Extending structures and classifying complements for left-symmetric algebras

TL;DR

This paper introduces a unified framework for classifying extensions and complements of left-symmetric and Novikov algebras using cohomological methods, providing a theoretical foundation and concrete examples.

Contribution

It defines the unified product for these algebras and constructs cohomological objects to classify all possible algebra structures and complements.

Findings

Existence of a left-symmetric algebra structure on E characterized by a unified product.

Development of cohomological objects for classification of extensions and complements.

Application of the theory to several detailed examples.

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.