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. Then 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

• Algebraic category
• Algebroid (disambiguation)
• Bialgebra
• Bicategory
• Convolution product
• Crossed module
• Double groupoid
• Higher-dimensional algebra
• Hopf algebra
• Module (mathematics)
• Ring (mathematics)

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

• 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. Vol. 124. Cambridge University Press. ISBN 978-0-521-34882-9.
• Mackenzie, Kirill C.H. (2005). General Theory of Lie Groupoids and Lie Algebroids. London Mathematical Society Lecture Note Series. Vol. 213. Cambridge University Press. ISBN 978-0-521-49928-6.
• 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. arXiv:math/9602220. Bibcode:1996math......2220W. CiteSeerX 10.1.1.29.5422.