Compatible Lie brackets related to elliptic curve
Abstract
For the direct sum of several copies of sln, a family of Lie brackets compatible with the initial one is constructed. The structure constants of these brackets are expressed in terms of θ functions associated with an elliptic curve. The structure of Casimir elements for these brackets is investigated. A generalization of this construction to the case of vectorvalued θfunctions is presented. A different procedure for constructing compatible Lie brackets based on the argument shift method for quadratic Poisson brackets is discussed.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 January 2006
 DOI:
 10.1063/1.2158434
 arXiv:
 arXiv:math/0506503
 Bibcode:
 2006JMP....47a3506O
 Keywords:

 02.10.Ud;
 02.30.Jr;
 02.60.Lj;
 Linear algebra;
 Partial differential equations;
 Ordinary and partial differential equations;
 boundary value problems;
 Mathematics  Quantum Algebra;
 Mathematical Physics;
 Mathematics  Mathematical Physics;
 High Energy Physics  Theory;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 17B80;
 17B63;
 32L81;
 14H70
 EPrint:
 18 pages, Latex