49 research outputs found
Quadratic algebras related to elliptic curves
We construct quadratic finite-dimensional Poisson algebras and their quantum
versions related to rank N and degree one vector bundles over elliptic curves
with n marked points. The algebras are parameterized by the moduli of curves.
For N=2 and n=1 they coincide with the Sklyanin algebras. We prove that the
Poisson structure is compatible with the Lie-Poisson structure on the direct
sum of n copies of sl(N). The derivation is based on the Poisson reduction from
the canonical brackets on the affine space over the cotangent bundle to the
groups of automorphisms of vector bundles.Comment: 21 page
Modeling of Slow Plasticity Waves
Quasi-static uniaxial loading of a bar with a length L is considered. Mechanical properties of a material in a
point are defined by the segment of negative slope on stress-strain diagram which follows the section of elastic
deformation. The deformation in specimen is uniform until the stress exceeds the peak yielding stress. The
analytical solution shows that stress-strain diagram of the specimen has a yielding plateau. It is shown that the
time for a slow wave to advance by a distance equal to the localized band width S is the same as it is required
for a plastic wave to run along the whole bar length
Elliptic Schlesinger system and Painlev{\'e} VI
We construct an elliptic generalization of the Schlesinger system (ESS) with
positions of marked points on an elliptic curve and its modular parameter as
independent variables (the parameters in the moduli space of the complex
structure). ESS is a non-autonomous Hamiltonian system with pair-wise commuting
Hamiltonians. The system is bihamiltonian with respect to the linear and the
quadratic Poisson brackets. The latter are the multi-color generalization of
the Sklyanin-Feigin-Odeskii classical algebras. We give the Lax form of the
ESS. The Lax matrix defines a connection of a flat bundle of degree one over
the elliptic curve with first order poles at the marked points.
The ESS is the monodromy independence condition on the complex structure for
the linear systems related to the flat bundle.
The case of four points for a special initial data is reduced to the
Painlev{\'e} VI equation in the form of the Zhukovsky-Volterra gyrostat,
proposed in our previous paper.Comment: 16 pages; Dedicated to the centenary of the publication of the
Painleve VI equation in the Comptes Rendus de l'Academie des Sciences de
Paris by Richard Fuchs in 190