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..
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.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:.
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.