Partial cohomology of groups

Abstract

We study the relations between partial and global group cohomology. We show, in particular, that given a unital partial action of a group G on a ring A, such that A is a direct product of indecomposable rings, then any partial n-cocycle with values in A is globalizable.

Partial cohomology of groups and extensions of semilattices of abelian groups

“…Thus, m ′ ∈ M(A). That m ′ ∈ C(M(A)) follows by (6) and Lemma 1.8. Clearly, the extension of m is unique, as each n ∈ M(A) satisfies na = n(aa −1 )a and an = a(a −1 a)n.…”

“…Influenced by R. Exel's notion of a continuous twisted partial group action on a C * -algebra [13] and its ring theoretic analogue in [4], a cohomology theory was introduced in [6], which suits unital twisted partial actions as well as the concept of their equivalence given in [5]. The cohomology from [6] is strongly related to the cohomology of inverse semigroups.…”

“…Given such m, we shall first prove the following equality m(aa −1 )a = a(a −1 a)m (6) and then use it to define an extension of m to a central multiplier on the whole A.…”

Globalization of group cohomology in the sense of Alvares-Alves-Redondo

“…A G-module over a group G is an action of G on an abelian group, and the starting point in [13] is the replacement of a G-module by a partial G-module. The latter means a unital partial action of G on a commutative monoid.…”

Representações parciais de grupos, seus domínios e o multiplicador de Schur parcial

“…Foram desenvolvidas ainda diversas aplicações à teoria de autômatos [22] e álgebras de Leavitt [29], além de ter sido publicada uma série de trabalhos relacionados ao assunto na área de * -álgebras (há uma discussão desses trabalhos em [14]). Também desenvolveu-se uma teoria de representações parciais projetivas [14], uma teoria cohomológica baseada em ações parciais em [18] e foram pesquisados outros assuntos sobre ações e co-ações de álgebras de Hopf [4,3,7,8,12].…”