R-algebroid

Inmathematics,R-algebroidsare constructed starting fromgroupoids.These are more abstract concepts than theLie algebroidsthat play a similar role in the theory ofLie groupoidsto that ofLie algebrasin the theory ofLie groups.(Thus, a Lie algebroid can be thought of as 'aLie algebrawithmany objects').

Definition

AnR-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 thefreeR-moduleon 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 acategorywith invertible morphisms. Then anR-categoryis defined as an extension of theR-algebroid concept by replacing the groupoid${\mathsf {G}}$in this construction with a general categoryCthat does not have all morphisms invertible.

R-algebroidsviaconvolution products

One can also define theR-algebroid,${\bar {R}}{\mathsf {G}}:=R{\mathsf {G}}(b,c)$,to be theset of functions${\mathsf {G}}(b,c){\longrightarrow }R$withfinite support,and with theconvolutionproductdefined 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 functionwithcompact support', and in this case$R\cong \mathbb {C}$.

Examples

• EveryLie algebrais a Lie algebroid over the one pointmanifold.
• The Lie algebroid associated to aLie 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 fromAlgebroid Structures and Algebroid Extended SymmetriesonPlanetMath,which is licensed under theCreative Commons Attribution/Share-Alike License.

Sources

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• Mackenzie, Kirill C.H. (2005).General Theory of Lie Groupoids and Lie Algebroids.London Mathematical Society Lecture Note Series. Vol. 213. Cambridge University Press.ISBN978-0-521-49928-6.
• Marle, Charles-Michel (2002). "Differential calculus on a Lie algebroid and Poisson manifolds".arXiv:0804.2451[math.DG].
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