\(*\)-polynomial identities of a nonsymmetric \(*\)-minimal algebra.
type: Item
instance of: scholarly article
title: \(*\)-polynomial identities of a nonsymmetric \(*\)-minimal algebra.
zbMATH Open document ID: 1305.16018
published in: Algebras and Representation Theory
publication date: 2014-06-23T00:00:00Z
review text: Historically polynomial identities with involution (or \(*\)-polynomial identities) are the first ``nonordinary'' polynomial identities to study, and they continue to be an object of intensive investigation. One of the main trends in the theory of \(*\)-polynomial identities is to follow the development of the theory of ordinary polynomial identities. The ``extremal'' algebras have the property that the \(T_*\)-ideals of their \(*\)-polynomial identities are essentially more complicated than the larger \(T_*\)-ideals. In particular, for finite dimensional algebras with involution in characteristic 0, the \(*\)-codimension sequence grows as in the ordinary case and its exponent is an integer. The related extremal algebras are subalgebras of block triangular matrices built on \(*\)-simple algebras, with some kind of symmetry of the blocks. Up to equivalence, there is a finite number of extremal algebras of a given exponent. The \(*\)-polynomial identities of the smallest algebra which is not so symmetric are the main subject of the paper under review. Namely, the authors consider the algebra \(UT_*(F\oplus F,F)\) over an infinite field \(F\) of characteristic different from 2. Here \(F\oplus F\) is equipped with the exchange involution \((a,b)^*=(b,a)\), \(a,b\in F\), and \(UT_*(F\oplus F,F)\) consists of \(4\times 4\) matrices of the form \[ \alpha e_{11}+ae_{12}+\beta(e_{22}+e_{33})+be_{34}+\gamma e_{44},\quad\alpha,\beta,\gamma,a,b\in F, \] with involution which is the flip along the ``other'' diagonal. The authors give a complete description of the polynomial identities of the algebra. They have found the basis of the identities (i.e., generators of the \(T_*\)-ideal) and a basis of the relatively free algebra. In characteristic 0, they describe also the \(GL\times GL\)-module structure of the relatively free algebra and compute the \(*\)-codimensions.
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16W10
zbMATH DE Number: 6306460
zbMATH Keywords: algebras with involution
zbMATH Keywords: polynomial identities
zbMATH Keywords: proper polynomials
zbMATH Keywords: cocharacters
zbMATH Keywords: \(*\)-codimension sequences
author: Vincenzo Nardozza
author: M. Onofrio Di Vincenzo
MaRDI profile type: Publication
full work available at URL: s10468-012-9393-8
OpenAlex ID: W2016269736
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DOI: 10.1007/S10468-012-9393-8
description: scientific article; zbMATH DE number 6306460