arXiv subjects

José L. Vilca-Rodríguez

Publications and source records attributed to José L. Vilca-Rodríguez.

3 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 ``$Λ$-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 $Λ$-construction behaves well with semidirect products of Lie algebras.

math.RA↗

arXiv2026

On Γ-embeddings and partial actions of function spaces

This paper deals with the extension of partial actions of topological groups on topological spaces. Within this framework, we introduce a class of topological embeddings defined via the inverse semigroup of homeomorphisms between open subsets of a topological space. We describe several embeddings of this type, referred to as $Γ$- embeddings, and we place particular emphasis on one of them. In particular, we prove that every topological space $Y$ admits a $Γ$-embedding into the space of continuous functions $C(X, Y )$, equipped with the compact-open topology, where $X$ is a compact space. Consequently, any partial action $θ$ of a topological group $G$ on $ Y$ naturally induces a partial action $\hatθ$ on $C(X, Y ).$ Throughout the paper, we investigate various relationships between these actions, as well as between their corresponding globalizations and enveloping spaces.

math.GN↗

arXiv2025

Inverse semialgebras and partial actions of Lie algebras

We introduce the concept of a non-associative (i.e. non-necessarily associtive) inverse semialgebra over a field, the Lie version of which is inspired by the set of all partially defined derivations of a non-associative algebra, whereas the associative case is based on such examples as the set of all partially defined linear maps of a vector space, the set of all sections of the structural sheaf of a scheme, the set of all regular functions defined on open subsets of an algebraic variety and the set of all smooth real valued functions defined on open subsets of a smooth manifold. Given a Lie algebra $L$ we define the notion of a partial action of $L$ on a non-associative algebra $A$ as an appropriate premorphism and introduce a Lie inverse semialgebra $E(L),$ which is a Lie analogue of R. Exel's inverse semigroup $S(G)$ that governs the partial actions of a group $G.$ We discuss how $E(L)$ controls the premorphisms from $L$ to $A,$ obtaining results on its total control. We define the concept of an $F$-inverse Lie semialgebra and obtain Lie theoretic analogues of some classical results of the theory of inverse semigroups, namely, we show that the category of partial representations of $L$ in meet semilattices is equivalent to the category ${\mathcal F}$ of $F$-inverse Lie semialgebras with morphisms that preserve the greatest elements of $σ$-classes. In addition, we establish an adjunction between the category of Lie algebras and the category ${\mathcal F}.$

math.RA↗