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

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

R-category

A groupoid G 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 G 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, R¯G:=RG(b,c), to be the set of functions G(b,c)R with finite support, and with the convolution product defined as follows: (f*g)(z)={(fx)(gy)z=xy} .[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 R.

Examples

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. Bibcode1996math......2220W.