Nijenhuis forms on L∞-algebras and Poisson geometry

Abstract

We investigate Nijenhuis deformations of L∞-algebras, a notion that unifies several Nijenhuis deformations, namely those of Lie algebras, Lie algebroids, Poisson structures and Courant structures. Additional examples, linked to Lie n-algebras and n-plectic manifolds, are included.

Nijenhuis forms on Lie-infinity algebras associated to Lie algebroids

“…The present article shows that two different kind of deformations can be explained by using Nijenhuis forms on L ∞ -algebras defined in [4]. The first one, detailed in Section 2, is inspired by Delgado [7], who gave an explicit construction showing that, for all Lie algebroid A, the graded space of sections of ∧A carries not only the Gerstenhaber bracket, but also an intriguing L ∞ -structure whose 2-ary bracket is the Gerstenhaber bracket, the 3-ary bracket is given by:…”

“…Recall from [4] the following proposition: Proposition 2.2. Let N be a Nijenhuis form in any of the senses above for a L ∞ -structure µ, then [N , µ] RN is an L ∞ -structure compatible with µ.…”

“…The present article is a continuation of [4] by the same authors, where Nijenhuis forms on Lie-infinity algebras [13] (from now on referred to as L ∞ -algebras) were introduced and various examples were given. Its purpose is to show that Nijenhuis forms on L ∞ -algebras are an efficient unification tool, which appears in contexts which are not obviously related neither between themselves, nor obviously related to Nijenhuis structures or higher structures.…”

“…The difficulty is that the square of a vector valued form can not be defined anymore. 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 reader used to higher structures maybe will not be surprised, but considering the wedge product as an analogous of a Nijenhuis tensor on a manifold may seem quite strange at first. By (a slightly modified version of) the general theory of [4], a pencil (indexed by N) of L ∞ -algebras can be derived: the L ∞ -algebra of Delgado [7] is among this pencil. In presence of a Nijenhuis tensor N , in the usual sense, of the Lie algebroid, the L ∞ -structures on this pencil can themselves be deformed by N .…”