Hierarchies and compatibility on Courant algebroids

Abstract

International audienc

Nijenhuis forms on Lie-infinity algebras associated to Lie algebroids

“…It follows from its definition that κ is a derivation on the graded vector space Γ(∧A) [2] and that it is a symmetric vector valued k-form of degree k−2 on Γ(∧A) [2]. The next two lemmas, proved in [4], give some results about the Richardson-Nijenhuis bracket of these κ's.…”

“…for all monomial multi-sections P = P 1 ∧ · · · ∧ P p ∈ Γ(∧ p A) [2]. The map N is called the extension of N by derivation on the graded vector space Γ(∧A) [2].…”

“…for all monomial multi-sections P = P 1 ∧ · · · ∧ P p ∈ Γ(∧ p A) [2]. The map N is called the extension of N by derivation on the graded vector space Γ(∧A) [2]. It is a derivation on the graded vector space Γ(∧A) [2], hence a symmetric vector valued 1-form on Γ(∧A) [2], and it has degree zero.…”

“…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 .…”

“…However, the equivalent of these squares are often quite natural, and [4] has suggested a solution which consists in defining altogether Nijenhuis forms N and their "square" K, with K a vector valued form of degree 0. By a square, we simply mean an other vector valued form such that deforming the L ∞ -structure twice by N is like deforming it by K. When this square K commutes with N , a hierarchy of compatible L ∞ -algebras arises naturally by deforming by N several times [12,2]. The purpose of this generalization, of course, is to obtain even more "unifications" in the process.…”