共查询到20条相似文献,搜索用时 46 毫秒
1.
《Comptes Rendus de l'Academie des Sciences Series IIA Earth and Planetary Science》1998,326(3):289-294
The purpose of this Note is to unify quantum groups and star-products under a general umbrella: quantum groupoids. It is shown that a quantum groupoid naturally gives rise to a Lie bialgebroid as a classical limit. The converse question, i.e.. the quantization problem, is posed. In particular, any regular triangular Lie bialgebroid is shown quantizable. For the Lie bialgebroid of a Poisson manifold, its quantization is equivalent to a star-product. 相似文献
2.
Fani Petalidou 《Differential Geometry and its Applications》2005,23(3):282-304
We first recall some basic definitions and facts about Jacobi manifolds, generalized Lie bialgebroids, generalized Courant algebroids and Dirac structures. We establish an one-one correspondence between reducible Dirac structures of the generalized Lie bialgebroid of a Jacobi manifold (M,Λ,E) for which 1 is an admissible function and Jacobi quotient manifolds of M. We study Jacobi reductions from the point of view of Dirac structures theory and we present some examples and applications. 相似文献
3.
《Comptes Rendus Mathematique》2008,346(3-4):193-198
We introduce the concept of Loday algebroids, a generalization of Courant algebroids. We define the naive cohomology and modular class of a Loday algebroid, and we show that the modular class of the double of a Lie bialgebroid vanishes. For Courant algebroids, we describe the relation between the naive and standard cohomologies and we conjecture that they are isomorphic when the Courant algebroid is transitive. To cite this article: M. Stiénon, P. Xu, C. R. Acad. Sci. Paris, Ser. I 346 (2008). 相似文献
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We study the relative modular classes of Lie algebroids, and we determine their relationship with the modular classes of Lie algebroids with a twisted Poisson structure. To cite this article: Y. Kosmann-Schwarzbach, A. Weinstein, C. R. Acad. Sci. Paris, Ser. I 341 (2005). 相似文献
6.
Yvette Kosmann-Schwarzbach 《Bulletin of the Brazilian Mathematical Society》2011,42(4):625-649
We study Nijenhuis structures on Courant algebroids in terms of the canonical Poisson bracket on their symplectic realizations.
We prove that the Nijenhuis torsion of a skew-symmetric endomorphism N of a Courant algebroid is skewsymmetric if N
2 is proportional to the identity, and only in this case when the Courant algebroid is irreducible. We derive a necessary and
sufficient condition for a skewsymmetric endomorphism to give rise to a deformed Courant structure. In the case of the double
of a Lie bialgebroid (A, A*), given an endomorphism N of A that defines a skew-symmetric endomorphism N of the double of A, we prove that the torsion ofN is the sum of the torsion of N and that of the transpose of N. 相似文献
7.
Y. Kosmann-Schwarzbach 《Acta Appl Math》1995,41(1-3):153-165
We show that to any Poisson manifold and, more generally, to any triangular Lie bialgebroid in the sense of Mackenzie and Xu, there correspond two differential Gerstenhaber algebras in duality, one of which is canonically equipped with an operator generating the graded Lie algebra bracket, i.e. with the structure of a Batalin-Vilkovisky algebra. 相似文献
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We characterize Poisson and Jacobi structures by means of complete lifts of the corresponding tensors: the lifts have to be
related to canonical structures by morphisms of corresponding vector bundles. Similar results hold for generalized Poisson
and Jacobi structures (canonical structures) associated with Lie algebroids and Jacobi algebroids. 相似文献
11.
We examine Lie (super)algebroids equipped with a homological section, i.e., an odd section that ‘self-commutes’, we refer to such Lie algebroids as inner Q-algebroids: these provide natural examples of suitably “superised” Q-algebroids in the sense of Mehta. Such Lie algebroids are a natural generalisation of Q-manifolds and Lie superalgebras equipped with a homological element. Amongst other results, we show that, via the derived bracket formalism, the space of sections of an inner Q-algebroid comes equipped with an odd Loday–Leibniz bracket. 相似文献
12.
YIN Yanbin & HE Longguang LMAM Capital Normal University Beijing China 《中国科学A辑(英文版)》2006,49(10):1341-1352
Protobialgebroids include several kinds of algebroid structures such as Lie algebroid, Lie bialgebroid, Lie quasi-bialgebroid, etc. In this paper, the Dirac theories are generalized from Lie bialgebroid to protobialgebroid. We give the integrable conditions for a maximally isotropic subbundle being a Dirac structure for a protobialgebroid by the notion of a characteristic pair. From the integrable conditions, we found out that the Dirac structure has closed relations with the twisting of a protobialgebroid. At last, some special cases of the Dirac structures for protobialgebroids are discussed. 相似文献
13.
