Related papers: Solvability for semisimple Hopf algebras via integ…
If H is a Hopf algebra whose square of the antipode is the identity, $v\in\l (V)\otimes H$ is a corepresentation, and $\pi :H\to\l (W)$ is a representation, then $u=(id\otimes\pi)v$ satisfies the equation $(t\otimes id)u^{-1}=((t\otimes…
Action of finite-dimensional Hopf algebra $H$ on commutative $k-$algebra $A$ is considered. As a generalization of the well-known fact for finite groups S. Montgomery raised a problem in 1993 whether $A$ is integral over subalgebra of…
Any finite-dimensional Hopf algebra has a left and a right integral. Conversely, Larsen and Sweedler showed that, if a finite-dimensional algebra with identity and a comultiplication with counit has a faithful left integral, it has to be a…
Let $H$ be a semisimple Hopf algebra over an algebraically closed field $\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$ and $p\nmid 2\dim_{\mathbbm{k}}(H)$. In this paper, we consider the smash product semisimple Hopf…
A criterion for subcoalgebras to be invariant under the adjoint action is given generalizing Masuoka's criterion for normal Hopf subalgebras. At the level of characters, the image of the induction functor from a normal Hopf subalgebra is…
We use braided groups to introduce a theory of $*$-structures on general inhomogeneous quantum groups, which we formulate as {\em quasi-$*$} Hopf algebras. This allows the construction of the tensor product of unitary representations up to…
The article is devoted to the describtion of quasitriangular structures (universal R-matrices) on cocommutative Hopf algebras. It is known that such structures are concentrated on finite dimensional Hopf subalgebras. In particular,…
We discuss algebraic and representation theoretic structures in braided tensor categories C which obey certain finiteness conditions. Much interesting structure of such a category is encoded in a Hopf algebra H in C. In particular, the Hopf…
The question of whether or not a Hopf algebra $H$ is faithfully flat over a Hopf subalgebra $A$ has received positive answers in several particular cases: when $H$ (or more generally, just $A$) is commutative, or cocommutative, or pointed,…
All rational semisimple braided tensor categories are representation categories of weak quasi Hopf algebras. To proof this result we construct for any given category of this kind a weak quasi tensor functor to the category of finite…
Let $\fS$ be an analytic free semigroup algebra. In this paper, we explore richer structures of $\fS$ and its predual $\fS_*$. We prove that $\fS$ and $\fS_*$ both are Hopf algebras. Moreover, the structures of $\fS$ and $\fS_*$ are closely…
In this work we study the induction (induced and coinduced)theory for Hopf group coalgebra. We define a substructure B of a Hopf group coalgebra $H$, called subHopf group coalgebra. Also, we have introduced the definition of Hopf group…
We investigate the splitting property of quasitriangular Hopf algebras through the lens of twisted tensor products. Specifically, we demonstrate that an infinite-dimensional quasitriangular Hopf algebra possesses the splitting property if…
Inspired by the work of Radford, for $H$ an arbitrary quasi-Hopf algebra we describe all the Hopf algebras of dimension $2$ within the braided category of left Yetter-Drinfeld modules over $H$ and determine the biproduct quasi-Hopf algebras…
We derive necessary and sufficient conditions for an ambiskew polynomial ring to have a Hopf algebra structure of a certain type. This construction generalizes many known Hopf algebras, for example U(sl2), U_q(sl2) and the enveloping…
Recently, S. Meljanac proposed a construction of a class of examples of an algebraic structure with properties very close to the Hopf algebroids $H$ over a noncommutative base $A$ of other authors. His examples come along with a subalgebra…
A basis B of a finite dimensional Hopf algebra H is said to be positive if all the structure constants of H relative to B are non-negative. A quasi-triangular structure $R\in H\otimes H$ is said to be positive with respect to B if it has…
Comodule algebras of a Hopf algebroid H with a bijective antipode, i.e. algebra extensions B\subseteq A by H, are studied. Assuming that a lifted canonical map is a split epimorphism of modules of the non-commutative base algebra of H,…
We develop a theory of right group-like projections in Hopf algebras linking them with the theory of left coideal subalgebras with two sided counital integrals. Every right group-like projection is associated with a left coideal subalgebra,…
In this paper we classify and give Kazhdan-Lusztig type character formulas for equivariantly irreducible representations of Lie algebras of reductive algebraic groups over a field of large positive characteristic. The equivariance is with…