2016/02/22 by Yuval Ginosar, Ginosar, Yuval, Ofir Schnabel +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1602.06694
openalex publication_date 2016/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Maximal connected grading classes of a finite-dimensional algebra A are in\none-to-one correspondence with Galois covering classes of A which admit no\nproper Galois covering and therefore are key in computing the intrinsic\nfundamental group \π1(A). Our first concern here is the algebras\nA=Mn(\ℂ). Their maximal connected gradings turn out to be in\none-to-one correspondence with the Aut(G)-orbits of non-degenerate classes in\nH2(G, C^*), where G runs over all groups of central type whose orders\ndivide n2. We show that there exist groups of central type G such that\nH2(G, C^*) admits more than one such orbit of non-degenerate classes. We\ncompute the family \Λ of positive integers n such that there is a\nunique group of central type of order n2, namely Cn\× Cn. The family\n\Λ is of square-free integers and contains all prime numbers. It is\nobtained by a full description of all groups of central type whose orders are\ncube-free. We establish the maximal connected gradings of all finite\ndimensional semisimple complex algebras using the fact that such gradings are\ndetermined by dimensions of complex projective representations of finite\ngroups. In some cases we give a description of the corresponding fundamental\ngroups.\n