The discriminant of a symplectic involution.
type: Item
instance of: scholarly article
title: The discriminant of a symplectic involution.
zbMATH Open document ID: 1057.16012
author: Grégory Berhuy
author: Marina Monsurrò
author: Jean-Pierre Tignol
published in: Pacific Journal of Mathematics
publication date: 2004-09-16T00:00:00Z
review text: 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 …
reviewed by: Ivan D. Chipchakov
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 11E72
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 11E04
zbMATH DE Number: 2101069
zbMATH Keywords: central simple algebras
zbMATH Keywords: symplectic involutions
zbMATH Keywords: invariants
zbMATH Keywords: discriminants
zbMATH Keywords: tensor products
zbMATH Keywords: orthogonal involutions
zbMATH Keywords: quaternion algebras
MaRDI profile type: Publication
full work available at URL: pjm.2003.209.201
OpenAlex ID: W1997671479
DOI: 10.2140/PJM.2003.209.201
description: scientific article; zbMATH DE number 2101069