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, ${\displaystyle R{\mathsf {G}}}$, is constructed from a groupoid ${\displaystyle {\mathsf {G}}}$ as follows. The object set of ${\displaystyle R{\mathsf {G}}}$ is the same as that of ${\displaystyle {\mathsf {G}}}$ and ${\displaystyle R{\mathsf {G}}(b,c)}$ is the free R-module on the set ${\displaystyle {\mathsf {G}}(b,c)}$, with composition given by the usual bilinear rule, extending the composition of ${\displaystyle {\mathsf {G}}}$.^[1]

R-category[]

A groupoid ${\displaystyle {\mathsf {G}}}$ 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 ${\displaystyle {\mathsf {G}}}$ 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, ${\displaystyle {\bar {R}}{\mathsf {G}}:=R{\mathsf {G}}(b,c)}$, to be the set of functions ${\displaystyle {\mathsf {G}}(b,c){\longrightarrow }R}$ with finite support, and with the convolution product defined as follows: ${\displaystyle \displaystyle (f*g)(z)=\sum \{(fx)(gy)\mid z=x\circ y\}}$ .^[2]

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 ${\displaystyle R\cong \mathbb {C} }$.

Examples[]

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

See also[]

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