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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.
Bhattacharjee et al. (Fri,) studied this question.