Английская Википедия: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 <math>E\to X</math> written as a Koszul connection on the <math>C^\infty(X)</math>-module of sections of <math>E\to X</math>.[1]
Commutative algebra
Let <math>A</math> be a commutative ring and <math>M</math> an A-module. There are different equivalent definitions of a connection on <math>M</math>.[2]
First definition
If <math>k \to A</math> is a ring homomorphism, a <math>k</math>-linear connection is a <math>k</math>-linear morphism
- <math> \nabla: M \to \Omega^1_{A/k} \otimes_A M </math>
which satisfies the identity
- <math> \nabla(am) = da \otimes m + a \nabla m </math>
A connection extends, for all <math>p \geq 0</math> to a unique map
- <math>\nabla: \Omega^p_{A/k} \otimes_A M \to \Omega^{p+1}_{A/k} \otimes_A M</math>
satisfying <math>\nabla(\omega \otimes f) = d\omega \otimes f + (-1)^p \omega \wedge \nabla f</math>. A connection is said to be integrable if <math>\nabla \circ \nabla = 0</math>, or equivalently, if the curvature <math> \nabla^2: M \to \Omega_{A/k}^2 \otimes M</math> vanishes.
Second definition
Let <math>D(A)</math> be the module of derivations of a ring <math>A</math>. A connection on an A-module <math>M</math> is defined as an A-module morphism
- <math> \nabla:D(A) \to \mathrm{Diff}_1(M,M); u \mapsto \nabla_u </math>
such that the first order differential operators <math>\nabla_u</math> on <math>M</math> obey the Leibniz rule
- <math>\nabla_u(ap)=u(a)p+a\nabla_u(p), \quad a\in A, \quad p\in
M.</math>
Connections on a module over a commutative ring always exist.
The curvature of the connection <math>\nabla</math> is defined as the zero-order differential operator
- <math>R(u,u')=[\nabla_u,\nabla_{u'}]-\nabla_{[u,u']} \, </math>
on the module <math>M</math> for all <math>u,u'\in D(A)</math>.
If <math>E\to X</math> is a vector bundle, there is one-to-one correspondence between linear connections <math>\Gamma</math> on <math>E\to X</math> and the connections <math>\nabla</math> on the <math>C^\infty(X)</math>-module of sections of <math>E\to X</math>. Strictly speaking, <math>\nabla</math> corresponds to the covariant differential of a connection on <math>E\to X</math>.
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 <math>A</math> 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 <math>P</math> is defined as a bimodule morphism
- <math> \nabla:D(A)\ni u\to \nabla_u\in \mathrm{Diff}_1(P,P)</math>
which obeys the Leibniz rule
- <math>\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.</math>
See also
- Connection (vector bundle)
- Connection (mathematics)
- Noncommutative geometry
- Supergeometry
- Differential calculus over commutative algebras
Notes
References
- Шаблон:Cite journal
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite journal
- Шаблон:Cite book
- Шаблон:Cite book
External links