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 ${\displaystyle E\to X}$ written as a Koszul connection on the ${\displaystyle C^{\infty }(X)}$-module of sections of ${\displaystyle E\to X}$.

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Commutative algebra

Let ${\displaystyle A}$ be a commutative ring and ${\displaystyle P}$ an A-module. There are different equivalent definitions of a connection on ${\displaystyle P}$. Let ${\displaystyle D(A)}$ be the module of derivations of a ring ${\displaystyle A}$. A connection on an A-module ${\displaystyle P}$ is defined as an A-module morphism

such that the first order differential operators ${\displaystyle \nabla _{u}}$ on ${\displaystyle P}$ obey the Leibniz rule

Connections on a module over a commutative ring always exist.

The curvature of the connection ${\displaystyle \nabla }$ is defined as the zero-order differential operator

on the module ${\displaystyle P}$ for all ${\displaystyle u,u'\in D(A)}$.

If ${\displaystyle E\to X}$ is a vector bundle, there is one-to-one correspondence between linear connections ${\displaystyle \Gamma }$ on ${\displaystyle E\to X}$ and the connections ${\displaystyle \nabla }$ on the ${\displaystyle C^{\infty }(X)}$-module of sections of ${\displaystyle E\to X}$. Strictly speaking, ${\displaystyle \nabla }$ corresponds to the covariant differential of a connection on ${\displaystyle E\to X}$.

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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.

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Noncommutative algebra

If ${\displaystyle A}$ 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 ${\displaystyle P}$ is defined as a bimodule morphism

which obeys the Leibniz rule

Source of the article : Wikipedia