Extending involutions on Frobenius algebras
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 \(\sigma _1\).
scientific article; zbMATH DE number 1837024