Representations up to homotopy of Lie algebroids

Abstract

We introduce and study the notion of representation up to homotopy of a Lie algebroid, paying special attention to examples. We use representations up to homotopy to define the adjoint representation of a Lie algebroid and show that the resulting cohomology controls the deformations of the structure. The Weil algebra of a Lie algebroid is defined and shown to coincide with Kalkman's BRST model for equivariant cohomology in the case of group actions. The relation of this algebra with the integration of Poisson and Dirac structures is explained in [Arias Abad, Crainic, Ann. Inst. Fourier].

Generalized complex geometry

“…The symplectic (nonnegatively) graded symplectic manifold M of degree 2 corresponds to vector bundle E over M with a fiberwise nondegenerate symmetric inner product , (it can be of arbitrary signature). For a given E, M is a symplectic submanifold of T * [2]E [1] corresponding to the isometric embedding E ֒→ E ⊕ E * with respect to the canonical pairing on E ⊕ E * , i.e.…”

“…, where g ab is the constant fiber metric , written in a local basis of sections for E. Indeed M is a minimal symplectic realization of E [1]. In local Darboux coordinates (x µ , p µ , e a ) of degree 0,2 and 1 respectively the symplectic structure is ω = dp µ dx…”

“…where M is a symplectic submanifold of T * [2]E[1] which provides a minimal symplectic realization of E [1]. The odd symplectic structure on the space of maps is…”

“…where L is a submanifold of N * [2]K [1] corresponding to the isometric embedding E ֒→ E ⊕ E * (see previous discussion) and K is a Dirac structure supported on a submanifold C [6]. A Dirac structure supported on a submanifold i : C ֒→ M is defined as a subbundle…”

“…Following Roytenberg [30] we describe a Courant algebroid E in terms of a graded symplectic manifold M of degree 2 with Hamiltonian S of degree 3. The manifold M is the minimal symplectic realization of E [1] and in local Darboux coordinates (x µ , p µ , e a ) of degree 0,2 and 1 respectively the symplectic structure is ω = dp µ dx µ + 1 2…”