# Connection is module structure over connection algebra

## Contents

## Statement

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 .

## Definitions used

### Connection

`Further information: Connection`

### Connection algebra

`Further information: Connection algebra`

## Proof

### 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:

as follows:

.

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 .

## References

### Textbook references

- Book:Globalcalculus
^{More info}, Page 64