Extending structures for Lie algebras

TL;DR

This paper introduces a unified approach to classify all Lie algebra structures on a vector space extension that contains a given Lie algebra as a subalgebra, using cohomological tools and explicit constructions.

Contribution

It develops the concept of a unified product for Lie algebras and provides a cohomological classification framework for extensions.

Findings

Classifies Lie algebra extensions via two cohomological objects.

Introduces the unified product as a key construction.

Provides detailed examples for flag extending structures.

Abstract

Let $\mathfrak{g}$ be a Lie algebra, $E$ a vector space containing $\mathfrak{g}$ as a subspace. The paper is devoted to the \emph{extending structures problem} which asks for the classification of all Lie algebra structures on $E$ such that $\mathfrak{g}$ is a Lie subalgebra of $E$. A general product, called the unified product, is introduced as a tool for our approach. Let $V$ be a complement of $\mathfrak{g}$ in $E$: the unified product $\mathfrak{g} \,\natural \, V$ is associated to a system $(\triangleleft, \, \triangleright, \, f, \{-, \, -\})$ consisting of two actions $\triangleleft$ and $\triangleright$, a generalized cocycle $f$ and a twisted Jacobi bracket $\{-, \, -\}$ on $V$. There exists a Lie algebra structure $[-,-]$ on $E$ containing $\mathfrak{g}$ as a Lie subalgebra if and only if there exists an isomorphism of Lie algebras $(E, [-,-]) \cong \mathfrak{g} \,\natural \, V$. All such Lie algebra structures on $E$ are classified by two cohomological type objects which are explicitly constructed. The first one ${\mathcal H}^{2}_{\mathfrak{g}} (V, \mathfrak{g})$ will classify all Lie algebra structures on $E$ up to an isomorphism that stabilizes $\mathfrak{g}$ while the second object ${\mathcal H}^{2} (V, \mathfrak{g})$ provides the classification from the view point ofthe extension problem. Several examples that compute both classifying objects ${\mathcal H}^{2}_{\mathfrak{g}} (V, \mathfrak{g})$ and ${\mathcal H}^{2} (V, \mathfrak{g})$ are worked out in detail in the case of flag extending structures.