R-algebroid explained

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

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

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,

{\barR}G:=RG(b,c)

, to be the set of functions

G(b,c){\longrightarrow}R

with finite support, and with the convolution product defined as follows:

\displaystyle(f*g)(z)=\sum\{(fx)(gy)\midz=x\circy\}

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\congC

Examples

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

References

Sources

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