R-algebroid
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
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
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
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
- Every Lie algebra is a Lie algebroid over the one point manifold.
- The Lie algebroid associated to a Lie groupoid.
See also
References
- 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. 124. Cambridge University Press. ISBN 978-0-521-34882-9. https://books.google.com/books?id=e4fyHTZr290C.
- Mackenzie, Kirill C.H. (2005). General Theory of Lie Groupoids and Lie Algebroids. London Mathematical Society Lecture Note Series. 213. Cambridge University Press. ISBN 978-0-521-49928-6. https://books.google.com/books?id=Sb50M26LGc4C.
- 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. Bibcode: 1996math......2220W.
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