Quantum differential calculus

Inquantum geometryornoncommutative geometryaquantum differential calculusornoncommutative differential structureon an algebra$A$over a field$k$means the specification of a space ofdifferential formsover the algebra. The algebra$A$here is regarded as acoordinate ringbut 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$A$-$A$-bimodule$\Omega ^{1}$over$A$,i.e. one can multiply elements of$\Omega ^{1}$by elements of$A$in an associative way:$$a(\omega b)=(a\omega )b,\ \forall a,b\in A,\ \omega \in \Omega ^{1}.$$
2. A linear map${\rm {d}}:A\to \Omega ^{1}$obeying the Leibniz rule$${\rm {d}}(ab)=a({\rm {d}}b)+({\rm {d}}a)b,\ \forall a,b\in A$$
3. $\Omega ^{1}=\{a({\rm {d}}b)\ |\ a,b\in A\}$
4. (optional connectedness condition)$\ker \ {\rm {d}}=k1$

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${\rm {d}}$are constant functions.

Anexterior algebraordifferentialgraded algebrastructure over$A$means a compatible extension of$\Omega ^{1}$to include analogues of higher order differential forms

$$\Omega =\oplus _{n}\Omega ^{n},\ {\rm {d}}:\Omega ^{n}\to \Omega ^{n+1}$$

obeying a graded-Leibniz rule with respect to an associative product on$\Omega$and obeying${\rm {d}}^{2}=0$.Here$\Omega ^{0}=A$and it is usually required that$\Omega$is generated by$A,\Omega ^{1}$.The product of differential forms is called theexterior or wedge productand often denoted$\wedge$.The noncommutative or quantumde Rham cohomologyis 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 theDirac operatorin the form of aspectral triple,and an exterior algebra can be constructed from this data. In thequantum groupsapproach 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 donotdemand that

$$a({\rm {d}}b)=({\rm {d}}b)a,\ \forall a,b\in A$$

since this would imply that${\rm {d}}(ab-ba)=0,\ \forall a,b\in A$,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 groupLie algebratheory).

Examples

1. For$A={\mathbb {C} }[x]$the algebra of polynomials in one variable the translation-covariant quantum differential calculi are parametrized by$\lambda \in \mathbb {C}$and take the form$$\Omega ^{1}={\mathbb {C} }.{\rm {d}}x,\quad ({\rm {d}}x)f(x)=f(x+\lambda )({\rm {d}}x),\quad {\rm {d}}f={f(x+\lambda )-f(x) \over \lambda }{\rm {d}}x$$This shows how finite differences arise naturally in quantum geometry. Only the limit$\lambda \to 0$has functions commuting with 1-forms, which is the special case of high school differential calculus.
2. For$A={\mathbb {C} }[t,t^{-1}]$the algebra of functions on an algebraic circle, the translation (i.e. circle-rotation)-covariant differential calculi are parametrized by$q\neq 0\in \mathbb {C}$and take the form$$\Omega ^{1}={\mathbb {C} }.{\rm {d}}t,\quad ({\rm {d}}t)f(t)=f(qt)({\rm {d}}t),\quad {\rm {d}}f={f(qt)-f(t) \over q(t-1)}\,{\rm {dt}}$$This shows how$q$-differentials arise naturally in quantum geometry.
3. For any algebra$A$one has auniversal differential calculusdefined by$$\Omega ^{1}=\ker(m:A\otimes A\to A),\quad {\rm {d}}a=1\otimes a-a\otimes 1,\quad \forall a\in A$$where$m$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

• Connes, A. (1994),Noncommutative geometry,Academic Press,ISBN0-12-185860-X
• Majid, S. (2002),A quantum groups primer,London Mathematical Society Lecture Note Series, vol. 292,Cambridge University Press,doi:10.1017/CBO9780511549892,ISBN978-0-521-01041-2,MR1904789