Globalization of Partial Group Actions on Not Necessarily Associative Algebras and Covariant Representations

TL;DR

This paper generalizes partial group actions to non-associative algebras, solves the globalization problem using the $ ext{ extsterling}$-construction, and explores covariant representations and their categorical properties.

Contribution

It introduces the $ ext{ extsterling}$-construction for non-associative algebras, addressing globalization and covariant representations in new algebraic contexts.

Findings

Successfully extends partial actions to non-associative algebras.

Provides a universal property for the globalization within the variety.

Shows the $ ext{ extsterling}$-construction's compatibility with semidirect products of Lie algebras.

Abstract

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 ``$\Lambda$-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 $\Lambda$-construction behaves well with semidirect products of Lie algebras.