Connection (algebraic framework)

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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

Summarize
Perspective

Let be a commutative ring and an A-module. There are different equivalent definitions of a connection on .[2]

First definition

If is a ring homomorphism, a -linear connection is a -linear morphism

which satisfies the identity

A connection extends, for all to a unique map

satisfying . A connection is said to be integrable if , or equivalently, if the curvature vanishes.

Second definition

Let be the module of derivations of a ring . A connection on an A-module is defined as an A-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

on the module for all .

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

Summarize
Perspective

If is a noncommutative ring, connections on left and right A-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 R-S-bimodule over noncommutative rings R and S. There are different definitions of such a connection.[5] Let us mention one of them. A connection on an R-S-bimodule is defined as a bimodule morphism

which obeys the Leibniz rule

See also

Notes

References

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