∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
The Atiyah Lie algebroid associated to a -principal bundle over is a Lie algebroid structure on the vector bundle , the quotient of the tangent bundle of the total space by the canonical induced -action.
The Lie groupoid that the Atiyah Lie algebroid integrates to is the Atiyah Lie groupoid. See there for more background and discussion.
Let be a Lie group with Lie algebra and let be a -principal bundle:
the Atiyah Lie algebroid sequence of is a sequence of Lie algebroids
where
is the adjoint bundle of Lie algebras, associated via the adjoint action of on its Lie algebra;
is the Atiyah Lie algebroid
is the tangent Lie algebroid of .
The Lie bracket on the sections of is that inherited from the tangent Lie algebroid of .
A splitting of the Atiyah Lie algebroid sequence in the category of Lie algebroids is precisely a flat connection on .
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 -algebroids, in terms of an horizontal categorification of nonabelian Lie algebra cohomology:
The -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 2-connections and Lie 2-algebras is on the first pages of
For Atiyah classes see:
Luc Illusie, Complexe cotangent et déformations (vol. 1) IV.2.3
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
Last revised on October 24, 2023 at 00:17:13. See the history of this page for a list of all contributions to it.