Nijenhuis forms on Lie-infinity algebras associated to Lie algebroids

ABSTRACT

Introducing Nijenhuis forms on L_∞-algebras gives a general frame to understand deformations of the latter. We give here a Nijenhuis interpretation of a deformation of an arbitrary Lie algebroid into an L_∞-algebra. Then we show that Nijenhuis forms on L_∞-algebras also give a short and efficient manner to understand Poisson-Nijenhuis structures and, more generally, the so-called exact Poisson quasi-Nijenhuis structures with background.

KEYWORDS:

• L_∞-algebra
• Nijenhuis
• Gerstenhaber algebra
• Lie algebroid

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16E35
• 17B63
• 53D17

Acknowledgments

The authors acknowledge C. Blohmann for sending us the manuscript of Delgado [8] which was the starting point of this study. They also want to thank the anonymous referee for suggestions which led to numerous improvements of the manuscript. The authors are grateful to N. L. Delgado and P. Antunes for their collaboration.

Notes

¹Note that not all the permutations in Sh(2,m+n−3) appear in (12) for a single σ. But, since for each unshuffle τ∈Sh(2,m+n−3) and 1≤j≤m, m+1≤i≤m+n−1, there exists an unshuffle σ∈Sh(m,n−1) such that τ(1) = σ(i) and τ(2) = σ(j), all elements in Sh(2,m+n−3) can be obtained by Equations (6), (7), (8), (9) and (10).