Extending structures and classifying complements for left-symmetric algebras

Abstract

Let $A$ be a left-symmetric (resp. Novikov) algebra, $E$ be a vector space containing $A$ as a subspace and $V$ be a complement of $A$ in $E$.The extending structures problem which asks for the classification of all left-symmetric (resp. Novikov) algebra structures on $E$ such that $A$ is a subalgebra of $E$ is studied. In this paper, the definition of the unified product of left-symmetric (resp. Novikov) algebras is introduced. It is shown that there exists a left-symmetric (resp. Novikov) algebra structure on $E$ such that $A$ is a subalgebra of $E$ if and only if $E$ is isomorphic to a unified product of $A$ and $V$. Two cohomological type objects $\mathcal{H}_A^2(V,A)$ and $\mathcal{H}^2(V,A)$ are constructed to give a theoretical answer to the extending structures problem. Furthermore, given an extension $A\subset E$ of left-symmetric (resp. Novikov) algebras, another cohomological type object is constructed to classify all complements of $A$ in $E$. Several special examples are provided in details.