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

Abstract

Let

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.