Signature of involutions of the second kind (Q1924562)

Let \(A\) be a central simple algebra over a field \(K\) of characteristic different from 2, and let \(\sigma\) be an involution on \(A\). If \(\sigma\) is \(K\)-linear, it is said to be of the first kind. In that case, \textit{D. Lewis} and \textit{J.-P. Tignol} have defined the signature of \(\sigma\) to be the square root of the signature of the quadratic form \(T_\sigma (x)= \text{Trd}_A (\sigma (x) x)\) [Arch. Math. 60, 128-135 (1993; Zbl 0739.11015)]. In this paper, we generalize this definition to the case of involutions of the second kind. The center \(K\) of \(A\) is then a quadratic extension of the field \(k\) fixed by \(\sigma\), and \(T_\sigma\) is a hermitian form. When the signature of such a hermitian form exists, the signature of \(\sigma\) can again be defined as the square root of the signature of \(T_\sigma\). We prove that this is a positive integer, and that in the split case, if \(\sigma\) is the adjoint involution with respect to some hermitian form \(H_\sigma\), then the signature of \(\sigma\) is the absolute value of the signature of \(H_\sigma\). This notion can be used to study the decomposability of an algebra with involution. In particular, we give examples of indecomposable involutions of the second kind, defined on the matrix algebra \(M_4 (K)\) or on \(M_2 (Q)\) for some quaternion algebra \(Q\).