Abstract
We show that a double Lie algebroid, together with a chosen decomposition, is equivalent to a pair of 2-term representations up to homotopy satisfying compatibility conditions which extend the notion of matched pair of Lie algebroids. We discuss in detail the double Lie algebroids arising from the tangent bundle of a Lie algebroid and the cotangent bundle of a Lie bialgebroid.
Highlights
Background and definitions2.1 Double vector bundles, decompositions and dualizationWe briefly recall the definitions of double vector bundles, of their linear and core sections, and of their linear splittings and lifts
Using the isomorphism defined by − ·, ·, the double vector bundle (D A; A, C∗; M) has a LA–vector bundle structure
We have seen above that (D A; A, C∗; M) has an induced LA–vector bundle structure, and we have shown that the decomposition induces a natural decomposition : A ×M C∗ → D B of derivative £ : C∗ (D A)
Summary
We briefly recall the definitions of double vector bundles, of their linear and core sections, and of their linear splittings and lifts. The core C of a double vector bundle is the intersection of the kernels of πA and πB. Given a double vector bundle (D; A, B; M), the space of sections B(D) is generated as a C∞(B)-module by two distinguished classes of sections (see [16]), the linear and the core sections which we describe. Definition 2.3 A section ξ ∈ B(D) is called linear if ξ : B → D is a bundle morphism from B → M to D → A over a section a ∈ (A). The fibered product A ×M B is a double vector bundle with side bundles A and B and core M × 0
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