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 E->X written as a Koszul connection on the C∞(X)-module of sections of E->X.[1]

## Contents

* 1 Commutative algebra
* 1.1 First definition
* 1.2 Second definition
* 2 Graded commutative algebra
* 3 Noncommutative algebra
* 4 See also
* 5 Notes
* 6 References
* 7 External links

## Commutative algebra

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

### First definition

If k->A is a ring homomorphism, a k-linear connection is a k-linear morphism

∇:M->ΩA/k1⊗AM

which satisfies the identity

∇(am)=da⊗m+a∇m

A connection extends, for all p≥0 to a unique map

∇:ΩA/kp⊗AM->ΩA/kp+1⊗AM

satisfying ∇(ω⊗f)=dω⊗f+(−1)pω∧∇f. A connection is said to be integrable if ∇∘∇=0, or equivalently, if the curvature ∇2:M->ΩA/k2⊗M vanishes.

### Second definition

Let D(A) be the module of derivations of a ring A. A connection on an A-module M is defined as an A-module morphism

∇:D(A)->Diff1(M,M);u↦∇u

such that the first order differential operators ∇u on M obey the Leibniz rule

∇u(ap)=u(a)p+a∇u(p),a∈A,p∈M.

Connections on a module over a commutative ring always exist.

The curvature of the connection ∇ is defined as the zero-order differential operator

R(u,u′)=[∇u,∇u′]−∇[u,u′]

on the module M for all u,u′∈D(A).

If E->X is a vector bundle, there is one-to-one correspondence between linear connections Γ on E->X and the connections ∇ on the C∞(X)-module of sections of E->X. Strictly speaking, ∇ corresponds to the covariant differential of a connection on E->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 A 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 P is defined as a bimodule morphism

∇:D(A)∋u->∇u∈Diff1(P,P)

which obeys the Leibniz rule

∇u(apb)=u(a)pb+a∇u(p)b+apu(b),a∈R,b∈S,p∈P.

## See also

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

## Notes

1. ↑ (Koszul 1950)
2. ↑ (Koszul 1950),(Mangiarotti Sardanashvily)
3. ↑ (Bartocci Bruzzo), (Mangiarotti Sardanashvily)
4. ↑ (Landi 1997)
5. ↑ (Dubois-Violette Michor),(Landi 1997)

## References

* Koszul, Jean-Louis (1950). "Homologie et cohomologie des algèbres de Lie". Bulletin de la Société Mathématique de France 78: 65–127. doi:10.24033/bsmf.1410. http://www.numdam.org/item/10.24033/bsmf.1410.pdf.
* Koszul, J. L. (1986). Lectures on Fibre Bundles and Differential Geometry (Tata University, Bombay, 1960). doi:10.1007/978-3-662-02503-1. ISBN 978-3-540-12876-2.
* Bartocci, Claudio; Bruzzo, Ugo; Hernández-Ruipérez, Daniel (1991). The Geometry of Supermanifolds. doi:10.1007/978-94-011-3504-7. ISBN 978-94-010-5550-5.
* Dubois-Violette, Michel; Michor, Peter W. (1996). "Connections on central bimodules in noncommutative differential geometry". Journal of Geometry and Physics 20 (2–3): 218–232. doi:10.1016/0393-0440(95)00057-7.
* Landi, Giovanni (1997). An Introduction to Noncommutative Spaces and their Geometries. Lecture Notes in Physics. 51. doi:10.1007/3-540-14949-X. ISBN 978-3-540-63509-3.
* Mangiarotti, L.; Sardanashvily, G. (2000). Connections in Classical and Quantum Field Theory. doi:10.1142/2524. ISBN 978-981-02-2013-6.

## External links

* Sardanashvily, G. (2009). Lectures on Differential Geometry of Modules and Rings.

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