Geometry of quantum systems (e.g.,
noncommutative geometry and supergeometry) is mainly
phrased in algebraic terms of modules and
algebras. Connections on modules are
generalization of a linear connection on a smooth vector bundle
E
→
X
{\displaystyle E\to X}
written as a Koszul connection on the
C
∞
(
X
)
{\displaystyle C^{\infty }(X)}
-module of sections of
E
→
X
{\displaystyle E\to X}
.
Contents
Commutative algebra
Let
A
{\displaystyle A}
be a commutative ring
and
M
{\displaystyle M}
an A-module. There are different equivalent definitions
of a connection on
M
{\displaystyle M}
.
First definition
If
k
→
A
{\displaystyle k\to A}
is a ring homomorphism, a
k
{\displaystyle k}
-linear connection is a
k
{\displaystyle k}
-linear morphism
∇
:
M
→
Ω
A
/
k
1
⊗
A
M
{\displaystyle \nabla :M\to \Omega _{A/k}^{1}\otimes _{A}M}
which satisfies the identity
∇
(
a
m
)
=
d
a
Second definition
Let
D
(
A
)
{\displaystyle D(A)}
be the module of derivations of a ring
A
{\displaystyle A}
. A
connection on an A-module
M
{\displaystyle M}
is defined
as an A-module morphism
∇
:
D
(
A
)
→
D
i
f
f
1
(
M
,
M
)
;
u
↦
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
A
{\displaystyle A}
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
P
{\displaystyle P}
is defined as a bimodule
morphism
∇
:
D
(
A
)
∋
u
→