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, $R𝖦$, is constructed from a groupoid $𝖦$ as follows. The object set of $R𝖦$ is the same as that of $𝖦$ 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 $𝖦$.

R-category

A groupoid $𝖦$ 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 $𝖦$ 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, $\overline{R}𝖦:=R𝖦(b,c)$, to be the set of functions $𝖦(b,c)⟶R$ with finite support, and with the convolution product defined as follows: $(f\ast g)(z)=\sum \{(fx)(gy)\mid z=x\circ y\}$ .

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 $R\cong ℂ$.

Examples

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

References

Sources

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