Extending involutions on Frobenius algebras
type: Item
instance of: scholarly article
title: Extending involutions on Frobenius algebras
zbMATH Open document ID: 1027.11026
author: Pascale Chuard-Koulmann
author: Jorge Morales
published in: Manuscripta Mathematica
publication date: 2002-12-01T00:00:00Z
review text: Let \(K\) be a field of characteristic not equal to \(2\), equipped with an involution \(x \to \bar x\), possibly the identity, and let \(k = \{\alpha \in K: \bar \alpha = \alpha \}\). Assume also that \(A\) is a central simple \(K\)-algebra, and \(B\) is a maximal commutative subalgebra of \(A\) with an involution \(\sigma \), extending the involution of \(K\). A \(K\)-vector subspace \(S\) of \(B\) is called \(\sigma \)-homogeneous, if \(K = k\) and \(\sigma \) or \(-\sigma \) acts on \(S\) as the identity. The paper under review deals with the problem of whether \(\sigma \) is extendable to an involution of \(A\). The answer to this question in the special case where \(B\) is semisimple has been obtained by \textit{M. Kneser} [see his ``Lectures on Galois cohomology of classical groups''. Tata Institute of Fundamental Research, Bombay (1969; Zbl 0246.14008)]. The present paper shows that if \(A\) possesses an involution extending \(\sigma \) and the socle of each local component of \(B\) is \(\sigma \)-homogeneous, then \(B\) is a Frobenius \(K\)-algebra of dimension equal to the degree of \(A\) (the fact that the homogeneity hypothesis is essential is illustrated by an example); this means that the dual \(B ^{\ast}= \text{Hom}_K (B, K)\), with the structure of a left \(B\)-module given by the rule \((b\lambda)(x)= \lambda(x.b): b,x \in B\), \(\lambda \in B ^{\ast }\), is free (of rank one) over \(B\). The authors obtain a similar result concerning the case in which \(A\) is a \({\mathbb Z}/2{\mathbb Z}\)-algebra, \(B\) is a graded subalgebra, and \(\sigma \) is determined by the grading of \(B\). They also characterize the central simple \(K\)-algebras of degree \(n\) including as a subalgebra a given Frobenius \(K\)-algebra \(B _1\) of dimension \(n\) with an involution \(\sigma _1\), and carrying involutions extending \(\sigma _1\).
reviewed by: Ivan D. Chipchakov
Mathematics Subject Classification ID: 11E57
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 11E72
zbMATH DE Number: 1837024
zbMATH Keywords: central simple \(K\)-algebra
zbMATH Keywords: involution
MaRDI profile type: Publication
full work available at URL: s002290200276
OpenAlex ID: W2080982930
DOI: 10.1007/S002290200276
description: scientific article; zbMATH DE number 1837024