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

Abstract

Among the involutions of a Clifford algebra, those induced by the involutions of the orthogonal group are the most natural ones. In this work, several basic properties of these involutions, such as the relations between their invariants, their occurrences and their decompositions are investigated.

Applications of the wall form to unipotent isometries of index two

“…Such an involution is indecomposable if and only if it is either a 2-dimensional reflection or a 4-dimensional basic null involution (see [18,Theorem 1]). Some properties of 4-dimensional basic null involutions, called interchange isometries are investigated in [13].…”

“…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)).…”