Journal of Lie Theory

no. 2
Compatible Lie Brackets: Towards a Classification
Journal of Lie Theory, Volume 24 (2014) no. 2, pp. 561-623
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.
DOI: 10.5802/jolt.797
Classification: 17B20, 17B22, 53Z05
Keywords: Semisimple Lie algebra, compatible Lie brackets, Lie pencil, bihamiltonian structure 
@article{JOLT_2014_24_2_a12,
     author = {A. Panasyuk},
     title = {Compatible {Lie} {Brackets:} {Towards} a {Classification}},
     journal = {Journal of Lie Theory},
     pages = {561--623},
     year = {2014},
     volume = {24},
     number = {2},
     doi = {10.5802/jolt.797},
     zbl = {1330.17013},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.797/}
}
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A. Panasyuk. Compatible Lie Brackets: Towards a Classification. Journal of Lie Theory, Volume 24 (2014) no. 2, pp. 561-623. doi: 10.5802/jolt.797

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