On representations of the crystallization of the quantized function algebra C(SUq(n + 1))
Saikat Das, Ayan Dey
TL;DR
The paper analyzes crystallization of the quantum SU$(n+1)$ function algebras by treating $A_q(n)=C(SU_q(n+1))$ and its crystallized limit $A_0(n)$. It proves that every irreducible representation of $A_0(n)$ arises as the norm limit of suitably scaled irreducible representations of $A_q(n)$, and that conversely, any non-degenerate (respectively faithful) representation of $A_0(n)$ comes from such a limit of representations of $A_q(n)$. Using Dixmier–Dixmier disintegration and Bruhat order arguments, it further shows that $A_0(n)$ can be realized as the C*-algebra generated by limit operators of faithful representations of $A_q(n)$, with a concrete instance given by the Soibelman representations of $C(SU_q(n+1))$. The work provides a detailed analysis of the spectrum topology, Borel structure, and the interrelation between $q$-deformed and crystallized algebras, yielding a robust framework for crystallization in the quantized function algebra setting.
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).