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

DOI:
https://doi.org/10.1090/S0002-9947-01-02741-6

Published electronically:
January 3, 2001

MathSciNet review:
1814065

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Abstract | References | Similar Articles | Additional Information

Let be a nilpotent Lie algebra, over a field of characteristic zero, and its universal enveloping algebra. In this paper we study: (1) the prime ideal structure of related to finitely generated -modules , and in particular the set of associated primes for such (note that now is equal to the set of annihilator primes for ); (2) the problem of nontriviality for the modules when is a (maximal) prime of , and in particular when is the augmentation ideal of . We define the support of , 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 . We prove the following generalization of a stability result given by W. Casselman and M. S. Osborne in the case when , as in the theorem, are abelian. We also present some of its interesting consequences.

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

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Additional Information

**Boris Sirola**

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

Email:
sirola@math.hr

DOI:
https://doi.org/10.1090/S0002-9947-01-02741-6

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