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On supports and associated primes of modules over the enveloping algebras of nilpotent Lie algebras

Author: Boris Sirola
Journal: Trans. Amer. Math. Soc. 353 (2001), 2131-2170
MSC (2000): Primary 17B35, 16P50
Published electronically: January 3, 2001
MathSciNet review: 1814065
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Let $\mathfrak n$ be a nilpotent Lie algebra, over a field of characteristic zero, and $\mathcal U$ its universal enveloping algebra. In this paper we study: (1) the prime ideal structure of $\mathcal U$ related to finitely generated $\mathcal U$-modules $V$, and in particular the set $\operatorname{Ass}V$ of associated primes for such $V$ (note that now $\operatorname{Ass}V$ is equal to the set $\operatorname{Annspec}V$ of annihilator primes for $V$); (2) the problem of nontriviality for the modules $V/\mathcal PV$ when $\mathcal P$ is a (maximal) prime of $\mathcal U$, and in particular when $\mathcal P$ is the augmentation ideal $\mathcal U\mathfrak n$ of $\mathcal U$. We define the support of $V$, as a natural generalization of the same notion from commutative theory, and show that it is the object of primary interest when dealing with (2). We also introduce and study the reduced localization and the reduced support, which enables to better understand the set $\operatorname{Ass}V$. We prove the following generalization of a stability result given by W. Casselman and M. S. Osborne in the case when $\mathfrak N$, $\mathfrak N$ as in the theorem, are abelian. We also present some of its interesting consequences.

Theorem. Let $\mathfrak Q$ be a finite-dimensional Lie algebra over a field of characteristic zero, and $\mathfrak N$ an ideal of $\mathfrak Q$; denote by $U(\mathfrak N)$ the universal enveloping algebra of $\mathfrak N$. Let $V$ be a $\mathfrak Q$-module which is finitely generated as an $\mathfrak N$-module. Then every annihilator prime of $V$, when $V$ is regarded as a $U(\mathfrak N)$-module, is $\mathfrak Q$-stable for the adjoint action of $\mathfrak Q$ on $U(\mathfrak N)$.

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Additional Information

Boris Sirola
Affiliation: Department of Mathematics, University of Zagreb, PP 335, 10002 Zagreb, Croatia

Keywords: Nilpotent Lie algebra, semisimple Lie algebra, parabolic subalgebra, universal enveloping algebra, prime ideal, augmentation ideal, associated prime, annihilator prime, foundation prime, localization, support, reduced localization, reduced support, (co)adjoint action, stable prime, coadjoint orbit
Received by editor(s): February 5, 1999
Received by editor(s) in revised form: June 3, 1999, and January 31, 2000
Published electronically: January 3, 2001
Additional Notes: The author was supported by Fulbright Grant No. 22676, and in part by the Ministry of Science and Technology, Republic of Croatia
Article copyright: © Copyright 2001 American Mathematical Society