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) 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
F of Lie F -inverse 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 F.

This is a joint work with Farangis Johari and José Luis Vilca-Rodríguez

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