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The definition of an action functional for the Jacobi sigma models, known for Jacobi brackets of functions, is generalized to Jacobi bundles, i.e. Lie brackets on sections of (possibly non-trivial) line bundles, with the particular case of contact manifolds. Different approaches are proposed, but all of them share a common feature: the presence of a homogeneity structure appearing as a principal action of the Lie group ${ℝ}^{\times }=\mathrm{\text{GL}}(1;ℝ)$. Consequently, solutions of the equations of motions are morphisms of certain Jacobi algebroids, i.e. principal ${ℝ}^{\times }$-bundles equipped additionally with a compatible Lie algebroid structure. Despite the different approaches we propose, there is a one-to-one correspondence between the space of solutions of the different models. The definition can be immediately extended to almost Poisson and almost Jacobi brackets, i.e. to brackets that do not satisfy the Jacobi identity. Our sigma models are geometric and fully covariant.

We introduce the definition of modular class of a Lie algebroid comorphism and exploit some of its properties.

We define hypersymplectic structures on Lie algebroids recovering, as particular cases, all the classical results and examples of hypersymplectic structures on manifolds. We prove a 1-1 correspondence theorem between hypersymplectic structures and hyperkähler structures. We show that the hypersymplectic framework is very rich in already known compatible pairs of tensors such as Poisson–Nijenhuis, ΩN and PΩ structures.

Given a 2-vector field on a manifold, first we briefly discuss the complete integrability of the distribution which is the image of the 2-vector field. Then we show that a new Lie algebroid is defined on such a maniold which is coincident with the cotangent Lie algebroid when the 2-vector field is Poisson. The result is extended to the case of Lie algebroids.

Let be a Gerstenhaber algebra generated by and . Given a degree -1 operator D on , we find the condition on D that makes a BV-algebra. Subsequently, we apply it to the Gerstenhaber or BV algebra associated to a Lie algebroid and obtain a global proof of the correspondence between BV-generators and flat connections.

This paper considers the Chern–Simons forms for $ℝ$-linear connections on Lie algebroids. A generalized Chern–Simons formula for such $ℝ$-linear connections is obtained. We apply it to define the Chern character and secondary characteristic classes for $ℝ$-linear connections of Lie algebroids.

When the vacuum Einstein equations are cast in the form of Hamiltonian evolution equations, the initial data lie in the cotangent bundle of the manifold of Riemannian metrics on a Cauchy hypersurface Σ. As in every Lagrangian field theory with symmetries, the initial data must satisfy constraints. But, unlike those of gauge theories, the constraints of general relativity do not arise as momenta of any Hamiltonian group action. In this paper, we show that the bracket relations among the constraints of general relativity are identical to the bracket relations in the Lie algebroid of a groupoid consisting of diffeomorphisms between space-like hypersurfaces in spacetimes. A direct connection is still missing between the constraints themselves, whose definition is closely related to the Einstein equations, and our groupoid, in which the Einstein equations play no role at all. We discuss some of the difficulties involved in making such a connection. In an appendix, we develop some aspects of diffeology, the basic framework for our treatment of function spaces.

We consider homotopy actions of a Lie algebroid on a graded manifold, defined as suitable $L_{\infty }$-algebra morphisms. On the “semi-direct product” we construct a homological vector field that projects to the Lie algebroid. Our main theorem states that this construction is a bijection. Since several classical geometric structures can be described by homological vector fields as above, we can display many explicit examples, involving Lie algebroids (including extensions, representations up to homotopy and their cocycles) as well as transitive Courant algebroids.

A natural geometric framework is proposed, based on ideas of W. M. Tulczyjew, for constructions of dynamics on general algebroids. One obtains formalisms similar to the Lagrangian and the Hamiltonian ones. In contrast with recently studied concepts of Analytical Mechanics on Lie algebroids, this approach requires much less than the presence of a Lie algebroid structure on a vector bundle, but it still reproduces the main features of the Analytical Mechanics, like the Euler–Lagrange-type equations, the correspondence between the Lagrangian and Hamiltonian functions (Legendre transform) in the hyperregular cases, and a version of the Noether Theorem.

The Hamilton–Jacobi equation for a Hamiltonian section on a Lie affgebroid is introduced and some examples are discussed.

We construct Hermitian representations of Lie algebroids and associated unitary representations of Lie groupoids by a geometric quantization procedure. For this purpose, we introduce a new notion of Hamiltonian Lie algebroid actions. The first step of our procedure consists of the construction of a prequantization line bundle. Next, we discuss a version of Kähler quantization suitable for this setting. We proceed by defining a Marsden–Weinstein quotient for our setting and prove a "quantization commutes with reduction" theorem. We explain how our geometric quantization procedure relates to a possible orbit method for Lie groupoids. Our theory encompasses the geometric quantization of symplectic manifolds, Hamiltonian Lie algebra actions, actions of bundles of Lie groups, and foliations, as well as some general constructions from differential geometry.

We review origins and main properties of the most important bracket operations appearing canonically in differential geometry and mathematical physics in the classical, as well as in the supergeometric setting. The review is supplemented by some new concepts and examples.

A hypersymplectic structure on a Lie algebroid determines several Poisson–Nijenhuis, ΩN and PΩ structures on that Lie algebroid. We show that these Poisson–Nijenhuis (respectively, ΩN, PΩ) structures on the Lie algebroid, are pairwise compatible.

In this paper, we extend the almost complex Poisson structures from almost complex manifolds to almost complex Lie algebroids. Examples of such structures are also given and the almost complex Poisson morphisms of almost complex Lie algebroids are studied.

We show that every hypersymplectic structure on a Lie algebroid A determines a hypersymplectic and a hyperkähler structure on the dual A*. This result is illustrated with an example on the Lie algebroid T(H³ × I), where H³ is the Heisenberg group.

We present a graded-geometric approach to modular classes of Lie algebroids and their generalizations, introducing in this setting an idea of relative modular class of a Dirac structure for certain type of Courant algebroids, called projectable. This novel approach puts several concepts related to Poisson geometry and its generalizations in a new light and simplifies proofs. It gives, in particular, a nice geometric interpretation of modular classes of twisted Poisson structures on Lie algebroids.

In a new approach, by using the exact sequences, semisprays on the prolongation of a Lie algebroids are introduced and many important results on the semisprays and sprays are obtained. Also, the horizontal endomorphisms, almost complex structures, vertical, horizontal and complete lifts on the prolongation of a Lie algebroid are considered. Then the distinguished connections on the prolongation of a Lie algebroid are introduced and the torsion and curvature tensors of these connections are considered. In particular, the Berwald-type and Yano-type connections are studied.

We explain and motivate Stefan–Sussmann singular foliations, and by replacing the tangent bundle of a manifold with an arbitrary Lie algebroid, we introduce singular subalgebroids. Both notions are defined using compactly supported sections. The main results of this note are an equivalent characterization, in which the compact support condition is removed, and an explicit description of the sheaf associated to any Stefan–Sussmann singular foliation or singular subalgebroid.

In this paper, we study inverse problem for sprays on Lie algebroids. We obtain necessary and sufficient conditions, based on semi-basic forms, for a spray to be Lagrangian. Then we discuss the Finsler metrizability of a spray and obtain some equations on the Jacobi endomorphism.

In this paper, we study generalized almost para-contact manifolds and obtain normality conditions in terms of classical tensor fields. We show that such manifolds naturally carry certain Lie bialgebroid/quasi-Lie algebroid structures on them and we relate these new generalized manifolds with classical almost para-contact manifolds. The paper contains several examples and a short review for relations between generalized geometry and string theory.