On centralizer algebras for spin representations

TL;DR

This paper presents a detailed algebraic description of centralizer algebras associated with spinor representations of quantum groups, highlighting differences between even and odd dimensions and classical versus quantum cases.

Contribution

It provides a new presentation of these algebras using generators and relations, including non-standard q-deformations for even dimensions.

Findings

Explicit generators and relations for the algebras

Description of the classical limit when q=1

Differences between even and odd-dimensional cases

Abstract

We give a presentation of the centralizer algebras for tensor products of spinor representations of quantum groups via generators and relations. In the even-dimensional case, this can be described in terms of non-standard q-deformations of orthogonal Lie algebras; in the odd-dimensional case only a certain subalgebra will appear. In the classical case q = 1 the relations boil down to Lie algebra relations.