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On supports and associated primes of modules over the enveloping algebras of nilpotent Lie algebras |
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| Authors: | Boris Sirola |
| Institution: | Department of Mathematics, University of Zagreb, PP 335, 10002 Zagreb, Croatia |
| Abstract: | Let
Theorem. Let |
| 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 |
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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. 
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 
.