Anchored vector bundles and Lie algebroids

Abstract

The derived generalized algebroid and the derived generalized Lie algebroid of an anchored vector bundle are defined. Some natural functors from the two categories of anchored vector bundles to the corresponding categories of generalized algebroids and generalized Lie algebroids respectively are also considered. A natural result is proved: the derived (Lie) algebroid of an anchored vector subbundle is a generalized (Lie) algebroid of the underlying bundle. Lifts of linear R-connections and skewsymmetric forms respectively are constructed and the modular class of an almost Lie structure is defined.

On Almost Complex Lie Algebroids

“…In the preliminary section we briefly recall some basic facts about Lie algebroids. For more, see for instance [9,11,14,24,27,37]. In the second section we define almost complex Lie algebroids over smooth manifolds, we present some examples and we obtain a Newlander-Nirenberg type theorem (Theorem 2.1).…”

Almost Lie Algebroids and Characteristic Classes

“…The Lie algebroids (as, for example [14]) are generalizations of Lie algebras and integrable regular distributions. In fact a Lie algebroid is an anchored vector bundle (see [17,18,20]) with a Lie bracket on the module of sections.…”

“…The derived anchored bundle constructed in [20], used for E 0 , gives a Lie algebroid structure (Theorem 2.9). The proof of this result needs some long computations, based on some auxiliary and technical results proved in Proposition 2.6.…”

“…As in the general case in [20], we can consider the derived anchored bundle E (1) = E ⊕(E ∧E), where the anchor is denoted as ρ (1) and the bracket by [·, ·] E (1) . Concretely, ρ (1) (X) = ρ(X),…”

Poisson structures on almost complex Lie algebroids

“…(Complete lift to the prolongation of a Lie algebroid). For a Lie algebroid (E, ρ E , [·, ·] E ) with rank E = m we can consider the prolongation of E over its vector bundle projection, see[14,29,36], which is a vector bundle p L : L p (E) → E of rank L p (E) = 2m which has a Lie algebroid structure over E. More exactly,L p (E) is the subset of E × T E defined by L p (E) = {(u, z) | ρ E (u) = p * (z)}, where p * : T E → T M is the canonical projection.The projection on the second factor ρ L p (E) : L p (E) → T E, given by ρ L p (E) (u, z) = z will be the anchor of the prolongation Lie algebroid L p (E), ρ L p (E) , [·, ·] L p (E) over E. According to[29,36], we can consider the vertical lift s v and the complete lift s c of a section s ∈ Γ(E) as sections of L p (E) as follows. The local basis of Γ(L p (E)) is given by X a (u) = e a (p(u)), ρ i a ∂∂x i | u , V a = 0, ∂ ∂y a ,where ∂ ∂x i , ∂ ∂y a , i = 1, .…”