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

R-category

A groupoid ${\mathsf {G}}$ can be regarded as a category with invertible morphisms. Than an R-category is defined as an extension of the R-algebroid concept by replacing the groupoid ${\mathsf {G}}$ 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, ${\bar {R}}{\mathsf {G}}:=R{\mathsf {G}}(b,c)$, to be the set of functions ${\mathsf {G}}(b,c){\longrightarrow }R$ with finite support, and with the convolution product defined as follows: $\displaystyle (f*g)(z)=\sum \{(fx)(gy)\mid z=x\circ y\}$ .

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 $R\cong \mathbb {C}$.

Examples

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

See also

2

References

This article incorporates material from Algebroid Structures and Algebroid Extended Symmetries on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

Sources

• R. Brown and G. H. Mosa. Double algebroids and crossed modules of algebroids, University of Wales-Bangor, Maths Preprint, 1986.
• G. H. Mosa: Higher dimensional algebroids and Crossed complexes, PhD Thesis, University of Wales, Bangor, (1986). (supervised by Ronald Brown).
• Kirill Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, Cambridge U. Press, 1987.
• Kirill Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, Cambridge U. Press, 2005
• Charles-Michel Marle, Differential calculus on a Lie algebroid and Poisson manifolds (2002). Also available in arXiv:0804.2451
• Alan Weinstein, Groupoids: unifying internal and external symmetry, AMS Notices, 43 (1996), 744-752. Also available as arXiv:math/9602220