On *-representations of polynomial algebras in quantum matrix spaces of rank 2

TL;DR

This paper classifies and constructs *-representations of polynomial algebras on quantum matrix spaces of rank 2, specifically for quantum complex matrices and symmetric matrices, providing a comprehensive understanding of their irreducible representations.

Contribution

It provides a complete classification of irreducible *-representations for these quantum polynomial algebras, including explicit construction methods.

Findings

Classified all irreducible *-representations of quantum symmetric matrix algebra.

Developed a construction method for *-representations of quantum matrix algebras.

Provided a full list of *-representations, including subrepresentations.

Abstract

In this paper we study of *-representations for polynomial algebras on quantum matrix spaces. We deal with two special cases of the polynomial algebras, namely the algebra of polynomials on quantum complex matrices $\mathrm{Mat_2}$ and on quantum complex symmetric matrices $\mathrm{Mat_2^{sym}}$. For the second algebra we classify all irreducible *-representations by bounded operators in a Hilbert space (up to a unitary equivalence). Moreover, we present a construction of *-representations of the above algebras which enables to obtain the full list of *-representations (sometimes by passing to subrepresentations).