Framework Of Lie Algebroids Research Articles

Forms and Chern Classes on Hermitian Lie Algebroids

In this paper, we study Hermitian Lie algebroids and introduce the notion of Chern A-connection on Hermitian vector bundles. Then, we prove that on every holomorphic vector bundle there exists a unique Chern A-connection and find its connection form and curvature form. Also, we generalize Weitzenbock’s formula to complex Lie algebroids and prove vanishing theorem for holomorphic sections. Moreover, we extend the Chern classes in complex geometry to Lie algebroids framework.

Mechanical Structures on Lie Algebroids

In this paper, we extend the study of mechanical systems in the general framework of Lie algebroids using the associated evolution nonlinear connections. In the case of the prolongations of a Lie algebroid over the vector bundle projections, we prove the equivalence between a regular section and a mechanical structure. Finally, we justify this construction with an example from optimal control theory.

Symmetries of second order differential equations on Lie algebroids

In this paper we investigate the relations between semispray, nonlinear connection, dynamical covariant derivative and Jacobi endomorphism on Lie algebroids. Using these geometric structures, we study the symmetries of second order differential equations in the general framework of Lie algebroids.

Weyl’s theory in the generalized Lie algebroids framework

The geometry of the Lie algebroid generalized tangent bundle of a generalized Lie algebroid is developed. Formulas of Ricci type and identities of Cartan and Bianchi type are presented. Introducing the notion of geodesic of a mechanical ρ,η-system with respect to a (ρ, η)-spray, the Berwald (ρ, η)-derivative operator, and its mixed curvature, we obtain main results to conceptualize the Weyl’s method in this general framework. Finally, we obtain two new results of Weyl type for the geometry of mechanical ρ,η-systems. In this way, it is proved that the projectively related sprays first have the same geodesics rather to an increasing parameter transformation and second their Berwald derivatives verify a respective relation.

Almost Analyticity on Almost (Para) Complex Lie Algebroids

The goal of this paper is to generalize the notion of almost analyticity in the almost (para)complex Lie algebroids framework. We use a global formalism which yields to generalizations of the main results of previous known almost (para)complex manifolds case.

Optimal control of affine connection control systems from the point of view of Lie algebroids

The purpose of this paper is to use the framework of Lie algebroids to study optimal control problems (OCPs) for affine connection control systems (ACCSs) on Lie groups. In this context, the equations for critical trajectories of the problem are geometrically characterized as a Hamiltonian vector field.

Mechanical systems in the generalized Lie algebroids framework

Mechanical systems, called by use, mechanical (ρ, η)-systems, Lagrange mechanical (ρ, η)-systems or Finsler mechanical (ρ, η)-systems, are presented. The canonical (ρ, η)-semi(spray) associated to a mechanical (ρ, η)-system is obtained. New and important results are obtained in the particular case of Lie algebroids. The Lagrange mechanical (ρ, η)-systems are the spaces necessary to develop a new Lagrangian formalism. We obtain the (ρ, η)-semispray associated to a regular Lagrangian L and external force Feand we derive the equations of Euler–Lagrange type. So, a new solution for the Weinstein's problem in the general framework of generalized Lie algebroids is presented.

Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids

This paper develops the notion of implicit Lagrangian systems on Lie algebroids and a Hamilton--Jacobi theory for this type of system. The Lie algebroid framework provides a natural generalization of classical tangent bundle geometry. We define the notion of an implicit Lagrangian system on a Lie algebroid $E$ using Dirac structures on the Lie algebroid prolongation $\mathcal{T}^E E^*$. This setting includes degenerate Lagrangian systems with nonholonomic constraints on Lie algebroids.

Reduction of Poisson–Nijenhuis Lie algebroids to symplectic-Nijenhuis Lie algebroids with a nondegenerate Nijenhuis tensor

We show how to reduce, under certain regularity conditions, a Poisson–Nijenhuis Lie algebroid to a symplectic-Nijenhuis Lie algebroid with a nondegenerate Nijenhuis tensor. We generalize the work done by Magri and Morosi for the reduction of Poisson–Nijenhuis manifolds. The choice of the more general framework of Lie algebroids is motivated by the geometrical study of some reduced bi-Hamiltonian systems. An explicit example of the reduction of a Poisson–Nijenhuis Lie algebroid is also provided.

Hochschild cohomology and Atiyah classes

In this paper we prove that on a smooth algebraic variety the HKR-morphism twisted by the square root of the Todd genus gives an isomorphism between the sheaf of poly-vector fields and the sheaf of poly-differential operators, both considered as derived Gerstenhaber algebras. In particular we obtain an isomorphism between Hochschild cohomology and the cohomology of poly-vector fields which is compatible with the Lie bracket and the cupproduct. The latter compatibility is an unpublished result by Kontsevich. Our proof is set in the framework of Lie algebroids and so applies without modification in much more general settings as well.

Nonholonomic Lagrangian systems on Lie algebroids

This paper presents a geometric description on Lie algebroids of Lagrangian systems subject to nonholonomic constraints. The Lie algebroid framework provides a natural generalization of classical tangent bundle geometry. We define the notion of nonholonomically constrained system, and characterize regularity conditions that guarantee that the dynamics of the system can be obtained as a suitable projection of the unconstrained dynamics. The proposed novel formalism provides new insights into the geometry of nonholonomic systems, and allows us to treat in a unified way a variety of situations, including systems with symmetry, morphisms, reduction, and nonlinearly constrained systems. Various examples illustrate the results.

Quasi-coordinates from the point of view of Lie algebroid structures

A geometrical description of the Lagrangian dynamics in quasi-coordinates on the tangent bundle, using the Lie algebroid framework, is given. Linear non-holonomic systems on Lie algebroids are solved in local coordinates adapted to the constraints, through Lagrangian multipliers and Gibbs–Appell generalized methods.

Lagrangian reduction by stages for non-holonomic systems in a Lie algebroid framework

The Lagrange-d'Alembert equations of a non-holonomic system with symmetry can be reduced to the Lagrange-d'Alembert-Poincarequations. In a previous contribution we have shown that both sets of equations fall in the category of the so-called 'Lagrangian systems on a subbundle of a Lie algebroid'. In this paper, we investigate the special case when the reduced system is again invariant under a new symmetry group (and so forth). Via Lie algebroid theory, we develop a geometric context in which successive reduction can be performed in an intrinsic way. We prove that, at each stage of the reduction, the reduced systems are part of the above mentioned category, and that the Lie algebroid structure in each new step is the quotient Lie algebroid of the previous step. We further show that that reduction in two stages is equivalent with direct reduction.

A Lie algebroid framework for non-holonomic systems

In order to obtain a framework in which both non-holonomic mechanical systems and non-holonomic mechanical systems with symmetry can be described, we introduce in this paper the notion of a Lagrangian system on a subbundle of a Lie algebroid.

Reduction in optimal control theory

A geometric setting for the Pontryagin maximum principle in optimal control theory is provided. The equations for critical trajectories are given in terms of a symplectic equation. These equations are generalized to the framework of Lie algebroids, giving thus a way to study reduction by symmetry groups of the maximum principle.

Toda Equations, bi-Hamiltonian Systems, and Compatible Lie Algebroids

We present the bi-Hamiltonian structure of Toda3, a dynamical system studied by Kupershmidt as a restriction of the discrete KP hierarchy. We derive this structure by a suitable reduction of the set of maps from Zd to GL(3,R), in the framework of Lie algebroids.