2012/08/08 by Andriy Panasyuk, Panasyuk, Andriy
Mathematics · Physics and Astronomy · #17B20 #17B22 #53Z05 #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1208.1642
openalex publication_date 2012/08/08 · openalex created_date 2022/09/11 · openalex updated_date 2026/07/28
Let mathfrakg be a vector space and [,],[,]' be a pair of Lie brackets\non mathfrakg. By definition they are compatible if [,]+[,]' is again a\nLie bracket. Such pairs play important role in bihamiltonian and r-matrix\nformalisms in the theory of integrable systems.\n We propose an approach to a long standing problem of classification of such\npairs in the case when one of them, say [,], is semisimple. It is known that\nany such pair is determined by a linear operator on ( mathfrakg,[,]), which\nis defined up to adding a derivation. We propose a special fixing of this\noperator to get rid of this ambiguity and consider the operators preserving the\nroot decomposition with respect to a Cartan subalgebra. The classification\nleads to two disjoint classes of pairs depending on the symmetry properties of\nthe corresponding operator with respect to the Killing form. Within each class\nwe recover known examples and obtain new ones. We present a list of examples in\neach case and conjecture the completeness of these lists.\n