共查询到20条相似文献,搜索用时 15 毫秒
1.
Henrique Bursztyn 《Advances in Mathematics》2007,211(2):726-765
We present a theory of reduction for Courant algebroids as well as Dirac structures, generalized complex, and generalized Kähler structures which interpolates between holomorphic reduction of complex manifolds and symplectic reduction. The enhanced symmetry group of a Courant algebroid leads us to define extended actions and a generalized notion of moment map. Key examples of generalized Kähler reduced spaces include new explicit bi-Hermitian metrics on CP2. 相似文献
2.
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
3.
We study the behavior of the modular class of a Lie algebroid under general Lie algebroid morphisms by introducing the relative modular class. We investigate the modular classes of pull-back morphisms and of base-preserving morphisms associated to Lie algebroid extensions. We also define generalized morphisms, including Morita equivalences, that act on the 1-cohomology, and observe that the relative modular class is a coboundary on the category of Lie algebroids and generalized morphisms with values in the 1-cohomology. 相似文献
4.
Mathieu Stiénon 《Comptes Rendus Mathematique》2009,347(9-10):545-550
Hypercomplex structures on Courant algebroids unify holomorphic symplectic structures and usual hypercomplex structures. In this Note, we prove the equivalence of two characterizations of hypercomplex structures on Courant algebroids, one in terms of Nijenhuis concomitants and the other in terms of (almost) torsionfree connections for which each of the three complex structures is parallel. To cite this article: M. Stiénon, C. R. Acad. Sci. Paris, Ser. I 347 (2009). 相似文献
5.
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
6.
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
7.
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
8.
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
9.
Jiefeng Liu Yunhe Sheng Chengming Bai Zhiqi Chen 《Mathematische Nachrichten》2016,289(14-15):1893-1908
In this paper, we introduce the notion of a left‐symmetric algebroid, which is a generalization of a left‐symmetric algebra from a vector space to a vector bundle. The left multiplication gives rise to a representation of the corresponding sub‐adjacent Lie algebroid. We construct left‐symmetric algebroids from ‐operators on Lie algebroids. We study phase spaces of Lie algebroids in terms of left‐symmetric algebroids. Representations of left‐symmetric algebroids are studied in detail. At last, we study deformations of left‐symmetric algebroids, which could be controlled by the second cohomology class in the deformation cohomology. 相似文献
10.
Janusz Grabowski 《Transformation Groups》2012,17(4):989-1010
Skew algebroid is a natural generalization of the concept of Lie algebroid. In this paper, for a skew algebroid E, its modular class mod(E) is defined in the classical as well as in the supergeometric formulation. It is proved that there is a homogeneous nowhere-vanishing 1-density on E * which is invariant with respect to all Hamiltonian vector fields if and only if E is modular, i.e., mod(E)?=?0. Further, the relative modular class of a subalgebroid is introduced and studied together with its application to holonomy, as well as the modular class of a skew algebroid relation. These notions provide, in particular, a unified approach to the concepts of a modular class of a Lie algebroid morphism and of a Poisson map. 相似文献
11.
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
12.
Xin ZHANG 《Frontiers of Mathematics in China》2018,13(5):1189-1214
We introduce two Hopf algebroids associated to a proper and holomorphic Lie group action on a complex manifold. We prove that the cyclic cohomology of each Hopf algebroid is equal to the Dolbeault cohomology of invariant differential forms. When the action is cocompact, we develop a generalized complex Hodge theory for the Dolbeault cohomology of invariant differential forms. We prove that every cyclic cohomology class of these two Hopf algebroids can be represented by a generalized harmonic form. This implies that the space of cyclic cohomology of each Hopf algebroid is finite dimensional. As an application of the techniques developed in this paper, we generalize the Serre duality and prove a Kodaira type vanishing theorem. 相似文献
13.
《Indagationes Mathematicae》2014,25(5):977-991
We introduce the notion of matched pairs of Courant algebroids and give several examples arising naturally from complex manifolds, holomorphic Courant algebroids, and certain regular Courant algebroids. We also consider the matched sum of two Dirac subbundles, one in each of two Courant algebroids forming a matched pair. 相似文献
14.
