Journal of Lie Theory 24 (2014), No. 2, 561--623
Copyright Heldermann Verlag 2014
Compatible Lie Brackets: Towards a Classification
Faculty of Mathematics and Computer Science, University of Warmia and Mazury, ul. Sloneczna 54, 10-710 Olsztyn, Poland
and: Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NASU, Naukova St. 3-b, 79060 L'viv, Ukraine
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.
Keywords: Semisimple Lie algebra, compatible Lie brackets, Lie pencil, bihamiltonian structure.
MSC: 17B20, 17B22, 53Z05
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