Braided quantum symmetries of graph C∗-algebras

We prove the existence of a universal braided compact quantum group acting on a graph [Formula: see text]-algebra in the category of [Formula: see text]-[Formula: see text]-algebras with a twisted monoidal structure, in the spirit of the seminal work of 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.