∞-Lie theory (higher geometry)
The Atiyah Lie algebroid associated to a $G$-principal bundle $P$ over $X$ is a Lie algebroid structure on the vector bundle $T P/ G$, the quotient of the tangent bundle of the total space $P$ by the canonical induced $G$-action.
The Lie groupoid that the Atiyah Lie algebroid integrates to is the Atiyah Lie groupoid. See there for more background and discussion.
Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and let $P \to X$ be a $G$-principal bundle:
the Atiyah Lie algebroid sequence of $P$ is a sequence of Lie algebroids
where
$ad(P) = P \times_G \mathfrak{g}$ is the adjoint bundle of Lie algebras, associated via the adjoint action of $G$ on its Lie algebra;
$at(P) := (T P)/G$ is the Atiyah Lie algebroid
$T X$ is the tangent Lie algebroid of $X$.
The Lie bracket on the sections of $at(P)$ is that inherited from the tangent Lie algebroid of $P$.
A splitting $\nabla_{flat} : T X \to at(P)$ of the Atiyah Lie algebroid sequence in the category of Lie algebroids is precisely a flat connection on $P$.
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 $L_\infty$-algebroids, in terms of an horizontal categorification of nonabelian Lie algebra cohomology:
The $Ext^1$-cohomology class corresponding to the Atiyah exact sequence (usually in a version for vector bundles/coherent sheaves) is the Atiyah class.
A discussion with an emphasis on the relation to connections and Lie 2-algebras is on the first pages of
For Atiyah classes see
115 (1999), 71–113.
Last revised on August 9, 2015 at 08:50:46. See the history of this page for a list of all contributions to it.