Gaston Andres Garcia

Finite-dimensional pointed Hopf algebras over \(\mathbb{S}_4\)

Let K be an algebraically closed field of characteristic 0. We conclude the classification of finite-dimensional pointed Hopf algebras whose group of group-likes is

On pointed Hopf algebras over dihedral groups

\mathbb{S}_4

Quantum subgroups of a simple quantum group at roots of one

. We also describe all pointed Hopf algebras over

Deformation by cocycles of pointed Hopf algebras over non-abelian groups

\mathbb{S}_5

Techniques for Classifying Hopf Algebras and Applications to Dimension p 3

whose infinitesimal braiding is associated to the rack of transpositions.

On Hopf algebras over quantum subgroups

Let k be an algebraically closed field of characteristic 0 and let D_m be the dihedral group of order 2m with m= 4t, with t bigger than 2. We classify all finite-dimensional Nichols algebras over D_m and all finite-dimensional pointed Hopf algebras whose group of group-likes is D_m, by means of the lifting method. Our main result gives an infinite family of non-abelian groups where the classification of finite-dimensional pointed Hopf algebras is completed. Moreover, it provides for each dihedral group infinitely many non-trivial new examples.

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

Let G be a connected, simply connected, simple complex algebraic group and let ϵ be a primitive l th root of one, l odd and 3∤ l if G is of type G 2 . We determine all Hopf algebra quotients of the quantized coordinate algebra 𝒪 ϵ ( G ).

Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type II: Unipotent classes in symplectic groups

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 formula to deform braided commutator-type relations. Using this construction, we show that all known finite dimensional pointed Hopf algebras over the dihedral groups D_m with m=4t > 11, over the symmetric group S_3 and some families over S_4 are cocycle deformations of bosonizations of Nichols algebras.

On Nichols algebras associated to simple racks

Classifying Hopf algebras of a given finite dimension n over ℂ is a challenging problem. If n is p, p2, 2p, or 2p2 with p prime, the classification is complete. If n = p3, the semisimple and the pointed Hopf algebras are classified, and much progress on the remaining cases was made by the second author but the general classification is still open. Here we outline some results and techniques which have been useful in approaching this problem and add a few new ones. We give some further results on Hopf algebras of dimension p3 and finish the classification for dimension 27.

Hopf Algebras of Dimension 16

Abstract Using the standard filtration associated with a generalized lifting method, we determine all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose coradical generates a Hopf subalgebra isomorphic to the smallest non-pointed non-cosemisimple Hopf algebra K and the corresponding infinitesimal module is an indecomposable object in YD K K (we assume that the diagrams are Nichols algebras). As a byproduct, we obtain new Nichols algebras of dimension 8 and new Hopf algebras of dimension 64.