We discuss sufficient conditions ensuring that certain endomorphisms of infinite factors arising from Cuntz algebras are braided. We analyze some explicit non-trivial examples associated to unitary solutions of the quantum Yang-Baxter equation on a Hilbert space of dimension 2. In particular we show the existence of endomorphisms of index 2 associated to representations of Hecke algebras at a primitive fourth root of unity. In this case we compute the associate fusion rules. These fusion rules define a finitely generated ~*-semiring which is not finite. Such a picture seems to be closely related to the description of (the dual of) a deformation, at a fourth root of unity, of some compact matrix group. This could be of some interest for the investigation of quantum symmetries naturally appearing in low-dimensional Quantum Field Theory.
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