Quantum differential calculus

In quantum geometry or noncommutative geometry 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 differential forms over the algebra. The algebra  A {\displaystyle A} 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 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:

1. 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}.}
2. 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}
3. Ω 1 = { a ( d b ) | a , b ∈ A } {\displaystyle \Omega ^{1}=\\{a({\rm {d}}b)\ |\ a,b\in A\\}}
4. (optional connectedness condition)  ker ⁡ d = k 1 {\displaystyle \ker \ {\rm {d}}=k1}

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 exterior algebra or differential graded algebra 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 exterior or wedge product and often denoted  ∧ {\displaystyle \wedge } . 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 groups 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

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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 not demand that

a ( d b ) = ( d b ) a , ∀ a , b ∈ A {\displaystyle a({\rm {d}}b)=({\rm {d}}b)a,\ \forall a,b\in A}

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 Lie algebra theory).

## Examples

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1. 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.
2. 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.
3. 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.

## See also

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* Quantum geometry
* Noncommutative geometry
* Quantum calculus
* Quantum group
* Quantum spacetime

## Further reading

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* Connes, A. (1994), Noncommutative geometry, Academic Press, ISBN 0-12-185860-X
* Majid, S. (2002), A quantum groups primer, London Mathematical Society Lecture Note Series, vol. 292, Cambridge University Press, doi:10.1017/CBO9780511549892, ISBN 978-0-521-01041-2, MR 1904789

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