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 $A$ are in one-to-one correspondence with Galois covering classes of $A$ which admit no proper Galois covering and therefore are key in computing the intrinsic fundamental group $\pi_1(A)$. Our first concern here is the algebras $A=M_n(\mathbb{C})$. Their maximal connected gradings turn out to be in one-to-one correspondence with the Aut$(G)$-orbits of non-degenerate classes in $H^2(G,\C^*)$, where $G$ runs over all groups of central type whose orders divide $n^2$. We show that there exist groups of central type $G$ such that $H^2(G,\C^*)$ admits more than one such orbit of non-degenerate classes. We compute the family $\Lambda$ of positive integers $n$ such that there is a unique group of central type of order $n^2$, namely $C_n\times C_n$. The family $\Lambda$ 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 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.