2001/10/02 by Alexander Odesskii, A. Odesskii, Odesskii, A. +2
Mathematics · Physics and Astronomy · #17B63 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #math-ph #math.AG #math.MP #math.QA #msc:17B63
paper · pdf · doi:10.48550/arxiv.math/0110032
21 pages, LaTeX
arxiv created 2001/10/02 · openalex publication_date 2001/10/02 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study polynomial Poisson algebras with some regularity conditions. Linear (Lie-Berezin-Kirillov) structures on dual spaces of semi-simple Lie algebras, quadratic Sklyanin elliptic algebras of \citeFO1,\citeFO2 as well as polynomial algebras recently described by Bondal-Dubrovin-Ugaglia (\citeBondal,\citeUg) belong to this class. We establish some simple determinantal relations between the brackets and Casimirs in this algebras. These relations imply in particular that for Sklyanin elliptic algebras the sum of Casimir degrees coincides with the dimension of the algebra. We are discussing some interesting examples of these algebras and in particular we show that some of them arise naturally in Hamiltonian integrable systems. Among these examples is a new class of two-body integrable systems admitting an elliptic dependence both on coordinates and momenta.