Partial actions and representations, cohomology and globalization

• Grant number: 12/01554-7
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Effective date (Start): July 01, 2012
• Effective date (End): June 30, 2015
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Mikhailo Dokuchaev
• Grantee: Mykola Khrypchenko
• Host 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

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.

Scientific publications

(8)

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

KHRYPCHENKO, MYKOLA; NOVIKOV, BORIS. REFLECTORS AND GLOBALIZATIONS OF PARTIAL ACTIONS OF GROUPS. JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY, v. 104, n. 3, p. 358-379, JUN 2018. (12/01554-7)

DOKUCHAEV, MIKHAILO; KHRYPCHENKO, MYKOLA. Partial cohomology of groups and extensions of semilattices of abelian groups. Journal of Pure and Applied Algebra, v. 222, n. 10, p. 2897-2930, OCT 2018. (12/01554-7, 15/09162-9)

KHRYPCHENKO, MYKOLA. Jordan derivations of finitary incidence rings. LINEAR & MULTILINEAR ALGEBRA, v. 64, n. 10, p. 2104-2118, OCT 2016. (12/01554-7)

DOKUCHAEV, MIKHAILO; KHRYPCHENKO, MYKOLA. Twisted partial actions and extensions of semilattices of groups by groups. INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION, v. 27, n. 7, p. 887-933, NOV 2017. (12/01554-7, 15/09162-9)

KHRYPCHENKO, MYKOLA. Partial actions and an embedding theorem for inverse semigroups. PERIODICA MATHEMATICA HUNGARICA, v. 78, n. 1, p. 47-57, MAR 2019. (12/01554-7)

DOKUCHAEV, M.; KHRYPCHENKO, M.. Partial cohomology of groups. Journal of Algebra, v. 427, p. 142-182, APR 1 2015. (12/01554-7)

DOKUCHAEV, MIKHAILO; KHRYPCHENKO, MYKOLA; JACOBO SIMON, JUAN. GLOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, v. 374, n. 3, p. 1863-1898, MAR 2021. (15/09162-9, 12/01554-7)

DOKUCHAEV, MIKHAILO; KHRYPCHENKO, MYKOLA. Partial cohomology of groups (vol 427, pg 142, 2015). Journal of Algebra, v. 506, p. 5-pg., 2018-07-15. (15/09162-9, 12/01554-7)