On supports and associated primes of modules over the enveloping algebras of nilpotent Lie algebras
| Autor: | Boris Širola |
|---|---|
| Rok vydání: | 2001 |
| Předmět: |
Pure mathematics
Quantum group Applied Mathematics General Mathematics nilpotent Lie algebra semisimple Lie algebra parabolic subalgebra universal enveloping algebra prime-ideal augmentation ideal associated prime localization support and coadjoint orbit Non-associative algebra Universal enveloping algebra Lie conformal algebra Nilpotent Lie algebra Algebra Adjoint representation of a Lie algebra Nilpotent group Semisimple Lie algebra Mathematics |
| Zdroj: | Transactions of the American Mathematical Society. 353:2131-2170 |
| ISSN: | 1088-6850 0002-9947 |
| Popis: | Let n \mathfrak n be a nilpotent Lie algebra, over a field of characteristic zero, and U \mathcal U its universal enveloping algebra. In this paper we study: (1) the prime ideal structure of U \mathcal U related to finitely generated U \mathcal U -modules V V , and in particular the set Ass V \operatorname {Ass}V of associated primes for such V V (note that now Ass V \operatorname {Ass}V is equal to the set Annspec V \operatorname {Annspec}V of annihilator primes for V V ); (2) the problem of nontriviality for the modules V / P V V/\mathcal PV when P \mathcal P is a (maximal) prime of U \mathcal U , and in particular when P \mathcal P is the augmentation ideal U n \mathcal U\mathfrak n of U \mathcal U . We define the support of V 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 Ass V \operatorname {Ass}V . We prove the following generalization of a stability result given by W. Casselman and M. S. Osborne in the case when N \mathfrak N , N \mathfrak N as in the theorem, are abelian. We also present some of its interesting consequences. Theorem. Let Q \mathfrak Q be a finite-dimensional Lie algebra over a field of characteristic zero, and N \mathfrak N an ideal of Q \mathfrak Q ; denote by U ( N ) U(\mathfrak N) the universal enveloping algebra of N \mathfrak N . Let V V be a Q \mathfrak Q -module which is finitely generated as an N \mathfrak N -module. Then every annihilator prime of V V , when V V is regarded as a U ( N ) U(\mathfrak N) -module, is Q \mathfrak Q -stable for the adjoint action of Q \mathfrak Q on U ( N ) U(\mathfrak N) . |
| Databáze: | OpenAIRE |
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