Partial actions and homology

• Ano de defesa: 2024
• Autor(a) principal: Usuga, Emmanuel Jerez
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Biblioteca Digitais de Teses e Dissertações da USP
• Palavras-chave em Português: Ações parciais; algebra homologica; Homological algebra; Partial actions; Partial representations; representação parciais
• Link de acesso: https://www.teses.usp.br/teses/disponiveis/45/45131/tde-19092024-184308/

We study the category of partial actions on groups, exploring (co)homological aspects of this category as well as some generalizations of partial group actions. We begin by analyzing its categorical structure and its relation with the category of groupoids. Subsequently, we study the (co)homological theory of partial group actions, motivated by their connection to groupoids and the simplicial structure that emerges from partial group actions. We conclude our study by examining the (co)homological structure of two generalizations of partial group actions: partial Hopf actions and twisted partial group actions. Initially, we consider the category of partial group actions, where the group and the set upon which the group acts can vary. Within this framework, we develop a theory of quotient partial actions and prove that this category is both (co)complete and encompasses the category of groupoids as a full subcategory. In particular, we establish the existence of a pair of adjoint functors, denoted as : Grpd PA and : PA Grpd, with the property that id. Next, for a given groupoid , we provide a characterization of all partial actions that allow the recovery of the groupoid through . This characterization is expressed in terms of certain normal subgroups of a universal group constructed from . Motivated by the simplicial structure that arises from the partial action groupoid of a partial group action, we employ simplicial methods to investigate the partial group (co)homology of partial group actions. We introduce the concept of universal globalization of a partial group action on a K-module and prove that, given a partial representation of G on M, the partial group homology of G with coefficients on M is naturally isomorphic to the usual group homology of K with coefficients in KG-module W, where M is the universal globalization of the partial group action associated with M. We dualize this result into a cohomological spectral sequence converging to partial group cohomology. The (co)homology of partial groups with coefficients in a KparG-module naturally arises as a component of a spectral sequence that calculates the Hochschild (co)homology of a partial crossed product. We extend this analysis to the Hopf case. Given a cocommutative Hopf algebra H over a commutative ring K and a symmetric partial action of H on a K-algebra A, we obtain a first quadrant Grothendieck spectral sequence converging to the Hochschild homology of the smash product A # H, involving the Hochschild homology of A and the partial homology of H. An analogous third quadrant cohomological spectral sequence is also obtained. The definition of the partial (co)homology of H under consideration is based on the category of the partial representations of H. A specific partial representation of H on a subalgebra B of the partial \"Hopf\" algebra Hpar is involved in the definition, and we construct a projective resolution of B. Finally, we conclude the work by considering twisted partial group actions. Given a group G and a partial factor set of G, we introduce the twisted partial group algebra, which governs the partial projective -representations of G into algebras over a field . Using the relation between partial projective representations and twisted partial actions, we endow the twisted partial group algebra with the structure of a crossed product by a twisted partial action of G on a commutative algebra. Then, we use twisted partial group algebras to obtain a first quadrant Grothendieck spectral sequence converging to the Hochschild homology of the crossed product A * G, involving the Hochschild homology of A and the partial homology of G, where is a unital twisted partial action of G on a -algebra A with a -based twist. An analogous third quadrant cohomological spectral sequence is also obtained.