INTRODUCTION The Cohomological Field Theory was propose by Kontsevich and Manin [5] for description of Gromov-Witten Classes. They prove that Cohomological Field Theory is equivalent to Formal Frobenius manifold. Formal Frobenius manifold is defined by a formal series F , satisfying to associative equations. In points of convergence the series F defines a Frobenius algebras. The set of these po...

We construct, for q a root of unity of odd order, an embedding of the projective special linear group PSL(n) into the group of bi-Galois objects over uq(sl(n)) ∗, the coordinate algebra of the Frobenius-Lusztig kernel of SL(n), which is shown to be an isomorphism at n = 2.

Our purpose is to search association schemes which are obtained by repeating an extension by a thin normal closed subset. To do this, we will focus the higher Frobenius-Schur indicators which are studied in many research areas recently. The classical Frobenius-Schur indicator for association schemes was introduced by Higman [3]. We will define the higher Frobenius-Schur indicators for associati...

The action of the symmetric group Sn on the set Parkn of parking functions of size n has received a great deal of attention in algebraic combinatorics. We prove that the action of Sn on Parkn extends to an action of Sn+1. More precisely, we construct a graded Sn+1-module Vn such that the restriction of Vn to Sn is isomorphic to Parkn. We describe the Sn-Frobenius characters of the module Vn in ...

We investigate functors between abelian categories having a left adjoint and a right adjoint that are similar (these functors are called quasi-Frobenius functors). We introduce the notion of a quasi-Frobenius bimodule and give a characterization of these bimodules in terms of quasi-Frobenius functors. Some applications to corings and graded rings are presented. In particular, the concept of qua...

We classify indecomposable commutative separable (special Frobenius) algebras and their local modules in (untwisted) group-theoretical modular categories. This gives a description of modular invariants for grouptheoretical modular data. As a bi-product we provide an answer to the question when (and in how many ways) two group-theoretical modular categories are equivalent as ribbon categories.

Constructions are given of families of loops which can be described in terms of the table obtained from the loop by using the operation of right division. The motivation comes from group representation theory and the group matrix which goes back to Frobenius.

We introduce a finiteness property for braided fusion categories, describe a conjecture that would characterize categories possessing this, and verify the conjecture in a number of important cases. In particular we say a category has property F if the associated braid group representations factor over a finite group, and suggest that categories of integral Frobenius-Perron dimension are precise...

Note. The determinant of a circulant matrix is an example of a group determinant, where the group is the cyclic group of order n. In 1880 Dedekind suggested generalizing the case of circulants (and more generally group de­ terminants for abelian groups) to arbitrary groups. It was this suggestion that led Frobenius to the creation group of representation theory. See [1] and the references therein.