Involutions and anti-automorphisms of central simple algebras. (Q1399146)

The asymmetry of a nonsingular bilinear pairing \(b\) on a finite-dimensional vector space \(V\) is the endomorphism \(a_b\) of \(V\) defined by the condition \(b(y,x)=b(x,a_b(y))\) for all \(x,y\in V\). This notion was extended to linear anti-automorphisms of central simple algebras by \textit{A.~Cortella} and \textit{J.-P.~Tignol} [J. Pure Appl. Algebra 167, No. 2-3, 175-193 (2002; Zbl 0988.11016)]. The paper under review follows on with observations in three directions. (1) Cortella and the reviewer established a necessary and sufficient condition for an endomorphism of a vector space to be the asymmetry of some nonsingular pairing. The author points out that the same conditions had been found (and expressed in a nicer form) by \textit{C. S. Ballantine} [Linear Multilinear Algebra 6, 201-217 (1978; Zbl 0392.15014)]. (2) The trace form of a linear anti-automorphism \(\sigma\) of a central simple algebra \(A\) is the quadratic form \(T_\sigma=T(\sigma(x)x)\), where \(T\) is the reduced trace on \(A\). Assuming that the characteristic is different from~\(2\), the author determines when the form \(T_\sigma\) is nonsingular and proves that its determinant is a square. (3) Let \(D\) be a central division algebra with an involutive anti-automorphism which does not restrict to the identity on the centre. Asymmetries of nonsingular sesquilinear forms on \(D\)-vector spaces are characterized, extending the Cortella-Tignol result from the linear to the semi-linear case.