nLab
Atiyah Lie algebroid

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

The Atiyah Lie algebroid associated to a GG-principal bundle PP over XX is a Lie algebroid structure on the vector bundle TP/GT P/ G, the quotient of the tangent bundle of the total space PP by the canonical induced GG-action.

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

Definition

Let GG be a Lie group with Lie algebra 𝔤\mathfrak{g} and let PXP \to X be a GG-principal bundle:

the Atiyah Lie algebroid sequence of PP is a sequence of Lie algebroids

ad(P)at(P)TX, ad(P) \to at(P) \to T X \,,

where

  • ad(P)=P× G𝔤ad(P) = P \times_G \mathfrak{g} is the adjoint bundle of Lie algebras, associated via the adjoint action of GG on its Lie algebra;

  • at(P):=(TP)/Gat(P) := (T P)/G is the Atiyah Lie algebroid

  • TXT X is the tangent Lie algebroid of XX.

The Lie bracket on the sections of at(P)at(P) is that inherited from the tangent Lie algebroid of PP.

Relation to connections

A splitting flat:TXat(P)\nabla_{flat} : T X \to at(P) of the Atiyah Lie algebroid sequence in the category of Lie algebroids is precisely a flat connection on PP.

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 L_\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

Revised on February 8, 2013 12:19:00 by Urs Schreiber (89.204.138.214)