577 research outputs found
Presymplectic representation of bi-Hamiltonian chain
Liouville integrable systems, which have bi-Hamiltonian representation of the
Gel'fand-Zakharevich type, are considered. Bi-presymplectic representation of
one-Casimir bi-Hamiltonian chains and weakly bi-presymplectic representation of
multi-Casimir bi-Hamiltonian chains are constructed. The reduction procedure
for Poisson and presymplectic structures is presented.Comment: 17 pages, to appear in J. Phys. A: Math. Ge
Blowing up generalized Kahler 4-manifolds
We show that the blow-up of a generalized Kahler 4-manifold in a
nondegenerate complex point admits a generalized Kahler metric. As with the
blow-up of complex surfaces, this metric may be chosen to coincide with the
original outside a tubular neighbourhood of the exceptional divisor. To
accomplish this, we develop a blow-up operation for bi-Hermitian manifolds.Comment: 16 page
The graded Jacobi algebras and (co)homology
Jacobi algebroids (i.e. `Jacobi versions' of Lie algebroids) are studied in
the context of graded Jacobi brackets on graded commutative algebras. This
unifies varios concepts of graded Lie structures in geometry and physics. A
method of describing such structures by classical Lie algebroids via certain
gauging (in the spirit of E.Witten's gauging of exterior derivative) is
developed. One constructs a corresponding Cartan differential calculus (graded
commutative one) in a natural manner. This, in turn, gives canonical generating
operators for triangular Jacobi algebroids. One gets, in particular, the
Lichnerowicz-Jacobi homology operators associated with classical Jacobi
structures. Courant-Jacobi brackets are obtained in a similar way and use to
define an abstract notion of a Courant-Jacobi algebroid and Dirac-Jacobi
structure. All this offers a new flavour in understanding the
Batalin-Vilkovisky formalism.Comment: 20 pages, a few typos corrected; final version to be published in J.
Phys. A: Math. Ge
Jacobi structures revisited
Jacobi algebroids, that is graded Lie brackets on the Grassmann algebra
associated with a vector bundle which satisfy a property similar to that of the
Jacobi brackets, are introduced. They turn out to be equivalent to generalized
Lie algebroids in the sense of Iglesias and Marrero and can be viewed also as
odd Jacobi brackets on the supermanifolds associated with the vector bundles.
Jacobi bialgebroids are defined in the same manner. A lifting procedure of
elements of this Grassmann algebra to multivector fields on the total space of
the vector bundle which preserves the corresponding brackets is developed. This
gives the possibility of associating canonically a Lie algebroid with any local
Lie algebra in the sense of Kirillov.Comment: 20 page
On classical finite and affine W-algebras
This paper is meant to be a short review and summary of recent results on the
structure of finite and affine classical W-algebras, and the application of the
latter to the theory of generalized Drinfeld-Sokolov hierarchies.Comment: 12 page
Complex of twistor operators in symplectic spin geometry
For a symplectic manifold admitting a metaplectic structure (a symplectic
analogue of the Riemannian spin structure), we construct a sequence consisting
of differential operators using a symplectic torsion-free affine connection.
All but one of these operators are of first order. The first order ones are
symplectic analogues of the twistor operators known from Riemannian spin
geometry. We prove that under the condition the symplectic Weyl curvature
tensor field of the symplectic connection vanishes, the mentioned sequence
forms a complex. This gives rise to a new complex for the so called Ricci type
symplectic manifolds, which admit a metaplectic structure.Comment: 18 pages, 1 figur
Nambu-Poisson Bracket and M-Theory Branes Coupled to Antisymmetric Fluxes
By using the recently proposed prescription arXiv:0804.3629 for obtaining the
brane action from multiple branes action in BLG theory, we examine
such transition when 11 Dimensional background antisymmetric fluxes couple to
the brane world volume. Such couplings was suggested in arXiv:0805.3427
where it was used the fact that various fields in BLG theory are valued in a
Lie 3-algebra. We argue that this action and promoting it by Nambu-Poisson
bracket gives the expected coupling of fluxes with brane at least at weak
coupling limit. We also study some other aspects of the action for example, the
gauge invariance of the theory.Comment: 14 page
Integrable Euler top and nonholonomic Chaplygin ball
We discuss the Poisson structures, Lax matrices, -matrices, bi-hamiltonian
structures, the variables of separation and other attributes of the modern
theory of dynamical systems in application to the integrable Euler top and to
the nonholonomic Chaplygin ball.Comment: 25 pages, LaTeX with AMS fonts, final versio
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