Irreducible representations of the crystallization of the quantized function algebras $C({\mathrm{SU}}_q(n+1))$

Abstract

The crystallization of the $C^{\ast }$-algebras $C({\mathrm{SU}}_q(n+1))$ was introduced by Giri and Pal in [Proc. Indian Acad. Sci. Math. Sci. 134 (2024), article no. 30] as a $C^{\ast }$-algebra $C({\mathrm{SU}}_0(n+1))$ given by a finite set of generators and relations. Here we study the representations of the $C^{\ast }$-algebra $C({\mathrm{SU}}_0(n+1))$ and prove a factorization theorem for its irreducible representations. This leads to a complete classification of all irreducible representations of this $C^{\ast }$-algebra. As an important consequence, we prove that all the irreducible representations of $C({\mathrm{SU}}_0(n+1))$ arise exactly as $q\to 0+$ limits of the irreducible representations of $C({\mathrm{SU}}_q(n+1))$. We also present a few other important corollaries of the classification theorem.