77 research outputs found
Compatible Lie brackets related to elliptic curve
For the direct sum of several copies of sl_n, a family of Lie brackets
compatible with the initial one is constructed. The structure constants of
these brackets are expressed in terms of theta-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
vector-valued theta-functions is presented. The brackets define a
multi-hamiltonian structure for the elliptic sl_n-Gaudin model. A different
procedure for constructing compatible Lie brackets based on the argument shift
method for quadratic Poisson brackets is discussed.Comment: 18 pages, Late
Double Poisson brackets on free associative algebras
We discuss double Poisson structures in sense of M. Van den Bergh on free
associative algebras focusing on the case of quadratic Poisson brackets. We
establish their relations with an associative version of Young-Baxter
equations, we study a bi-hamiltonian property of the linear-quadratic pencil of
the double Poisson structure and propose a classification of the quadratic
double Poisson brackets in the case of the algebra with two free generators.
Many new examples of quadratic double Poisson brackets are proposed.Comment: 19 pages, late
Functional realization of some elliptic Hamiltonian structures and bosonization of the corresponding quantum algebras
We introduce a functional realization of the Hamiltonian structure on the
moduli space of P-bundles on the elliptic curve E. Here P is parabolic subgroup
in SL_n. We also introduce a construction of the corresponding quantum
algebras.Comment: 20 pages, Amstex, minor change
Classification of integrable Vlasov-type equations
Classification of integrable Vlasov-type equations is reduced to a functional
equation for a generating function. A general solution of this functional
equation is found in terms of hypergeometric functions.Comment: latex, 15 pages, to appear in Theoretical and Mathematical Physic
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