30 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
Two-component generalizations of the Camassa-Holm equation
A classification of integrable two-component systems of non-evolutionary partial differential equations that are analogous to the Camassa-Holm equation is carried out via the perturbative symmetry approach. Independently, a classification of compatible pairs of Hamiltonian operators is carried out, which leads to bi-Hamiltonian structures for the same systems of equations. Some exact solutions and Lax pairs are also constructed for the systems considered
Special symplectic Lie groups and hypersymplectic Lie groups
A special symplectic Lie group is a triple such that
is a finite-dimensional real Lie group and is a left invariant
symplectic form on which is parallel with respect to a left invariant
affine structure . In this paper starting from a special symplectic Lie
group we show how to ``deform" the standard Lie group structure on the
(co)tangent bundle through the left invariant affine structure such
that the resulting Lie group admits families of left invariant hypersymplectic
structures and thus becomes a hypersymplectic Lie group. We consider the affine
cotangent extension problem and then introduce notions of post-affine structure
and post-left-symmetric algebra which is the underlying algebraic structure of
a special symplectic Lie algebra. Furthermore, we give a kind of double
extensions of special symplectic Lie groups in terms of post-left-symmetric
algebras.Comment: 32 page
Topological superfluid 3He-B in magnetic field and Ising variable
The topological superfluid 3He-B provides many examples of the interplay of
symmetry and topology. Here we consider the effect of magnetic field on
topological properties of 3He-B. Magnetic field violates the time reversal
symmetry. As a result, the topological invariant supported by this symmetry
ceases to exist; and thus the gapless fermions on the surface of 3He-B are not
protected any more by topology: they become fully gapped. Nevertheless, if
perturbation of symmetry is small, the surface fermions remain relativistic
with mass proportional to symmetry violating perturbation -- magnetic field.
The 3He-B symmetry gives rise to the Ising variable I=+/- 1, which emerges in
magnetic field and which characterizes the states of the surface of 3He-B. This
variable also determines the sign of the mass term of surface fermions and the
topological invariant describing their effective Hamiltonian. The line on the
surface, which separates the surface domains with different I, contains 1+1
gapless fermions, which are protected by combined action of symmetry and
topology.Comment: 5 pages, JETP Letters style, no figures, version submitted to JETP
Letter
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