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

We prove the existence of a universal braided compact quantum group acting on a graph ${\mathrm{C}}^{\ast }$-algebra in the category of $\mathbb{T}$-${\mathrm{C}}^{\ast }$-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.