588 research outputs found
Coupling Poisson and Jacobi structures on foliated manifolds
Let M be a differentiable manifold endowed with a foliation F. A Poisson
structure P on M is F-coupling if the image of the annihilator of TF by the
sharp-morphism defined by P is a normal bundle of the foliation F. This notion
extends Sternberg's coupling symplectic form of a particle in a Yang-Mills
field. In the present paper we extend Vorobiev's theory of coupling Poisson
structures from fiber bundles to foliations and give simpler proofs of
Vorobiev's existence and equivalence theorems of coupling Poisson structures on
duals of kernels of transitive Lie algebroids over symplectic manifolds. Then
we discuss the extension of the coupling condition to Jacobi structures on
foliated manifolds.Comment: LateX, 38 page
On invariants of almost symplectic connections
We study the irreducible decomposition under Sp(2n, R) of the space of
torsion tensors of almost symplectic connections. Then a description of all
symplectic quadratic invariants of torsion-like tensors is given. When applied
to a manifold M with an almost symplectic structure, these instruments give
preliminary insight for finding a preferred linear almost symplectic connection
on M . We rediscover Ph. Tondeur's Theorem on almost symplectic connections.
Properties of torsion of the vectorial kind are deduced
Twistor theory on a finite graph
We show how the description of a shear-free ray congruence in Minkowski space
as an evolving family of semi-conformal mappings can naturally be formulated on
a finite graph. For this, we introduce the notion of holomorphic function on a
graph. On a regular coloured graph of degree three, we recover the space-time
picture. In the spirit of twistor theory, where a light ray is the more
fundamental object from which space-time points should be derived, the line
graph, whose points are the edges of the original graph, should be considered
as the basic object. The Penrose twistor correspondence is discussed in this
context
A class of Poisson-Nijenhuis structures on a tangent bundle
Equipping the tangent bundle TQ of a manifold with a symplectic form coming
from a regular Lagrangian L, we explore how to obtain a Poisson-Nijenhuis
structure from a given type (1,1) tensor field J on Q. It is argued that the
complete lift of J is not the natural candidate for a Nijenhuis tensor on TQ,
but plays a crucial role in the construction of a different tensor R, which
appears to be the pullback under the Legendre transform of the lift of J to
co-tangent manifold of Q. We show how this tangent bundle view brings new
insights and is capable also of producing all important results which are known
from previous studies on the cotangent bundle, in the case that Q is equipped
with a Riemannian metric. The present approach further paves the way for future
generalizations.Comment: 22 page
Cohomology of skew-holomorphic Lie algebroids
We introduce the notion of skew-holomorphic Lie algebroid on a complex
manifold, and explore some cohomologies theories that one can associate to it.
Examples are given in terms of holomorphic Poisson structures of various sorts.Comment: 16 pages. v2: Final version to be published in Theor. Math. Phys.
(incorporates only very minor changes
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
A broad spectrum of genomic changes in Latinamerican patients with EXT1/EXT2-CDG
Multiple osteochondromatosis (MO), or EXT1/EXT2-CDG, is an autosomal dominant O-linked glycosylation disorder characterized by the formation of multiple cartilage-capped tumors (osteochondromas). In contrast, solitary osteochondroma (SO) is a non-hereditary condition. EXT1 and EXT2, are tumor suppressor genes that encode glycosyltransferases involved in heparan sulfate elongation. We present the clinical and molecular analysis of 33 unrelated Latin American patients (27 MO and 6 SO). Sixty-three percent of all MO cases presented severe phenotype and two malignant transformations to chondrosarcoma (7%). We found the mutant allele in 78% of MO patients. Ten mutations were novel. The disease-causing mutations remained unknown in 22% of the MO patients and in all SO patients. No second mutational hit was detected in the DNA of the secondary chondrosarcoma from a patient who carried a nonsense EXT1 mutation. Neither EXT1 nor EXT2 protein could be detected in this sample. This is the first Latin American research program on EXT1/EXT2-CDG
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
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
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