Inverse semialgebras and partial actions of Lie algebras

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} .