ar X iv : 0 80 8 . 41 25 v 1 [ m at h . D G ] 2 9 A ug 2 00 8 Poisson quasi - Nijenhuis structures with background

We define the Poisson quasi-Nijenhuis structures with background on Lie algebroids and we prove that to any generalized complex structure on a Courant algebroid which is the double of a Lie algebroid is associated such a structure. We prove that any Lie algebroid with a Poisson quasi-Nijenhuis structure with background constitutes, with its dual, a quasi-Lie bialgebroid. We also prove that any pair (π,ω) of a Poisson bivector and a 2-form induces a Poisson quasi-Nijenhuis structure with background and we observe that particular cases correspond to already known compatibilities between π and ω . Introduction The aim of this work1 is to define the notion of Poisson quasi-Nijenhuis structure on a Lie algebroid with a (closed) 3-form background. The Poisson quasi-Nijenhuis structures (without background) were introduced by Stiénon and Xu in [14] on the tangent Lie algebroid and then on any Lie algebroid by Caseiro et al. in [1]. The case with background was already studied by Zucchini [19] but we remarked that a condition is missing in the definition proposed there. This extra condition was already in [14] and appears naturally in this work when we ask for some structures to be integrable (or some brackets to verify the Jacobi identity). In this work we will use a supermanifold approach [16, 12] to describe Lie algebroid structures. Let us consider a vector bundle A → M and change the parity of the fiber coordinates (considering them odd), then we obtain a supermanifold denoted by ΠA. The algebra of functions on ΠA, which are polynomial in the fibre coordinates, is denoted by C∞(ΠA) and coincides with Ω(A) := Γ( •A∗), the exterior algebra of A-forms. Let consider a Lie algebroid structure on A given by d, a degree 1 derivation of Ω(A) such that d2 = 0. In this supermanifold setting, d is a vector field on ΠA and can be seen as the derivation defined by a hamiltonian on ΠA, i.e. an element μ ∈C∞(T ∗ΠA). Then d = {μ, .}), where the so-called big bracket [5], {., .}, is the canonical Poisson bracket on the symplectic supermanifold T ∗ΠA. The condition d2 = 0 is equivalent to {μ,μ} = 0. To each f ∈C∞(T ∗ΠA) is associated a bidegree (ε,δ ). In fact, since using Legendre transform (see [10]) T ∗(ΠA) ∼= T ∗(ΠA∗), we can define ε (resp. δ ) as the polynomial degree of 1This paper was presented as a poster at the “Poisson 2008” conference at the EPFL in Lausanne in July 2008.