Quantum differential calculus

In quantum geometry or noncommutative geometry a quantum differential calculus or noncommutative differential structure on an algebra ${\displaystyle A}$ over a field ${\displaystyle k}$ means the specification of a space of differential forms over the algebra. The algebra ${\displaystyle A}$ 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 ${\displaystyle A}$-${\displaystyle A}$-bimodule ${\displaystyle \Omega ^{1}}$ over ${\displaystyle A}$, i.e. one can multiply elements of ${\displaystyle \Omega ^{1}}$ by elements of ${\displaystyle A}$ in an associative way: ${\displaystyle a(\omega b)=(a\omega )b,\ \forall a,b\in A,\ \omega \in \Omega ^{1}.}$
2. A linear map ${\displaystyle {\rm {d}}:A\to \Omega ^{1}}$ obeying the Leibniz rule ${\displaystyle {\rm {d}}(ab)=a({\rm {d}}b)+({\rm {d}}a)b,\ \forall a,b\in A}$
3. ${\displaystyle \Omega ^{1}=\{a({\rm {d}}b)\ |\ a,b\in A\}}$
4. (optional connectedness condition) ${\displaystyle \ker \ {\rm {d}}=k1}$

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

An exterior algebra or differential graded algebra structure over ${\displaystyle A}$ means a compatible extension of ${\displaystyle \Omega ^{1}}$ to include analogues of higher order differential forms

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

obeying a graded-Leibniz rule with respect to an associative product on ${\displaystyle \Omega }$ and obeying ${\displaystyle {\rm {d}}^{2}=0}$. Here ${\displaystyle \Omega ^{0}=A}$ and it is usually required that ${\displaystyle \Omega }$ is generated by ${\displaystyle A,\Omega ^{1}}$. 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

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

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

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

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