Subproduct systems with quantum group symmetry. II

TL;DR

This paper extends the analysis of Temperley-Lieb subproduct systems with quantum group symmetry, establishing their algebraic properties, K-theory, and relations, highlighting the role of quantum symmetry groups and monoidal equivalence.

Contribution

It generalizes previous results to the full parameter case, providing a comprehensive understanding of the associated Toeplitz and Cuntz-Pimsner algebras with quantum group symmetry.

Findings

Toeplitz algebras are nuclear

Complete relations for the algebras are found

K-theory of the Cuntz-Pimsner algebras is computed

Abstract

We complete our analysis of the Temperley-Lieb subproduct systems, which define quantum analogues of Arveson's $2$-shift, by extending the main results of the previous paper to the general parameter case. Specifically, we show that the associated Toeplitz algebras are nuclear, find complete sets of relations for them, prove that they are equivariantly $KK$-equivalent to $\mathbb C$ and compute the $K$-theory of the associated Cuntz-Pimsner algebras. A key role is played by quantum symmetry groups, first studied by Mrozinski, preserving Temperley-Lieb polynomials up to rescaling, and their monoidal equivalence to $U_q(2)$.