Quantum integrals associated to quantum Hom-Yetter-Drinfel’d modules are defined, and the affineness criterion for quantum Hom-Yetter-Drinfel’d modules is proved in the following form. Let (H,α) be a monoidal Hom-Hopf algebra, (A,β) an (H,α)-Hom-bicomodule algebra and $B=A^{coH}$. Under the assumption that there exists a total quantum integral γ: H → Hom(H,A) and the canonical map $\beta :A{\otimes }_{B}A\to A\otimes H$, $a{\otimes }_{B}b\mapsto S^{-1}\left(b_{\left[1\right]}\right)\alpha \left(b_{\left[0\right][-1]}\right)\otimes {\beta }^{-1}\left(a\right)\beta \left(b_{\left[0\right]\left[0\right]}\right)$, is surjective, we prove that the induction functor $A{\otimes }_B-:̃{{(}_k)}_B{\to }_A^H$ is an equivalence of categories.

Let $H_4$ and $H_8$ be the Sweedler’s and Kac-Paljutkin Hopf algebras, respectively. We prove that any Hopf algebra which factorizes through $H_8$ and $H_4$ (equivalently, any bicrossed product between the Hopf algebras $H_8$ and $H_4$) must be isomorphic to one of the following four Hopf algebras: $H_8\otimes H_4,H_{32,1},H_{32,2},H_{32,3}$. The set of all matched pairs $(H_8,H_4,▹,◃)$ is explicitly described, and then the associated bicrossed product is given by generators and relations.

Let $A_{T}H$ denote the twisted smash product of an arbitrary algebra A and a Hopf algebra H over a field. We present an analogue of the celebrated Blattner-Montgomery duality theorem for $A_{T}H$, and as an application we establish the relationship between the homological dimensions of $A_{T}H$ and A if H and its dual H* are both semisimple.

We introduce the concept of relative Hom-Hopf modules and investigate their structure in a monoidal category $̃\left({ℳ}_k\right)$. More particularly, the fundamental theorem for relative Hom-Hopf modules is proved under the assumption that the Hom-comodule algebra is cleft. Moreover, Maschke’s theorem for relative Hom-Hopf modules is established when there is a multiplicative total Hom-integral.

We introduce the concept of geometrically reductive quantum group which is a generalization of the Mumford definition of geometrically reductive algebraic group. We prove that if G is a geometrically reductive quantum group and acts rationally on a commutative and finitely generated algebra A, then the algebra of invariants $A^G$ is finitely generated. We also prove that in characteristic 0 a quantum group G is geometrically reductive if and only if every rational G-module is semisimple, and that in...

We investigate the structures of Hopf $*$-algebra on the Radford algebras over $ℂ$. All the $*$-structures on $H$ are explicitly given. Moreover, these Hopf $*$-algebra structures are classified up to equivalence.