Inverse semialgebras and partial actions of Lie algebras

TL;DR

This paper introduces the concept of inverse semialgebras in the Lie algebra context, exploring partial actions, and establishing categorical equivalences and adjunctions with classical Lie algebra structures.

Contribution

It defines Lie inverse semialgebras and partial actions, extending inverse semigroup theory to Lie algebras, and establishes categorical equivalences and adjunctions.

Findings

Defined Lie inverse semialgebras as analogues of inverse semigroups.

Established the category equivalence between partial representations and F-inverse Lie semialgebras.

Proved the existence of an adjunction between Lie algebras and F-inverse Lie semialgebras.

Abstract

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 $\sigma$-classes. In addition, we establish an adjunction between the category of Lie algebras and the category ${\mathcal F}.$