‎The Mod $2$ Steenrod algebra is a Hopf algebra that consists of the primary cohomology operations‎, ‎denoted by $Sq^n$‎, ‎between the cohomology groups with $mathbb{Z}_2$ coefficients of any topological space‎. ‎Regarding to its vector space structure over $mathbb{Z}_2$‎, ‎it has many base systems and some of the base systems can also be restricted to its sub algebras‎. ‎On the contrary‎, ‎in ...

A classical result in the theory of Hopf algebras concerns the uniqueness and existence of inte-grals: for an arbitrary Hopf algebra, the integral space has dimension ≤ 1, and for a finite dimensional Hopf algebra, this dimension is exaclty one. We generalize these results to quasi-Hopf algebras and dual quasi-Hopf algebras. In particular, it will follow that the bijectivity of the antipode fol...

A classical result in the theory of Hopf algebras concerns the uniqueness and existence of inte-grals: for an arbitrary Hopf algebra, the integral space has dimension ≤ 1, and for a finite dimensional Hopf algebra, this dimension is exaclty one. We generalize these results to quasi-Hopf algebras and dual quasi-Hopf algebras. In particular, it will follow that the bijectivity of the antipode fol...

Let $H$ be a Hopf algebra and $A$ an $H$-bimodule algebra‎. ‎In this paper‎, ‎we investigate Gorenstein global dimensions for Hopf‎ ‎algebras and twisted smash product algebras $Astar H$‎. ‎Results from‎ ‎the literature are generalized‎. 

In this paper, we associate canonically a precyclic mod- ule to a regular multiplier Hopf algebra endowed with a group-like projection and a modular pair in involution satisfying certain con- dition

We develop a theory which unifies the universal (co)acting bi/Hopf algebras as studied by Sweedler, Manin and Tambara with recently introduced \cite{AGV1} bi/Hopf-algebras that are among all support equivalent algebras. Our approach uses vector spaces endowed family of linear maps between tensor powers $A$, called $\Omega$-algebras. This allows us to treat algebras, coalgebras, braided many oth...

1. Introduction Hopf algebras are achieving prominence in combinatorics through the innuence of G.-C. Rota and his school, who developed the theory of incidence Hopf algebras (see 7], 15], 16]). The aim of this paper is to show that incidence Hopf algebras of partition lattices provide an eecient combinatorial framework for formal group theory and algebraic topology. We start by showing that th...

Let H be a bialgebra. An H-module-algebra is an associative algebra A that is also an H-module such that the multiplication map on A becomes an H-module morphism. This algebraic structure arises often in algebraic topology, quantum groups [11, Chapter V.6], Lie and Hopf algebras theory [3, 16, 19], and group representations [1, Chapter 3]. For example, in algebraic topology, the complex cobordi...

The connected covers of the classifying space BO induce a decreasing filtration {B,) of H, ( B O ;212) by sub-Hopf algebras over the Steenrod algebra A. We describe a multiplicative grading on H*( B O ;212) inducing a direct sum splitting of B, over A,, where {A,) is the usual (increasing) filtration of A. The pieces in the splittings are finite, and the grading extends that of H,R2S3 which spl...