Trivial Central Extensions of Lie Bialgebras

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.

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