Lie structure of smash products
type: Item
instance of: scholarly article
title: Lie structure of smash products
zbMATH Open document ID: 1407.16026
author: Salvatore Siciliano
author: Hamid Usefi
published in: Israel Journal of Mathematics
publication date: 2017-04-28T00:00:00Z
review text: An associative algebra \(A\) is called Lie solvable (respectively Lie nilpotent) if the associated Lie algebra with the Lie bracket \([a,b]=ab-ba\) is solvable (respectively nilpotent). Let \(L\) be either a restricted Lie algebra over a field \(\mathbb{F}\) of positive characteristic, or an ordinary Lie algebra over an arbitrary field \(\mathbb{F}\), and let \(G\) be a group acting on \(L\). The authors discuss when is the smash product \(u(L)\# \mathbb{F}G\) in the first situation, or the smash product \(U(L)\# \mathbb{F}G\) in the second one, Lie solvable or Lie nilpotent. Here \(u(L)\) denotes the restricted enveloping algebra of \(L\) in the first case, and the universal enveloping algebra of \(L\) in the second case. For the Lie solvable problem, the authors work under the assumption that the characteristic of \(\mathbb{F}\) is \(\neq 2\). Using the Cartier-Kostant-Milnor-Moore Theorem, it is obtained as a consequence that a cocommutative Hopf algebra \(H\) over a field of characteristic zero is Lie solvable if and only if \(H\) is commutative.
reviewed by: Sorin Dascalescu
Mathematics Subject Classification ID: 16S40
Mathematics Subject Classification ID: 16T05
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 17B50
zbMATH DE Number: 6709001
zbMATH Keywords: Lie algebra
zbMATH Keywords: restricted Lie algebra
zbMATH Keywords: universal enveloping algebra
zbMATH Keywords: restricted enveloping algebra
zbMATH Keywords: smash product
zbMATH Keywords: Lie solvable (nilpotent) associative algebra
zbMATH Keywords: cocommutative Hopf algebra
MaRDI profile type: Publication
full work available at URL: s11856-017-1439-5
OpenAlex ID: W2596131805
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DOI: 10.1007/S11856-017-1439-5
description: scientific article; zbMATH DE number 6709001