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Representation Theory

ISSN 1088-4165



The enveloping algebra of the Lie
superalgebra $osp(1,2r)$

Author: Ian M. Musson
Journal: Represent. Theory 1 (1997), 405-423
MSC (1991): Primary 17B35
Published electronically: November 17, 1997
MathSciNet review: 1479886
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Abstract: Let $\mathfrak{g}$ be the Lie superalgebra $osp(1,2r)$ and $U(\mathfrak{g})$ the enveloping algebra of $\mathfrak{g}$.

In this paper we obtain a description of the set of primitive ideals Prim $\! U(\mathfrak{g})$ as an ordered set. We also obtain the multiplicities of composition factors of Verma modules over $U(\mathfrak{g})$, and of simple highest weight modules for $\! U(\mathfrak{g})$ when regarded as a $U(\mathfrak{g}_{0})$-module by restriction.

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  • [AL] M. Aubry and J. M. Lemaire, Zero divisors in enveloping algebras of graded Lie algebras, J. Pure Appl. Algebra 38 (1985), 159-166. MR 87a:17022
  • [B] E. Behr, Enveloping algebras of Lie superalgebras, Pacific J. Math. 130 (1987), 9-25. MR 89b:17023
  • [BB] A. Beilinson and J. Bernstein, Localisation de ${\mathfrak g}$-modules, C.R. Acad. Sci. Paris 292 (1981), 15-18. MR 82k:14015
  • [BK] J.-L. Brylinski and M. Kashiwara, Kazdhan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), 387-410. MR 83e:22020
  • [D] J. Dixmier, Enveloping Algebras, North Holland, Amsterdam, 1977. MR 58:16803b
  • [J1] J. C. Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Mathematics, 750, Springer, Berlin 1979. MR 81m:17011
  • [J2] J. C. Jantzen, Einhüllende Algebren halbeinfacher Lie Algebren, Springer, Berlin, 1983. MR 86m:17012
  • [Jak] H. P. Jakobsen, The Full Set of Unitarizable Highest Weight Modules of Basic Classical Lie Superalgebras, Mem. of the Amer. Math. Soc., No. 532, Amer. Math. Soc., Providence, RI, 1994. MR 95c:17013
  • [K1] V. G. Kac, Lie Superalgebras, Adv. in Math. 16 (1977), 8-96. MR 58:5803
  • [K2] V. G. Kac, Representations of classical Lie superalgebras, Lecture Notes in Mathematics, 676, Springer, Berlin 1978 pp. 579-626. MR 80f:17006
  • [K3] V. G. Kac, Highest weight representations of conformal current algebras, Symposium on Topological and Geometrical Methods in Field Theory, pages 3-15, Espoo, Finland, World Scientific, 1986. MR 91d:17031
  • [KK] V. G. Kac and D. Kazdhan, Structure of representations with highest weight of infinite dimensional Lie algebras, Adv. in Math. 34 (1979), 97-108. MR 81d:17004
  • [L1] E. S. Letzter, Finite correspondence of spectra in Noetherian ring extensions, Proc. Amer. Math. Soc. 116 (1992), 645-652. MR 93a:16003
  • [L2] E. S. Letzter, A bijection of primitive spectra for classical Lie superalgebras of Type I, J. London Math. Soc. 53 (1996), 39-49. MR 96k:17016
  • [M1] I. M. Musson, A classification of primitive ideals in the enveloping algebra of a classical simple Lie superalgebra, Adv. in Math., 91 (1992), 252-268. MR 93c:17022
  • [M2] I. M. Musson, On the center of the enveloping algebra of a classical simple Lie superalgebra, J. Algebra 193 (1997), 75-101. CMP 97:14
  • [M3] I. M. Musson, Some Lie superalgebras related to the Weyl algebras, in preparation.
  • [P] G. Pinczon, The enveloping algebra of the Lie superalgebra $osp(1,2)$, J. Algebra 132 (1990), 219-242. MR 91j:17014
  • [S] N.N. Shapovalov, On a bilinear form on the universal enveloping algebra of a complex semisimple Lie algebra, Funct. Anal. Appl. 6 (1972), 307-312.
  • [Sch] M. Scheunert, The theory of Lie superalgebras, Lecture Notes in Mathematics, 716, Springer-Verlag, Berlin, 1979. MR 80i:17005

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Additional Information

Ian M. Musson
Affiliation: Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI 53201-0413

Received by editor(s): January 27, 1997
Received by editor(s) in revised form: July 25, 1997
Published electronically: November 17, 1997
Additional Notes: Research partially supported by National Science Foundation grant DMS 9500486.
Article copyright: © Copyright 1997 American Mathematical Society

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