Poisson Quasi-Nijenhuis Structures with Background

Abstract

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.

Nijenhuis and compatible tensors on Lie and Courant algebroids

“…In section 4, we show how Poisson-Nijenhuis, ΩN and P Ω structures and also Hitchin pairs on a Lie algebroid (A, µ) can be seen either as Nijenhuis tensors or compatible pairs of tensors on the Courant algebroid (A ⊕ A * , µ). Considering, in section 5, the Courant algebroid with background (A ⊕ A * , µ + H), we see exact Poisson quasi-Nijenhuis structures with background as Nijenhuis tensors on this Courant algebroid, recovering a result in [1]. For Poisson quasi-Nijenhuis structures (without background) a special case where two 3-forms involved are exact is also considered.…”

“…Let us describe locally the Poisson bracket of the algebra F . Fix local coordinates [1], where x i , ξ a are local coordinates on A [1] and p i , θ a are their associated moment coordinates. In these local coordinates, the Poisson bracket is given by {p i , x i } = {θ a , ξ a } = 1, i = 1, .…”

“…For P Ω structures, we have the following: Poisson quasi-Nijenhuis structures with background on Lie algebroids were introduced in [1]. We recall that a Poisson quasi-Nijenhuis structure with background on (A, µ) is a quadruple (π, N, φ, H), where π is a bivector, N is a (1, 1)-tensor and φ and H are closed 3-forms such that N • π # = π # • N * and (i) π is Poisson,…”

“…In [1] it is proved 2 that if J N + J π + J ω is a Nijenhuis tensor on (A ⊕ A * , µ + H) and satisfies (J N + J π + J ω ) 2 = λ id A⊕A * , with λ ∈ {−1, 0, 1}, then the quadruple (π, N, d µ ω, H) is a Poisson quasi-Nijenhuis structure with background on (A, µ). It is easy to see that the same result holds for any λ ∈ R. It is worth noticing that (J N + J π + J ω ) 2 = λ id A⊕A * , λ ∈ R, is equivalent to the three conditions:…”

Nijenhuis forms on Lie-infinity algebras associated to Lie algebroids

“…It is well known that the Schouten-Nijenhuis bracket is a graded skew-symmetric bracket of degree zero on A ′ = ⊕ i≥−1 A ′ i that defines a graded Lie algebra bracket on A ′ = Γ(∧A) [1] and that the differential d A is a derivation of Γ(∧A * ) that squares to zero. It is also well known that A ′ = Γ(∧A) [1], [·, ·] SN , ∧ is a Gerstenhaber algebra.…”