Subproduct systems with quantum group symmetry. II
Erik Habbestad, Sergey Neshveyev
TL;DR
The paper extends the analysis of Temperley-Lieb subproduct systems with quantum group symmetry to general parameters, showing nuclearity of the Toeplitz algebras and providing complete relations and $KK$-equivalence to $\mathbb{C}$. It leverages the quantum groups $\tilde{O}_P^+$, monoidally equivalent to $U_q(2)$, and a bi-Galois object framework to connect $\mathcal{O}_P$ with a crossed product $\mathcal{O}_P\rtimes_{\bar{\beta}} \mathbb{Z}$, enabling explicit $K$-theory computations. A key technical step is proving that the embedding ${\mathbb C}\to\mathcal{T}_P$ is a $KK^{\tilde{O}_P^+}$-equivalence, pursued via a Baum–Connes argument for the dual of $U_q(2)$. Consequently, the paper derives the $K$-theory of $\mathcal{O}_P$: $K_0(\mathcal{O}_P) \cong \mathbb{Z}/(m-2)\mathbb{Z}$ and $K_1(\mathcal{O}_P) \cong \mathbb{Z}$ for $m=2$ (and $0$ for $m\ge 3$), situating $\mathcal{O}_P$ as a quantum deformation of function algebras on classical spaces like $S^3$ and Podleś spheres within the $D(U_q(2))$-equivariant KK-theory framework.
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 $U_q(2)$.