On the sheaf of smooth forms on Lie algebroids over triangulated spaces

TL;DR

This paper develops a sheaf of differential forms on Lie algebroids over triangulated spaces, proving its fineness and establishing a cohomology framework for compatible families of Lie algebroids.

Contribution

It introduces a sheaf of smooth forms on Lie algebroids over triangulated spaces and proves its fineness, advancing the understanding of their cohomology.

Findings

Sheaf of differential forms on Lie algebroids is constructed.

The sheaf is proven to be fine.

Cohomology of compatible Lie algebroid families is defined.

Abstract

Cohomology of a compatible family of Lie algebroids defined on a family of transverse manifolds is defined. A sheaf of differential forms on a compatible family of Lie algebroids defined over regular open subsets of a simplicial complex is constructed. It is proved that sheaf is fine.