On finite-dimensional Hopf algebras

This is a survey on the state-of-the-art of the classification of finite-dimensional complex Hopf algebras. This general question is addressed through the consideration of different classes of such Hopf algebras. Pointed Hopf algebras constitute the class best understood; the classification of those with abelian group is expected to be completed soon and there is substantial progress in the non-abelian case.Comment: 25 pages. To be presented at the algebra session of ICM 2014. Submitted versio

On pointed Hopf algebras associated with alternating and dihedral groups

We classify finite-dimensional complex pointed Hopf algebra with group of group-like elements isomorphic to A_5. We show that any pointed Hopf algebra with infinitesimal braiding associated with the conjugacy class of $\pi$ \in $A_n$ is infinite-dimensional if the order of $\pi$ is odd except for $\pi=(1 2 3)$ in $A_4$. We also study pointed Hopf algebras over the dihedral groups.Comment: v2: minor corrections, we remove Table 1 and change Remark 3.4; v3: minor corrections, we modify reference [FGV]; final version to appear in Rev. Uni\'on Mat. Argent., 17 page

Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type I. Non-semisimple classes in PSL(n,q)

We show that Nichols algebras of most simple Yetter-Drinfeld modules over the projective special linear group over a finite field, corresponding to non-semisimple orbits, have infinite dimension. We spell out a new criterium to show that a rack collapses.Comment: minor changes, to appear in Journal of Algebr

On infinite-dimensional Hopf algebras

This is a survey on pointed Hopf algebras with finite Gelfand-Kirillov dimension and related aspects of the theory of infinite-dimensional Hopf algebras.Comment: Comments are welcome! Version 2: references and a few details are adde

Twisting Hopf algebras from cocycle deformations

Let $H$ be a Hopf algebra. Any finite-dimensional lifting of $V\in {}^{H}_{H}\mathcal{YD}$ arising as a cocycle deformation of $A=\mathfrak{B}(V)\#H$ defines a twist in the Hopf algebra $A^*$, via dualization. We follow this recipe to write down explicit examples and show that it extends known techniques for defining twists. We also contribute with a detailed survey about twists in braided categories.Comment: 20 page

Quantum subgroups of a simple quantum group at roots of 1

Let G be a connected, simply connected, simple complex algebraic group and let e be a primitive l-th root of 1, with l odd and 3 does not divide l if G is of type G_{2}. We determine all Hopf algebra quotients of the quantized coordinate algebra of G at e.Comment: 29 pages, accepted in Compositio Mathematic