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 are 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 $H^2(G,\mathbb {C}^*)$, 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 $H^2(G,\mathbb {C}^*)$ 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 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.