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 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]- Connection (vector bundle)
- Connection (mathematics)
- Noncommutative geometry
- Supergeometry
- Differential calculus over commutative algebras
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.