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, [math]\displaystyle{ R\mathsf{G} }[/math], is constructed from a groupoid [math]\displaystyle{ \mathsf{G} }[/math] as follows. The object set of [math]\displaystyle{ R\mathsf{G} }[/math] is the same as that of [math]\displaystyle{ \mathsf{G} }[/math] and [math]\displaystyle{ R\mathsf{G}(b,c) }[/math] is the free R-module on the set [math]\displaystyle{ \mathsf{G}(b,c) }[/math], with composition given by the usual bilinear rule, extending the composition of [math]\displaystyle{ \mathsf{G} }[/math].

R-category

A groupoid [math]\displaystyle{ \mathsf{G} }[/math] 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 [math]\displaystyle{ \mathsf{G} }[/math] 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, [math]\displaystyle{ {\bar R}\mathsf{G}:=R\mathsf{G}(b,c) }[/math], to be the set of functions [math]\displaystyle{ \mathsf{G}(b,c){\longrightarrow}R }[/math] with finite support, and with the convolution product defined as follows: [math]\displaystyle{ \displaystyle (f*g)(z)= \sum \{(fx)(gy)\mid z=x\circ y \} }[/math] .

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 [math]\displaystyle{ R\cong \mathbb{C} }[/math].

Examples

• Every Lie algebra is a Lie algebroid over the one point manifold.
• The Lie algebroid associated to a Lie groupoid.

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.