18 research outputs found
A Unified Algebraic Approach to Classical Yang-Baxter Equation
In this paper, the different operator forms of classical Yang-Baxter equation
are given in the tensor expression through a unified algebraic method. It is
closely related to left-symmetric algebras which play an important role in many
fields in mathematics and mathematical physics. By studying the relations
between left-symmetric algebras and classical Yang-Baxter equation, we can
construct left-symmetric algebras from certain classical r-matrices and
conversely, there is a natural classical r-matrix constructed from a
left-symmetric algebra which corresponds to a parak\"ahler structure in
geometry. Moreover, the former in a special case gives an algebraic
interpretation of the ``left-symmetry'' as a Lie bracket ``left-twisted'' by a
classical r-matrix.Comment: To appear in Journal of Physics A: Mathematical and Theoretica
Weakly-nonlocal Symplectic Structures, Whitham method, and weakly-nonlocal Symplectic Structures of Hydrodynamic Type
We consider the special type of the field-theoretical Symplectic structures
called weakly nonlocal. The structures of this type are in particular very
common for the integrable systems like KdV or NLS. We introduce here the
special class of the weakly nonlocal Symplectic structures which we call the
weakly nonlocal Symplectic structures of Hydrodynamic Type. We investigate then
the connection of such structures with the Whitham averaging method and propose
the procedure of "averaging" of the weakly nonlocal Symplectic structures. The
averaging procedure gives the weakly nonlocal Symplectic Structure of
Hydrodynamic Type for the corresponding Whitham system. The procedure gives
also the "action variables" corresponding to the wave numbers of -phase
solutions of initial system which give the additional conservation laws for the
Whitham system.Comment: 64 pages, Late
From Rota-Baxter Algebras to Pre-Lie Algebras
Rota-Baxter algebras were introduced to solve some analytic and combinatorial
problems and have appeared in many fields in mathematics and mathematical
physics. Rota-Baxter algebras provide a construction of pre-Lie algebras from
associative algebras. In this paper, we give all Rota-Baxter operators of
weight 1 on complex associative algebras in dimension and their
corresponding pre-Lie algebras.Comment: 23 pages, appear in Journal of Physics A; Mathematical and
Theoretica