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《Indagationes Mathematicae》2022,33(2):372-387
In this note we compute the cohomology of the elliptic tangent bundle, a Lie algebroid introduced in Cavalcanti and Gualtieri (2018), Cavalcanti et al. (2020) used to describe singular symplectic forms arising from generalised complex geometry. 相似文献
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The aim of this note is to communicate a simple example of a Lie–Rinehart algebra whose enveloping algebra is not a Hopf algebroid, neither in the sense of Böhm and Szlachányi, nor in the sense of Lu. 相似文献
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《Indagationes Mathematicae》2014,25(5):1019-1053
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J. GRABOWSKI D. IGLESIAS J. C. MARRERO E. PADRON P. URBANSKI 《数学学报(英文版)》2007,23(5):769-788
Abstract We study affine Jacobi structures (brackets) on an affine bundle π : A → M, i.e. Jacobi brackets that close on affine functions. We prove that if the rank of A is non-zero, there is a one-toone correspondence between affine Jacobi structures on A and Lie algebroid structures on the vector bundle A^+ = ∪p∈M Aff(Ap, R) of affine functionals. In the case rank A = 0, it is shown that there is a one-to-one correspondence between affine Jacobi structures on A and local Lie algebras on A^+. Some examples and applications, also for the linear case, are discussed. For a special type of affine Jacobi structures which are canonically exhibited (strongly-affine or affine-homogeneous Jacobi structures) over a real vector space of finite dimension, we describe the leaves of its characteristic foliation as the orbits of an affine representation. These affine Jacobi structures can be viewed as an analog of the Kostant-Arnold-Liouville linear Poisson structure on the dual space of a real finite-dimensional Lie algebra. 相似文献
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In this paper, we generalize all the results obtained on para‐Kähler Lie algebras in [3] to para‐Kähler Lie algebroids. In particular, we study exact para‐Kähler Lie algebroids as a generalization of exact para‐Kähler Lie algebras. This study leads to a natural generalization of pseudo‐Hessian manifolds, we call them contravariant pseudo‐Hessian manifolds. Contravariant pseudo‐Hessian manifolds have many similarities with Poisson manifolds. We explore these similarities which, among others, leads to a powerful machinery to build examples of non trivial pseudo‐Hessian structures. Namely, we will show that given a finite dimensional commutative and associative algebra , the orbits of the action Φ of on given by are pseudo‐Hessian manifolds, where . We illustrate this result by considering many examples of associative commutative algebras and show that the resulting pseudo‐Hessian manifolds are very interesting. 相似文献
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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
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Christian Nassau 《Transactions of the American Mathematical Society》2002,354(5):1749-1757
We show that for with its geometrically induced structure maps is not an Hopf algebroid because neither the augmentation nor the coproduct are multiplicative. As a consequence the algebra structure of is slightly different from what was supposed to be the case. We give formulas for and and show that the inversion of the formal group of is induced by an antimultiplicative involution . Some consequences for multiplicative and antimultiplicative automorphisms of for are also discussed.
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We introduce hom-Lie-Rinehart algebras as an algebraic analogue of hom-Lie algebroids, and systematically describe a cohomology complex by considering coefficient modules. We define the notion of extensions for hom-Lie-Rinehart algebras. In the sequel, we deduce a characterization of low dimensional cohomology spaces in terms of the group of automorphisms of certain abelian extension and the equivalence classes of those abelian extensions in the category of hom-Lie-Rinehart algebras, respectively. We also construct a canonical example of hom-Lie-Rinehart algebra associated to a given Poisson algebra and an automorphism. 相似文献
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David Iglesias Juan C. Marrero David Martín de Diego Eduardo Martínez 《Journal of Nonlinear Science》2008,18(3):221-276
This paper studies the construction of geometric integrators for nonholonomic systems. We develop a formalism for nonholonomic
discrete Euler–Lagrange equations in a setting that permits to deduce geometric integrators for continuous nonholonomic systems
(reduced or not). The formalism is given in terms of Lie groupoids, specifying a discrete Lagrangian and a constraint submanifold
on it. Additionally, it is necessary to fix a vector subbundle of the Lie algebroid associated to the Lie groupoid. We also
discuss the existence of nonholonomic evolution operators in terms of the discrete nonholonomic Legendre transformations and
in terms of adequate decompositions of the prolongation of the Lie groupoid. The characterization of the reversibility of
the evolution operator and the discrete nonholonomic momentum equation are also considered. Finally, we illustrate with several
classical examples the wide range of application of the theory (the discrete nonholonomic constrained particle, the Suslov
system, the Chaplygin sleigh, the Veselova system, the rolling ball on a rotating table and the two wheeled planar mobile
robot).
This work was partially supported by MEC (Spain) Grants MTM 2006-03322, MTM 2007-62478, MTM 2006-10531, project “Ingenio Mathematica”
(i-MATH) No. CSD 2006-00032 (Consolider-Ingenio 2010) and S-0505/ESP/0158 of the CAM. 相似文献
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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
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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
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The simplified methodology for obtaining the Lie algebra structures of nonlinear evolution equations
The technique for obtaining the prolongation structure of differential equations is simplified. This new simplified method is used to obtain the Lie algebra structure of the Burgers–KdV equation. 相似文献
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Alexander Usvyatsov 《Topology and its Applications》2008,155(14):1607-1617
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湍流相干结构与小尺度结构之间的相互作用 总被引:1,自引:0,他引:1
本文首先对切变湍流场中存在大尺度相干结构与小尺度结构和不同尺度结构之间的相互作用进行了试验研究,得到了一些特征值,其次,在试验的基础上,建立了考虑不同尺度结构之间的相互作用后相干结构的数学描述和模式识别方法,并对光滑壁面与均匀密集加糙壁面条件下湍流边界层中的相干结构进行了模式识别。结果表明,文中所建立的计算方法是可行的。 相似文献
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Maria L. Barberis Isabel G. Dotti Miatello Roberto J. Miatello 《Annals of Global Analysis and Geometry》1995,13(3):289-301
Given a manifoldM, a Clifford structure of orderm onM is a family ofm anticommuting complex structures generating a subalgebra of dimension 2
m
of End(T(M)). In this paper we investigate the existence of locally invariant Clifford structures of orderm2 on a class of locally homogeneous manifolds. We study the case of solvable extensions ofH-type groups, showing in particular that the solvable Lie groups corresponding to the symmetric spaces of negative curvature carry invariant Clifford structures of orderm2. We also show that for eachm and any finite groupF, there is a compact flat manifold with holonomy groupF and carrying a Clifford structure of orderm.Partially supported by Conicor (Argentina)Partially supported by grants from Conicet, Conicor, SECYTUNg (Argentina), and I.C.T.P. (Trieste)Partially supported by grants from Conicet, Conicor, SECYTUNC (Argentina), T.W.A.S and I.C.T.P. (Trieste) 相似文献