This kind of structures naturally comes in CFT models relation to quantum groups a root of unity: the full symmetry algebra is not quite a quantum group at root of unity, because if it were one would have to include the nonphysical quantum dimension zero finite-dimensional quantum group representations into the (pre)Hilbert space; those are the zero norm states which do not contribute to physics (like ghosts). If one quotients by these states then the true unit of a quantum group becomes an idempotent (projector), hence one deals with weak Hopf algebras instead as a price of dealing with true, physical, Hilbert space.
G. Mack, Volker Schomerus, Quasi Hopf quantum symmetry in quantum theory, Nucl. Phys. B370(1992) 185.
where also weak quasi-bialgebras are considered and physical motivation is discussed in detail. Further work in this vain is in
G. Böhm, K. Szlachányi, A coassociative -quantum group with non-integral dimensions, Lett. Math. Phys. 35 (1996) 437–456, arXiv:q-alg/9509008g/abs/q-alg/9509008); Weak -Hopf algebras: the coassociative symmetry of non-integral dimensions, in: Quantum groups and quantum spaces (Warsaw, 1995), 9-19, Banach Center Publ. 40, Polish Acad. Sci., Warszawa 1997.