Mitja Mastnak

Cohomology of finite-dimensional pointed Hopf algebras

We prove finite generation of the cohomology ring of any finite dimensional pointed Hopf algebra, having abelian group of grouplike elements, under some mild restrictions on the group order. The proof uses the recent classification by Andruskiewitsch and Schneider of such Hopf algebras. Examples include all of Lusztigs small quantum groups, whose cohomology was first computed explicitly by Ginzburg and Kumar, as well as many new pointed Hopf algebras. We also show that in general the cohomology ring of a Hopf algebra in a braided category is braided commutative. As a consequence we obtain some further information about the structure of the cohomology ring of a finite dimensional pointed Hopf algebra and its related Nichols algebra.

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

Hopf Algebra Extensions Arising from Semi-Direct Products of Groups

The Hopf algebra extensions arising from a semi-direct product of groups is a special case of Hopf algebra extensions arising from a matched pair of groups. Such a construction was first considered by G. I. Kac in the 1960s (see [Kac]). He established an exact sequence (now called Kac sequence) that connects group cohomology and Hopf algebra extensions. Later this sequence was revisited and generalized by A. Masuoka [Ma1, Ma2]. The methods used in establishing the Kac sequence involve some powerful homological algebra, but do not provide explicit descriptions of the differentials. Our computational approach describes the homomorphisms involved explicitly and this makes it possible to present some nice examples of groups of Hopf algebra extensions (Opext). It proves useful to introduce the multiplication and the comultiplication parts of the second cohomology group of Hopf algebras Hm and H 2 c . By adapting some methods from the theory of group extensions, a couple of (surprising) sufficient conditions, each of which ensures the equality

On semitransitive Lie algebras of minimal dimension

Abstract Let be an -dimensional vector space over . Some structural results on Lie subalgebras of acting semitransitively and of minimal possible dimension are obtained.

An approximate, multivariable version of Specht's theorem

In this article we provide generalizations of Spechts theorem which states that two n × n matrices A and B are unitarily equivalent if and only if all traces of words in two non-commuting variables applied to the pairs (A, A*) and (B, B*) coincide. First, we obtain conditions which allow us to extend this to simultaneous similarity or unitary equivalence of families of operators, and secondly, we show that it suffices to consider a more restricted family of functions when comparing traces. Our results do not require the traces of words in (A, A*) and (B, B*) to coincide, but only to be close.