The project will be concentrated on questions related to the development of a (co)homological theory based on partial actions and on globalization of such actions. More precisely, we will consider partial projective representations of groups, partial Schur Multipliers, Brauer groups based on partial actions and, more generally, partial cohomology groups. In this direction we plan to advance in calculation of the partial Schur Multipliers for concrete group G, in particular, to determine the component pMG×G(G) of the partial Schur Multiplier pM(G) which consists of the equivalence classes of the totally defined partial factor sets. Since all other components of the semigroup pM(G) are homomorphic images of pMG×G(G), this gives a rather satisfactory information about pM(G). Notice that the usual Schur multiplier M(G) is a subgroup of pMG×G(G), but, in general, M(G) is essentially smaller than pMG×G(G). The candidate will also participate in the elaboration for the case of partial actions of the concept analogous to that of the Brauer group. For this purpose we shall use partial crossed products. Obtaining enough results in this research line we shall proceed with the elaboration of the main ingrediemts of a (co)homological theory based on partial actions. In a related direction we will study the problem of gobalization of partial actions on sets with a binary relation, such as a partial order. In particular we plan to find conditions which guarantee the existence of binary relation on the set under the global action compatible with the initial one.
News published in Agência FAPESP Newsletter about the scholarship: