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Geometry of [[Quantum mechanics|quantum systems]] (e.g.,
[[noncommutative geometry]] and [[supergeometry]]) is mainly
phrased in algebraic terms of [[module (mathematics)|modules]] and
[[algebras]]. '''Connections''' on modules are
generalization of a linear [[connection (vector bundle)|connection]] on a smooth [[vector bundle]] <math>E\to
X</math> written as a [[Koszul connection]] on the
<math>C^\infty(X)</math>-module of sections of <math>E\to
X</math>.<ref>Koszul (1950)</ref>

== Commutative algebra ==

Let <math>A</math> be a commutative [[ring (mathematics)|ring]]
and <math>P</math> a <math>A</math>-[[module (mathematics)|module]]. There are different equivalent definitions
of a connection on <math>P</math>.<ref>Koszul (1950), Mangiarotti
(2000)</ref> Let <math>D(A)</math> be the module of [[derivation (abstract algebra)|derivations]] of a ring <math>A</math>. A
connection on an <math>A</math>-module <math>P</math> is defined
as an <math>A</math>-module morphism

: <math> \nabla:D(A)\ni u\to \nabla_u\in \mathrm{Diff}_1(P,P)</math>

such that the first order [[differential calculus over commutative algebras|differential operator]]s <math>\nabla_u</math> on
<math>P</math> obey the Leibniz rule

: <math>\nabla_u(ap)=u(a)p+a\nabla_u(p), \quad a\in A, \quad p\in
P.</math>

Connections on a module over a commutative ring always exist.

The curvature of the connection <math>\nabla</math> is defined as
the zero-order differential operator

: <math>R(u,u')=[\nabla_u,\nabla_{u'}]-\nabla_{[u,u']} \, </math>

on the module <math>P</math> for all <math>u,u'\in D(A)</math>.

If <math>E\to X</math> is a vector bundle, there is one-to-one
correspondence between [[connection (vector bundle)|linear
connections]] <math>\Gamma</math> on <math>E\to X</math> and the
connections <math>\nabla</math> on the
<math>C^\infty(X)</math>-module of sections of <math>E\to
X</math>. Strictly speaking, <math>\nabla</math> corresponds to
the [[covariant derivative|covariant differential]] of a
connection on <math>E\to X</math>.

== Graded commutative algebra ==

The notion of a connection on modules over commutative rings is
straightforwardly extended to modules over a [[superalgebra|graded
commutative algebra]].<ref>Bartocci (1991), Mangiarotti
(2000)</ref> This is the case of
[[supergeometry|superconnections]] in [[supergeometry]] of
[[graded manifold]]s and [[supergeometry|supervector bundles]].
Superconnections always exist.

== Noncommutative algebra ==

If <math>A</math> is a noncommutative ring, connections on left
and right <math>A</math>-modules are defined similarly to those on
modules over commutative rings.<ref>Landi (1997)</ref> 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
<math>R-S</math>-[[bimodule]] over noncommutative rings
<math>R</math> and <math>S</math>. There are different definitions
of such a connection.<ref>Dubois-Violette
(1996), Landi (1997)</ref> Let us mention one of them. A connection on an
<math>R-S</math>-bimodule <math>P</math> is defined as a bimodule
morphism

: <math> \nabla:D(A)\ni u\to \nabla_u\in \mathrm{Diff}_1(P,P)</math>

which obeys the Leibniz rule

: <math>\nabla_u(apb)=u(a)pb+a\nabla_u(p)b +apu(b), \quad a\in R,
\quad b\in S, \quad p\in P.</math>

== See also ==

*[[Connection (vector bundle)]] 
*[[Connection (mathematics)]]
*[[Noncommutative geometry]]
*[[Supergeometry]] 
*[[Differential calculus over commutative algebras]]

== Notes ==

{{reflist}}

== References ==

* Koszul, J., Homologie et cohomologie des algebres de Lie,''Bulletin de la Societe Mathematique'' '''78''' (1950) 65
* Koszul, J., ''Lectures on Fibre Bundles and Differential Geometry'' (Tata University, Bombay, 1960)
* Bartocci, C., Bruzzo, U., Hernandez Ruiperez, D., ''The Geometry of Supermanifolds'' (Kluwer Academic Publ., 1991) ISBN 0-7923-1440-9
* Dubois-Violette, M., Michor, P., Connections on central bimodules in noncommutative differential geometry, ''J. Geom. Phys.'' '''20''' (1996) 218. [http://arxiv.org/abs/q-alg/9503020 arXiv:q-alg/9503020v2]
* Landi, G., ''An Introduction to Noncommutative Spaces and their Geometries'', Lect. Notes Physics, New series m: Monographs, '''51''' (Springer, 1997) ArXiv [http://arxiv.org/abs/hep-th/9701078 eprint], iv+181 pages.
* Mangiarotti, L., [[Gennadi Sardanashvily|Sardanashvily, G.]], ''Connections in Classical and Quantum Field Theory'' (World Scientific, 2000) ISBN 981-02-2013-8

== External links ==
* [[Gennadi Sardanashvily|Sardanashvily, G.]], ''Lectures on Differential Geometry of Modules and Rings'' (Lambert Academic Publishing, Saarbrücken, 2012); [http://xxx.lanl.gov/abs/0910.1515 arXiv: 0910.1515]

[[Category:Connection (mathematics)]]
[[Category:Noncommutative geometry]]