Quantum differential calculus

In <a href="/facts/Quantum_geometry/u2kx8MJY">quantum geometry</a> or <a href="/facts/Noncommutative_geometry/WSwggL1p">noncommutative geometry</a> a quantum differential calculus or noncommutative differential structure on an algebra 
 
 
 
 A
 
 
 {\displaystyle A}
 
 over a field 
 
 
 
 k
 
 
 {\displaystyle k}
 
 means the specification of a space of <a href="/facts/Differential_form/T9gyHtMv">differential forms</a> over the algebra. The algebra 
 
 
 
 A
 
 
 {\displaystyle A}
 
 here is regarded as a <a href="/facts/Coordinate_ring/WVynz2Kn">coordinate ring</a> but it is important that it may be noncommutative and hence not an actual algebra of coordinate functions on any actual space, so this represents a point of view replacing the specification of a differentiable structure for an actual space. In ordinary differential geometry one can multiply differential 1-forms by functions from the left and from the right, and there exists an exterior derivative. Correspondingly, a first order quantum differential calculus means at least the following:

<ol><li>An 
 
 
 
 A
 
 
 {\displaystyle A}
 
-
 
 
 
 A
 
 
 {\displaystyle A}
 
-bimodule 
 
 
 
 
 Ω
 
 1
 
 
 
 
 {\displaystyle \Omega ^{1}}
 
 over 
 
 
 
 A
 
 
 {\displaystyle A}
 
, i.e. one can multiply elements of 
 
 
 
 
 Ω
 
 1
 
 
 
 
 {\displaystyle \Omega ^{1}}
 
 by elements of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 in an associative way: 
 
 
 
 a
 (
 ω
 b
 )
 =
 (
 a
 ω
 )
 b
 ,
  
 ∀
 a
 ,
 b
 ∈
 A
 ,
  
 ω
 ∈
 
 Ω
 
 1
 
 
 .
 
 
 {\displaystyle a(\omega b)=(a\omega )b,\ \forall a,b\in A,\ \omega \in \Omega ^{1}.}
 
</li>
<li>A linear map 
 
 
 
 
 
 d
 
 
 :
 A
 →
 
 Ω
 
 1
 
 
 
 
 {\displaystyle {\rm {d}}:A\to \Omega ^{1}}
 
 obeying the Leibniz rule 
 
 
 
 
 
 d
 
 
 (
 a
 b
 )
 =
 a
 (
 
 
 d
 
 
 b
 )
 +
 (
 
 
 d
 
 
 a
 )
 b
 ,
  
 ∀
 a
 ,
 b
 ∈
 A
 
 
 {\displaystyle {\rm {d}}(ab)=a({\rm {d}}b)+({\rm {d}}a)b,\ \forall a,b\in A}
 
</li>
<li>
 
 
 
 
 Ω
 
 1
 
 
 =
 {
 a
 (
 
 
 d
 
 
 b
 )
  
 
 |
 
  
 a
 ,
 b
 ∈
 A
 }
 
 
 {\displaystyle \Omega ^{1}=\{a({\rm {d}}b)\ |\ a,b\in A\}}
 
</li>
<li>(optional connectedness condition) 
 
 
 
 ker
 ⁡
  
 
 
 d
 
 
 =
 k
 1
 
 
 {\displaystyle \ker \ {\rm {d}}=k1}
 
</li></ol>
The last condition is not always imposed but holds in ordinary geometry when the manifold is connected. It says that the only functions killed by 
 
 
 
 
 
 d
 
 
 
 
 {\displaystyle {\rm {d}}}
 
 are constant functions.
An <a href="/facts/Exterior_algebra/rdN4TvpR">exterior algebra</a> or differential <a href="/facts/Graded_algebra/AIUjkSaP">graded algebra</a> structure over 
 
 
 
 A
 
 
 {\displaystyle A}
 
 means a compatible extension of 
 
 
 
 
 Ω
 
 1
 
 
 
 
 {\displaystyle \Omega ^{1}}
 
 to include analogues of higher order differential forms

 
 
 
 Ω
 =
 
 ⊕
 
 n
 
 
 
 Ω
 
 n
 
 
 ,
  
 
 
 d
 
 
 :
 
 Ω
 
 n
 
 
 →
 
 Ω
 
 n
 +
 1
 
 
 
 
 {\displaystyle \Omega =\oplus _{n}\Omega ^{n},\ {\rm {d}}:\Omega ^{n}\to \Omega ^{n+1}}

obeying a graded-Leibniz rule with respect to an associative product on 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
 and obeying 
 
 
 
 
 
 
 d
 
 
 
 2
 
 
 =
 0
 
 
 {\displaystyle {\rm {d}}^{2}=0}
 
. Here 
 
 
 
 
 Ω
 
 0
 
 
 =
 A
 
 
 {\displaystyle \Omega ^{0}=A}
 
 and it is usually required that 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
 is generated by 
 
 
 
 A
 ,
 
 Ω
 
 1
 
 
 
 
 {\displaystyle A,\Omega ^{1}}
 
. The product of differential forms is called the <a href="/facts/Wedge_product/rdN4TvpR">exterior or wedge product</a> and often denoted 
 
 
 
 ∧
 
 
 {\displaystyle \wedge }
 
. The noncommutative or quantum <a href="/facts/De_Rham_cohomology/4qGKeJq0">de Rham cohomology</a> is defined as the cohomology of this complex.
A higher order differential calculus can mean an exterior algebra, or it can mean the partial specification of one, up to some highest degree, and with products that would result in a degree beyond the highest being unspecified.
The above definition lies at the crossroads of two approaches to noncommutative geometry. In the Connes approach a more fundamental object is a replacement for the <a href="/facts/Dirac_operator/1kdGnoTt">Dirac operator</a> in the form of a <a href="/facts/Spectral_triple/jsvcMVGj">spectral triple</a>, and an exterior algebra can be constructed from this data. In the <a href="/facts/Quantum_group/huNTuaXs">quantum groups</a> approach to noncommutative geometry one starts with the algebra and a choice of first order calculus but constrained by covariance under a quantum group symmetry.

Quantum differential calculus open-in-new

Quantum differential calculus