On the hyperbolic Kac-Moody Lie algebra 𝐻𝐴₁⁽¹⁾

Kang, Seok-Jin

doi:10.1090/S0002-9947-1994-1120776-X

On the hyperbolic Kac-Moody Lie algebra $HA_ 1^ {(1)}$
HTML articles powered by AMS MathViewer

by Seok-Jin Kang

Trans. Amer. Math. Soc. 341 (1994), 623-638

DOI: https://doi.org/10.1090/S0002-9947-1994-1120776-X

PDF | Request permission

Abstract:

In this paper, using a homological theory of graded Lie algebras and the representation theory of $A_1^{(1)}$, we compute the root multiplicities of the hyperbolic Kac-Moody Lie algebra $HA_1^{(1)}$ up to level $4$ and deduce some interesting combinatorial identities.

References

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

Paths, Maya diagrams, and representations of

19

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 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 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, DOI 10.1007/978-1-4684-8765-7
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
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, 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
Victor G. Kac and Dale H. Peterson, Infinite-dimensional Lie algebras, theta functions and modular forms, Adv. in Math. 53 (1984), no. 2, 125–264. MR 750341, DOI 10.1016/0001-8708(84)90032-X

Gradations and structure of Kac-Moody Lie algebras

Seok-Jin Kang, Kac-Moody Lie algebras, spectral sequences, and the Witt formula, Trans. Amer. Math. Soc. 339 (1993), no. 2, 463–493. MR 1102889, DOI 10.1090/S0002-9947-1993-1102889-0
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
J. Lepowsky and S. Milne, Lie algebraic approaches to classical partition identities, Adv. in Math. 29 (1978), no. 1, 15–59. MR 501091, DOI 10.1016/0001-8708(78)90004-X

Kostant’s formula in Kac-Moody Lie algebras

Formules de caractères pour les algèbres de Kac-Moody générales

Jean-Pierre Serre, Lie algebras and Lie groups, W. A. Benjamin, Inc., New York-Amsterdam, 1965. Lectures given at Harvard University, 1964. MR 0218496