Abelian Group Clifford Algebras
Tim Neijens, Fred Van Oystaeyen

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), 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 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.
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Taxonomy
TopicsFinite Group Theory Research · Algebraic and Geometric Analysis · Advanced Topics in Algebra
