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Minimal prime ideals in enveloping algebras of Lie superalgebras


Authors: Ellen Kirkman and James Kuzmanovich
Journal: Proc. Amer. Math. Soc. 124 (1996), 1693-1702
MSC (1991): Primary 16S30; Secondary 16D30, 17B35, 17A70
DOI: https://doi.org/10.1090/S0002-9939-96-03230-3
MathSciNet review: 1307538
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Abstract: Let ${\mathfrak g}$ be a finite dimensional Lie superalgebra over a field of characteristic zero. Let $U({\mathfrak g})$ be the enveloping algebra of ${\mathfrak g}$. We show that when ${\mathfrak g} = b(n)$, then $U({\mathfrak g})$ is not semiprime, but it has a unique minimal prime ideal; it follows then that when ${\mathfrak g}$ is classically simple, $U({\mathfrak g})$ has a unique minimal prime ideal. We further show that when ${\mathfrak g}$ is a finite dimensional nilpotent Lie superalgebra, then $U({\mathfrak g})$ has a unique minimal prime ideal.


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  • [AF] F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Second Edition. Springer-Verlag, New York, 1992. MR 94i:16001
  • [B] E. J. Behr, `Enveloping algebras of Lie superalgebras', Pacific J. Math. 130 (1987), 9-25. MR 89b:17023
  • [Be] A. D. Bell, `A criterion for primeness of enveloping algebras of Lie superalgebras', J. Pure Appl. Algebra 69 (1990), 111-120. MR 92b:17014
  • [BM] A. D. Bell and I. M. Musson, `Primitive factors of enveloping algebras of nilpotent Lie superalgebras', J. London Math. Soc. (2) 42 (1990), 401-408. MR 92b:17013
  • [CH] A. W. Chatters and C. R. Hajarnavis, Rings with Chain Conditions. Pitman, Boston, 1980. MR 82k:16020
  • [K] V. Kac, `Lie superalgebras', Adv. Math. 26 (1977), 8-96. MR 58:5803
  • [KKS] E. Kirkman, J. Kuzmanovich, and L. Small, `Finitistic dimensions of Noetherian rings', J. Algebra 147 (1992), 350-364. MR 93h:16008
  • [Le] E. Letzter, `Prime and primitive ideals in enveloping algebras of solvable Lie superalgebras', in Abelian Groups and Noncommutative Rings: A Collection of Papers in Memory of Robert B. Warfield, Jr. Contemporary Mathematics 130, American Mathematical Society, Providence, 1992. MR 93i:17003
  • [LM] E. Letzter and I. Musson, `Complete sets of representations of classical Lie superalgebras', Lett. Math. Phys. 31 (1994), 247-253. CMP 94:14
  • [L] T. Levasseur, `Some properties of non-commutative regular graded rings', Glasgow Math. J. 34 (1992), 277-300. MR 93k:16045
  • [MR] J.C. McConnell and J.C. Robson, Noncommutative Noetherian Rings. Wiley Series in Pure and Appl. Math., New York, 1987. MR 89j:16023
  • [S] M. Scheunert, The Theory of Lie Superalgebras. Lecture Notes in Math. 716, Springer-Verlag, Berlin, 1979. MR 80i:17005
  • [SZ] J. T. Stafford and J. J. Zhang, `Homological properties of (graded) Noetherian PI rings', preprint. CMP 95:01
  • [W] M. C. Wilson, `Primeness of the enveloping algebra of a Cartan type Lie superalgebra', preprint. CMP 95:03

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Additional Information

Ellen Kirkman
Affiliation: Department of Mathematics Wake Forest University Winston-Salem, North Carolina 27109
Email: kirkman@mthcsc.wfu.edu

James Kuzmanovich
Affiliation: Department of Mathematics Wake Forest University Winston-Salem, North Carolina 27109
Email: kuz@mthcsc.wfu.edu

DOI: https://doi.org/10.1090/S0002-9939-96-03230-3
Keywords: Enveloping algebra, Lie superalgebra, minimal prime ideals
Received by editor(s): August 12, 1994
Received by editor(s) in revised form: December 13, 1994
Additional Notes: The first author was supported in part by a grant from the National Security Agency.
Communicated by: Ken Goodearl
Article copyright: © Copyright 1996 American Mathematical Society

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