Radford's biproduct for quasi-Hopf algebras and bosonization

Yetter-Drinfeld Categories for Quasi-Hopf Algebras

“…H YD, and the inverse of the braiding is given by (see [6]): (1) such that the following conditions hold, for all h ∈ H and m ∈ M :…”

“…We prove also that the category H H YD fd of finite dimensional left Yetter-Drinfeld modules is rigid and then we compute explicitly the canonical isomorphisms in H H YD fd . Finally, we show that certain duals of H 0 , the braided Hopf algebra introduced in [6,7], are isomorphic as braided Hopf algebras if H is a finite dimensional triangular quasi-Hopf algebra. …”

“…In addition, in [4] it was shown that H 0 is actually quantum commutative as an algebra in H H YD. If (H, R) is quasitriangular, then H 0 is a Hopf algebra with bijective antipode in H M, with the additional structure (see [6]):…”

On Quasi-Hopf Smash Products and Twisted Tensor Products of Quasialgebras

“…The next result is a generalization of the fact, proved in [11], Proposition 2.5, that H 0 is an algebra in H H YD; the proof is similar to the one in [11] and will be omitted. Proposition 4.4 If H is a quasi-Hopf algebra, (B, λ, Φ λ ) a left H-comodule algebra and v : H → B a morphism of H-comodule algebras, then B v becomes an algebra in H H YD with coaction…”

Hopf cyclic cohomology in braided monoidal categories

“…One knows that a quasitriangular quasi-Hopf algebra (H, R, Φ, α, β) is a Hopf algebra in the braided monoidal category of (left) H-modules [2,19]. This braided Hopf algebra H has the following structure.…”