Geodesic
Compatible Lie Brackets: Towards a Classification
Journal of Lie theory, Tome 24 (2014) no. 2, pp. 561-623.

Voir la notice de l'article provenant de la source Heldermann Verlag

We propose an approach to a long-standing problem of classification of pairs of compatible Lie-algebra structures, one of which is semisimple. Any such pair is determined by a linear operator which is defined up to the addition of a derivation. We introduce a special fixing of this operator to get rid of this ambiguity and consider the operators preserving the root decomposition with respect to a Cartan subalgebra. The classification leads to two disjoint classes of pairs depending on the symmetry properties of the corresponding operator with respect to the Killing form. We present a list of known and new examples in each case and conjecture the completeness of these lists.
Classification : 17B20, 17B22, 53Z05
Mots-clés : Semisimple Lie algebra, compatible Lie brackets, Lie pencil, bihamiltonian structure 
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     author = {A. Panasyuk },
     title = {Compatible {Lie} {Brackets:} {Towards} a {Classification}},
     journal = {Journal of Lie theory},
     pages = {561--623},
     publisher = {mathdoc},
     volume = {24},
     number = {2},
     year = {2014},
     url = {http://geodesic.mathdoc.fr/item/JLT_2014_24_2_JLT_2014_24_2_a12/}
}
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A. Panasyuk . Compatible Lie Brackets: Towards a Classification. Journal of Lie theory, Tome 24 (2014) no. 2, pp. 561-623. http://geodesic.mathdoc.fr/item/JLT_2014_24_2_JLT_2014_24_2_a12/