On the decomposition of metabolic involutions

Abstract

The problem of whether a metabolic idempotent of a central simple algebra with involution is contained in an invariant quaternion subalgebra is investigated. As an application, the similar problem is studied for skewsymmetric elements whose squares lie in the square of the underlying field.Mathematics Subject Classification: 16W10, 16K20, 16K50.

Symmetric Square-Central Elements in Products of Orthogonal Involutions in Characteristic Two

“…In the case where char F 2, deg F A = 8 and σ has a trivial discriminant, the index of A is not 2 and one of the components of the Clifford c 2017 Australian Mathematical Publishing Association Inc. 0004-9727/2017 $16.00 algebra C(A, σ) splits, it was shown in [17, (3.14)] that every skew-symmetric squarecentral element of A lies in a σ-invariant quaternion subalgebra. In [14], some criteria were obtained for symmetric and skew-symmetric elements whose squares lie in F 2 to be contained in a σ-invariant quaternion subalgebra. Also, a sufficient condition was obtained in [12, (6.3)] for symmetric square-central elements in a totally decomposable algebra with orthogonal involution in characteristic two, to be contained in a stable quaternion subalgebra.…”

“…In this work we study some properties of symmetric square-central elements in totally decomposable algebras with orthogonal involution in characteristic two. Let (A, σ) be a totally decomposable algebra with orthogonal involution over a field F of characteristic two and let x ∈ A \ F be a symmetric element with α := x 2 ∈ F. Since the case where α ∈ F 2 was investigated in [14], we assume that α ∈ F × \ F ×2 . First, in Section 3, we study some properties of inseparable subalgebras, introduced in [12].…”