On representations of the crystallization of the quantized function algebra C(SUq(n + 1))

TL;DR

This paper proves that the representations of the crystal limit algebra C(K0) of the quantized function algebra C(SUq(n+1)) can be obtained as limits of representations of the quantum algebras C(Kq), addressing a question posed by Giri & Pal.

Contribution

It establishes a correspondence between representations of the quantum algebras and their crystal limits, showing that all representations of C(K0) arise as limits of those of C(Kq).

Findings

Representations of C(Kq) induce representations of C(K0) via limits.

Every representation of C(K0) is realized as a limit of C(Kq) representations.

C(K0) can be generated by limit operators of faithful C(Kq) representations.

Abstract

The crystal limit C(K0) of the $q$-family of C*-algebras C(Kq) was introduced by Giri & Pal for all K=SU(n+1), n\geq 2. This article aims to prove that the crystal limits C(K0) have the property that the representations of C(Kq) give rise to the representations of the crystallized algebra C(K0) by sending generators of C(K0) to the limit of (scaled) generators of C(Kq)$ and every representation of C(K0) occurs in this way. This work addresses a question raised by Giri & Pal in \cite{GirPal-2024}. As a consequence, one can realize C(K0) as the C*-algebra generated by the limit operators of faithful representations of C(Kq).