Nicolás Andruskiewitsch

From racks to pointed Hopf algebras

Abstract A fundamental step in the classification of finite-dimensional complex pointed Hopf algebras is the determination of all finite-dimensional Nichols algebras of braided vector spaces arising from groups. The most important class of braided vector spaces arising from groups is the class of braided vector spaces ( C X,c q ) , where X is a rack and q is a 2-cocycle on X with values in C × . Racks and cohomology of racks appeared also in the work of topologists. This leads us to the study of the structure of racks, their cohomology groups and the corresponding Nichols algebras. We will show advances in these three directions. We classify simple racks in group-theoretical terms; we describe projections of racks in terms of general cocycles; we introduce a general cohomology theory of racks containing properly the existing ones. We introduce a “Fourier transform” on racks of certain type; finally, we compute some new examples of finite-dimensional Nichols algebras.

Finite quantum groups over abelian groups of prime exponent

Abstract We classify pointed finite-dimensional complex Hopf algebras whose group of group-like elements is abelian of prime exponent p, p>17. The Hopf algebras we find are members of a general family of pointed Hopf algebras we construct from Dynkin diagrams. As special cases of our construction we obtain all the Frobenius–Lusztig kernels of semisimple Lie algebras and their parabolic subalgebras. An important step in the classification result is to show that all these Hopf algebras are generated by group-like and skew-primitive elements.

Finite-dimensional pointed Hopf algebras with alternating groups are trivial

It is shown that Nichols algebras over alternating groups

On Nichols Algebras with Generic Braiding

{\mathbb A_m}

On the structure of (co-Frobenius) Hopf algebras

(m ≥ 5) are infinite dimensional. This proves that any complex finite dimensional pointed Hopf algebra with group of group-likes isomorphic to

Quantum subgroups of a simple quantum group at roots of one

{\mathbb A_m}