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

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

First definition

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

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

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 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

Notes

References

  • 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.
  • Sardanashvily, G. (2009). "Lectures on Differential Geometry of Modules and Rings". arXiv: 0910.1515 [ math-ph].
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

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

First definition

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

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

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 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

Notes

References

  • 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.
  • Sardanashvily, G. (2009). "Lectures on Differential Geometry of Modules and Rings". arXiv: 0910.1515 [ math-ph].

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