DISCRIMINANT AND CLIFFORD ALGEBRAS

Abstract:  Abstract. The centralizer of a square-central skew-symmetric unit in a central simple algebra with orthogonal involution carries a unitary involution. The discriminant algebra of this unitary involution is shown to be an orthogonal summand in one of the components of the Clifford algebra of the orthogonal involution. As an application, structure theorems for orthogonal involutions on central simple algebras of degree 8 are obtained. Throughout this paper, F denotes a field of characteristic different from 2. Let A be a central simple F-algebra of degree n = 4m, for some integer m, endowed with an involution s of orthogonal type. In the first two sections, we assume that the algebra A contains an element ? such that s(?) = -? and ? 2 = a ? F ?. We denote by ? the centralizer of ? in A. Since ? is skew-symmetric, s induces an involution ?s on ?, and (?, ?s) is a central simple algebra with unitary involution of degree 2m over the ?tale quadratic extension F (?) of F. With this data, we may associate two different central simple algebras with involution, namely the Clifford algebra of (A, s), and the discriminant algebra of ( ?, ?s),