Crystal limits of compact semisimple quantum groups as higher-rank graph algebras

TL;DR

This paper studies the crystal limit of compact semisimple quantum groups, showing that their associated algebras form a continuous field of higher-rank graph algebras with explicit descriptions.

Contribution

It introduces an $A_0$-subalgebra of the quantized coordinate ring and proves its limit as $q o 0$ forms a Kumjian-Pask algebra, linking quantum groups to higher-rank graph algebras.

Findings

The family of operators $\pi_q(a)$ admits a norm-limit as $q o 0$.

The limit algebra $O[K_0]$ is a Kumjian-Pask algebra.

A continuous field of $C^*$-algebras is constructed with explicit fibers.

Abstract

Let $O_q[K]$ denote the quantized coordinate ring over the field $\mathbb{C}(q)$ of rational functions corresponding to a compact semisimple Lie group $K$, equipped with its *-structure. Let $A_0$ in $\mathbb{C}(q)$ denote the subring of regular functions at $q=0$. We introduce an $A_0$-subalgebra $O_q^{A_0}[K]$ of $O_q[K]$ which is stable with respect to the *-structure, and which has the following properties with respect to the crystal limit $q \to 0$. The specialization of $O_q[K]$ at each $q$ in $(0,\infty)\setminus\{1\}$ admits a faithful *-representation $\pi_q$ on a fixed Hilbert space, a result due to Soibelman. We show that for every element $a$ in $O_q^{A_0}K$, the family of operators $\pi_q(a)$ admits a norm-limit as $q \to 0$. These limits define a *-representation $\pi_0$ of $O_q^{A_0}K$. We show that the resulting *-algebra $O[K_0]=\pi_0(O_q^{A_0}[K])$ is a Kumjian-Pask algebra, in the sense of Aranda Pino, Clark, an Huef and Raeburn. We give an explicit description of the underlying higher-rank graph in terms of crystal basis theory. As a consequence, we obtain a continuous field of $C^*$-algebras $(C(K_q))_{q\in[0,\infty]}$, where the fibres at $q = 0$ and $\infty$ are explicitly defined higher-rank graph algebras.