Isometries of Clifford Algebras I

2017/01/25 by Patrick Eberlein, Eberlein, Patrick
Mathematics · #15A66 #22F99 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.1701.07421

openalex publication_date 2017/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let V be a finite dimensional vector space over a field F of characteristic different from 2, and let Q be a nondegenerate, symmetric, bilinear form on V. Let Cℓ(V,Q) be the Clifford algebra determined by V and Q. The bilinear form Q extends in a natural way to a nondegenerate, symmetric, bilinear form Q on Cℓ(V,Q). Let G be the group of isometries of Cℓ(V,Q) relative to Q, and let LG be the Lie algebra of infinitesimal isometries of Cℓ(V,Q) relative to Q. We derive some basic structural information about LG, and we compute G in the case that F = R, V = Rⁿ and Q is positive definite on Rⁿ. In a sequel to this paper we determine LG in the case that F = R, V = Rⁿ and Q is nondegenerate on Rⁿ.

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