Determinant, Characteristic Polynomial, and Inverse in Commutative Analogues of Clifford Algebras

2024/12/28 by Sharma, Heerak, Dmitry Shirokov, Shirokov, Dmitry
Mathematics · #15A15 #15A66 #Advanced Algebra and Geometry #Algebraic and Geometric Analysis #FOS: Mathematics #FOS: Physical sciences #Finite Group Theory Research #Mathematical Physics (math-ph) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2412.20135

openalex publication_date 2024/12/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Commutative analogues of Clifford algebras are algebras defined in the same way as Clifford algebras except that their generators commute with each other, in contrast to Clifford algebras in which the generators anticommute. In this paper, we solve the problem of finding multiplicative inverses in commutative analogues of Clifford algebras by introducing a matrix representation for these algebras and the notion of determinant in them. We give a criteria for checking if an element has a multiplicative inverse or not and, for the first time, explicit formulas for multiplicative inverses in the case of arbitrary dimension. The new theorems involve only operations of conjugation and do not involve matrix operations. We also consider notions of trace and other characteristic polynomial coefficients and give explicit formulas for them without using matrix representations.

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