Irreducible representations of the crystallization of the quantized function algebras $C(S{U}_q(n+1))$

Abstract

Crystallization of the $C^{\ast }$-algebras $C(S{U}_q(n+1))$ was introduced by Giri \& Pal as a $C^{\ast }$-algebra $C(S{U}_0(n+1))$ given by a finite set of generators and relations. Here we study representations of the $C^{\ast }$-algebra $C(S{U}_0(n+1))$ and prove a factorization theorem for its irreducible representations. This leads to a complete classification of all irreducible representations of this $C^{\ast }$-algebra. As an important consequence, we prove that all the irreducible representations of $C(S{U}_0(n+1))$ arise exactly as $q\to 0+$ limits of irreducible representations of $C(S{U}_q(n+1))$. We also present a few other important corollaries of the classification theorem.

v1: 42 pages, 6 diagrams. v2: 45 pages, 6 diagrams (first section expanded; some references added). v3: 46 pages, one section added at the end, few other small changes made. v4: 48 pages, few minor errors and typos corrected; some parts rewritten to improve the exposition. Final version, to appear in the Journal of NCG