94 research outputs found
Quantization of canonical cones of algebraic curves
We introduce a quantization of the graded algebra of functions on the
canonical cone of an algebraic curve C, based on the theory of formal
pseudodifferential operators. When C is a complex curve with Poincar\'e
uniformization, we propose another, equivalent construction, based on the work
of Cohen-Manin-Zagier on Rankin-Cohen brackets. We give a presentation of the
quantum algebra when C is a rational curve, and discuss the problem of
constructing algebraically "differential liftings"
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
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
Double Elliptic Dynamical Systems From Generalized Mukai - Sklyanin Algebras
We consider the double-elliptic generalisation of dynamical systems of
Calogero-Toda-Ruijsenaars type using finite-dimensional Mukai-Sklyanin
algebras. The two-body system, which involves an elliptic dependence both on
coordinates and momenta, is investigated in detail and the relation with Nambu
dynamics is mentioned. We identify the 2D complex manifold associated with the
double elliptic system as an elliptically fibered rational ("1/2K3 ") surface.
Some generalisations are suggested which provide the ground for a description
of the N-body systems. Possible applications to SUSY gauge theories with
adjoint matter in with two compact dimensions are discussed.Comment: 31 pages, Late
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