Trivial Central Extensions of Lie Bialgebras

Details Trivial Central Extensions of Lie Bialgebras Trivial Central Extensions of Lie Bialgebras

2011-10-05

arxiv.org/abs/1110.1072v1

Mathematics

Quantum Algebra

23 pages

Scientific paper

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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