On the structure and representation theory of $q$-deformed Clifford algebras

TL;DR

This paper generalizes the structure and representation theory of $q$-deformed Clifford algebras by introducing a twist parameter, analyzing their basis, decomposition, center, semisimplicity, and irreducible representations, and connecting classical and quantum cases.

Contribution

It introduces a new generalized definition of quantum Clifford algebras with a twist parameter, providing basis, decomposition, center, semisimplicity criteria, and irreducible representations, linking classical and quantum frameworks.

Findings

The algebra can be decomposed into rank 1 components.

The center is a classical Clifford algebra over a group algebra.

Semisimplicity depends on the cyclic group's properties.

Abstract

We provide a generalized definition for the quantized Clifford algebra introduced by Hayashi using another parameter $k$ that we call the twist. For a field of characteristic not equal to $2$, we provide a basis for our quantized Clifford algebra, show that it can be decomposed into rank $1$ components, and compute its center to show it is a classical Clifford algebra over the group algebra of a product of cyclic groups of order $2k$. In addition, we characterize the semisimplicity of our quantum Clifford algebra in terms of the semisimplicity of a cyclic group of order $2k$ and give a complete set of irreducible representations. We construct morphisms from quantum groups and explain various relationships between the classical and quantum Clifford algebras. By changing our generators, we provide a further generalization to allow $k$ to be a half integer, where we recover certain quantum Clifford algebras introduced by Fadeev, Reshetikhin, and Takhtajan as a special case.