R-algebroid

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In mathematics, R-algebroids are constructed starting from groupoids. These are more abstract concepts than the Lie algebroids that play a similar role in the theory of Lie groupoids to that of Lie algebras in the theory of Lie groups. (Thus, a Lie algebroid can be thought of as 'a Lie algebra with many objects ').

Definition[edit]

An R-algebroid, , is constructed from a groupoid as follows. The object set of is the same as that of and is the free R-module on the set , with composition given by the usual bilinear rule, extending the composition of .[1]

R-category[edit]

A groupoid can be regarded as a category with invertible morphisms. Than an R-category is defined as an extension of the R-algebroid concept by replacing the groupoid in this construction with a general category C that does not have all morphisms invertible.

R-algebroids via convolution products[edit]

One can also define the R-algebroid, , to be the set of functions with finite support, and with the convolution product defined as follows: .[2]

Only this second construction is natural for the topological case, when one needs to replace 'function' by 'continuous function with compact support', and in this case .

Examples[edit]

See also[edit]

References[edit]

  1. ^ G. H. Mosa: Higher dimensional algebroids and Crossed complexes, PhD Thesis, University of Wales, Bangor, (1986). (supervised by Ronald Brown)
  2. ^ R. Brown and G. H. Mosa. Double algebroids and crossed modules of algebroids, University of Wales-Bangor, Maths Preprint, 1986.

This article incorporates material from Algebroid Structures and Algebroid Extended Symmetries on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

Sources
  • R. Brown and G. H. Mosa. Double algebroids and crossed modules of algebroids, University of Wales-Bangor, Maths Preprint, 1986.
  • G. H. Mosa: Higher dimensional algebroids and Crossed complexes, PhD Thesis, University of Wales, Bangor, (1986). (supervised by Ronald Brown).
  • Kirill Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, Cambridge U. Press, 1987.
  • Kirill Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, Cambridge U. Press, 2005
  • Charles-Michel Marle, Differential calculus on a Lie algebroid and Poisson manifolds (2002). Also available in arXiv:0804.2451
  • Alan Weinstein, Groupoids: unifying internal and external symmetry, AMS Notices, 43 (1996), 744-752. Also available as arXiv:math/9602220