Generalized Clifford Algebras as Algebras in Suitable Symmetric Linear Gr-Categories

TL;DR

This paper explores generalized Clifford algebras within symmetric linear Gr-categories, extending previous results by leveraging gauge transformations to derive decomposition theorems and weak Hopf structures in a more conceptual framework.

Contribution

It extends the categorical approach to Clifford algebras to their generalized forms, providing new decomposition theorems and weak Hopf algebra structures.

Findings

Derived decomposition theorem for generalized Clifford algebras

Established categorical weak Hopf structures in symmetric Gr-categories

Simplified proofs using gauge transformations

Abstract

By viewing Clifford algebras as algebras in some suitable symmetric Gr-categories, Albuquerque and Majid were able to give a new derivation of some well known results about Clifford algebras and to generalize them. Along the same line, Bulacu observed that Clifford algebras are weak Hopf algebras in the aforementioned categories and obtained other interesting properties. The aim of this paper is to study generalized Clifford algebras in a similar manner and extend the results of Albuquerque, Majid and Bulacu to the generalized setting. In particular, by taking full advantage of the gauge transformations in symmetric linear Gr-categories, we derive the decomposition theorem and provide categorical weak Hopf structures for generalized Clifford algebras in a conceptual and simpler manner.