Lie algebra extensions related with linear bundles of Lie brackets

TL;DR

This paper explores special Lie algebra extensions connected to Lie-Poisson structures, demonstrating how certain classes can be constructed via linear bundles of Lie algebras, enriching the understanding of algebraic structures in geometric contexts.

Contribution

It introduces a novel approach to constructing Lie algebra extensions using linear bundles, linking algebraic and geometric structures in Lie theory.

Findings

Certain Lie algebra extensions can be naturally constructed from linear bundles.

The work connects Lie-Poisson structures with algebraic extensions.

Provides a framework for understanding extensions in the context of dual Lie algebras.

Abstract

We consider some special type extensions of an arbitrary Lie algebra ${\cal G}$, arising in the theory of Lie-Poisson structures over $({\cal G}^*)^n$, where ${\cal G}^*$ is the dual of ${\cal G}$. We show that some classes of these extensions can be constructed in a natural way using some linear bundles of Lie algebras.