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

Abstract

Let Oq [K] denote the quantized coordinate ring over the field C(q) of rational functions corresponding to a compact semisimple Lie group K, equipped with its ∗-structure. Let A0 ⊂ C(q) denote the subring of regular functions at q = 0. We introduce an A0-subalgebra OA0q [K] ⊂ Oq [K] which is stable with respect to the ∗-structure, and which has the following properties with respect to the crystal limit The specialization of Oq [K] at each q ∈ (0,∞) \ {1} admits a faithful ∗-representation πq on a fixed Hilbert space, a result due to Soibelman. We show that for every element a ∈ OA0 q [K], the family of operators πq (a) admits a norm-limit as q → 0. These limits define a ∗-representation π0 of OA0 q [K]. We show that the resulting ∗-algebra O[K0] = π0 (OA0 q [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 ofC∗-algebras (C(Kq ))q∈[0,∞], where the fibres at q = 0 and∞are explicitly defined higher-rank graph algebras.