Connection (algebraic framework)

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 formula_1 written as a Koszul connection on the
formula_2-module of sections of formula_1.
Commutative algebra.
Let formula_4 be a commutative ring
and formula_5 an "A"-module. There are different equivalent definitions
of a connection on formula_5. Let formula_7 be the module of derivations of a ring formula_4. A
connection on an "A"-module formula_5 is defined
as an "A"-module morphism
such that the first order differential operators formula_11 on
formula_5 obey the Leibniz rule
Connections on a module over a commutative ring always exist.
The curvature of the connection formula_14 is defined as
the zero-order differential operator
on the module formula_5 for all formula_17.
If formula_18 is a vector bundle, there is one-to-one
correspondence between linear
connections formula_19 on formula_18 and the
connections formula_14 on the
formula_2-module of sections of formula_1. Strictly speaking, formula_14 corresponds to
the covariant differential of a
connection on formula_18.
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 formula_4 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 formula_5 is defined as a bimodule
morphism
which obeys the Leibniz rule