Strong 1-Boundedness of Unimodular Free Orthogonal Quantum Groups

TL;DR

This paper extends the strong 1-boundedness property to unimodular free orthogonal quantum groups with symplectic parameters, showing these factors are not isomorphic to free group factors and establishing new bounds for their generators.

Contribution

It proves strong 1-boundedness for unimodular free orthogonal quantum groups with symplectic parameters, generalizing previous results and introducing techniques for adding elements without losing boundedness.

Findings

Proves strong 1-boundedness for new class of quantum groups.

Computes free derivatives using Pauli matrices decomposition.

Shows inclusion of the fundamental character preserves 1-boundedness.

Abstract

Recently, Brannan and Vergnioux showed that the free orthogonal quantum group factors $\mathcal{L}\mathbb{F}O_M$ have Jung's strong 1-boundedness property, and hence are not isomorphic to free group factors. We prove an analogous result for the other unimodular case, where the parameter matrix is the standard symplectic matrix in 2N dimensions $J_{2N}$. We compute free derivatives of the defining relations by introducing self-adjoint generators through a decomposition of the fundamental representation in terms of Pauli matrices, resulting in 1-boundedness of these generators. Moreover, we prove that under certain conditions, one can add elements to a 1-bounded set without losing 1-boundedness. In particular this allows us to include the character of the fundamental representation, proving strong 1-boundedness.