On the identical relations of associative and Lie algebras equipped with an action
type: Item
instance of: scholarly article
title: On the identical relations of associative and Lie algebras equipped with an action
author: M. Cárdenas Montoya
author: David Riley
published in: Journal of Algebra
publication date: 2022-11-23T00:00:00Z
review text: This article deals with identities of algebras with actions. The very general type of action is as follows: Let \(R\) and \(A\) be associative \(k\)-algebras, with \(R\rightarrow \mathrm{End}_k(A)\). Then \(A\) is a generalized \(R\)-algebra if for each \(r\in R\) there exists two finite sets of pairs \(r^+=(r_{1,i}^+,r_{2,i}^+)\) and \(r^-=(r_{1,j}^-,r_{2,j}^-)\) such that for all \(r\in R\) and \(a_1, a_2\in A\) \[ r(a_1a_2)=\sum_i r_{1,i}^+(a_1)r_{2,i}^+(a_2)+ \sum_j r_{1,j}^-(a_2)r_{2,j}^-(a_1). \] The main theorem concerns actions with two properties: The first is that the image of \(R\) in \(\mathrm{End}_k (A)\) is finite dimensional, say of dimension~\(m\); the second is that \(A\) satisfies an identity of the form \[ x_1\cdots x_d=\sum_{\sigma\ne1}\sum r_1(x_{\sigma(1)}) \cdots r_d(x_{\sigma(d)}), \] where the inner sum ranges over a finite set of \(d\)-tuples of elements of \(R\). In this case the authors' prove that the algebra \(A\) must satisfy an ordinary polynomial identity. Moreover, they bound the degree of the minimal identity as an explicit function of \(m\) and \(d\).
reviewed by: Allan Berele
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16T05
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16W22
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 17B01
Mathematics Subject Classification ID: 17B40
Mathematics Subject Classification ID: 17B70
zbMATH DE Number: 7622329
zbMATH Keywords: polynomial identities
zbMATH Keywords: PI-algebras
zbMATH Keywords: generalized polynomial identities
zbMATH Keywords: identical relations with actions
zbMATH Keywords: Hopf algebra actions
zbMATH Keywords: generalized Hopf algebra actions
zbMATH Keywords: automorphisms
zbMATH Keywords: anti-automorphisms
zbMATH Keywords: involutions
zbMATH Keywords: derivations
zbMATH Keywords: group-graded algebras
Wikidata QID: Q115350260
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2022.10.015
OpenAlex ID: W4307647266
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DOI: 10.1016/J.JALGEBRA.2022.10.015
description: scientific article; zbMATH DE number 7622329