Determinant, Characteristic Polynomial, and Inverse in Commutative Analogues of Clifford Algebras
Heerak Sharma, Dmitry Shirokov
TL;DR
This work establishes a complete framework for determinants, traces, characteristic polynomials, and multiplicative inverses in the commutative analogues K_{p,q} of Clifford algebras. By constructing a faithful tensor-product matrix representation β and defining Det and Tr via det(β(U)) and tr(β(U)), the authors derive matrix-free, conjugation-based formulas for these invariants, including an explicit product formula Det(U) = ∏_{A⊆{1,...,n}} U^{(A)}. They also provide explicit expressions for the adjoint and the inverse when Det(U) ≠ 0, and prove that Det(U) = 0 exactly characterizes zero divisors. The results generalize known cases for low n (e.g., complex, split-complex, and commutative quaternions) and offer a unified, algebraic route to characteristic polynomials and inverse problems, with potential applications to multicomplex spaces and insights into Clifford algebras.
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.