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We classify all real three dimensional Lie bialgebras. In each case, their automorphism group as ... more We classify all real three dimensional Lie bialgebras. In each case, their automorphism group as Lie bialgebras is also given. Comment: 27 pages
Journal of Algebra, 2013
From a Lie algebra g satisfying Z(g) = 0 and Λ 2 (g) g = 0 (in particular, for g semisimple) we d... more From a Lie algebra g satisfying Z(g) = 0 and Λ 2 (g) g = 0 (in particular, for g semisimple) we describe explicitly all Lie bialgebra structures on extensions of the form L = g × K in terms of Lie bialgebra structures on g (not necessarily factorizable nor quasi-triangular) and its biderivations, for any field K with char K = 0. If moreover, [g, g] = g, then we describe also all Lie bialgebra structures on extensions L = g × K n . In interesting cases we characterize the Lie algebra of biderivations.
The theory of Poisson-Lie groups occupies a central place in the theory of Poisson man- ifolds. T... more The theory of Poisson-Lie groups occupies a central place in the theory of Poisson man- ifolds. The category of connected, simply connected Poisson-Lie groups is equivalent to the category of Lie bialgebras [8]. Therefore, a basic problem in the theory of Poisson manifolds is the classification of Lie bialgebras. A fundamental contribution to this question is the theorem of Belavin and Drinfeld [2] which contains the classification of all the simple factorizable complex Lie bialgebras.
We classify all real three dimensional Lie bialgebras. In each case, their automorphism group as ... more We classify all real three dimensional Lie bialgebras. In each case, their automorphism group as Lie bialgebras is also given. Comment: 27 pages
Journal of Algebra, 2013
From a Lie algebra g satisfying Z(g) = 0 and Λ 2 (g) g = 0 (in particular, for g semisimple) we d... more From a Lie algebra g satisfying Z(g) = 0 and Λ 2 (g) g = 0 (in particular, for g semisimple) we describe explicitly all Lie bialgebra structures on extensions of the form L = g × K in terms of Lie bialgebra structures on g (not necessarily factorizable nor quasi-triangular) and its biderivations, for any field K with char K = 0. If moreover, [g, g] = g, then we describe also all Lie bialgebra structures on extensions L = g × K n . In interesting cases we characterize the Lie algebra of biderivations.
The theory of Poisson-Lie groups occupies a central place in the theory of Poisson man- ifolds. T... more The theory of Poisson-Lie groups occupies a central place in the theory of Poisson man- ifolds. The category of connected, simply connected Poisson-Lie groups is equivalent to the category of Lie bialgebras [8]. Therefore, a basic problem in the theory of Poisson manifolds is the classification of Lie bialgebras. A fundamental contribution to this question is the theorem of Belavin and Drinfeld [2] which contains the classification of all the simple factorizable complex Lie bialgebras.