Filtered multiplicative bases of restricted enveloping algebras
type: Item
instance of: scholarly article
title: Filtered multiplicative bases of restricted enveloping algebras
zbMATH Open document ID: 1257.17027
author: Salvatore Siciliano
author: Victor Bovdi
author: Alexandre N. Grishkov
published in: Algebras and Representation Theory
publication date: 2012-07-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/0912.0387
full work available at URL: 1277494.pdf
review text: A vector space basis \(\mathfrak{B}\) of a finite-dimensional associative algebra \(A\) is called filtered multiplicative if the product of two basis vectors in \(\mathfrak{B}\) is either zero or belongs to \(\mathfrak{B}\) and the basis vectors in \(\mathfrak{B}\) belonging to the Jacobson radical \(\mathfrak{rad}(A)\) of \(A\) form a basis of \(\mathfrak{rad}(A)\). In the paper under review the authors investigate the existence of filtered multiplicative bases for the restricted universal enveloping algebras of certain classes of finite-dimensional \(p\)-nilpotent restricted Lie algebras. Let \(\mathfrak{F}_p\) denote the class of finite-dimensional \(p\)-nilpotent restricted Lie algebras. Then the authors prove that for abelian \(L\in\mathfrak{F}_p\) the restricted universal enveloping algebra \(u(L)\) of \(L\) has a filtered multiplicative basis if and only if \(L\) is a direct sum of cyclic \(p\)-subalgebras. In particular, over perfect ground fields \(u(L)\) has always a filtered multiplicative basis, and contrary to commutative group algebras, commutative restricted universal enveloping algebras need not to have a filtered multiplicative basis. Moreover, it is proved that restricted universal enveloping algebras of certain generalizations of abelian \(p\)-nilpotent restricted Lie algebras, namely powerful restricted Lie algebras and \(p\)-nilpotent restricted Lie algebras of nilpotency class \(2\) over fields of characteristic \(p\neq 2\), do not have a filtered multiplicative basis. The \(2\)-nilpotent three-dimensional Heisenberg algebra shows that the last result fails in characteristic \(2\).
reviewed by: Jörg Feldvoss
Mathematics Subject Classification ID: 17B50
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 16S30
zbMATH DE Number: 6054170
zbMATH Keywords: Filtered multiplicative basis
zbMATH Keywords: restricted Lie algebra
zbMATH Keywords: restricted universal enveloping algebra
zbMATH Keywords: \(p\)-nilpotent restricted Lie algebra
zbMATH Keywords: cyclic restricted Lie algebra
zbMATH Keywords: abelian restricted Lie algebra
zbMATH Keywords: powerful restricted Lie algebra
zbMATH Keywords: nilpotency class \(2\)
zbMATH Keywords: three-dimensional Heisenberg algebra
MaRDI profile type: Publication
OpenAlex ID: W2051784437
arXiv ID: 0912.0387
cites work: Q4510753
cites work: Q4672959
cites work: Q5446356
cites work: Representation-finite algebras and multiplicative bases
cites work: On a filtered multiplicative basis of group algebras
cites work: On a filtered multiplicative basis of group algebras. II.
cites work: Restricted Lie algebras with bounded cohomology and related classes of algebras
cites work: Symmetrische Algebren mit endlich vielen unzerlegbaren Darstellungen. I.
cites work: Frattini theory for restricted Lie algebras
cites work: Restricted Lie Algebras and Their Envelopes
cites work: Existence of a multiplicative basis for a finitely spaced module over an aggregate
cites work: On powerful andp-central restricted Lie algebras
cites work: Q4039778
DOI: 10.1007/S10468-009-9203-0
description: scientific article; zbMATH DE number 6054170