
Journal of Lie Theory 24 (2014), No. 2, 561623 Copyright Heldermann Verlag 2014 Compatible Lie Brackets: Towards a Classification Andriy Panasyuk Faculty of Mathematics and Computer Science, University of Warmia and Mazury, ul. Sloneczna 54, 10710 Olsztyn, Poland and: Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NASU, Naukova St. 3b, 79060 L'viv, Ukraine panas@matman.uwm.edu.pl We propose an approach to a longstanding problem of classification of pairs of compatible Liealgebra 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 [ Fulltextpdf (713 KB)] for subscribers only. 