Groups of central-type, maximal connected gradings and intrinsic fundamental groups of complex semisimple algebras

Abstract

Maximal connected grading classes of a finite-dimensional algebra A A are in one-to-one correspondence with Galois covering classes of A A which admit no proper Galois covering and therefore are key in computing the intrinsic fundamental group π 1 ( A ) \pi _1(A) . Our first concern here are the algebras A = M n ( C ) A=M_n(\mathbb {C}) . Their maximal connected gradings turn out to be in one-to-one correspondence with the Aut ( G ) (G) -orbits of non-degenerate classes in H 2 ( G , C ∗ ) H^2(G,\mathbb {C}^*) , where G G runs over all groups of central-type whose orders divide n 2 n^2 . We show that there exist groups of central-type G G such that H 2 ( G , C ∗ ) H^2(G,\mathbb {C}^*) admits more than one such orbit of non-degenerate classes. We compute the family Λ \Lambda of positive integers n n such that there is a unique group of central-type of order n 2 n^2 , namely C n × C n C_n\times C_n . The family Λ \Lambda is of square-free integers and contains all prime numbers. It is obtained by a full description of all groups of central-type whose orders are cube-free. We then establish the maximal connected gradings of all finite-dimensional semisimple complex algebras using the fact that such gradings are determined by dimensions of complex projective representations of finite groups. In some cases we give a description of the corresponding fundamental groups.