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 $T P/ G$, 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.

Definition

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

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

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 $P$.

Relation to connections

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:

Atiyah class

The $Ext^1$-cohomology class corresponding to the Atiyah exact sequence (usually in a version for vector bundles/coherent sheaves) is the Atiyah class.

Related concepts

• Courant Lie 2-algebroid

References

• Michael Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181–207, doi,MR0086359
• Pietro Tortella, Representations of Atiyah algebroids and logarithmic connections, arxiv/1505.04763

A discussion with an emphasis on the relation to 2-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 [pdf, pdf]

For Atiyah classes see:

• Luc Illusie, Complexe cotangent et déformations (vol. 1) IV.2.3

• MO:atiyah-class-for-non-locally-free-sheaf

• M. Kapranov, Rozansky–Witten invariants via Atiyah classes, Compositio Math. 115 (1999), 71–113.

• Ugo Bruzzo, Igor Mencattini, Vladimir Rubtsov,

  Nonabelian holomorphic Lie algebroid extensions, Intern. J. Math. 26, No. 05, 1550040 (2015) doi arXiv:1305.2377

• Zhuo Chen, Mathieu Stiénon, Ping Xu, From Atiyah classes to homotopy Leibniz algebras, arXiv/1204.1075; A Hopf algebra associated to a Lie pair, arxiv/1409.6803

• R. A. Mehta, M. Stiénon, P. Xu, The Atiyah class of a dg-vector bundle, arxiv/1502.03119

• Nikita Markarian, The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem, J. Lond. Math. Soc. (2) 79 (2009), no. 1, 129–143 doi

• F. Bottacin, Atiyah classes for Lie algebroids, pdf

• Ajay C. Ramadoss, The big Chern classes and the Chern character, Internat. J. Math. 19 (2008), no. 6, 699–746.

• Zhuo Chen, Mathieu Stiénon, Ping Xu, From Atiyah classes to homotopy Leibniz algebras, Commun. Math. Phys. 341 (2016) 309-349 [arXiv:1204.1075, doi:10.1007/s00220-015-2494-6]

• Stack Project 92.19 (tag/09DF) The Atiyah class of a sheaf of modules