Braided quantum symmetries of graph $\mathrm{C}^*$-algebras

TL;DR

This paper constructs a universal braided quantum symmetry group acting on graph C*-algebras, extending quantum group theory with a twisted monoidal structure, and provides explicit examples including the Cuntz algebra.

Contribution

It introduces a braided analogue of the free unitary quantum group and computes the universal braided quantum symmetry for specific graph C*-algebras.

Findings

Existence of a universal braided compact quantum group for graph C*-algebras.

Construction of a braided free unitary quantum group and its bosonization.

Explicit computation for the Cuntz algebra case.

Abstract

We prove the existence of a universal braided compact quantum group acting on a graph $\mathrm{C}^*$-algebra in the category of $\mathbb{T}$-$\mathrm{C}^*$-algebras with a twisted monoidal structure, in the spirit of the seminal work of S. Wang. To achieve this, we construct a braided analogue of the free unitary quantum group and study its bosonization. As a concrete example, we compute this universal braided compact quantum group for the Cuntz algebra.