# 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:
* 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: $a(\omega b)=(a\omega )b,\ \forall a,b\in A,\ \omega \in \Omega ^{1}.$
* A linear map ${\rm {d}}:A\to \Omega ^{1}$ obeying the Leibniz rule ${\rm {d}}(ab)=a({\rm {d}}b)+({\rm {d}}a)b,\ \forall a,b\in A$
* $\Omega ^{1}=\{a({\rm {d}}b)\ |\ a,b\in A\}$
* (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

* 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.
* 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.
* 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.