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 E\to
X written as a Koszul connection on the C^\infty(X)-module of sections of E\to
X.[1]

Commutative algebra[edit]

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

 \nabla:D(A)\ni u\to \nabla_u\in \mathrm{Diff}_1(P,P)

such that the first order differential operators \nabla_u on P obey the Leibniz rule

\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 \nabla is defined as the zero-order differential operator

R(u,u')=[\nabla_u,\nabla_{u'}]-\nabla_{[u,u']} \,

on the module P for all u,u'\in D(A).

If E\to X is a vector bundle, there is one-to-one correspondence between linear connections \Gamma on E\to X and the connections \nabla on the C^\infty(X)-module of sections of E\to
X. Strictly speaking, \nabla corresponds to the covariant differential of a connection on E\to X.

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 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

 \nabla:D(A)\ni u\to \nabla_u\in \mathrm{Diff}_1(P,P)

which obeys the Leibniz rule

\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.

See also[edit]

Notes[edit]

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

References[edit]

  • 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

External links[edit]