Abelian Group Clifford Algebras

TL;DR

This paper introduces a matrix-based method to represent Abelian Group Clifford Algebras and provides a way to determine if their center is trivial, building on prior work relating to projective representations and twisted group rings.

Contribution

It offers a novel matrix representation for Abelian Group Clifford Algebras and a procedure to check the triviality of their center, extending previous theoretical frameworks.

Findings

Matrix representation of Abelian Group Clifford Algebras

Method to determine if the center is trivial

Connection to projective representations and twisted group rings

Abstract

In "A note on generalized Clifford algebras and representations" (Caenepeel, S.; Van Oystaeyen, F., Comm. Algebra 17 (1989) no. 1, 93--102.) generalized Clifford algebras were introduced via Clifford representations; these correspond to projective representations of a finite group (Abelian), $G$ say, such that the corresponding twisted group ring has minimal center. The latter then translates to the fact that the corresponding 2-cocycle allows a minimal (none!) number of ray classes and this forces a decomposition of $G$ in cyclic components in a suitable way, cf. Zmud, M., Symplectic geometries and projective representations of finite Abelian groups, (Russian) Mat. Sb. (N.S.) 87(129) (1972), 3--17.. In this small paper, I will provide a way to represent an Abelian Group Clifford Algebra using a matrix, and then give a way to calculate whether or not the center is trivial.