# 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 ${\displaystyle E\to X}$.[1]

## Commutative algebra

Let ${\displaystyle A}$ be a commutative ring and ${\displaystyle P}$ a ${\displaystyle A}$-module. There are different equivalent definitions of a connection on ${\displaystyle P}$.[2] Let ${\displaystyle D(A)}$ be the module of derivations of a ring ${\displaystyle A}$. A connection on an ${\displaystyle A}$-module ${\displaystyle P}$ is defined as an ${\displaystyle A}$-module morphism

${\displaystyle \nabla :D(A)\ni u\to \nabla _{u}\in \mathrm {Diff} _{1}(P,P)}$

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

${\displaystyle \nabla _{u}(ap)=u(a)p+a\nabla _{u}(p),\quad a\in A,\quad p\in P.}$

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

${\displaystyle R(u,u')=[\nabla _{u},\nabla _{u'}]-\nabla _{[u,u']}\,}$

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 ${\displaystyle E\to X}$.

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

${\displaystyle \nabla :D(A)\ni u\to \nabla _{u}\in \mathrm {Diff} _{1}(P,P)}$

which obeys the Leibniz rule

${\displaystyle \nabla _{u}(apb)=u(a)pb+a\nabla _{u}(p)b+apu(b),\quad a\in R,\quad b\in S,\quad p\in P.}$

## Notes

1. Koszul (1950)
2. Koszul (1950), Mangiarotti (2000)
3. Bartocci (1991), Mangiarotti (2000)
4. Landi (1997)
5. Dubois-Violette (1996), Landi (1997)

## 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