Irreducible representations of the crystallisation of the $C^{*}$-algebra $C(SU_{q}(n+1))$

Manabendra Giri, Arup Kumar Pal

Published 2024-02-14, updated 2024-04-06Version 2

Crystallization of the $C^*$-algebras $C(SU_{q}(n+1))$ was introduced by Giri \& Pal in arXiv:2203.14665 [math.QA] as a $C^*$-algebra $A_{n}(0)$ given by a finite set of generators and relations. Here we study the irreducible representations of the $C^*$-algebra $A_{n}(0)$ and prove a factorization theorem for its irreducible representations. This leads to a complete classification of all irreducible representations of $A_{n}(0)$. As an important consequence, we prove that all the irreducible representations arise exactly as $q\to 0+$ limits of the irreducible representations of $C (SU_{q}(n+1))$ given by a result of Soibelman. As a consequence, we show that $A_{n}(0)$ is a type I $C^{*}$-algebra.