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

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.