Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the hyperbolic Kac-Moody Lie algebra $ HA\sb 1\sp {(1)}$

Author: Seok-Jin Kang
Journal: Trans. Amer. Math. Soc. 341 (1994), 623-638
MSC: Primary 17B67; Secondary 11B65
MathSciNet review: 1120776
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [B-M] S. Berman and R. V. Moody, Multiplicities in Lie algebras, Proc. Amer. Math. Soc. 76 (1979), 223-228. MR 537078 (80h:17013)
  • [B] N. Bourbaki, Groupes et algèbres de Lie, Hermann, Paris, 1968, Chaps. 4-6. MR 0240238 (39:1590)
  • [D-J-K-M-O] E. Date, M. Jimbo, A. Kuniba, T. Miwa, and M. Okado, Paths, Maya diagrams, and representations of $ \hat sl(r,C)$, Adv. Stud. Pure Math. 19 (1989), 149-191.
  • [Fe] A. J. Feingold, Tensor products of certain modules for the generalized Cartan matrix Lie algebra $ A_1^{(1)}$, Comm. Algebra 9 (1981), 1323-1341. MR 618906 (83f:17005)
  • [F-F] A. J. Feingold and I. B. Frenkel, A hyperbolic Kac-Moody algebra and the theory of Siegel modular forms of genus $ 2$, Math. Ann. 263 (1983), 87-144. MR 697333 (86a:17006)
  • [F-L] A. J. Feingold and J. Lepowsky, The Weyl-Kac character formula and power series identities, Adv. Math. 29 (1978), 271-309. MR 509801 (83a:17015)
  • [F-K] I. B. Frenkel and V. G. Kac, Basic representations of affine Lie algebras and dual resonance models, Invent. Math. 62 (1980), 23-66. MR 595581 (84f:17004)
  • [Fu] D. B. Fuks, Cohomology of infinite-dimensional Lie algebras, Consultants Bureau, New York; Plenum, New York, 1986. MR 874337 (88b:17001)
  • [G-K] O. Gabber and V. G. Kac, On defining relations of certain infinite dimensional Lie algebras, Bull. Amer. Math. Soc. 5 (1981), 185-189. MR 621889 (84b:17011)
  • [H-S] G. Hochschild and J. P. Serre, Cohomology of Lie algebras, Ann. of Math. (2) 57 (1953), 591-603. MR 0054581 (14:943c)
  • [J] N. Jacobson, Lie algebras, 2nd ed., Dover, New York, 1979. MR 559927 (80k:17001)
  • [Kac1] V. G. Kac, Simple irreducible graded Lie algebras of finite growth, Math. USSR-Izv. 2 (1968), 1271-1311. MR 0259961 (41:4590)
  • [Kac2] -, Infinite dimensional Lie algebras, 2nd ed., Cambridge Univ. Press, London and New York, 1985. MR 823672 (87c:17023)
  • [K-M-W] V. G. Kac, R. V. Moody, and M. Wakimoto, On $ {E_{10}}$, Differential Geometrical Methods in Theoretical Physics (K. Bleuler and M. Werner, eds.), Kluwer, Dordrecht and Boston, 1988, pp. 109-128. MR 981374 (90e:17031)
  • [K-P] V. G. Kac and D. H. Peterson, Infinite dimensional Lie algebras, theta functions, and modular forms, Adv. Math. 53 (1984), 125-264. MR 750341 (86a:17007)
  • [Kang1] S.-J. Kang, Gradations and structure of Kac-Moody Lie algebras, Yale Univ. dissertation, 1990.
  • [Kang2] -, Kac-Moody Lie algebras, spectral sequences, and the Witt formula, Trans. Amer. Math. Soc. 339 (1993), 463-495. MR 1102889 (93m:17013)
  • [L] J. Lepowsky, Application of the numerator formula to $ k$-rowed plane partitions, Adv. Math. 35 (1980), 175-194. MR 560134 (81f:17011)
  • [L-M] J. Lepowsky and S. Milne, Lie algebraic approaches to classical partition function identities, Adv. Math. 29 (1978), 15-59. MR 501091 (82f:17005)
  • [Liu] L.-S. Liu, Kostant's formula in Kac-Moody Lie algebras, Yale Univ. dissertation, 1990.
  • [Ma] O. Mathieu, Formules de caractères pour les algèbres de Kac-Moody générales, Astérisque 159-160 (1988).
  • [J] J. P. Serre, Lie algebras and Lie groups, Benjamin, New York, 1965. MR 0218496 (36:1582)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 17B67, 11B65

Retrieve articles in all journals with MSC: 17B67, 11B65

Additional Information

Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society