Representation theory of the Reflection Equation Algebra II: Theory of shapes

TL;DR

This paper advances the understanding of the Reflection Equation Algebra by classifying its irreducible representations through a novel shape parameterization, connecting classical Poisson structures with quantum algebra representations.

Contribution

It introduces a new shape matrix concept for the REA and proves each irreducible representation has a unique shape, linking classical and quantum structures.

Findings

Explicit parametrization of symplectic leaves via shape matrices

Introduction of a quantized shape matrix for REA

Uniqueness of shape for each irreducible representation

Abstract

We continue our study of the representations of the Reflection Equation Algebra (=REA) on Hilbert spaces, focusing again on the REA constructed from the $R$-matrix associated to the standard $q$-deformation of $GL(N,\mathbb{C})$ for $0<q<1$. We consider the Poisson structure appearing as the classical limit of the $R$-matrix, and parametrize the symplectic leaves explicitly in terms of a type of matrix we call a shape matrix. We then introduce a quantized version of the shape matrix for the REA, and show that each irreducible representation of the REA has a unique shape.