Connection (algebraic framework)

<h2 id="commutative-algebra">Commutative algebra</h2>
<p>Let 
  
    
      
        A
      
    
    {\displaystyle A}
  
 be a commutative <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a>
and 
  
    
      
        M
      
    
    {\displaystyle M}
  
 an <i>A</i>-<a href="/facts/Module_(mathematics)/jP0LIhFL">module</a>. There are different equivalent definitions
of a connection on 
  
    
      
        M
      
    
    {\displaystyle M}
  
.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>
<h3>First definition</h3>
<p>If 
  
    
      
        k
        →
        A
      
    
    {\displaystyle k\to A}
  
 is a ring homomorphism, a 
  
    
      
        k
      
    
    {\displaystyle k}
  
-linear connection is a 
  
    
      
        k
      
    
    {\displaystyle k}
  
-linear morphism
</p>

∇
        :
        M
        →
        
          Ω
          
            A
            
              /
            
            k
          
          
            1
          
        
        
          ⊗
          
            A
          
        
        M
      
    
    {\displaystyle \nabla :M\to \Omega _{A/k}^{1}\otimes _{A}M}

<p>which satisfies the identity
</p>

∇
        (
        a
        m
        )
        =
        d
        a
        ⊗
        m
        +
        a
        ∇
        m
      
    
    {\displaystyle \nabla (am)=da\otimes m+a\nabla m}

<p>A connection extends, for all 
  
    
      
        p
        ≥
        0
      
    
    {\displaystyle p\geq 0}
  
 to a unique map
</p>

∇
        :
        
          Ω
          
            A
            
              /
            
            k
          
          
            p
          
        
        
          ⊗
          
            A
          
        
        M
        →
        
          Ω
          
            A
            
              /
            
            k
          
          
            p
            +
            1
          
        
        
          ⊗
          
            A
          
        
        M
      
    
    {\displaystyle \nabla :\Omega _{A/k}^{p}\otimes _{A}M\to \Omega _{A/k}^{p+1}\otimes _{A}M}

<p>satisfying 
  
    
      
        ∇
        (
        ω
        ⊗
        f
        )
        =
        d
        ω
        ⊗
        f
        +
        (
        −
        1
        
          )
          
            p
          
        
        ω
        ∧
        ∇
        f
      
    
    {\displaystyle \nabla (\omega \otimes f)=d\omega \otimes f+(-1)^{p}\omega \wedge \nabla f}
  
. A connection is said to be integrable if 
  
    
      
        ∇
        ∘
        ∇
        =
        0
      
    
    {\displaystyle \nabla \circ \nabla =0}
  
, or equivalently, if the curvature 
  
    
      
        
          ∇
          
            2
          
        
        :
        M
        →
        
          Ω
          
            A
            
              /
            
            k
          
          
            2
          
        
        ⊗
        M
      
    
    {\displaystyle \nabla ^{2}:M\to \Omega _{A/k}^{2}\otimes M}
  
 vanishes.
</p>
<h3>Second definition</h3>
<p>Let 
  
    
      
        D
        (
        A
        )
      
    
    {\displaystyle D(A)}
  
 be the module of <a href="/facts/Derivation_(abstract_algebra)/vcSqpxzu">derivations</a> of a ring 
  
    
      
        A
      
    
    {\displaystyle A}
  
. A
connection on an <i>A</i>-module 
  
    
      
        M
      
    
    {\displaystyle M}
  
 is defined
as an <i>A</i>-module morphism
</p>

∇
        :
        D
        (
        A
        )
        →
        
          
            D
            i
            f
            f
          
          
            1
          
        
        (
        M
        ,
        M
        )
        ;
        u
        ↦
        
          ∇
          
            u
          
        
      
    
    {\displaystyle \nabla :D(A)\to \mathrm {Diff} _{1}(M,M);u\mapsto \nabla _{u}}

<p>such that the first order <a href="/facts/Differential_calculus_over_commutative_algebras/cUKSP9H9">differential operators</a> 
  
    
      
        
          ∇
          
            u
          
        
      
    
    {\displaystyle \nabla _{u}}
  
 on

M
      
    
    {\displaystyle M}
  
 obey the Leibniz rule
</p>

∇
          
            u
          
        
        (
        a
        p
        )
        =
        u
        (
        a
        )
        p
        +
        a
        
          ∇
          
            u
          
        
        (
        p
        )
        ,
        
        a
        ∈
        A
        ,
        
        p
        ∈
        M
        .
      
