The Serre spectral sequence of a Lie subalgebroid

TL;DR

This paper introduces a spectral sequence for Lie subalgebroids that generalizes classical spectral sequences in differential geometry, converging to Lie algebroid cohomology and applicable in Poisson geometry.

Contribution

It develops a unified spectral sequence framework for Lie subalgebroids, extending classical constructions and analyzing convergence properties in different geometric contexts.

Findings

Spectral sequence converges to Lie algebroid cohomology for wide subalgebroids.

Converges to formal cohomology over proper submanifolds.

Recovers several known constructions in Poisson geometry.

Abstract

We study a spectral sequence approximating Lie algebroid cohomology associated to a Lie subalgebroid. This is a simultaneous generalisation of several classical constructions in differential geometry, including the Leray-Serre spectral sequence for de Rham cohomology associated to a fibration, the Hochschild-Serre spectral sequence for Lie algebras, and the Mackenzie spectral sequence for Lie algebroid extensions. We show that, for wide Lie subalgebroids, the spectral sequence converges to the Lie algebroid cohomology, and that, for Lie subalgebroids over proper submanifolds, the spectral sequence converges to the formal Lie algebroid cohomology. We discuss applications and recover several constructions in Poisson geometry in which this spectral sequence has appeared naturally in the literature.