Generalized Lie bialgebroids and Jacobi structures

Iglesias, David; Marrero, Juan Carlos

doi:10.1016/s0393-0440(01)00032-8

Cited by 75 publications

(220 citation statements)

References 26 publications

(60 reference statements)

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“…Mathematically it is a version of the deformation of the de Rham differential considered by E. Witten [25] and similar to the calculus for Jacobi algebroids as developed in [11,8,9]. With this calculus, gradients and divergences, so (generalized) Laplace-Beltrami operators, associated with pseudo-Riemannian metrics are naturally defined.…”

Section: Introductionmentioning

confidence: 99%

The Schrödinger operator as a generalized Laplacian

Grabowska¹,

Grabowski²,

Urbański³

2008

J. Phys. A: Math. Theor.

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The Schrödinger operators on the Newtonian space-time are defined in a way which make them independent on the class of inertial observers. In this picture the Schrödinger operators act not on functions on the space-time but on sections of certain one-dimensional complex vector bundle -the Schrödinger line bundle. This line bundle has trivializations indexed by inertial observers and is associated with an U (1)-principal bundle with an analogous list of trivializations -the Schrödinger principal bundle. If an inertial frame is fixed, the Schrödinger bundle can be identified with the trivial bundle over space-time, but as there is no canonical trivialization (inertial frame), these sections interpreted as 'wave-functions' cannot be viewed as actual functions on the space-time. In this approach the change of an observer results not only in the change of actual coordinates in the space-time but also in a change of the phase of wave functions. For the Schrödinger principal bundle a natural differential calculus for 'wave forms' is developed that leads to a natural generalization of the concept of Laplace-Beltrami operator associated with a pseudoRiemannian metric. The free Schrödinger operator turns out to be the Laplace-Beltrami operator associated with a naturally distinguished invariant pseudo-Riemannian metric on the Schrödinger principal bundle. The presented framework does not involve any ad hoc or axiomatically introduced geometrical structures. It is based on the traditional understanding of the Schrödinger operator in a given reference frame -which is supported by producing right physics predictions -and it is proven to be strictly related to the frame-independent formulation of analytical Newtonian mechanics and Hamilton-Jacobi equations, that makes a bridge between the classical and quantum theory. MSC 2000: 35J10, 70G45.

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Section: Introductionmentioning

confidence: 99%

The Schrödinger operator as a generalized Laplacian

Grabowska¹,

Grabowski²,

Urbański³

2008

J. Phys. A: Math. Theor.

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“…The generalized Lie algebroids are in one-one correspondence with pairs consisting of a Lie algebroid A and a 1-cocycle φ 0 ∈ Γ(A * ) relative to the Lie algebroid exterior derivate d, i.e., dφ 0 = 0, (cf. [11,12,17]). …”

Section: Jacobi Manifolds and Lie Algebroidsmentioning

confidence: 99%

“…, ρ) is a Lie algebroid structure on A, φ 0 ∈ Γ(A * ) is a 1-cocycle and P ∈ Γ(∧ 2 A) a bisection on A satisfying [[P, P ]] + 2P ∧ i(φ 0 )P = 0 (see [17]). …”

Section: − Affine Jacobi Structures and Triangular Generalized Lie mentioning

confidence: 99%

“…The Schouten bracket is graded anticommutative, satisfies the graded Jacobi identity and the graded Leibniz rule. One can interpret a Poisson structure on M as a canonical structure for the Schouten bracket of multivector fields on M , i.e., as an element Λ of Lie degree −1 satisfying the master equation This point of view provides an easy passage to the theory of Jacobi structures and Jacobi algebroids [11] (or generalized Lie algebroids in the sense of [17]). It is enough to replace the (graded) Leibniz rule by the generalized Leibniz rule which is valid for the first-order differential operators.…”

Section: Introductionmentioning

confidence: 99%

“…The obtained bracket is called a Schouten-Jacobi bracket on A. Its restriction to sections of A defines a Lie algebroid structure on A, but its restriction to sections and functions defines a Jacobi algebroid ( [11,12]) structure on A (a generalized Lie algebroid in [17]). …”

Section: Introductionmentioning

confidence: 99%

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Jacobi Structures on Affine Bundles

et al. 2006

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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-to-one correspondence between affine Jacobi structures on A and Lie algebroid structures on the vector bundle A + = p∈M Af f (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 finitedimensional Lie algebra.Mathematics Subject Classification (2000): 53D17, 53D05, 81S10.

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From momentum maps and dual pairs to symplectic and Poisson groupoids

Marle

Progress in Mathematics

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Generalized Lie bialgebroids and Jacobi structures

Cited by 75 publications

References 26 publications

The Schrödinger operator as a generalized Laplacian

The Schrödinger operator as a generalized Laplacian

Jacobi Structures on Affine Bundles

From momentum maps and dual pairs to symplectic and Poisson groupoids

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