Jan Kubarski 《Czechoslovak Mathematical Journal》2006,56(2):359-376
This paper is a continuation of [19], [21], [22]. We study flat connections with isolated singularities in some transitive
Lie algebroids for which either ℝ or sl(2, ℝ) or so(3) are isotropy Lie algebras. Under the assumption that the dimension
of the isotropy Lie algebra is equal to n + 1, where n is the dimension of the base manifold, we assign to any such isolated singularity a real number called an index. For ℝ-Lie
algebroids, this index cannot be an integer. We prove the index theorem (the Euler-Poincaré-Hopf theorem for flat connections)
saying that the index sum is independent of the choice of a connection. Multiplying this index sum by the orientation class
of M, we get the Euler class of this Lie algebroid. Some integral formulae for indices are given. 相似文献
15.
This note studies invariant generators for a certain class of invariant smooth generalized subbundles of exact Courant algebroids, which generalizes existing results for invariant vector subbundles of exact Courant algebroids. As a result, we provide a simple and geometric proof for Theorem 1 in [1]. 相似文献
16.
In this paper we extend the theory of last multipliers as solutions of the Liouville’s transport equation to Lie algebroids
with their top exterior power as trivial line bundle (previously developed for vector fields and multivectors). We define
the notion of exact section and the Liouville equation on Lie algebroids. The aim of the present work is to develop the theory
of this extension from the tangent bundle algebroid to a general Lie algebroid (e.g. the set of sections with a prescribed
last multiplier is still a Gerstenhaber subalgebra). We present some characterizations of this extension in terms of Witten
and Marsden differentials. 相似文献
17.
《Journal of Algebra》2005,283(1):367-398
We study the family of vertex algebras associated with vertex algebroids, constructed by Gorbounov, Malikov, and Schechtman. As the main result, we classify all the (graded) simple modules for such vertex algebras and we show that the equivalence classes of graded simple modules one-to-one correspond to the equivalence classes of simple modules for the Lie algebroids associated with the vertex algebroids. To achieve our goal, we construct and exploit a Lie algebra from a given vertex algebroid. 相似文献
18.
Sophie Chemla 《Mathematische Zeitschrift》1999,232(2):367-388
Interpreting Lie algebroid theory in terms of -modules, we define a duality functor for a Lie algebroid as well as a direct image functor for a morphism of Lie algebroids.
Generalizing the work of Schneiders (see also the work of Schapira-Schneiders) and making assumptions analog to his, we show
that the duality functor and the direct image functor commute. As an application, we extend to Lie algebroids some duality
properties already known for Lie algebras.
Received December 12, 1997; in final form April 8, 1998 相似文献
19.
The main purpose of the paper is the study of the total space of a holomorphic Lie algebroid E. The paper is structured in three parts. In the first section, we briefly introduce basic notions on holomorphic Lie algebroids. The local expressions are written and the complexified holomorphic bundle is introduced. The second section presents two approaches on the study of the geometry of the complex manifold E. The first part contains the study of the tangent bundle \(T_{\mathbb {C}}E=T'E\oplus T''E\) and its link, via the tangent anchor map, with the complexified tangent bundle \(T_{\mathbb {C}}(T'M)=T'(T'M)\oplus T''(T'M)\). A holomorphic Lie algebroid structure is emphasized on \(T'E\). A special study is made for integral curves of a spray on \(T'E\). Theorem 2.8 gives the coefficients of a spray, called canonical, obtained from a complex Lagrangian on \(T'E\). In the second part of section two, we study the holomorphic prolongation \(\mathcal {T}'E\) of the Lie algebroid E. In the third section, we study how a complex Lagrange (Finsler) structure on \(T'M\) induces a Lagrangian structure on E. Three particular cases are analysed by the rank of the anchor map, the dimensions of manifold M, and those of the fibres. We obtain the correspondent on E of the Chern–Lagrange nonlinear connection from \(T'M\). 相似文献