Some Properties and Examples of Triangular Pointed Hopf Algebras

This paper studies the duals of some finite dimensional pointed Hopf algebras, with abelian group of grouplikes, over an algebraically closed field of characteristic 0, which are either Radford biproducts or else nontrivial liftings of a biproduct.

Since the discovery of quantum groups (Drinfeld, Jimbo) and finite dimensional variations thereof (Lusztig, Manin), these objects were studied from different points of view and had many applications. The present paper is part of a series where we intend to show that important classes of Hopf algebras are quantum groups and therefore belong to Lie theory. One of our main results is the explicit construction of a general family of pointed Hopf algebras from Dynkin diagrams. All...

In a previous work \cite{AS2} we showed how to attach to a pointed Hopf algebra A with coradical $\k\Gamma$, a braided strictly graded Hopf algebra R in the category $_{\Gamma}^{\Gamma}\Cal{YD}$ of Yetter-Drinfeld modules over $\Gamma$. In this paper, we consider a further invariant of A, namely the subalgebra R' of R generated by the space V of primitive elements. Algebras of this kind are known since the pioneering work of Nichols. It turns out that R' is completely determi...

We explain that a new theorem of Deligne on symmetric tensor categories implies, in a straightforward manner, that any finite dimensional triangular Hopf algebra over an algebraically closed field of characteristic zero has Chevalley property, and in particular the list of finite dimensional triangular Hopf algebras over such a field given in math.QA/0008232, math.QA/0101049 is complete. We also use Deligne's theorem to settle a number of questions about triangular Hopf algeb...

We propose the following principle to study pointed Hopf algebras, or more generally, Hopf algebras whose coradical is a Hopf subalgebra. Given such a Hopf algebra A, consider its coradical filtration and the associated graded coalgebra grad(A). Then grad(A) is a graded Hopf algebra, since the coradical A_0 of A is a Hopf subalgebra. In addition, there is a projection \pi: grad(A) \to A_0; let R be the algebra of coinvariants of \pi. Then, by a result of Radford and Majid, R ...

Let $\mathbb{k}$ be an algebraically closed field. We give a complete classification of non-connected pointed Hopf algebras of dimension $16$ with char$\,\mathbb{k}=2$ that are generated by group-like elements and skew-primitive elements. It turns out that there are infinitely many classes (up to isomorphism) of pointed Hopf algebras of dimension 16. In particular, we obtain infinitely many new examples of non-commutative non-cocommutative finite-dimensional pointed Hopf alge...

Let $p$ be a prime. We complete the classification on pointed Hopf algebras of dimension $p^2$ over an algebraically closed field $k$. When $\text{char}k \neq p$, our result is the same as the well-known result for $\text{char}k=0$. When $\text{char}k=p$, we obtain 14 types of pointed Hopf algebras of dimension $p^2$, including a unique noncommutative and noncocommutative type.

We give a presentation in terms of generators and relations of Hopf algebras generated by skew-primitive elements and abelian group of group-like elements with action given via characters. This class of pointed Hopf algebras has shown great importance in the classification theory and can be seen as generalized quantum groups. As a consequence we get an analog presentation of Nichols algebras of diagonal type.

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 also obtain the structure theorems for semisimple Hopf algebras of dimension $9p^2$ and $25q^2$ for all primes $3\neq p$ and $5\neq q$.

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 the second cohomology group of extensions of $\k C_p$ by $\k^G$ where $C_p$ is a cyclic group of order $p$ and $G$ a finite abelian group. We carry out an explicit classification for Hopf algebras of this kind of dimension $p^4$ for any odd p...