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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. Then 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]

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
  • Brown, R.; Mosa, G. H. (1986). "Double algebroids and crossed modules of algebroids". Maths Preprint. University of Wales-Bangor.
  • Mosa, G.H. (1986). Higher dimensional algebroids and Crossed complexes (PhD). University of Wales. uk.bl.ethos.815719.
  • Mackenzie, Kirill C.H. (1987). Lie Groupoids and Lie Algebroids in Differential Geometry. London Mathematical Society Lecture Note Series. Vol. 124. Cambridge University Press. ISBN 978-0-521-34882-9.
  • Mackenzie, Kirill C.H. (2005). General Theory of Lie Groupoids and Lie Algebroids. London Mathematical Society Lecture Note Series. Vol. 213. Cambridge University Press. ISBN 978-0-521-49928-6.
  • Marle, Charles-Michel (2002). "Differential calculus on a Lie algebroid and Poisson manifolds". arXiv:0804.2451 [math.DG].
  • Weinstein, Alan (1996). "Groupoids: unifying internal and external symmetry". AMS Notices. 43: 744–752. arXiv:math/9602220. Bibcode:1996math......2220W. CiteSeerX 10.1.1.29.5422.