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{{Expert-subject|Mathematics|date=March 2012}}
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In [[quantum geometry]] or [[noncommutative geometry]] a '''quantum differential calculus''' or '''noncommutative differential structure''' on an algebra <math>A</math> over a field <math>k</math> means the specification of a space of [[differential form]]s over the algebra.  The algebra <math>A</math> here is regarded as a [[coordinate ring]] 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 for an actual space of a differentiable structure. In ordinary differential geometry one can multiply differential 1-forms by functions form the left and the right and has an exterior derivative. Correspondingly, a first order quantum differential calculus means at least the following:
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1. An <math>A-A</math>-bimodule <math>\Omega^1</math> over <math>A</math>, i.e. one can multiply elements of <math>\Omega^1</math> by elements of <math>A</math> in an associative way:
 
: <math> a(\omega b)=(a\omega)b,\ \forall a,b\in A,\ \omega\in\Omega^1</math>.
 
2. A linear map <math>{\rm d}:A\to\Omega^1</math> obeying the Leibniz rule
: <math>{\rm d}(ab)=a({\rm d}b)+({\rm d}a)b,\ \forall a,b\in A</math>
 
3. <math>\Omega^1=\{a({\rm d}b)\ |\ a,b\in A\}</math>
 
4. (optional connectedness condition) <math>\ker\ {\rm d}=k1</math>
 
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 <math>{\rm d}</math> are constant functions.
 
An ''[[exterior algebra]]'' or ''differential [[graded algebra]]'' structure over <math>A</math> means a compatible extension of <math>\Omega^1</math> to include analogues of higher order differential forms
 
: <math>\Omega=\oplus_n\Omega^n,\ {\rm d}:\Omega^n\to\Omega^{n+1}</math>
 
obeying a graded-Leibniz rule with respect to an associative product on <math>\Omega</math> and obeying <math>{\rm d}^2=0</math>. Here <math>\Omega^0=A</math> and it is usually required that <math>\Omega</math> is generated by <math>A,\Omega^1</math>. The product of differential forms is called the [[wedge product|exterior or wedge product]] and often denoted <math>\wedge</math>. The noncommutative or quantum [[de Rham cohomology]] 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 [[Dirac operator]] in the form of a [[spectral triple]], and an exterior algebra can be constructed from this data. In the [[quantum group]]s 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.
 
==Note==
The above definition is minimal and gives something more general than classical differential calculus even when the algebra <math>A</math> is commutative or functions on an actual space. This is because we do ''not'' demand that
 
: <math>a({\rm d}b)=({\rm db})a,\ \forall a,b\in A</math>
 
since this would imply that <math>{\rm d}(ab-ba)=0,\ \forall a,b\in A</math>, 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 [[Lie algebra]] theory).
 
==Examples==
1. For <math>A={\Bbb C}[x]</math> the algebra of polynomials in one variable the translation-covariant quantum differential calculi are parametrized by <math>\lambda\in \Bbb C</math> and take the form
 
: <math> \Omega^1={\Bbb 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</math>
 
This shows how finite differences arise naturally in quantum geometry. Only the limit <math>\lambda\to 0</math> has functions commuting with 1-forms, which is the special case of high school differential calculus.
 
2. For <math>A={\Bbb C}[t,t^{-1}]</math> the algebra of functions on an algebraic circle, the translation (i.e. circle-rotation)-covariant differential calculi are  parametrized by <math>q\ne 0\in \Bbb C</math> and take the form
 
: <math> \Omega^1={\Bbb 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}</math>
 
This shows how <math>q</math>-differentials arise naturally in quantum geometry.
 
3. For any algebra <math>A</math> one has a '''universal differential calculus''' defined by
 
: <math>\Omega^1=\ker(m:A\otimes A\to A),\quad {\rm d}a=1\otimes a-a\otimes 1,\quad\forall a\in A</math>
 
where <math>m</math> is the algebra product. By axiom 3., any first order calculus is a quotient of this.
 
==See also==
*[[Quantum geometry]]
*[[Noncommutative geometry]]
*[[Quantum calculus]]
*[[Quantum group]]
*[[Quantum spacetime]]
 
==Further reading==
*{{citation| first=A.|last=Connes|title=Noncommutative geometry|publisher=[[Academic Press]]| year=1994| isbn=0-12-185860-X}}
*{{citation | last1=Majid | first1=S. | title=A quantum groups primer | publisher=[[Cambridge University Press]] | series=London Mathematical Society Lecture Note Series | isbn=978-0-521-01041-2 | id={{MathSciNet | id = 1904789}} | year=2002 | volume=292}}
 
[[Category:Noncommutative geometry]]
[[Category:Algebraic structures]]

Latest revision as of 09:57, 1 March 2014

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