arXiv subjects

Emmanuel Jerez

Publications and source records attributed to Emmanuel Jerez.

7 recordsLinked to original sources

arXiv2026

Globalization of Partial Group Actions on Not Necessarily Associative Algebras and Covariant Representations

We extend the concept of a partial group action to non-associative algebras in a variety $\mathcal{V}(I)$, solve the globalization problem within $\mathcal{V}(I)$ and examine its universal property. It is achieved using what we call the ``$\Lambda$-construction'', which we also apply to deal with covariant representations in the associative and Lie algebra settings, considering related categories and constructing an adjoint pair of functors between them. We also show that the $\Lambda$-construction behaves well with semidirect products of Lie algebras.

math.RA↗

arXiv2025

Homology of Epsilon-Strongly Graded Algebras

Let $G$ be a group and $S$ a unital epsilon-strongly $G$-graded algebra. We construct spectral sequences converging to the Hochschild (co)homology of $S$. Each spectral sequence is expressed in terms of the partial group (co)homology of $G$ with coefficients in the Hochschild (co)homology of the degree-one component of $S$. Moreover, we show that the homology spectral sequence decomposes according to the conjugacy classes of $G$, and, by means of the globalization functor, its $E^2$-page can be identified with the ordinary group homology of suitable centralizers.

math.KT↗

arXiv2024

Twisted partial group algebra and related topological partial dynamical system

Given a group $ G $, a field $ \kappa $, and a factor set $ \sigma $ arising from a partial projective $ \kappa $-representation of $ G $. This leads to the construction of a topological partial dynamical system $ (\Omega_\sigma, G, \hat{\theta}) $, where $ \Omega_\sigma $ is a compact, totally disconnected Hausdorff space, and $ \sigma $ acts as a twist for $ \hat{\theta} $. We show that the twisted partial group algebra $ \kappa_{par}^{\sigma} G $ can be realized as a crossed product $ {\mathscr L}(\Omega_\sigma) \rtimes_{(\hat{\theta}, \sigma)} G $, with $ {\mathscr L}(\Omega_\sigma) $ denoting the $ \kappa $-algebra of locally constant functions $ \Omega_\sigma \to \kappa $. The space $ \Omega_\sigma $ corresponds to the spectrum of a unital commutative subalgebra in $ \kappa_{par}^{\sigma} G $, generated by idempotents. By describing $ \Omega_\sigma $ as a subspace of the Bernoulli space $ 2^G $, we examine conditions under which the spectral partial action $ \hat{\theta} $ is topologically free, impacting the ideal structure of $ \kappa_{par}^{\sigma} G $. We further explore generating idempotent factor sets of $ G $ and present conditions on them to ensure the topological freeness of $ \hat{\theta} $. Inspired by Exel's semigroup $ \mathcal{S}(G) $, which governs partial actions and representations of $ G $ and relates to $ \kappa_{par}G $, we characterize the twisted partial group algebra $ \kappa_{par}^{\sigma}G $ as generated by a $ \kappa $-cancellative inverse semigroup constructed from elements of $ \Omega_\sigma $. When $ \Omega_\sigma $ is discrete, we demonstrate that $ \kappa_{par}^{\sigma} G $ decomposes into a product of matrix algebras over twisted subgroup algebras, generalizing known results for finite $ G $.

math.RA↗

arXiv2024

On the homology of partial group representations

We study how the partial group (co)homology of a group $G$ with coefficient in a partial representation $M$ can be described using the usual group (co)homology. To address this, we introduce the concept of the \textit{universal globalization} $\Lambda(M)$ of a partial group representation $M$ of $G$. Our main result shows that the partial group homology $H^{\text{par}}_{\bullet}(G, M)$ is naturally isomorphic to the classical group homology $H_{\bullet}(G, \Lambda(M))$. We extend this result to the cohomological framework, obtaining a spectral sequence involving the classical group cohomology that converges to the partial group cohomology. Notably, when $G$ is countable, the spectral sequence collapses, resulting in a natural isomorphism $H^{\bullet}_{\text{par}}(G, M) \cong H^{\bullet}(G, \operatorname{Hom}_{K_{\text{par}} G}(\Lambda(K_{par}G), M))$, where $K_{par}G$ stands for the partial group algebra of $G$.

math.AT↗

arXiv2023

(Co)Homology of Partial Smash Products

Given a cocommutative Hopf algebra $\mathcal{H}$ over a commutative ring $K$ and a symmetric partial action of $\mathcal{H}$ on a $K$-algebra $A,$ we obtain a first quadrant Grothendieck spectral sequence converging to the Hochschild homology of the smash product $A \# \mathcal{H},$ involving the Hochschild homology of $A$ and the partial homology of $\mathcal{H}.$ An analogous third quadrant cohomological spectral sequence is also obtained. The definition of the partial (co)homology of $\mathcal{H}$ under consideration is based on the category of the partial representations of $\mathcal{H}.$ A specific partial representation of $\mathcal{H}$ on a subalgebra $\mathcal{B}$ of the partial ``Hopf" algebra $\mathcal{H}_{par} $ is involved in the definition and we construct a projective resolution of $\mathcal{B}.$

math.RA↗

arXiv2023

The twisted partial group algebra and (co)homology of partial crossed products

Given a group $G$ and a partial factor set $\sigma $ of $G,$ we introduce the twisted partial group algebra $\kappa_{par}^{\sigma}G,$ which governs the partial projective $\sigma$-representations of $G$ into algebras over a filed $\kappa.$ Using the relation between partial projective representations and twisted partial actions we endow $\kappa_{par}^\sigma G$ with the structure of a crossed product by a twisted partial action of $G$ on a commutative subalgebra of $\kappa_{par}^{\sigma} G.$ Then, we use twisted partial group algebras to obtain a first quadrant Grothendieck spectral sequence converging to the Hochschild homology of the crossed product $A\ast_{\Theta} G,$ involving the Hochschild homology of $A$ and the partial homology of $G,$ where ${\Theta}$ is a unital twisted partial action of $G$ on a $\kappa$-algebra $A$ with a $\kappa $-based twist. An analogous third quadrant cohomological spectral sequence is also obtained.

math.RA↗

arXiv2023

The category of partial group actions: quotients, (co)limits and groupoids

We consider the category of partial 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 $\Phi : \textbf{Grpd} \to \textbf{PA}$ and $\Psi : \textbf{PA} \to \textbf{Grpd}$, with the property that $\Psi \Phi \cong 1_{\textbf{Grpd} }$. Next, for a given groupoid $\Gamma$, we provide a characterization of all partial actions that allow the recovery of the groupoid $\Gamma$ through $\Psi$. This characterization is expressed in terms of certain normal subgroups of a universal group constructed from $\Gamma.$

math.GR↗