2001
DOI: 10.1007/pl00004833
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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 … Show more

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Cited by 10 publications
(14 citation statements)
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“…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.…”
Section: Totally Decomposable Orthogonal Involutions Of Degreementioning
confidence: 96%
See 1 more Smart Citation
“…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.…”
Section: Totally Decomposable Orthogonal Involutions Of Degreementioning
confidence: 96%
“…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).…”
Section: Totally Decomposable Orthogonal Involutions Of Degreementioning
confidence: 99%
“…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.…”
Section: Ii])mentioning
confidence: 99%
“…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.…”
Section: Special Necessary and Sufficient Conditionsmentioning
confidence: 99%
“…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]). …”
Section: Introductionmentioning
confidence: 99%