Lie algebroids generated by cohomology operators

TL;DR

This paper explores how cohomology operators and their Fr"olicher-Nijenhuis decomposition can generate new Lie algebroid structures on tangent bundles, linking existing geometric structures with algebraic frameworks.

Contribution

It introduces novel Lie algebroid examples derived from cohomology operators and their decompositions, expanding the understanding of geometric and algebraic structures on tangent bundles.

Findings

New Lie algebroid structures on tangent bundles from cohomology operators

Construction of Lie algebroids associated with idempotent endomorphisms

Connection between geometric structures and algebraic Lie algebroids

Abstract

By studying the Fr\"olicher-Nijenhuis decomposition of cohomology operators (that is, derivations $D$ of the exterior algebra $\Omega (M)$ with $\mathbb{Z}-$degree $1$ and $D^2=0$), we describe new examples of Lie algebroid structures on the tangent bundle $TM$ (and its complexification $T^{\mathbb{C}}M$) constructed from pre-existing geometric ones such as foliations, complex, product or tangent structures. We also describe a class of Lie algebroids on tangent bundles associated to idempotent endomorphisms with nontrivial Nijenhuis torsion.