We give a complete classification of right coideal subalgebras that contain all grouplike elements for the quantum group $U_q^{+}\left({\mathrm{𝔰𝔬}}_{2n+1}\right)$ provided that $q$ is not a root of 1. If $q$ has a finite multiplicative order $t>4$; this classification remains valid for homogeneous right coideal subalgebras of the Frobenius–Lusztig kernel $u_q^{+}\left({\mathrm{𝔰𝔬}}_{2n+1}\right)$. In particular, the total number of right coideal subalgebras that contain the coradical equals $\left(2n\right)!!$; the order of the Weyl group defined by the root system of type $B_n$.

We define and investigate Ringel-Hall algebras of coalgebras (usually infinite-dimensional). We extend Ringel's results [Banach Center Publ. 26 (1990) and Adv. Math. 84 (1990)] from finite-dimensional algebras to infinite-dimensional coalgebras.

As generalizations of separable and Frobenius algebras, separable and Frobenius monoidal Hom-algebras are introduced. They are all related to the Hom-Frobenius-separability equation (HFS-equation). We characterize these two Hom-algebraic structures by the same central element and different normalizing conditions, and the structure of these two types of monoidal Hom-algebras is studied. The Nakayama automorphisms of Frobenius monoidal Hom-algebras are considered.

Let $̃{(}_k){\left(H\right)}_A^C$ be the category of Doi Hom-Hopf modules, $̃{{(}_k)}_A$ be the category of A-Hom-modules, and F be the forgetful functor from $̃{(}_k){\left(H\right)}_A^C$ to $̃{{(}_k)}_A$. The aim of this paper is to give a necessary and suffcient condition for F to be separable. This leads to a generalized notion of integral. Finally, applications of our results are given. In particular, we prove a Maschke type theorem for Doi Hom-Hopf modules.

We call a monoidal category C a Serre category if for any C, D ∈ C such that C ⊗ D is semisimple, C and D are semisimple objects in C. Let H be an involutory Hopf algebra, M, N two H-(co)modules such that M ⊗ N is (co)semisimple as a H-(co)module. If N (resp. M) is a finitely generated projective k-module with invertible Hattory-Stallings rank in k then M (resp. N) is (co)semisimple as a H-(co)module. In particular, the full subcategory of all finite dimensional modules, comodules or Yetter-Drinfel’d...

The goal of our work is to study the spaces of primitive elements of some combinatorial Hopf algebras, whose underlying vector spaces admit linear basis labelled by subsets of the set of maps between finite sets. In order to deal with these objects we introduce the notion of shuffle algebras, which are coloured algebras where composition is not always defined. We define bialgebras in this framework and compute the subpaces of primitive elements associated to them. These spaces of primitive elements...

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...

Let $𝕜$ be an algebraically closed field of characteristic $p\ne 2$, and let $Q_8$ be the quaternion group. We describe the structures of all simple modules over the quantum double $D\left(𝕜Q_8\right)$ of group algebra $𝕜Q_8$. Moreover, we investigate the tensor product decomposition rules of all simple $D\left(𝕜Q_8\right)$-modules. Finally, we describe the Grothendieck ring $G_0\left(D\left(𝕜Q_8\right)\right)$ by generators with relations.

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.