The discriminant of a symplectic involution.
Let \(F\) be a field of characteristic different from \(2\), and \(A\) a finite-dimensional central simple \(F\)-algebra with a symplectic involution \(\theta\), i.e. an involution which, after a scalar extension to a splitting field, is adjoint to an alternating form [see \textit{M.-A. Knus}, \textit{A. Merkurjev}, \textit{M. Rost} and \textit{J.-P. Tignol}, The Book of Involutions, Am. Math. Soc. Colloq. Publ. 44 (1998; Zbl 0955.16001), Section 2]. It is well-known that then the degree \(\deg(A)\) is necessarily an even integer \(n=2m\). In contrast to central simple algebras with orthogonal involutions, for which invariants corresponding to the discriminant and Clifford algebras of quadratic forms are defined, no ``classical'' invariant, besides the signature, is known for the symplectic case. When \(m\) is even, i.e. \(n\) is a multiple of \(4\), and \(\sigma\) is also a symplectic involution of \(A\), the paper under review introduces an invariant \(\Delta_\theta(\sigma)\) of the pair \((\theta,\sigma)\) with value in the third Galois cohomology group \(H^3(F,\mu_2)\), where \(\mu_2=\{-1,+1\}\). It shows that \(\Delta_\theta(\sigma)\) depends only on the conjugacy classes of \(\theta\) and \(\sigma\). The authors call \(\Delta_\theta(\sigma)\) a discriminant of the pair \((\theta,\sigma)\) because it is directly linked to the discriminant of Hermitian forms (see Example 2). They prove that the discriminant is the only cohomological invariant of degree \(3\), and establish its relations with trace forms. The concluding part of the paper is devoted to structural questions in the special case where \(m=4\) and the Schur index \(\text{ind}(A)\) is \(2\) or \(4\) (which ensures the existence of hyperbolic symplectic involutions of \(A\)); the case of \(\text{ind}(A)=1\) is omitted, since then all symplectic involutions are hyperbolic. It is shown that \((A,\sigma)\) decomposes into a tensor product \((Q,\gamma )\otimes_F(A_0,\sigma_0)\), where \(Q\) is a quaternion subalgebra of \(A\), \(\gamma\) is the conjugation in \(Q\), \(A_0\) is the centralizer of \(Q\) in \(A\), and \(\sigma_0\) is an orthogonal involution. In addition, it is proved that if \(\theta\) is hyperbolic, then \(\Delta_\theta(\sigma)=0\) if and only if there is a decomposition \((A,\sigma)=(A_1,\sigma_1)\otimes_F(A_2,\sigma_2)\otimes _F(A_3,\gamma_3)\), where \(A_1,A_2,A_3\) are quaternion \(F\)-subalgebras of \(A\), \(A_1\) is isomorphic to the matrix algebra \(M_2(F)\), \(\sigma_1,\sigma_2\) are orthogonal involutions of \(A_1\) and \(A_2\), respectively, and \(\gamma_3\) is the conjugation in \(A_3\).
scientific article; zbMATH DE number 2101069