Ph.D Thesis | |

Ph.D Student | Ben David Nir |
---|---|

Subject | On Groups of Central Type and Involutive Yang-Baxter Groups: a Cohomological Approach |

Department | Department of Mathematics |

Supervisors | PROF. Eli Aljadeff |

DR. Yuval Ginosar |

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 *Q**−*action. 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 *c**π* 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.