Ph.D Thesis
Ph.D Student Nir Ben David On Groups of Central Type and Involutive Yang-Baxter Groups: a Cohomological Approach Department of Mathematics Full Professor Aljadeff Eli Dr. Ginosar Yuval

Abstract

A finite group G is of central type (in the non-classical sense) if it admits a non-degenerate 2-cocycle c with values in the multiplicative group of the field of complex numbers (G acts trivially on this group).

Groups of central type play a fundamental role in the classification of semisimple triangular complex Hopf algebras and can be determined by their representation-theoretical properties.

Suppose that a finite group Q is involutive Yang Baxter group namely Q acts on an abelian group A such that there exists a bijective 1-cocycle π  from Q to  A , where A is the  dual group of A and endowed with the diagonal Qaction. Under this assumption, Etingof and Gelaki gave an explicit formula for a non-degenerate 2-cocycle for the semidirect product of Q  and A. Hence, this semidirect product is of central type.

Our first result is a more general correspondence between bijective and non-degenerate cohomology classes. In particular, given a bijective 1-cocycle π as above, we construct non-degenerate 2-cocycle cπ for certain extensions 1 A G Q 1 which are not necessarily split.

We thus strictly extend the above family of central type groups.

Our second result is a contruction of involutive Yang Baxter groups by cohomological methods.

More precisely, given an involutive Yang Baxter group Q, we construct extensions of Q by an abelian group which are involutive Yang Baxter group. By this construction, we can reprove  that  certain families of groups are involutive Yang Baxter groups, among them semidirect products  of an involutive Yang Baxter group with an abelian group, A-type groups and finite nilpotent groups of class 2.