Английская Википедия:Connection (algebraic framework)

Содержание

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

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>.

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>.

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. 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. 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. 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

References

• Шаблон:Cite journal
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite journal
• Шаблон:Cite book
• Шаблон:Cite book

External links

• Шаблон:Cite arxiv