Lie algebra extensions related with linear bundles of Lie brackets

Abstract

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

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