Quantum differential calculus

<h2 id="note">Note</h2>
<p>The above definition is minimal and gives something more general than classical differential calculus even when the algebra 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is commutative or functions on an actual space. This is because we do <i>not</i> demand that
</p><p>
  
    
      
        a
        (
        
          
            d
          
        
        b
        )
        =
        (
        
          
            d
          
        
        b
        )
        a
        ,
         
        ∀
        a
        ,
        b
        ∈
        A
      
    
    {\displaystyle a({\rm {d}}b)=({\rm {d}}b)a,\ \forall a,b\in A}

</p><p>since this would imply that 
  
    
      
        
          
            d
          
        
        (
        a
        b
        −
        b
        a
        )
        =
        0
        ,
         
        ∀
        a
        ,
        b
        ∈
        A
      
    
    {\displaystyle {\rm {d}}(ab-ba)=0,\ \forall a,b\in A}
  
, which would violate axiom 4 when the algebra was noncommutative. As a byproduct, this enlarged definition includes finite difference calculi and quantum differential calculi on finite sets and finite groups (finite group <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> theory).
</p>
<h2 id="examples">Examples</h2>
<ol><li>For 
  
    
      
        A
        =
        
          
            C
          
        
        [
        x
        ]
      
    
    {\displaystyle A={\mathbb {C} }[x]}
  
 the algebra of polynomials in one variable the translation-covariant quantum differential calculi are parametrized by 
  
    
      
        λ
        ∈
        
          C
        
      
    
    {\displaystyle \lambda \in \mathbb {C} }
  
 and take the form 
  
    
      
        
          Ω
          
            1
          
        
        =
        
          
            C
          
        
        .
        
          
            d
          
        
        x
        ,
        
        (
        
          
            d
          
        
        x
        )
        f
        (
        x
        )
        =
        f
        (
        x
        +
        λ
        )
        (
        
          
            d
          
        
        x
        )
        ,
        
        
          
            d
          
        
        f
        =
        
          
            
              f
              (
              x
              +
              λ
              )
              −
              f
              (
              x
              )
            
            λ
          
        
        
          
            d
          
        
        x
      
    
    {\displaystyle \Omega ^{1}={\mathbb {C} }.{\rm {d}}x,\quad ({\rm {d}}x)f(x)=f(x+\lambda )({\rm {d}}x),\quad {\rm {d}}f={f(x+\lambda )-f(x) \over \lambda }{\rm {d}}x}
  
 This shows how finite differences arise naturally in quantum geometry. Only the limit 
  
    
      
        λ
        →
        0
      
    
    {\displaystyle \lambda \to 0}
  
 has functions commuting with 1-forms, which is the special case of high school differential calculus.</li>
<li>For 
  
    
      
        A
        =
        
          
            C
          
        
        [
        t
        ,
        
          t
          
            −
            1
          
        
        ]
      
    
    {\displaystyle A={\mathbb {C} }[t,t^{-1}]}
  
 the algebra of functions on an algebraic circle, the translation (i.e. circle-rotation)-covariant differential calculi are  parametrized by 
  
    
      
        q
        ≠
        0
        ∈
        
          C
        
      
    
    {\displaystyle q\neq 0\in \mathbb {C} }
  
 and take the form 
  
    
      
        
          Ω
          
            1
          
        
        =
        
          
            C
          
        
        .
        
          
            d
          
        
        t
        ,
        
        (
        
          
            d
          
        
        t
        )
        f
        (
        t
        )
        =
        f
        (
        q
        t
        )
        (
        
          
            d
          
        
        t
        )
        ,
        
        
          
            d
          
        
        f
        =
        
          
            
              f
              (
              q
              t
              )
              −
              f
              (
              t
              )
            
            
              q
              (
              t
              −
              1
              )
            
          
        
        
        
          
            d
            t
          
        
      
    
    {\displaystyle \Omega ^{1}={\mathbb {C} }.{\rm {d}}t,\quad ({\rm {d}}t)f(t)=f(qt)({\rm {d}}t),\quad {\rm {d}}f={f(qt)-f(t) \over q(t-1)}\,{\rm {dt}}}
  
 This shows how 
  
    
      
        q
      
    
    {\displaystyle q}
  
-differentials arise naturally in quantum geometry.</li>
<li>For any algebra 
  
    
      
        A
      
    
    {\displaystyle A}
  
 one has a universal differential calculus defined by 
  
    
      
        
          Ω
          
            1
          
        
        =
        ker
        ⁡
        (
        m
        :
        A
        ⊗
        A
        →
        A
        )
        ,
        
        
          
            d
          
        
        a
        =
        1
        ⊗
        a
        −
        a
        ⊗
        1
        ,
        
        ∀
        a
        ∈
        A
      
    
    {\displaystyle \Omega ^{1}=\ker(m:A\otimes A\to A),\quad {\rm {d}}a=1\otimes a-a\otimes 1,\quad \forall a\in A}
  
 where 
  
    
      
        m
      
    
    {\displaystyle m}
  
 is the algebra product. By axiom 3., any first order calculus is a quotient of this.</li></ol>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Quantum_geometry/u2kx8MJY">Quantum geometry</a></li>
<li><a href="/facts/Noncommutative_geometry/WSwggL1p">Noncommutative geometry</a></li>
<li><a href="/facts/Quantum_calculus/oeuJkhgE">Quantum calculus</a></li>
<li><a href="/facts/Quantum_group/huNTuaXs">Quantum group</a></li>
<li><a href="/facts/Quantum_spacetime/ExKmzFzQ">Quantum spacetime</a></li></ul>
<h2 id="further-reading">Further reading</h2>
<ul><li>Connes, A. (1994), <a href="https://archive.org/details/noncommutativege0000conn"><i>Noncommutative geometry</i></a>, <a href="/facts/Academic_Press/qxoKjzFy">Academic Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-12-185860-X</li>
<li>Majid, S. (2002), <i>A quantum groups primer</i>, London Mathematical Society Lecture Note Series, vol. 292, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1017%2FCBO9780511549892">10.1017/CBO9780511549892</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-01041-2, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1904789">1904789</a></li></ul>

Quantum differential calculus open-in-new

Quantum differential calculus