Nijenhuis and compatible tensors on Lie and Courant algebroids

Abstract

Abstract. We show that well known structures on Lie algebroids can be viewed as Nijenhuis tensors or pairs of compatible tensors on Courant algebroids. We study compatibility and construct hierarchies of these structures.

Nijenhuis forms on Lie-infinity algebras associated to Lie algebroids

“…For instance, a Poisson structure on a Lie algebroid A, being a tensor from A * to A, becomes itself a tensor on the Courant algebroid A ⊕ A * which can be shown to be Nijenhuis. Hence, both Poisson structures and Nijenhuis tensors become, eventually, Nijenhuis tensors, and so do closed 2-forms, ΩN and Poisson-Nijenhuis structures [3].…”

“…These so-called Poisson-Nijenhuis structures can be defined on an arbitrary Lie algebroid [12,10], give hierachies of Poisson structures, and may even describe entirely the initial integrable system [6]. Several authors [8,11,3,2] have also studied Nijenhuis tensors on Loday algebras and Courant algebroids. These extensions share a common idea: a tensor N is defined to be Nijenhuis for the structure X (with X = Poisson, Lie algebroid, Courant algebroid and so on) when deforming X twice by N is like deforming X by N 2 .…”

“…Interesting "unifications" happen while generalizing the notion of Nijenhuis to Courant algebroids [3]. For instance, a Poisson structure on a Lie algebroid A, being a tensor from A * to A, becomes itself a tensor on the Courant algebroid A ⊕ A * which can be shown to be Nijenhuis.…”

“…[3] An exact Poisson quasi-Nijenhuis structure with background on a Lie algebroid (A, [·, ·] , ρ) is a quadruple (π,…”

Nijenhuis forms on L∞-algebras and Poisson geometry

“…We discuss how Nijenhuis tensors on Courant algebroids [5,13,2,3] fit in our defintion of Nijenhuis forms on some L ∞ -algebras. In order to include Lie algebroids in our examples, we recall the concept of multiplicative L ∞ -algebras (related to P ∞algebras in [6]).…”