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In quantum geometry or noncommutative geometry a quantum differential calculus or noncommutative differential structure on an algebra A over a field k means the specification of a space of differential forms over the algebra. The algebra 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 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:

1. An AA-bimodule Ω1 over A, i.e. one can multiply elements of Ω1 by elements of A in an associative way:

a(ωb)=(aω)b,a,bA,ωΩ1.

2. A linear map d:AΩ1 obeying the Leibniz rule

d(ab)=a(db)+(da)b,a,bA

3. Ω1={a(db)|a,bA}

4. (optional connectedness condition) kerd=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 are constant functions.

An exterior algebra or differential graded algebra structure over A means a compatible extension of Ω1 to include analogues of higher order differential forms

Ω=nΩn,d:ΩnΩn+1

obeying a graded-Leibniz rule with respect to an associative product on Ω and obeying d2=0. Here Ω0=A and it is usually required that Ω is generated by A,Ω1. The product of differential forms is called the exterior or wedge product and often denoted . 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

The above definition is minimal and gives something more general than classical differential calculus even when the algebra A is commutative or functions on an actual space. This is because we do not demand that

a(db)=(db)a,a,bA

since this would imply that d(abba)=0,a,bA, 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 A=[x] the algebra of polynomials in one variable the translation-covariant quantum differential calculi are parametrized by λ and take the form

Ω1=.dx,(dx)f(x)=f(x+λ)(dx),df=f(x+λ)f(x)λdx

This shows how finite differences arise naturally in quantum geometry. Only the limit λ0 has functions commuting with 1-forms, which is the special case of high school differential calculus.

2. For A=[t,t1] the algebra of functions on an algebraic circle, the translation (i.e. circle-rotation)-covariant differential calculi are parametrized by q0 and take the form

Ω1=.dt,(dt)f(t)=f(qt)(dt),df=f(qt)f(t)q(t1)dt

This shows how q-differentials arise naturally in quantum geometry.

3. For any algebra A one has a universal differential calculus defined by

Ω1=ker(m:AAA),da=1aa1,aA

where m is the algebra product. By axiom 3., any first order calculus is a quotient of this.

See also

Further reading

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