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We develop a mechanism for classication of isomorphism types of non-trivial semisimple Hopf algebras whose group of grouplikes $G(H)$ is abelian of prime index $p$ which is the smallest prime divisor of $|G(H)|$. We describe structure of…

In this paper we consider some properties of semisimple Hopf algebras of dimension pq where p and q are distinct primes. These properties are useful for classification of such Hopf algebras. In particular, we show that for such a Hopf…

Let $k$ be an algebraically closed field of characteristic $0$. In this paper, we obtain the structure theorems for semisimple Hopf algebras of dimension $p^2q^2$ over $k$, where $p,q$ are prime numbers with $p^2<q$. As an application, we…

Let G be an arbitrary finite group and let S and T be two subsets such that |S|>1, |T|>1, and |TS|< |T|+|S|< |G|-1. We show that if |S|< |G|-4|G|^{1/2}+1 then either S is a geometric progression or there exists a non-trivial subgroup H such…

Let H be a Hopf algebra of dimension pq over an algebraically closed field of characteristic zero, where p, q are odd primes with p < q < 4p+12. We prove that H is semisimple and thus isomorphic to a group algebra, or the dual of a group…

In this paper we classify triangular semisimple and cosemisimple Hopf algebras over any algebraically closed field k. Namely, we construct, for each positive integer N, relatively prime to the characteristic of k if it is positive, a…

Let $p,q$ be prime numbers with $p>q^3$, and $k$ an algebraically closed field of characteristic 0. In this paper, we obtain the structure theorems for semisimple Hopf algebras of dimension $pq^3$.

In this paper, we prove that a non-semisimple Hopf algebra H of dimension 4p with p an odd prime over an algebraically closed field of characteristic zero is pointed provided H contains more than two group-like elements. In particular, we…

Let $q$ be a prime number, $k$ an algebraically closed field of characteristic 0, and $H$ a semisimple Hopf algebra of dimension $2q^3$. This paper proves that $H$ is always semisolvable. That is, such Hopf algebras can be obtained by (a…

Let $q$ be a prime number, $k$ an algebraically closed field of characteristic 0, and $H$ a non-trivial semisimple Hopf algebra of dimension $2q^3$. This paper proves that $H$ can be constructed either from group algebras and their duals by…

If $H$ is a numerical semigroup (that is, a cofinite subset of the non-negative integers closed under addition), then the non-empty subsets of $H$ form a semigroup $\mathcal P(H)$ under the sumset operation induced by addition in $H$.…

Given a finite abelian group $G$ and a subset $S\subseteq G$, we let $N_{G,\ S}$ be the smallest integer $N$ such that for any subset $A\subseteq G$ with $N$ elements, we have $g+S\subseteq A$ for some $g\in G$. Using the probabilistic…

The aim of this paper is to study quasitriangular structures on a class of semisimple Hopf algebras constructed through abelian extensions of $Z_2$ for an abelian group $G$. We prove that there are only two forms of them. Using such…

A subalgebra pair of semisimple complex algebras B < A with inclusion matrix M is depth two if MM^t M < nM for some positive integer n and all corresponding entries. If A and B are the group algebras of finite group-subgroup pair H < G, the…

Let H be a finite dimensional non-semisimple Hopf algebra over an algebraically closed field k of characteristic 0. If H has no nontrivial skew-primitive elements, we find some bounds for the dimension of H_1, the second term in the…

In this paper, we study finite dimensional non-cosemisimple Hopf algebras through its underlying coalgebra. We decompose such a Hopf algebra as a direct sum of `blocks'. Blocks are closely related to each other by `rules' and form the…

Masuoka proved that for a prime p, semisimple Hopf algebras of dimension 2p over an algebraically closed field k of characteristic 0, are trivial (i.e. are either group algebras or the dual of group algebras). Westreich and the second…

The aim of this paper is to give all quasitriangular structures on a class of semisimple Hopf algebras constructed through abelian extensions of $\Bbbk\mathbb{Z}_{2}$ by $\Bbbk^G$ for an abelian group $G$. We first introduce the concept of…

Let p be a prime, and denote the class of radically graded finite dimensional quasi-Hopf algebras over C, whose radical has codimension p, by RG(p). The purpose of this paper is to continue the structure theory of finite dimensional…

This is a survey paper about representation theory and noncommutative geometry of reductive p-adic groups G. The main focus points are: 1. The structure of the Hecke algebra H(G), the Harish-Chandra-Schwartz algebra S(G) and the reduced…