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

Abstract

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 π 1 (A). Our first concern here is the algebras A = Mn(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 Λ of positive integers n such that there is a unique group of central type of order n 2 , namely Cn × Cn. The family Λ 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.

Non-existence of integral Hopf orders for twists of several simple groups of Lie type

“…We will use the terminology central type instead, which is the standard nowadays in this setting and applies to arbitrary groups, not necessarily abelian. See, for instance, the introductions of [3] and [18]. We stress that the property of being non-degenerate for a 2-cocycle is preserved under multiplication by coboundaries; so it is indeed a property of its cohomology class.…”

Mackey's obstruction map for discrete graded algebras

“…The first claim is proven under the more general setting of Theorem 4.12 in the sequel. For the proof of (2) see, e.g., [15,Prop. 2.4], whereas in order to prove (3) note that the homogeneous components of G-graded F-algebras, whose grading class is invertible, should be one-dimensional over F and admit homogeneous units.…”

Realization-obstruction exact sequences for Clifford system extensions

“…Let us mention two of them. A graded isomorphism (see, e.g., [8,Definition 2.3]) between (4.1) and another strongly G-graded k-algebra…”