Connection is module structure over connection algebra
Let be a vector bundle over a differential manifold . Then, a connection on is equivalent to giving (the vector space of sections of ) the structure of a module over the connection algebra of . Equivalently, it gives (the sheaf of sections of ) the structure of a module over the sheaf of connection algebras over .
Further information: Connection
Further information: Connection algebra
From a connection to a module structure
The outline of the proof is as follows:
- We first show that a connection gives an action of the first-order differentiable operators on the space of sections.
- Next, we show that the Leibniz rule property of connections allows us to extend this to a well-defined action of the connection algebra.
Given: A manifold , a vector bundle over , a connection on . is the algebra of smooth fiber-preserving maps from to . is the Lie algebra of first-order differential operators on and is the connection algebra on .
To prove: gives rise to a homomorphism from to .
Proof: gives rise to a map:
First observe that the map sends to , and is the identity restricted to that subset. In other words, the differential operator of multiplication by a function , goes to the operator of multiplication by the function .
We now prove that the map is a -bimodule map from to , i.e., left and right multiplication by can be pulled out of the :
- is -bilinear: This is obvious.
- Left module map property: For any element in and any , we have . This essentially follows from the fact that a connection is tensorial in the direction of differentiation:
- For any element in and any , we have . This essentially follows from the Leibniz rule property.
Since is a -bimodule map, it extends to a unique -bimodule map from the -tensor algebra over . By definition, the element induces the zero map on , so the map descends to a homomorphism , as desired.
From a module structure to a connection
Given: A manifold , a vector bundle over . is the algebra of smooth fiber-preserving maps from to . is the Lie algebra of first-order differential operators on and is the connection algebra on . A module structure of over .