Connection (algebraic framework)


Geometry of quantum systems is mainly
phrased in algebraic terms of modules and
algebras. Connections on modules are
generalization of a linear connection on a smooth vector bundle written as a Koszul connection on the
-module of sections of.

Commutative algebra

Let be a commutative ring
and an A-module. There are different equivalent definitions
of a connection on. Let be the module of derivations of a ring. A
connection on an A-module is defined
as an A-module morphism
such that the first order differential operators on
obey the Leibniz rule
Connections on a module over a commutative ring always exist.
The curvature of the connection is defined as
the zero-order differential operator
on the module for all.
If is a vector bundle, there is one-to-one
correspondence between linear
connections on and the
connections on the
-module of sections of. Strictly speaking, corresponds to
the covariant differential of a
connection on.

Graded commutative algebra

The notion of a connection on modules over commutative rings is
straightforwardly extended to modules over a graded
commutative algebra. This is the case of
superconnections in supergeometry of
graded manifolds and supervector bundles.
Superconnections always exist.

Noncommutative algebra

If is a noncommutative ring, connections on left
and right A-modules are defined similarly to those on
modules over commutative rings. However
these connections need not exist.
In contrast with connections on left and right modules, there is a
problem how to define a connection on an
R-S-bimodule over noncommutative rings
R and S. There are different definitions
of such a connection. Let us mention one of them. A connection on an
R-S-bimodule is defined as a bimodule
morphism
which obeys the Leibniz rule