# 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 written as a Koszul connection on the
-module of sections of .^{[1]}

## Commutative algebra

Let be a commutative ring
and a -module. There are different equivalent definitions
of a connection on .^{[2]} Let be the module of derivations of a ring . A
connection on an -module is defined
as an -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

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.^{[3]} 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 -modules are defined similarly to those on
modules over commutative rings.^{[4]} 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
-bimodule over noncommutative rings
and . There are different definitions
of such a connection.^{[5]} Let us mention one of them. A connection on an
-bimodule is defined as a bimodule
morphism

which obeys the Leibniz rule

## See also

- Connection (vector bundle)
- Connection (mathematics)
- Noncommutative geometry
- Supergeometry
- Differential calculus over commutative algebras

## Notes

## 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. 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 eprint, iv+181 pages. - Mangiarotti, L., Sardanashvily, G.,
*Connections in Classical and Quantum Field Theory*(World Scientific, 2000) ISBN 981-02-2013-8

## External links

- Sardanashvily, G.,
*Lectures on Differential Geometry of Modules and Rings*(Lambert Academic Publishing, Saarbrücken, 2012); arXiv: 0910.1515