Trivial Central Extensions of Lie Bialgebras

TL;DR

This paper classifies all Lie bialgebra structures on trivial central extensions of certain Lie algebras, especially semisimple ones, in terms of structures on the original algebra and its biderivations.

Contribution

It explicitly describes Lie bialgebra structures on extensions of Lie algebras under specific conditions, including semisimple cases and higher-dimensional extensions.

Findings

Complete classification for extensions of semisimple Lie algebras.

Characterization of biderivations in key cases.

Extension results applicable over any field of characteristic zero.

Abstract

From a Lie algebra $\mathfrak{g}$ satisfying $\mathcal{Z}(\mathfrak{g})=0$ and $\Lambda^2(\mathfrak{g})^\mathfrak{g}=0$ (in particular, for $\g$ semisimple) we describe explicitly all Lie bialgebra structures on extensions of the form $\mathfrak{L} =\mathfrak{g}\times \mathbb{K}$ in terms of Lie bialgebra structures on $\mathfrak{g}$ (not necessarily factorizable nor quasi-triangular) and its biderivations, for any field $\mathbb{K}$ with char $\mathbb{K}=0$. If moreover, $[\mathfrak{g},\mathfrak{g}]=\mathfrak{g}$, then we describe also all Lie bialgebra structures on extensions $\mathfrak{L} =\mathfrak{g}\times \mathbb{K}^n$. In interesting cases we characterize the Lie algebra of biderivations.