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

TL;DR

This paper introduces a matrix representation and determinant concept for commutative analogues of Clifford algebras, providing explicit formulas for inverses and characteristic polynomial coefficients without matrix operations.

Contribution

It develops a novel approach to find inverses and characteristic polynomials in commutative Clifford analogues using conjugation-based formulas, avoiding matrix computations.

Findings

Explicit formulas for inverses in arbitrary dimensions.

Criteria for invertibility of elements.

Formulas for trace and characteristic polynomial coefficients.

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.