577 research outputs found
Algebraic quantum Hamiltonians on the plane
We consider second order differential operators with polynomial
coefficients that preserve the vector space of polynomials of degrees not
greater then . We assume that the metric associated with the symbol of
is flat and that the operator is potential. In the case of two independent
variables we obtain some classification results and find polynomial forms for
the elliptic and Calogero-Moser Hamiltonians and for the elliptic
Inosemtsev model.Comment: 14 page
Bi-Hamiltonian ODEs with matrix variables
We consider a special class of linear and quadratic Poisson brackets related
to ODE systems with matrix variables. We investigate general properties of such
brackets, present an example of a compatible pair of quadratic and linear
brackets and found the corresponding hierarchy of integrable models, which
generalizes the two-component Manakov's matrix system in the case of arbitrary
number of matrices.Comment: 9 pages, late
Parameter-dependent associative Yang-Baxter equations and Poisson brackets
We discuss associative analogues of classical Yang-Baxter equation
meromorphically dependent on parameters. We discover that such equations enter
in a description of a general class of parameter-dependent Poisson structures
and double Lie and Poisson structures in sense of M. Van den Bergh. We propose
a classification of all solutions for one-dimensional associative Yang-Baxter
equations.Comment: 18 pages, LATEX2, ws-ijgmmp style. Few typos corrected,
aknowledgements adde
On (2+1)-dimensional hydrodynamic type systems possessing pseudopotential with movable singularities
A certain class of integrable hydrodynamic type systems with three
independent and N dependent variables is considered. We choose the existence of
a pseudopotential as a criterion of integrability. It turns out that the class
of integrable systems having pseudopotentials with movable singularities is
described by a functional equation, which can be solved explicitly. This allows
us to construct interesting examples of integrable hydrodynamic systems for
arbitrary N.Comment: 12 pages, late
Algebraic structures connected with pairs of compatible associative algebras
We study associative multiplications in semi-simple associative algebras over
C compatible with the usual one or, in other words, linear deformations of
semi-simple associative algebras over C. It turns out that these deformations
are in one-to-one correspondence with representations of certain algebraic
structures, which we call M-structures in the matrix case and PM-structures in
the case of direct sums of several matrix algebras. We also investigate various
properties of PM-structures, provide numerous examples and describe an
important class of PM-structures. The classification of these PM-structures
naturally leads to affine Dynkin diagrams of A, D, E-type.Comment: 29 pages, Latex. The case of semi-simple algebras A and B is
completed (Chapter 4). A construction of compatible products is added
(Chapter 1
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