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:GlobalcalculusMore info, Page 64