391 research outputs found
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
Jacobi-Nijenhuis algebroids and their modular classes
Jacobi-Nijenhuis algebroids are defined as a natural generalization of
Poisson-Nijenhuis algebroids, in the case where there exists a Nijenhuis
operator on a Jacobi algebroid which is compatible with it. We study modular
classes of Jacobi and Jacobi-Nijenhuis algebroids
Symmetry classification of third-order nonlinear evolution equations. Part I: Semi-simple algebras
We give a complete point-symmetry classification of all third-order evolution
equations of the form
which admit semi-simple symmetry algebras and extensions of these semi-simple
Lie algebras by solvable Lie algebras. The methods we employ are extensions and
refinements of previous techniques which have been used in such
classifications.Comment: 53 page
Leibniz 2-algebras and twisted Courant algebroids
In this paper, we give the categorification of Leibniz algebras, which is
equivalent to 2-term sh Leibniz algebras. They reveal the algebraic structure
of omni-Lie 2-algebras introduced in \cite{omniLie2} as well as twisted Courant
algebroids by closed 4-forms introduced in \cite{4form}.
We also prove that Dirac structures of twisted Courant algebroids give rise
to 2-term -algebras and geometric structures behind them are exactly
-twisted Lie algebroids introduced in \cite{Grutzmann}.Comment: 22 pages, to appear in Comm. Algebr
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
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
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
Lie algebroid foliations and -Dirac structures
We prove some general results about the relation between the 1-cocycles of an
arbitrary Lie algebroid over and the leaves of the Lie algebroid
foliation on associated with . Using these results, we show that a
-Dirac structure induces on every leaf of its
characteristic foliation a -Dirac structure , which comes
from a precontact structure or from a locally conformal presymplectic structure
on . In addition, we prove that a Dirac structure on can be obtained from and we discuss the relation between the leaves of
the characteristic foliations of and .Comment: 25 page
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