Atiyah algebroid

"Atiyah sequence" redirects here. For the spectral sequence of Atiyah and Hirzebruch, see Atiyah–Hirzebruch spectral sequence.

In mathematics, the Atiyah algebroid, or Atiyah sequence, of a principal G-bundle P over a manifold M is the Lie algebroid of the gauge groupoid of P. Explicitly, it is given by the following short exact sequence of vector bundles over M:

$0\to P\times _{G}{\mathfrak {g}}\to TP/G\to TM\to 0.$

It is named after Michael Atiyah, who introduced the construction to study the existence theory of complex analytic connections, and it has applications in gauge theory and mechanics.

References

• Michael F. Atiyah (1957), "Complex analytic connections in fibre bundles", Trans. Amer. Math. Soc., 85: 181–207.
• "Geometric structures encoded in the lie structure of an Atiyah algebroid", Transformation Groups, 16: 137–160, 2011, doi:10.1007/s00031-011-9126-9 {{citation}}: Unknown parameter |authors= ignored (help), available as arXiv:0905.1226.
• Kirill Mackenzie (1987), Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society lecture notes, vol. 124, CUP, ISBN 978-0521348829.
• Kirill Mackenzie (2005), General theory of lie groupoids and lie algebroids, London Mathematical Society lecture notes, vol. 213, CUP, ISBN 978-0521499286.
• "A Lie algebroid framework for non-holonomic systems", J. Phys. A: Math. Gen., 38: 1097–1111, 2005, doi:10.1088/0305-4470/38/5/011 {{citation}}: Unknown parameter |authors= ignored (help).