Connection (algebraic framework)

Let $A$ be a commutative ring

and $M$ an A-module. There are different equivalent definitions

of a connection on $M$.{{harv|Koszul|1950}},{{harv|Mangiarotti

|Sardanashvily|2000}}

If $k\; \backslash to\; A$ is a ring homomorphism, a $k$-linear connection is a $k$-linear morphism

: $\backslash nabla:\; M\; \backslash to\; \backslash Omega^1\_\{A/k\}\; \backslash otimes\_A\; M$

which satisfies the identity

: $\backslash nabla(am)\; =\; da\; \backslash otimes\; m\; +\; a\; \backslash nabla\; m$

A connection extends, for all $p\; \backslash geq\; 0$ to a unique map

: $\backslash nabla:\; \backslash Omega^p\_\{A/k\}\; \backslash otimes\_A\; M\; \backslash to\; \backslash Omega^\{p+1\}\_\{A/k\}\; \backslash otimes\_A\; M$

satisfying $\backslash nabla(\backslash omega\; \backslash otimes\; f)\; =\; d\backslash omega\; \backslash otimes\; f\; +\; (-1)^p\; \backslash omega\; \backslash wedge\; \backslash nabla\; f$. A connection is said to be integrable if $\backslash nabla\; \backslash circ\; \backslash nabla\; =\; 0$, or equivalently, if the curvature $\backslash nabla^2:\; M\; \backslash to\; \backslash Omega\_\{A/k\}^2\; \backslash otimes\; M$ vanishes.

Let $D(A)$ be the module of derivations of a ring $A$. A

connection on an A-module $M$ is defined

as an A-module morphism

: $\backslash nabla:D(A)\; \backslash to\; \backslash mathrm\{Diff\}\_1(M,M);\; u\; \backslash mapsto\; \backslash nabla\_u$

such that the first order differential operators $\backslash nabla\_u$ on

$M$ obey the Leibniz rule

: $\backslash nabla\_u(ap)=u(a)p+a\backslash nabla\_u(p),\; \backslash quad\; a\backslash in\; A,\; \backslash quad\; p\backslash in$

M.

Connections on a module over a commutative ring always exist.

The curvature of the connection $\backslash nabla$ is defined as

the zero-order differential operator

: $R(u,u\text{'})=[\backslash nabla\_u,\backslash nabla\_\{u\text{'}\}]-\backslash nabla\_\{[u,u\text{'}]\}\; \backslash ,$

on the module $M$ for all $u,u\text{'}\backslash in\; D(A)$.

If $E\backslash to\; X$ is a vector bundle, there is one-to-one

correspondence between [[connection (vector bundle)|linear

connections]] $\backslash Gamma$ on $E\backslash to\; X$ and the

connections $\backslash nabla$ on the

$C^\backslash infty(X)$-module of sections of $E\backslash to$

X. Strictly speaking, $\backslash nabla$ corresponds to

the covariant differential of a

connection on $E\backslash to\; X$.

The notion of a connection on modules over commutative rings is

straightforwardly extended to modules over a [[superalgebra|graded

commutative algebra]].{{harv|Bartocci|Bruzzo|Hernández-Ruipérez|1991}}, {{harv|Mangiarotti

|Sardanashvily|2000}} This is the case of

superconnections in supergeometry of

graded manifolds and supervector bundles.

Superconnections always exist.

If $A$ is a noncommutative ring, connections on left

and right A-modules are defined similarly to those on

modules over commutative rings.{{harv|Landi|1997}} 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.{{harv|Dubois-Violette|Michor|1996}},{{harv|Landi |1997}} Let us mention one of them. A connection on an

R-S-bimodule $P$ is defined as a bimodule

morphism

: $\backslash nabla:D(A)\backslash ni\; u\backslash to\; \backslash nabla\_u\backslash in\; \backslash mathrm\{Diff\}\_1(P,P)$

which obeys the Leibniz rule

: $\backslash nabla\_u(apb)=u(a)pb+a\backslash nabla\_u(p)b\; +apu(b),\; \backslash quad\; a\backslash in\; R,$

\quad b\in S, \quad p\in P.

• Connection (vector bundle)
• Connection (mathematics)
• Noncommutative geometry
• Supergeometry
• Differential calculus over commutative algebras

{{reflist}}

{{ref begin}}

• {{cite journal |last1=Koszul |first1=Jean-Louis |title=Homologie et cohomologie des algèbres de Lie |journal=Bulletin de la Société Mathématique de France |year=1950 |volume=78 |pages=65–127 |doi=10.24033/bsmf.1410 |url=http://www.numdam.org/item/10.24033/bsmf.1410.pdf}}
• {{cite book |s2cid=51020097 |doi=10.1007/978-3-662-02503-1 |title=Lectures on Fibre Bundles and Differential Geometry (Tata University, Bombay, 1960) |year=1986 |last1=Koszul |first1=J. L. |doi-broken-date=1 November 2024 |isbn=978-3-540-12876-2|zbl=0244.53026 }}
• {{cite book |doi=10.1007/978-94-011-3504-7|title=The Geometry of Supermanifolds |year=1991 |last1=Bartocci |first1=Claudio |last2=Bruzzo |first2=Ugo |last3=Hernández-Ruipérez |first3=Daniel |isbn=978-94-010-5550-5 }}
• {{cite journal |arxiv=q-alg/9503020|doi=10.1016/0393-0440(95)00057-7 |title=Connections on central bimodules in noncommutative differential geometry |year=1996 |last1=Dubois-Violette |first1=Michel |last2=Michor |first2=Peter W. |journal=Journal of Geometry and Physics |volume=20 |issue=2–3 |pages=218–232 |s2cid=15994413 }}
• {{cite book |doi=10.1007/3-540-14949-X|title=An Introduction to Noncommutative Spaces and their Geometries |series=Lecture Notes in Physics |year=1997 |volume=51 |isbn=978-3-540-63509-3 |s2cid=14986502|arxiv=hep-th/9701078|first1=Giovanni |last1=Landi}}
• {{cite book |doi=10.1142/2524|title=Connections in Classical and Quantum Field Theory |year=2000 |last1=Mangiarotti |first1=L. |last2=Sardanashvily |first2=G. |isbn=978-981-02-2013-6 }}

{{ref end}}

• {{cite arXiv |eprint=0910.1515|last1=Sardanashvily |first1=G. |title=Lectures on Differential Geometry of Modules and Rings |class=math-ph|year=2009 }}

Category:Connection (mathematics)

Category:Noncommutative geometry