Subproduct systems with quantum group symmetry

Abstract

We introduce a class of subproduct systems of finite dimensional Hilbert spaces whose fibers are defined by the Jones-Wenzl projections in Temperley-Lieb algebras. The quantum symmetries of a subclass of these systems are the free orthogonal quantum groups. For this subclass, we show that the corresponding Toeplitz algebras are nuclear C${}^{\ast }$-algebras that are $KK$-equivalent to $\mathbb{C}$ and obtain a complete list of generators and relations for them. We also show that their gauge-invariant subalgebras coincide with the algebras of functions on the end compactifications of the duals of the free orthogonal quantum groups. Along the way we prove a few general results on equivariant subproduct systems, in particular, on the behavior of the Toeplitz and Cuntz-Pimsner algebras under monoidal equivalence of quantum symmetry groups.

arXiv comment22 pages; v2: minor changes, final version

References in corpus (6)

• The Baum-Connes conjecture for free orthogonal quantum groups
• A Remark on Quantum Group Actions and Nuclearity
• The Martin boundary of a discrete quantum group
• Cuntz-Pimsner algebras for subproduct systems
• Gysin sequences and SU(2)-symmetries of C*-algebras
• Berezin quantization of noncommutative projective varieties

Cited by in corpus (2)

• The Motzkin subproduct system
• Subproduct systems with quantum group symmetry. II