Orthogonal symmetries and Clifford algebras

Abstract

Abstract. Involutions of the Clifford algebra of a quadratic space induced by orthogonal symmetries are investigated.

Surjectivity of the total Clifford invariant and Brauer dimension

“…A similar universal property for algebras with involution over a field is stated in [Mah,§3]. The even Clifford algebra has the following additional properties.…”

Involutions of a Clifford Algebra Induced by Involutions of Orthogonal Group in Characteristic 2

“…In [9] it has been shown that if char F = 2, then there exists a 2-dimensional quadratic space (V, q) and an involution τ in O(V, q) such that (Q, σ) ≃ (C(V ), J τ ) (see [9, (6.2)]). Here we state this result for a field F of characteristic 2.…”

“…Also they were considered in [15] in connection with the Pfister Factor Conjecture, which was finally settled in [1]. In characteristic = 2, some properties of these involutions were also investigated in [9].…”

“…In §6, analogous to [9], a geometric characterization of an involution of a quaternion algebra is obtained. It turns out that if (Q, σ) is a quaternion algebra with involution then there exists a quadratic plane (E, ϕ) and an involution τ ∈ O(E, ϕ) such that (Q, σ) ≃ (C(E), J τ ).…”

Applications of the wall form to unipotent isometries of index two

“…Every involution τ in O(V, q) induces a natural involution J τ , introduced by D. B. Shapiro [14], on the Clifford algebra C(q) satisfying J τ (v) = τ (v) for v ∈ V . In [11], it was shown that every totally decomposable algebra with involution over a field of characteristic different from 2 can be expressed as the Clifford algebra of a quadratic space with a natural involution induced by an involution in the orthogonal group (see [13] for a characteristic 2 counterpart). In characteristic 2, it is readily seen that the converse is also true (see (4.5)).…”