Inverse semialgebras and partial actions of Lie algebras

Abstract

We introduce the concept of a nonassociative (i.e., not necessarily associative) inverse semialgebra over a field, the Lie version of which is inspired by the set of all partially defined derivations of a nonassociative 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 nonassociative 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 a Lie $F$-inverse 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 Lie $F$-inverse semialgebras with morphisms that preserve the greatest elements of $\sigma$-classes. In addition, we establish an adjunction between the category of Lie algebras and the category $\mathcal{F}$.