    
    {\displaystyle \nabla _{u}(ap)=u(a)p+a\nabla _{u}(p),\quad a\in A,\quad p\in M.}

<p>Connections on a module over a commutative ring always exist.
</p><p>The curvature of the connection 
  
    
      
        ∇
      
    
    {\displaystyle \nabla }
  
 is defined as
the zero-order differential operator
</p>

R
        (
        u
        ,
        
          u
          ′
        
        )
        =
        [
        
          ∇
          
            u
          
        
        ,
        
          ∇
          
            
              u
              ′
            
          
        
        ]
        −
        
          ∇
          
            [
            u
            ,
            
              u
              ′
            
            ]
          
        
        
      
    
    {\displaystyle R(u,u')=[\nabla _{u},\nabla _{u'}]-\nabla _{[u,u']}\,}

<p>on the module 
  
    
      
        M
      
    
    {\displaystyle M}
  
 for all 
  
    
      
        u
        ,
        
          u
          ′
        
        ∈
        D
        (
        A
        )
      
    
    {\displaystyle u,u'\in D(A)}
  
.
</p><p>If 
  
    
      
        E
        →
        X
      
    
    {\displaystyle E\to X}
  
 is a vector bundle, there is one-to-one
correspondence between <a href="/facts/Connection_(vector_bundle)/Rxv5S2Af">linear
connections</a> 
  
    
      
        Γ
      
    
    {\displaystyle \Gamma }
  
 on 
  
    
      
        E
        →
        X
      
    
    {\displaystyle E\to X}
  
 and the
connections 
  
    
      
        ∇
      
    
    {\displaystyle \nabla }
  
 on the

C
          
            ∞
          
        
        (
        X
        )
      
    
    {\displaystyle C^{\infty }(X)}
  
-module of sections of 
  
    
      
        E
        →
        X
      
    
    {\displaystyle E\to X}
  
. Strictly speaking, 
  
    
      
        ∇
      
    
    {\displaystyle \nabla }
  
 corresponds to
the <a href="/facts/Covariant_derivative/sRGN0Xh6">covariant differential</a> of a
connection on 
  
    
      
        E
        →
        X
      
    
    {\displaystyle E\to X}
  
.
</p>
<h2 id="graded-commutative-algebra">Graded commutative algebra</h2>
<p>The notion of a connection on modules over commutative rings is
straightforwardly extended to modules over a <a href="/facts/Superalgebra/96vSHP12">graded
commutative algebra</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> This is the case of
<a href="/facts/Supergeometry/ucKekXjf">superconnections</a> in <a href="/facts/Supergeometry/ucKekXjf">supergeometry</a> of
<a href="/facts/Graded_manifold/cr99oc5R">graded manifolds</a> and <a href="/facts/Supergeometry/ucKekXjf">supervector bundles</a>.
Superconnections always exist.
</p>
<h2 id="noncommutative-algebra">Noncommutative algebra</h2>
<p>If 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is a noncommutative ring, connections on left
and right <i>A</i>-modules are defined similarly to those on
modules over commutative rings.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> However
these connections need not exist.
</p><p>In contrast with connections on left and right modules, there is a
problem how to define a connection on an
<i>R</i>-<i>S</i>-<a href="/facts/Bimodule/NskmNGQ1">bimodule</a> over noncommutative rings
<i>R</i> and <i>S</i>. There are different definitions
of such a connection.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> Let us mention one of them. A connection on an
<i>R</i>-<i>S</i>-bimodule 
  
    
      
        P
      
    
    {\displaystyle P}
  
 is defined as a bimodule
morphism
</p>

∇
        :
        D
        (
        A
        )
        ∋
        u
        →
        
          ∇
          
            u
          
        
        ∈
        
          
            D
            i
            f
            f
          
          
            1
          
        
        (
        P
        ,
        P
        )
      
    
    {\displaystyle \nabla :D(A)\ni u\to \nabla _{u}\in \mathrm {Diff} _{1}(P,P)}

<p>which obeys the Leibniz rule
</p>

∇
          
            u
          
        
        (
        a
        p
        b
        )
        =
        u
        (
        a
        )
        p
        b
        +
        a
        
          ∇
          
            u
          
        
        (
        p
        )
        b
        +
        a
        p
        u
        (
        b
        )
        ,
        
        a
        ∈
        R
        ,
        
        b
        ∈
        S
        ,
        
        p
        ∈
        P
        .
      
