Quantum differential calculus

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 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.

An exterior algebra or differential graded algebra structure 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 the exterior or wedge product and often denoted $$\wedge$$. 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
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

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

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 group Lie algebra theory).

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 $$q\ne 0\in \mathbb C$$ 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 a universal differential calculus defined 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.