Local description of generalized forms on transitive Lie algebroids and applications

TL;DR

This paper develops a local description framework for forms on transitive Lie algebroids, enabling the construction of global geometric structures and spectral sequences, with applications to Atiyah algebroids and derivations.

Contribution

It introduces a novel local description of forms on transitive Lie algebroids and constructs associated global structures and spectral sequences.

Findings

Established a ch-de Rham bicomplex for transitive Lie algebroids

Defined metrics, st-Hodge, and integration on these structures

Applied theory to Atiyah Lie algebroids and derivations

Abstract

In this paper we study the local description of spaces of forms on transitive Lie algebroids. We use this local description to introduce global structures like metrics, $\ast$-Hodge operation and integration along the algebraic part of the transitive Lie algebroid (its kernel). We construct a \v{C}ech-de Rham bicomplex with a Leray-Serre spectral sequence. We apply the general theory to Atiyah Lie algebroids and to derivations on a vector bundle.