Jump to content

Connection (algebraic framework)

From Wikipedia, the free encyclopedia

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

[edit]

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

First definition

[edit]

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

[edit]

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

[edit]

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

[edit]

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

[edit]

Notes

[edit]

References

[edit]
  • Koszul, Jean-Louis (1950). "Homologie et cohomologie des algèbres de Lie" (PDF). Bulletin de la Société Mathématique de France. 78: 65–127. doi:10.24033/bsmf.1410.
  • Koszul, J. L. (1986). Lectures on Fibre Bundles and Differential Geometry (Tata University, Bombay, 1960). doi:10.1007/978-3-662-02503-1 (inactive 2024-04-26). ISBN 978-3-540-12876-2. S2CID 51020097. Zbl 0244.53026.{{cite book}}: CS1 maint: DOI inactive as of April 2024 (link)
  • 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. arXiv:q-alg/9503020. doi:10.1016/0393-0440(95)00057-7. S2CID 15994413.
  • Landi, Giovanni (1997). An Introduction to Noncommutative Spaces and their Geometries. Lecture Notes in Physics. Vol. 51. arXiv:hep-th/9701078. doi:10.1007/3-540-14949-X. ISBN 978-3-540-63509-3. S2CID 14986502.
  • Mangiarotti, L.; Sardanashvily, G. (2000). Connections in Classical and Quantum Field Theory. doi:10.1142/2524. ISBN 978-981-02-2013-6.
[edit]
  • Sardanashvily, G. (2009). "Lectures on Differential Geometry of Modules and Rings". arXiv:0910.1515 [math-ph].