71 research outputs found
Algebraic theories of brackets and related (co)homologies
A general theory of the Frolicher-Nijenhuis and Schouten-Nijenhuis brackets
in the category of modules over a commutative algebra is described. Some
related structures and (co)homology invariants are discussed, as well as
applications to geometry.Comment: 14 pages; v2: minor correction
Infinitely many local higher symmetries without recursion operator or master symmetry: integrability of the Foursov--Burgers system revisited
We consider the Burgers-type system studied by Foursov, w_t &=& w_{xx} + 8 w
w_x + (2-4\alpha)z z_x, z_t &=& (1-2\alpha)z_{xx} - 4\alpha z w_x +
(4-8\alpha)w z_x - (4+8\alpha)w^2 z + (-2+4\alpha)z^3, (*) for which no
recursion operator or master symmetry was known so far, and prove that the
system (*) admits infinitely many local generalized symmetries that are
constructed using a nonlocal {\em two-term} recursion relation rather than from
a recursion operator.Comment: 10 pages, LaTeX; minor changes in terminology; some references and
definitions adde
Secondary Calculus and the Covariant Phase Space
The covariant phase space of a Lagrangian field theory is the solution space
of the associated Euler-Lagrange equations. It is, in principle, a nice
environment for covariant quantization of a Lagrangian field theory. Indeed, it
is manifestly covariant and possesses a canonical (functional) "presymplectic
structure" w (as first noticed by Zuckerman in 1986) whose degeneracy
(functional) distribution is naturally interpreted as the Lie algebra of gauge
transformations. We propose a fully rigorous approach to the covariant phase
space in the framework of secondary calculus. In particular we describe the
degeneracy distribution of w. As a byproduct we rederive the existence of a Lie
bracket among gauge invariant functions on the covariant phase space.Comment: 40 pages, typos correcte
Iterated Differential Forms II: Riemannian Geometry Revisited
A natural extension of Riemannian geometry to a much wider context is
presented on the basis of the iterated differential form formalism developed in
math.DG/0605113 and an application to general relativity is given.Comment: 12 pages, extended version of the published note Dokl. Math. 73, n. 2
(2006) 18
On the variational noncommutative Poisson geometry
We outline the notions and concepts of the calculus of variational
multivectors within the Poisson formalism over the spaces of infinite jets of
mappings from commutative (non)graded smooth manifolds to the factors of
noncommutative associative algebras over the equivalence under cyclic
permutations of the letters in the associative words. We state the basic
properties of the variational Schouten bracket and derive an interesting
criterion for (non)commutative differential operators to be Hamiltonian (and
thus determine the (non)commutative Poisson structures). We place the
noncommutative jet-bundle construction at hand in the context of the quantum
string theory.Comment: Proc. Int. workshop SQS'11 `Supersymmetry and Quantum Symmetries'
(July 18-23, 2011; JINR Dubna, Russia), 4 page
Geometry of jet spaces and integrable systems
An overview of some recent results on the geometry of partial differential
equations in application to integrable systems is given. Lagrangian and
Hamiltonian formalism both in the free case (on the space of infinite jets) and
with constraints (on a PDE) are discussed. Analogs of tangent and cotangent
bundles to a differential equation are introduced and the variational Schouten
bracket is defined. General theoretical constructions are illustrated by a
series of examples.Comment: 54 pages; v2-v6 : minor correction
A unified approach to computation of integrable structures
We expose (without proofs) a unified computational approach to integrable
structures (including recursion, Hamiltonian, and symplectic operators) based
on geometrical theory of partial differential equations. We adopt a coordinate
based approach and aim to provide a tutorial to the computations.Comment: 19 pages, based on a talk on the SPT 2011 conference,
http://www.sptspt.it/spt2011/ ; v2, v3: minor correction
Contact Integrable Extensions of Symmetry Pseudo-Group and Coverings of the r-th Modified Dispersionless Kadomtsev -- Petviashvili Equation
We apply the technique of integrable extensions to the symmetry pseudo-group
of the r-th mdKP equation. This gives another look on deriving known coverings
and allows us to find new coverings for this equation
- …