nLab
Atiyah Lie algebroid

Idea

The Atiyah Lie algebroid associated to a $G$-principal bundle $P$ over $X$ is a Lie algebroid structure on the vector bundle $TP/G$.

The Lie groupoid that the Atiyah Lie algebroid integrates to is the Atiyah Lie groupoid. See there for more background and discussion.

Definition

Let $G$ be a Lie group with Lie algebra $𝔤$ and let $P\to X$ be a $G$-principal bundle:

the Atiyah Lie algebroid sequence of $P$ is a sequence of Lie algebroids

$$\mathrm{ad}(P)\to \mathrm{at}(P)\to TX,$$ad(P) \to at(P) \to T X \,,

where

• $\mathrm{ad}(P)=P{\times }_G𝔤$ is the adjoint bundle? of Lie algebras, associated via the adjoint action? of $G$ on its Lie algebra;

• $\mathrm{at}(P):=(TP)/G$ is the Atiyah Lie algebroid

• $TX$ is the tangent Lie algebroid of $X$.

The bracket on the sections of $\mathrm{at}(P)$ is that inherited from the tangent Lie algebroid of $P$.

Relation to connections

A splitting ${\nabla }_{\mathrm{flat}}:TX\to \mathrm{at}(P)$ of the Atiyah Lie algebroid sequence in the category of Lie algebroids is precisely a flat connection on $P$.

To get non-flat connections in the literature one often sees discussed splittings of the Atiyah Lie algebroid sequence in the category just of vector bundles. In that case one finds the curvature of the connection precisely as the obstruction to having a splitting even in Lie algebroids.

One can describe non-flat connections without leaving the context of Lie algebroids by passing to higher Lie algebroids, namely $L_{\mathrm{\infty }}$-algebroids, in terms of an horizontal categorification of nonabelian Lie algebra cohomology:

…

References

A discussion with an emphasis on the relation to connections and Lie 2-algebras is on the first pages of

• Danny Stevenson, Lie 2-algebras and the geometry of gerbes, Unni Namboodiri Lectures 2006 slides