# Obstructions to representations up to homotopy and ideals

## TL;DR

This paper explores obstructions to representations up to homotopy in Lie algebroids, showing Pontryagin characters vanish under such representations and deriving new criteria for their existence.

## Contribution

It introduces Pontryagin character obstructions to representations up to homotopy and generalizes Bott's vanishing theorem to Lie algebroid contexts.

## Key findings

- Pontryagin characters vanish for bundles with representations up to homotopy
- Coincidence of classical Pontryagin classes in 2-term representations
- New obstructions to infinitesimal ideal systems in Lie algebroids

## Abstract

This paper considers the Pontryagin characters of graded vector bundles of finite rank, in the cohomology vector spaces of a Lie algebroid over the same base. These Pontryagin characters vanish if the graded vector bundle carries a representation up to homotopy of the Lie algebroid. As a consequence, this gives a strong obstruction to the existence of a representation up to homotopy on a graded vector bundle of finite rank. In particular, if a graded vector bundle $E[0]\oplus F[1]\to M$ carries a $2$-term representation up to homotopy of a Lie algebroid $A\to M$, then all the (classical) $A$-Pontryagin classes of $E$ and $F$ must coincide. This paper generalises as well Bott's vanishing theorem to the setting of Lie algebroid representations (up to homotopy) on arbitrary vector bundles. As an application, the main theorems induce new obstructions to the existence of infinitesimal ideal systems in a given Lie algebroid.