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)



Local projective resolutions and translation functors for Kac-Moody algebras

Author: Wayne Neidhardt
Journal: Trans. Amer. Math. Soc. 305 (1988), 221-245
MSC: Primary 17B67; Secondary 17B10
MathSciNet review: 920156
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \mathfrak{g}$ be a Kac-Moody algebra defined by a not necessarily symmetrizable generalized Cartan matrix. We define translation functors and use them to show that the multiplicities $ (M({w_1} \cdot \lambda ):L({w_2} \cdot \lambda ))$ are independent of the dominant integral weight $ \lambda $, depending only on the elements of the Weyl group. In order to define the translation functors, we introduce the notion of local projective resolutions and use them to develop the machinery of homological algebra in certain categories of $ \mathfrak{g}$-modules.

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

  • [1] I. N. Bernstein, I. M. Gelfand, and S. I. Gelfand, Differential operators on the base affine space and a study of $ \mathfrak{g}$-modules, Lie Groups and Their Representations, Proc. Summer School on Group Representations, János Bolyai Math. Soc., Wiley, 1975, pp. 35-69.
  • [2] V. V. Deodhar, O. Gabber, and V. G. Kac, Structure of some categories of representations of infinite-dimensional Lie algebras, Adv. in Math. 45 (1982), 92-116. MR 663417 (83i:17012)
  • [3] T. Enright, On the fundamental series of a real semisimple Lie algebra: their irreducibility, resolutions and multiplicity formulae, Ann. of Math. (2) 110 (1979), 1-82. MR 541329 (81a:17003)
  • [4] H. Garland and J. Lepowsky, Lie algebra homology and the Macdonald-Kac formulas, Invent. Math. 34 (1976), 37-76. MR 0414645 (54:2744)
  • [5] J. C. Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Math., vol. 750, Springer, Berlin, 1979. MR 552943 (81m:17011)
  • [6] V. G. Kac, Simple irreducible graded Lie algebras of finite growth, Izv. Akad. Nauk SSSR 32 (1968), 1323-1367; English transl., Math. USSR-Izv. 2 (1968), 1271-1311. MR 0259961 (41:4590)
  • [7] R. V. Moody, A new class of Lie algebras, J. Algebra 10 (1968), 211-230. MR 0229687 (37:5261)
  • [8] W. Neidhardt, The $ BGG$ resolution, character and denominator formulas, and related results for Kac-Moody algebras, Ph.D. thesis, Univ. of Wisconsin, Madison, Wisc., 1985. MR 854079 (87k:17025)
  • [9] -, The $ BGG$ resolution, character and denominator formulas, and related results for Kac-Moody algebras, Trans. Amer. Math. Soc. 297 (1987), 487-504. MR 854079 (87k:17025)
  • [10] -, A translation principle for Kac-Moody algebras, Proc. Amer. Math. Soc. 100 (1987), 395-400. MR 891132 (88m:17006)
  • [11] A. Rocha-Caridi and N. Wallach, Projective modules over graded Lie algebras. I, Math. Z. 180 (1982), 151-177. MR 661694 (83h:17018)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 17B67, 17B10

Retrieve articles in all journals with MSC: 17B67, 17B10

Additional Information

Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society