The Universal $C^*$-Algebra of the Quantum Matrix Ball and its Irreducible $*$-Representations

TL;DR

This paper establishes the universal $C^*$-algebra structure for the quantum matrix ball, classifies its irreducible $*$-representations, and connects these to the Fock representation, advancing understanding of quantum matrix spaces.

Contribution

It proves the universal enveloping $C^*$-algebra exists for the quantum matrix ball and classifies all irreducible $*$-representations using a diagram approach.

Findings

Universal $C^*$-algebra is isomorphic to the closure of the Fock representation.

Irreducible $*$-representations can be lifted to those of $ ext{SU}_{2n}$ quantum groups.

Complete classification of irreducible $*$-representations of $ ext{Pol}( ext{Mat}_n)_q$.

Abstract

We prove that any irreducible $*$-representation of $\mathrm{Pol}(\mathrm{Mat}_n)_q$ can be 'lifted' to an irreducible *-representation of $\mathbb{C}[SU_{2n}]_q$, this result is then used to show the existence of the universal enveloping $C^*$- algebra of $\mathrm{Pol}(\mathrm{Mat}_n)_q$ and to prove that it is isomorphic to the closure of the image of the Fock representation. Moreover, we also classify all irreducible $*$-representations of $\mathrm{Pol}(\mathrm{Mat}_n)_q$ using a diagram approach.