2019/09/08 by Andriy Panasyuk, Panasyuk, Andriy, Vsevolod Shevchishin +1 · 1 citation
Mathematics · #17B80 #37K10 #70H06 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.1909.03561
openalex publication_date 2019/09/08 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28
The paper is devoted to quadratic Poisson structures compatible with the\ncanonical linear Poisson structures on trivial 1-dimensional central extensions\nof semisimple Lie algebras. In particular, we develop the general theory of\nsuch structures and study related families of functions in involution. We also\nshow that there exists a 10-parametric family of quadratic Poisson structures\non gl(3)^* compatible with the canonical linear Poisson structure and\ncontaining the 3-parametric family of quadratic bivectors recently introduced\nby Vladimir Sokolov. The involutive family of polynomial functions related to\nthe corresponding Poisson pencils contains the hamiltonian of the polynomial\nform of the elliptic Calogero--Moser system.\n