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.

Commutative algebra

be a commutative ring an A-module.

First definition

-linear morphism

which satisfies the identity

to a unique map

vanishes.

Second definition

be the module of derivations. A connection on an A is defined as an A-module morphism

such that the first order differential operators obey the Leibniz rule

Connections on a module over a commutative ring always exist.

is defined as the zero-order differential operator

.

is a vector bundle, there is one-to-one correspondence between linear connections corresponds to the covariant differential.

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

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 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

External links