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

Abstract

Abstract Let O q ⁢ [ K ] \mathcal{O}_{q}[K] be the quantized coordinate ring over the field C ⁢ ( q ) \mathbb{C}(q) of rational functions corresponding to a compact semisimple Lie group 𝐾, equipped with its ∗-structure. Let A 0 ⊂ C ⁢ ( q ) {\mathbf{A}_{0}}\subset\mathbb{C}(q) denote the subring of regular functions at q = 0 q=0 . We introduce an A 0 \mathbf{A}_{0} -subalgebra O q A 0 ⁢ [ K ] ⊂ O q ⁢ [ K ] \mathcal{O}_{q}^{{\mathbf{A}_{0}}}[K]\subset\mathcal{O}_{q}[K] which is stable with respect to the ∗-structure and which has the following properties with respect to the crystal limit q → 0 q\to 0 . The specialization of O q ⁢ [ K ] \mathcal{O}_{q}[K] at each q ∈ ( 0 , ∞ ) ∖ { 1 } q\in(0,\infty)\setminus\{1\} admits a faithful ∗-representation π q \pi_{q} on a fixed Hilbert space, a result due to Soibelman. We show that, for every element a ∈ O q A 0 ⁢ [ K ] a\in\mathcal{O}_{q}^{{\mathbf{A}_{0}}}[K] , the family of operators π q ⁢ ( a ) \pi_{q}(a) admits a norm limit as q → 0 q\to 0 . These limits define a ∗-representation π 0 \pi_{0} of O q A 0 ⁢ [ K ] \mathcal{O}_{q}^{{\mathbf{A}_{0}}}[K] . We show that the resulting ∗-algebra O ⁢ [ K 0 ] = π 0 ⁢ ( O q A 0 ⁢ [ K ] ) \mathcal{O}[K_{0}]=\pi_{0}(\mathcal{O}_{q}^{{\mathbf{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 * C^{*} -algebras ( C ⁢ ( K q ) ) q ∈ [ 0 , ∞ ] (C(K_{q}))_{q\in[0,\infty]} , where the fibres at q = 0 q=0 and ∞ are explicitly defined higher-rank graph algebras.