Quantized function algebras at \({\varvec{q=0}}\): Type \({\varvec{A}}_{{\varvec{n}}}\) case

We define the notion of quantized function algebras at \(q=0\) $q=0$ or crystallization of the q deformations of the type \(A_{n}\) $A_n$ compact Lie groups at the \(C^*\) $C^{\ast }$ -algebra level. The \(C^{*}\) $C_{}^{\ast }$ -algebra \(A_{n}(0)\) $A_n(0)$ is defined as a universal \(C^*\) $C^{\ast }$ -algebra given by a finite set of generators and relations. We obtain these relations by looking at the irreducible representations of the quantized function algebras for \(q>0\) $q>0$ and taking limit as \(q\rightarrow 0+\) $q\to 0+$ after rescaling the generating elements appropriately. We then prove that in the \(n=2\) $n=2$ case the irreducible representations \(A_{2}(0)\) $A_2(0)$ are precisely the \(q\rightarrow 0+\) $q\to 0+$ limits of the irreducible representations of the \(C^*\) $C^{\ast }$ -algebras \(A_{2}(q)\) $A_2(q)$ .