Related papers: On pointed Hopf algebras associated with the symme…

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…

We formulate the generation of finite dimensional pointed Hopf algebras by group-like elements and skew-primitives in geometric terms. This is done through a more general study of connected and coconnected Hopf algebras inside a braided…

Let $U$ and $A$ be algebras over a field $k$. We study algebra structures $H$ on the underlying tensor product $U{\otimes}A$ of vector spaces which satisfy $(u{\otimes}a)(u'{\otimes}a') = uu'{\otimes}aa'$ if $a = 1$ or $u' = 1$. For a pair…

Any finite-dimensional complex pointed Hopf algebra with group of group-likes isomorphic to a sporadic group, with the possible exception of the Fischer groups Fi22, the Baby Monster B and the Monster M, is a group algebra.

We show that all unipotent classes in finite simple Chevalley or Steinberg groups, different from PSL_n(q) and PSp_{2n}(q), collapse (i.e. are never the support of a finite-dimensional Nichols algebra), with a possible exception on one…

We present techniques that allow to decide that the dimension of some pointed Hopf algebras associated with non-abelian groups is infinite. These results are consequences of arXiv:0803.2430v1. We illustrate each technique with applications.

Let $\mathds{k}$ be an algebraically closed field of characteristic zero. We determine all finite-dimensional Hopf algebras over $\mathds{k}$ whose Hopf coradical is isomorphic to the unique $12$-dimensional Hopf algebra $\mathcal{C}$…

In this paper, we study the representations of the new finite-dimensional pointed Hopf algebras in positive characteristic given in \cite{Cib09}. We find that these Hopf algebras are symmetric algebras. We determine the simple modules and…

We introduce a method to construct explicitly multiplicative 2-cocycles for bosonizations of Nichols algebras B(V) over Hopf algebras H. These cocycles arise as liftings of H-invariant linear functionals on V tensor V and give a close…

We study deformations of graded braided bialgebras using cohomological methods. In particular, we show that many examples of Nichols algebras, including the finite-dimensional ones arising in the Andruskiewitsch-Schneider program of…

For $n,m\in \mathbb{N}$, let $H_{n,m}$ be the dual of the Radford algebra of dimension $n^{2}m$. We present new finite-dimensional Nichols algebras arising from the study of simple Yetter-Drinfeld modules over $H_{n,m}$. Along the way, we…

In this paper we construct and study two new families of finite dimensional pointed Hopf algebras which generalize Radford's families. We show that over any infinite field which contains a primitive nth root of unity, one of the families…

We classify all non-abelian groups G such that there exists a pair (V,W) of absolutely simple Yetter-Drinfeld modules over G such that the Nichols algebra of the direct sum of V and W is finite-dimensional under two assumptions: the square…

Let A be a finite dimensional algebra over an algebraically closed field with the radical nilpotent of index 2. It is shown that A has finitely many conjugacy classes of left ideals if and only if A is of finite representation type provided…

We parameterize the finite-dimensional irreducible representations of a class of pointed Hopf algebras over an algebraically closed field of characteristic zero by dominant characters. The Hopf algebras we are considering arise in the work…

We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, GK-dimension for short, through the study of Nichols algebras over Q 8 ,the quaternion group . We find all the irreducible Yetter-Drinfeld modules…

The classification of Nichols algebras is an essential step in the classification theory of pointed Hopf algebras by lifting method of N. Andruskiewitsch and H.-J. Schneider. Arithmetic root systems are invariants of Nichols algebras of…

We construct finite-dimensional Hopf algebras whose coradical is the group algebra of a central extension of an abelian group. They fall into families associated to a semisimple Lie algebra together with a Dynkin diagram automorphism. We…

Let $H$ be the $16$-dimensional nontrivial (namely, noncommutative and noncocommutative) semisimple Hopf algebra $H_{b:x^2y}$ classified by Kashina. We figure out all simple Yetter-Drinfeld $H$-modules, and then determine all…

We establish the relationship among Nichols algebras, Nichols braided Lie algebras and Nichols Lie algebras. We prove two results: (i) Nichols algebra $\mathfrak B(V)$ is finite-dimensional if and only if Nichols braided Lie algebra…