On the l^2-Betti numbers of universal quantum groups

TL;DR

This paper computes the first -Betti number of duals of free unitary quantum groups as one, and shows all -Betti numbers vanish for duals of quantum automorphism groups of full matrix algebras, advancing understanding of quantum group invariants.

Contribution

It provides explicit calculations of -Betti numbers for specific classes of dual quantum groups, revealing new structural properties.

Findings

First -Betti number of duals of free unitary quantum groups is one.

All -Betti numbers vanish for duals of quantum automorphism groups of full matrix algebras.

Advances understanding of -Betti numbers in quantum group theory.

Abstract

We show that the first $\ell^2$-Betti number of the duals of the free unitary quantum groups is one, and that all $\ell^2$-Betti numbers vanish for the duals of the quantum automorphism groups of full matrix algebras.