Clifford algebras of hyperbolic involutions

Abstract

For (A, σ) a central simple algebra of even degree with hyperbolic orthogonal involution, we describe the canonically induced involution σ on the even Clifford algebra (C 0 (A, σ), σ) of (A, σ). When deg A ≡ 0 mod 8, A ∼ = M 2 (B) and the interesting part of σ is isomorphic to the canonical involution on an exterior power algebra of B. As a corollary, we get some properties of the involution on the exterior power algebra.Associated to any central simple algebra (A, σ) of even degree with orthogonal involution over a field F of characteristic / = 2 is its even Clifford algebra (C 0 (A, σ), σ). The even Clifford algebra is also an algebra (although not F -central) with involution (not necessarily orthogonal), see [KMRT98, 8.10], [Jac64, §III], or [Tit68]. Even when given a very specific description of (A, σ), a description of σ is generally unknown. P. Morandi pointed out to me that this is so even in the simplest case, which is when σ is hyperbolic. (Note that A has a hyperbolic involution if and only if it is isomorphic to M 2 (B) for some B. When this occurs, there is up to isomorphism only one hyperbolic involution on A.) In this paper, we describe σ in that case.We begin by looking at the case where A is split and σ is isotropic. In 1.1 we show that σ restricts to be hyperbolic on each component of C 0 (A, σ). This is quite easy to prove, but is very surprising and seems not to be in the literature (except in the case where σ is hyperbolic). As a consequence, we get that whenever σ is orthogonal, it has trivial discriminant.The result when A is split allows us to focus on the situation when A is nonsplit and σ is hyperbolic. Then deg A ≡ 0 mod 4 and it was proven by Tits [Tit68, p. 40] and, independently, Allen [All68] that (C 0 (A, σ), σ) ∼ =

Algebras with involution that become hyperbolic over the function field of a conic

“…As was observed in Proposition 1.3, (C + (A, σ), σ + ) FQ is isotropic. If it is hyperbolic, then (A, σ) is hyperbolic by the main theorem of [6], a contradiction. Therefore, the anisotropic kernel of (C + (A, σ), σ + ) has degree 4.…”

“…For the rest of the proof, we may thus assume A is not split . If (A, σ) is hyperbolic, then it follows from [6] that the split component of (C(A, σ), σ) is isotropic, hence (b)⇒(c). Conversely, if (c) holds, then [11, (8.5)] shows that (C 0 (ϕ), τ 0 ) is hyperbolic, hence (C + (ϕ), τ + ) also is hyperbolic, proving (c)⇒(b).…”

On Hyperbolic Clifford Algebras with Involution

“…The Book of Involutions [5] and the articles [2], [3], [4] which have been sources of inspiration for our work, contain some special cases of these theorems, either implicitly or as a result of more general theorems. Our approach even for these known cases is different.…”

“…Consider an element δ ∈ K such that δ 2 = abcd. Note that −abcdA = cd(λ 12 ab + λ 34 δ) 2 + bd(λ 13 ac − λ 24 δ) 2 + bc(λ 14 ad + λ 23 δ) 2 Proof. Let q ≃ a, b, c, d, e be a diagonalization of q and let {e 1 , e 2 , e 3 , e 4 , e 5 } be an orthogonal basis of V with respect to q such that q(e 1 ) = a, q(e 2 ) = b, q(e 3 ) = c, q(e 4 ) = d and q(e 5 ) = e.…”

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

“…It is relevant to mention that in characteristic = 2, the behavior of J τ on simple components of C 0 (V ), even for the simplest case τ = id, has been of importance in many applications of Clifford algebras in the literature (see, e.g., [3], [16], [11], [13]). …”