Quantized function algebras at $q=0$: type $A_{n}$ case

TL;DR

This paper introduces a new framework for quantized function algebras at q=0 for type A_n, defining their structure via generators and relations derived from limits of q>0 cases, with detailed analysis for n=2.

Contribution

It constructs the q=0 quantized function algebras at the C*-algebra level for type A_n and proves their irreducible representations are limits of q>0 representations in the n=2 case.

Findings

Defined the universal C*-algebra for quantized function algebras at q=0

Derived relations from irreducible representations for q>0

Proved representation limits for n=2 case

Abstract

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