Connection (algebraic framework) explained
written as a
Koszul connection on the
-module of sections of
.
Commutative algebra
Let
be a commutative
ringand
an
A-
module. There are different equivalent definitionsof a connection on
.
[1] First definition
If
is a ring homomorphism, a
-linear connection is a
-linear morphism
which satisfies the identity
\nabla(am)=da ⊗ m+a\nablam
A connection extends, for all
to a unique map
satisfying
\nabla(\omega ⊗ f)=d\omega ⊗ f+(-1)p\omega\wedge\nablaf
. 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
. Aconnection on an
A-module
is definedas an
A-module morphism
\nabla:D(A)\toDiff1(M,M);u\mapsto\nablau
such that the first order differential operators
on
obey the Leibniz rule
\nablau(ap)=u(a)p+a\nablau(p), a\inA, p\in
M.
Connections on a module over a commutative ring always exist.
The curvature of the connection
is defined asthe zero-order differential operator
R(u,u')=[\nablau,\nablau']-\nabla[u,u']
on the module
for all
.
If
is a vector bundle, there is one-to-onecorrespondence between
linearconnections
on
and theconnections
on the
-module of sections of
. Strictly speaking,
corresponds tothe
covariant differential of aconnection on
.
Graded commutative algebra
The notion of a connection on modules over commutative rings isstraightforwardly extended to modules over a gradedcommutative algebra.[2] This is the case ofsuperconnections in supergeometry ofgraded manifolds and supervector bundles.Superconnections always exist.
Noncommutative algebra
If
is a noncommutative ring, connections on leftand right
A-modules are defined similarly to those onmodules over commutative rings. Howeverthese connections need not exist.
In contrast with connections on left and right modules, there is aproblem how to define a connection on anR-S-bimodule over noncommutative ringsR and S. There are different definitionsof such a connection.[3] Let us mention one of them. A connection on anR-S-bimodule
is defined as a bimodulemorphism
\nabla:D(A)\niu\to\nablau\inDiff1(P,P)
which obeys the Leibniz rule
\nablau(apb)=u(a)pb+a\nablau(p)b+apu(b), a\inR,
b\inS, p\inP.
See also
References
- Koszul . Jean-Louis . Homologie et cohomologie des algèbres de Lie . Bulletin de la Société Mathématique de France . 1950 . 78 . 65–127 . 10.24033/bsmf.1410 .
- Book: 51020097 . 10.1007/978-3-662-02503-1 . Lectures on Fibre Bundles and Differential Geometry (Tata University, Bombay, 1960) . 1986 . Koszul . J. L. . 1 November 2024 . 978-3-540-12876-2. 0244.53026 .
- Book: 10.1007/978-94-011-3504-7. The Geometry of Supermanifolds . 1991 . Bartocci . Claudio . Bruzzo . Ugo . Hernández-Ruipérez . Daniel . 978-94-010-5550-5 .
- q-alg/9503020. 10.1016/0393-0440(95)00057-7 . Connections on central bimodules in noncommutative differential geometry . 1996 . Dubois-Violette . Michel . Michor . Peter W. . Journal of Geometry and Physics . 20 . 2–3 . 218–232 . 15994413 .
- Book: 10.1007/3-540-14949-X. An Introduction to Noncommutative Spaces and their Geometries . Lecture Notes in Physics . 1997 . 51 . 978-3-540-63509-3 . 14986502. hep-th/9701078. Giovanni . Landi.
- Book: 10.1142/2524. Connections in Classical and Quantum Field Theory . 2000 . Mangiarotti . L. . Sardanashvily . G. . 978-981-02-2013-6 .
External links
- 0910.1515. Sardanashvily . G. . Lectures on Differential Geometry of Modules and Rings . math-ph. 2009 .
Notes and References
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- ,
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