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Partial Schur Multiplier, cohomology and related topics

Grant number: 12/07610-6
Support type:Scholarships in Brazil - Post-Doctorate
Effective date (Start): September 01, 2012
Effective date (End): August 31, 2014
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Algebra
Principal researcher:Mikhailo Dokuchaev
Grantee:Hector Edonis Pinedo Tapia
Home Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
Associated research grant:09/52665-0 - Groups, rings and algebras: interactions and applications, AP.TEM

Abstract

This project is related to questions about the partial Schur multiplier pM(G) of a group G, with the main purpose to develope a new cohomological theory based on partial actions. The concept of partial Schur multiplier was introduced in order to study partial projective representations. Unlike the classical case, pM(G) is not a group, but it is an inverse commutative semigroup, and can be written as a union of abelian groups called components. Each component is formed by the equivalence classesof partially defined factor sets, whose domains are invariant sets under a natural action of a semigroup T which does not depend on the structure of G. Hence, in order to describe pM(G) completely, we need to find all the domains and study all the components. Following this line, we plan to calculate some families of relevant components of pM(G); in particular, to advance in the study of the total componentpM_GxG(G) which consists of the equivalence classes of the totally defined partialfactor sets. We already know from previous investigations that, in particular, pM_GxG(G) contains as a subgroup the classical Schur Multiplier M(G) of G; but pM_GxG(G) is essentially bigger than M(G); and moreover, anyother component of the semigroup pM(G) is a homomorphic image of pM_GxG(G).Then by knowing the total component it is possible to determine pM(G) completely.In a related direction, we shall study elementary partial representations of infinitegroups, observe that for the finite case any partial factor set is exactly a union of theelementary ones, we want to give a version of this result for the infinite case. Thecandidate will also participate in the elaboration of the concept of the partial Brauergroup, which will be a generalization of the classical Brauer group, and which is animportant step towards the elaboration of the inicial ingredients of a cohomologicaltheory based on partial actions.