-categories and -functors in the representation theory of Lie algebras

Author:
Ben Cox

Journal:
Trans. Amer. Math. Soc. **343** (1994), 433-453

MSC:
Primary 17B67; Secondary 17B55

DOI:
https://doi.org/10.1090/S0002-9947-1994-1191610-7

MathSciNet review:
1191610

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The fields of algebra and representation theory contain abundant examples of functors on categories of modules over a ring. These include of course Horn, Ext, and Tor as well as the more specialized examples of completion and localization used in the setting of representation theory of a semisimple Lie algebra. In this article we let be a Lie subalgebra of a Lie algebra and be a functor on some category of -modules. We then consider the following general question: For a -module *E* what hypotheses on and *E* are sufficient to insure that admits a canonical structure as a -module? The article offers an answer through the introduction of the notion of -categories and -functors. The last section of the article treats various examples of this theory.

**[D]**V. Deodhar,*On a construction of representations and a problem of Enright*, Invent. Math.**57**(1908), 101-118. MR**567193 (81f:17004)****[E]**T. J. Enright,*On the fundamental series of a real semisimple Lie algebra and their irreducibility, resolution and multiplicity formulae*, Ann. of Math. (2)**110**(1979), 1-82. MR**541329 (81a:17003)****[ES]**T. J. Enright and B. Shelton,*Categories of highest weight modules*;*application to classical Hermitian symmetric pairs*, Mem. Amer. Math. Soc.**67**(1987). MR**888703 (88f:22052)****[EW]**T. J. Enright and N. R. Wallach,*Notes homological algebra and representations of Lie algebras*, Duke Math. J.**47**(1980), 1-15. MR**563362 (81c:17013)****[GJ]**O. Gabber and A. Joseph,*On the Bernstein-Gelfand-Gelfand resolution and the Duflo sum formula*, Compositio Math.**43**(1981), 107-131. MR**631430 (82k:17009)****[HS]**P. Hilton and U. Stammbach,*A course in homological algebra*, Springer-Verlag, Berlin and New York, 1971. MR**0346025 (49:10751)****[Jo1]**A. Joseph,*The Enright functor in the Bernstein-Gelfand-Gelfand category*, Invent. Math.**57**(1980), 101-118.**[Jo2]**-,*Completion functors in the category*, Lecture Notes in Math., vol. 1020, Springer-Verlag, Berlin and New York, 1982, pp. 80-106.**[Jo3]**-,*On the Demazure character formula*, Ann. Sci. Ecole Norm. Sup.**18**(1985), 389-419. MR**826100 (87g:17006a)****[Ka]**V. Kac,*Infinite dimensional Lie algebras*, Cambridge Univ. Press, London and New York, 1985. MR**823672 (87c:17023)****[Ki]**W. Kim,*On the concept of completions and explicit determination of invariant pairing for general Kac-Moody Lie algebras*, Comm. Algebra**16**(1988), 185-222. MR**921949 (89d:17025)****[Kn]**A. Knapp,*Lie groups, Lie algebras, and cohomology*, Math. Notes, vol. 34, Princeton Univ. Press, Princeton, NJ, 1988. MR**938524 (89j:22034)****[RW]**A. Rocha-Caridi and N. Wallach,*Projective modules over graded Lie algebras*. I, Invent. Math.**180**(1982), 151-177. MR**661694 (83h:17018)****[Wa]**N. Wallach,*Real reductive groups*I, Academic Press, San Diego, Calif., 1988. MR**929683 (89i:22029)****[Z]**G. Zuckerman,*Tensor products of finite and infinite dimensional representations of semisimple Lie groups*, Ann. of Math. (2)**106**(1972), 295-308. MR**0457636 (56:15841)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
17B67,
17B55

Retrieve articles in all journals with MSC: 17B67, 17B55

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1994-1191610-7

Article copyright:
© Copyright 1994
American Mathematical Society