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

Musson, Ian

doi:10.1090/S1088-4165-97-00020-4

The enveloping algebra of the Lie superalgebra $osp(1,2r)$
HTML articles powered by AMS MathViewer

by Ian M. Musson PDF

Represent. Theory 1 (1997), 405-423 Request permission

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.

References

Marc Aubry and Jean-Michel Lemaire, Zero divisors in enveloping algebras of graded Lie algebras, J. Pure Appl. Algebra 38 (1985), no. 2-3, 159–166. MR 814174, DOI 10.1016/0022-4049(85)90006-4
Erazm J. Behr, Enveloping algebras of Lie superalgebras, Pacific J. Math. 130 (1987), no. 1, 9–25. MR 910651, DOI 10.2140/pjm.1987.130.9
Alexandre Beĭlinson and Joseph Bernstein, Localisation de $g$-modules, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 1, 15–18 (French, with English summary). MR 610137
J.-L. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), no. 3, 387–410. MR 632980, DOI 10.1007/BF01389272
Jacques Dixmier, Algèbres enveloppantes, Cahiers Scientifiques, Fasc. XXXVII, Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1974 (French). MR 0498737
Jens Carsten Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Mathematics, vol. 750, Springer, Berlin, 1979 (German). MR 552943, DOI 10.1007/BFb0069521
J. C. Jantzen, Einhüllende Algebren halbeinfacher Lie-Algebren, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 393–401 (German). MR 804695, DOI 10.1007/978-3-642-68955-0
Hans Plesner Jakobsen, The full set of unitarizable highest weight modules of basic classical Lie superalgebras, Mem. Amer. Math. Soc. 111 (1994), no. 532, vi+116. MR 1214730, DOI 10.1090/memo/0532
V. G. Kac, Lie superalgebras, Advances in Math. 26 (1977), no. 1, 8–96. MR 486011, DOI 10.1016/0001-8708(77)90017-2
V. Kac, Representations of classical Lie superalgebras, Differential geometrical methods in mathematical physics, II (Proc. Conf., Univ. Bonn, Bonn, 1977) Lecture Notes in Math., vol. 676, Springer, Berlin, 1978, pp. 597–626. MR 519631, DOI 10.1007/BFb0063691
Victor G. Kac, Highest weight representations of conformal current algebras, Topological and geometrical methods in field theory (Espoo, 1986) World Sci. Publ., Teaneck, NJ, 1986, pp. 3–15. MR 1026476
V. G. Kac and D. A. Kazhdan, Structure of representations with highest weight of infinite-dimensional Lie algebras, Adv. in Math. 34 (1979), no. 1, 97–108. MR 547842, DOI 10.1016/0001-8708(79)90066-5
Edward S. Letzter, Finite correspondence of spectra in Noetherian ring extensions, Proc. Amer. Math. Soc. 116 (1992), no. 3, 645–652. MR 1098402, DOI 10.1090/S0002-9939-1992-1098402-1
Edward S. Letzter, A bijection of primitive spectra for classical Lie superalgebras of type I, J. London Math. Soc. (2) 53 (1996), no. 1, 39–49. MR 1362685, DOI 10.1112/jlms/53.1.39
Ian M. Musson, A classification of primitive ideals in the enveloping algebra of a classical simple Lie superalgebra, Adv. Math. 91 (1992), no. 2, 252–268. MR 1149625, DOI 10.1016/0001-8708(92)90018-G
I. M. Musson, On the center of the enveloping algebra of a classical simple Lie superalgebra, J. Algebra 193 (1997), 75–101.
I. M. Musson, Some Lie superalgebras related to the Weyl algebras, in preparation.
Georges Pinczon, The enveloping algebra of the Lie superalgebra $\textrm {osp}(1,2)$, J. Algebra 132 (1990), no. 1, 219–242. MR 1060845, DOI 10.1016/0021-8693(90)90265-P
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.
Manfred Scheunert, The theory of Lie superalgebras, Lecture Notes in Mathematics, vol. 716, Springer, Berlin, 1979. An introduction. MR 537441, DOI 10.1007/BFb0070929