Domenico Perrone 《Differential Geometry and its Applications》2012,30(1):49-58
The purpose of this paper is to classify all simply connected homogeneous almost cosymplectic three-manifolds. We show that each such three-manifold is either a Lie group G equipped with a left invariant almost cosymplectic structure or a Riemannian product of type R×N, where N is a Kähler surface of constant curvature. Moreover, we find that the Reeb vector field of any homogeneous almost cosymplectic three-manifold, except one case, defines a harmonic map. 相似文献
14.
《Indagationes Mathematicae》2014,25(5):846-871
We introduce the notion of tropicalization for Poisson structures on with coefficients in Laurent polynomials. To such a Poisson structure we associate a polyhedral cone and a constant Poisson bracket on this cone. There is a version of this formalism applicable to viewed as a real Poisson manifold. In this case, the tropicalization gives rise to a completely integrable system with action variables taking values in a polyhedral cone and angle variables spanning a torus.As an example, we consider the canonical Poisson bracket on the dual Poisson–Lie group for in the cluster coordinates of Fomin–Zelevinsky defined by a certain choice of solid minors. We prove that the corresponding integrable system is isomorphic to the Gelfand–Zeitlin completely integrable system of Guillemin–Sternberg and Flaschka–Ratiu. 相似文献
15.
Grabowska Katarzyna Grabowski Janusz Urbaski Pawe 《Annals of Global Analysis and Geometry》2003,24(2):101-130
Natural affine analogs of Lie brackets on affine bundles are studied.In particular, a close relation to Lie algebroids and a duality withcertain affine analog of Poisson structure is established as well asaffine versions of complete lifts and Cartan exterior calculi. 相似文献
16.
Izu Vaisman 《Mediterranean Journal of Mathematics》2010,7(4):415-444
If A is a Lie algebroid over a foliated manifold (M, F){(M, {\mathcal {F}})}, a foliation of A is a Lie subalgebroid B with anchor image TF{T{\mathcal {F}}} and such that A/B is locally equivalent with Lie algebroids over the slice manifolds of F{\mathcal F}. We give several examples and, for foliated Lie algebroids, we discuss the following subjects: the dual Poisson structure
and Vaintrob's supervector field, cohomology and deformations of the foliation, integration to a Lie groupoid. In the last
section, we define a corresponding notion of a foliation of a Courant algebroid A as a bracket–closed, isotropic subbundle B with anchor image TF{T{\mathcal {F}}} and such that B ^ /B{B^{ \bot } /B} is locally equivalent with Courant algebroids over the slice manifolds of F{\mathcal F}. Examples that motivate the definition are given. 相似文献
17.
M. A. Olshanetsky 《Theoretical and Mathematical Physics》2007,150(3):301-314
We construct a quadratic Poisson algebra of Hamiltonian functions on a two-dimensional torus compatible with the canonical
Poisson structure. This algebra is an infinite-dimensional generalization of the classical Sklyanin-Feigin-Odesskii algebras.
It yields an integrable modification of the two-dimensional hydrodynamics of an ideal fluid on the torus. The Hamiltonian
of the standard two-dimensional hydrodynamics is defined by the Laplace operator and thus depends on the metric. We replace
the Laplace operator with a pseudodifferential elliptic operator depending on the complex structure. The new Hamiltonian becomes
a member of a commutative bi-Hamiltonian hierarchy. In conclusion, we construct a Lie bialgebroid of vector fields on the
torus.
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 3, pp. 355–370, March, 2007. 相似文献
18.
Domenico Perrone 《Acta Mathematica Hungarica》2013,138(1-2):102-126
We investigate almost contact metric manifolds whose Reeb vector field is a harmonic unit vector field, equivalently a harmonic section. We first consider an arbitrary Riemannian manifold and characterize the harmonicity of a unit vector field ??, when ??? is symmetric, in terms of Ricci curvature. Then, we show that for the class of locally conformal almost cosymplectic manifolds whose Reeb vector field ?? is geodesic, ?? is a harmonic section if and only if it is an eigenvector of the Ricci operator. Moreover, we build a large class of locally conformal almost cosymplectic manifolds whose Reeb vector field is a harmonic section. Finally, we exhibit several classes of almost contact metric manifolds where the associated almost contact metric structures ?? are harmonic sections, in the sense of Vergara-Diaz and Wood?[25], and in some cases they are also harmonic maps. 相似文献
19.
A VB-algebroid is essentially defined as a Lie algebroid object in the category of vector bundles. There is a one-to-one correspondence between VB-algebroids and certain flat Lie algebroid superconnections, up to a natural notion of equivalence. In this setting, we are able to construct characteristic classes, which in special cases reproduce characteristic classes constructed by Crainic and Fernandes. We give a complete classification of regular VB-algebroids, and in the process we obtain another characteristic class of Lie algebroids that does not appear in the ordinary representation theory of Lie algebroids. 相似文献
20.
We prove a general integrability result for matched pairs of Lie algebroids. (Matched pairs of Lie algebras are also known as double Lie algebras or twilled extensions of Lie algebras.) The method used is an extension of a method introduced by Lu and Weinstein in the case of Poisson Lie groups, and yields double groupoids which satisfy an étale form of the vacancy condition. 相似文献