Residually finite quantum group algebras

TL;DR

This paper proves that for all dimensions except three, certain quantum group algebras have enough finite-dimensional representations to distinguish non-zero elements, showing they are maximally almost periodic.

Contribution

It establishes the residual finiteness and maximal almost periodicity of specific quantum group algebras, answering a previously posed question.

Findings

Quantum group algebra $A_u(n)$ is residually finite for $n eq 3$.

Discrete quantum group with algebra $A_u(n)$ embeds into its quantum Bohr compactification.

Analogous residual finiteness results for algebra $B_u(n)$ with bilinear form.

Abstract

We show that provided $n\ne 3$, the involutive Hopf *-algebra $A_u(n)$ coacting universally on an $n$-dimensional Hilbert space has enough finite-dimensional representations in the sense that every non-zero element acts non-trivially in some finite-dimensional *-representation. This implies that the discrete quantum group with group algebra $A_u(n)$ is maximal almost periodic (i.e. it embeds in its quantum Bohr compactification), answering a question posed by P. So tan. We also prove analogous results for the involutive Hopf *-algebra $B_u(n)$ coacting universally on an $n$-dimensional Hilbert space equipped with a non-degenerate bilinear form.