1988, pp. 109–128. MR 981374

Gradations and structure of Kac-Moody Lie algebras

Bertram Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329–387. MR 142696, DOI 10.2307/1970237
J. Lepowsky, Application of the numerator formula to $k$-rowed plane partitions, Adv. in Math. 35 (1980), no. 2, 179–194. MR 560134, DOI 10.1016/0001-8708(80)90047-X
James Lepowsky and Robert V. Moody, Hyperbolic Lie algebras and quasiregular cusps on Hilbert modular surfaces, Math. Ann. 245 (1979), no. 1, 63–88. MR 552580, DOI 10.1007/BF01420431

Kostant’s formula in Kac-Moody Lie algebras

I. G. Macdonald, Affine root systems and Dedekind’s $\eta$-function, Invent. Math. 15 (1972), 91–143. MR 357528, DOI 10.1007/BF01418931

Formules de caractères pour les algèbres de Kac-Moody générales

159-160

John McCleary, User’s guide to spectral sequences, Mathematics Lecture Series, vol. 12, Publish or Perish, Inc., Wilmington, DE, 1985. MR 820463
Robert V. Moody, A new class of Lie algebras, J. Algebra 10 (1968), 211–230. MR 229687, DOI 10.1016/0021-8693(68)90096-3
Robert V. Moody, Root systems of hyperbolic type, Adv. in Math. 33 (1979), no. 2, 144–160. MR 544847, DOI 10.1016/S0001-8708(79)80003-1
Robert E. Mosher and Martin C. Tangora, Cohomology operations and applications in homotopy theory, Harper & Row, Publishers, New York-London, 1968. MR 0226634
Jean-Pierre Serre, Lie algebras and Lie groups, W. A. Benjamin, Inc., New York-Amsterdam, 1965. Lectures given at Harvard University, 1964. MR 0218496