    
    {\displaystyle \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.}

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Connection_(vector_bundle)/Rxv5S2Af">Connection (vector bundle)</a></li>
<li><a href="/facts/Connection_(mathematics)/KURsCmIu">Connection (mathematics)</a></li>
<li><a href="/facts/Noncommutative_geometry/WSwggL1p">Noncommutative geometry</a></li>
<li><a href="/facts/Supergeometry/ucKekXjf">Supergeometry</a></li>
<li><a href="/facts/Differential_calculus_over_commutative_algebras/cUKSP9H9">Differential calculus over commutative algebras</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Koszul, Jean-Louis (1950). <a href="http://www.numdam.org/item/10.24033/bsmf.1410.pdf">"Homologie et cohomologie des algèbres de Lie"</a> (PDF). <i>Bulletin de la Société Mathématique de France</i>. 78: 65–127. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.24033%2Fbsmf.1410">10.24033/bsmf.1410</a>.</li>
<li>Koszul, J. L. (1986). <i>Lectures on Fibre Bundles and Differential Geometry (Tata University, Bombay, 1960)</i>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-662-02503-1">10.1007/978-3-662-02503-1</a> (inactive 1 November 2024). <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-12876-2. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:51020097">51020097</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0244.53026">0244.53026</a>.{{cite book}}:  CS1 maint: DOI inactive as of November 2024 (link)</li>
<li>Bartocci, Claudio; Bruzzo, Ugo; Hernández-Ruipérez, Daniel (1991). <i>The Geometry of Supermanifolds</i>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-94-011-3504-7">10.1007/978-94-011-3504-7</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-94-010-5550-5.</li>
<li>Dubois-Violette, Michel; Michor, Peter W. (1996). "Connections on central bimodules in noncommutative differential geometry". <i>Journal of Geometry and Physics</i>. 20 (2–3): 218–232. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/q-alg/9503020">q-alg/9503020</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0393-0440%2895%2900057-7">10.1016/0393-0440(95)00057-7</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:15994413">15994413</a>.</li>
<li>Landi, Giovanni (1997). <i>An Introduction to Noncommutative Spaces and their Geometries</i>. Lecture Notes in Physics. Vol. 51. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/hep-th/9701078">hep-th/9701078</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F3-540-14949-X">10.1007/3-540-14949-X</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-63509-3. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:14986502">14986502</a>.</li>
<li>Mangiarotti, L.; Sardanashvily, G. (2000). <i>Connections in Classical and Quantum Field Theory</i>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1142%2F2524">10.1142/2524</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-981-02-2013-6.</li></ul>

<h2 id="external-links">External links</h2>
<ul><li>Sardanashvily, G. (2009). "Lectures on Differential Geometry of Modules and Rings". <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/0910.1515">0910.1515</a> [<a href="https://arxiv.org/archive/math-ph">math-ph</a>].</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>(Koszul 1950) - Koszul, Jean-Louis (1950). "Homologie et cohomologie des algèbres de Lie" (PDF). Bulletin de la Société Mathématique de France. 78: 65–127. doi:10.24033/bsmf.1410. <a href="http://www.numdam.org/item/10.24033/bsmf.1410.pdf" target="_blank">http://www.numdam.org/item/10.24033/bsmf.1410.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>(Koszul 1950),(Mangiarotti & Sardanashvily 2000) - Koszul, Jean-Louis (1950). "Homologie et cohomologie des algèbres de Lie" (PDF). Bulletin de la Société Mathématique de France. 78: 65–127. doi:10.24033/bsmf.1410. <a href="http://www.numdam.org/item/10.24033/bsmf.1410.pdf" target="_blank">http://www.numdam.org/item/10.24033/bsmf.1410.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>(Bartocci, Bruzzo & Hernández-Ruipérez 1991), (Mangiarotti & Sardanashvily 2000) - Bartocci, Claudio; Bruzzo, Ugo; Hernández-Ruipérez, Daniel (1991). The Geometry of Supermanifolds. doi:10.1007/978-94-011-3504-7. ISBN 978-94-010-5550-5. <a href="https://doi.org/10.1007%2F978-94-011-3504-7" target="_blank">https://doi.org/10.1007%2F978-94-011-3504-7</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>(Landi 1997) - Landi, Giovanni (1997). An Introduction to Noncommutative Spaces and their Geometries. Lecture Notes in Physics. Vol. 51. arXiv:hep-th/9701078. doi:10.1007/3-540-14949-X. ISBN 978-3-540-63509-3. S2CID 14986502. <a href="https://arxiv.org/abs/hep-th/9701078" target="_blank">https://arxiv.org/abs/hep-th/9701078</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>(Dubois-Violette & Michor 1996),(Landi 1997) - Dubois-Violette, Michel; Michor, Peter W. (1996). "Connections on central bimodules in noncommutative differential geometry". Journal of Geometry and Physics. 20 (2–3): 218–232. arXiv:q-alg/9503020. doi:10.1016/0393-0440(95)00057-7. S2CID 15994413. <a href="https://arxiv.org/abs/q-alg/9503020" target="_blank">https://arxiv.org/abs/q-alg/9503020</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
</ol>

Connection (algebraic framework) open-in-new

Connection (algebraic framework)