51 research outputs found
Quantum differential forms
Formalism of differential forms is developed for a variety of Quantum and
noncommutative situations
What a classical r-matrix really is
The notion of classical -matrix is re-examined, and a definition suitable
to differential (-difference) Lie algebras, -- where the standard definitions
are shown to be deficient, -- is proposed, the notion of an -operator. This notion has all the natural properties one would expect form
it, but lacks those which are artifacts of finite-dimensional isomorpisms such
as not true in differential generality relation \mbox{End}\, (V) \simeq V^*
\otimes V for a vector space . Examples considered include a quadratic
Poisson bracket on the dual space to a Lie algebra; generalized
symplectic-quadratic models of such brackets (aka Clebsch representations); and
Drinfel'd's 2-cocycle interpretation of nondegenate classical -matrices
On the Moyal quantized BKP type hierarchies
Quantization of BKP type equations are done through the Moyal bracket and the
formalism of pseudo-differential operators. It is shown that a variant of the
dressing operator can also be constructed for such quantized systems
On the nature of the Virasoro algebra
The multiplication in the Virasoro algebra comes from the commutator in a quasiassociative algebra with the multiplication
\renewcommand{\theequation}{} \be \ba{l} \ds e_p * e_q = - {q (1 + \epsilon
q) \over 1 + \epsilon (p + q)} e_{p+q} + {1 \over 2} \theta \left[p^3 - p +
\left(\epsilon - \epsilon^{-1} \right) p^2 \right] \delta^0_{p+q},
\vspace{3mm}\\ \ds e_p * \theta = \theta* e_p = 0. \ea \ee The multiplication
in a quasiassociative algebra satisfies the property
\renewcommand{\theequation}{} \be a * (b * c) - (a * b) * c = b * (a * c) -
(b * a) * c, \qquad a, b, c \in {\cal R}. \ee This property is necessary and
sufficient for the Lie algebra {\it Lie} to have a phase space. The
above formulae are put into a cohomological framework, with the relevant
complex being different from the Hochschild one even when the relevant
quasiassociative algebra becomes associative. Formula above
also has a differential-variational counterpart
Hamiltonian structures for general PDEs
We sketch out a new geometric framework to construct Hamiltonian operators
for generic, non-evolutionary partial differential equations. Examples on how
the formalism works are provided for the KdV equation, Camassa-Holm equation,
and Kupershmidt's deformation of a bi-Hamiltonian system.Comment: 12 pages; v2, v3: minor correction
Form-invariance upon relativisation of the Hamiltonian structures of fluids is not universal
AbstractThe principle of form-invariance upon relativisation of the Hamiltonian structures of fluids and plasmas is an empirical observation that the (special) relativistic version of a given non-relativistic Hamiltonian system of classical Physica (a) is also a Hamiltonian system, and moreover, (b) has the same noncanonical Hamiltonian structure as its nonrelativistic counterpart. It is shown that one-dimensional gas dynamics of a polytropic gas violates this principle
r-matrices for relativistic deformations of integrable systems
We include the relativistic lattice KP hierarchy, introduced by Gibbons and
Kupershmidt, into the -matrix framework. An -matrix account of the
nonrelativistic lattice KP hierarchy is also provided for the reader's
convenience. All relativistic constructions are regular one-parameter
perturbations of the nonrelativistic ones. We derive in a simple way the linear
Hamiltonian structure of the relativistic lattice KP, and find for the first
time its quadratic Hamiltonian structure. Amasingly, the latter turns out to
coincide with its nonrelativistic counterpart (a phenomenon, known previously
only for the simplest case of the relativistic Toda lattice)
On algebraic models of dynamical systems
We describe a universal algebraic model which, being read appropriately, yields (periodic and infinite) discrete dynamical systems, as well as their ‘continuous limits’, which cover all differential scalar Lax systems. For this model we give: Two different constructions of an infinity of integrals; modified equations; deformations; infinitesimal automorphisms. The basic tools are supplied by symbolic calculus and the abstract Hamiltonian formalism.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43214/1/11005_2004_Article_BF00401731.pd
On integrability of a (2+1)-dimensional perturbed Kdv equation
A (2+1)-dimensional perturbed KdV equation, recently introduced by W.X. Ma
and B. Fuchssteiner, is proven to pass the Painlev\'e test for integrability
well, and its 44 Lax pair with two spectral parameters is found. The
results show that the Painlev\'e classification of coupled KdV equations by A.
Karasu should be revised
Symplectic Geometries on , Hamiltonian Group Actions and Integrable Systems
Various Hamiltonian actions of loop groups \wt G and of the algebra
of first order differential operators in one variable are
defined on the cotangent bundle T^*\wt G of a Loop Group. The moment maps
generating the actions are shown to factorize through those
generating the loop group actions, thereby defining commuting diagrams of
Poisson maps to the duals of the corresponding centrally extended algebras. The
maps are then used to derive a number of infinite commuting families of
Hamiltonian flows that are nonabelian generalizations of the dispersive water
wave hierarchies. As a further application, sets of pairs of generators of the
nonabelian mKdV hierarchies are shown to give a commuting hierarchy on T^*\wt
G that contain the WZW system as its first element.Comment: 49 page
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