Finite groups with the same character tables, Drinfel’d algebras and Galois algebras

Abstract

We prove that finite groups have the same complex character tables iff the group algebras are twisted forms of each other as Drinfel’d quasi-bialgebras or iff there is non-associative bi-Galois algebra over these groups. The interpretations of class-preserving automorphisms and permutation representations with the same character in terms of Drinfel’d algebras are also given. 1. Introduction. The theory of quasi-Hopf algebras was developed by V.G.Drinfel’d for the description of quantizations of Lie groups and algebras or so-called quantum groups. Althought the deformational quantization approach which is so useful in the theory of quantum groups can’t be applied for the the case of finite groups, the idea of twisting seems to be very suitible for reformulating of various problems from representation theory of finite groups. The key observations of this article is that any bijection between character tables of finite groups corresponds to the quasi-isomorphism of the group algebras considered as quasi-Hopf algebras and any two homomorphisms of the group algebras define the same map of character tables iff they are twisted forms. In particular, we can give the definitions in terms of (quasi-)Hopf algebras of such objects as class-preserving automorphisms, permutation representations with the same character, groups with the same character tables. Namely, any class-preserving automorphism is twisted form of identity maps as homomorphisms of Hopf algebras. Two permutation representations have the same complex character iff the corresponding homomorphisms into symmetric group are twisted forms as homomorphisms of Hopf algebras. Two groups have the same character tables iff their group algebras are twisted forms as quasi-Hopf algebras. This point of view allows to select the subclass of pairs of groups with the same character tables. This subclass consists of pairs of groups whose group algebas are twisted forms as Hopf algebras.