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R algebroid

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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 ').

Contents

Definition

An R-algebroid, R G , is constructed from a groupoid G as follows. The object set of R G is the same as that of G and R G ( 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. Than 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, R ¯ G := R G ( b , c ) , to be the set of functions G ( b , c ) ⟶ R with finite support, and with the convolution product defined as follows: ( f ∗ g ) ( z ) = ∑ { ( f x ) ( g y ) ∣ z = x ∘ 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 ≅ C .

Examples

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

    • References

    R-algebroid Wikipedia
    (Text) CC BY-SA