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 ${\mathrm{\Omega }}^1$ over $A$, i.e. one can multiply elements of ${\mathrm{\Omega }}^1$ by elements of $A$ in an associative way: $$a(\omega b)=(a\omega )b,\mathrm{\forall }a,b\in A,\omega \in {\mathrm{\Omega }}^1.$$
2. A linear map $\mathrm{d}:A\to {\mathrm{\Omega }}^1$ obeying the Leibniz rule $$\mathrm{d}(ab)=a(\mathrm{d}b)+(\mathrm{d}a)b,\mathrm{\forall }a,b\in A$$
3. ${\mathrm{\Omega }}^1=\{a(\mathrm{d}b)|a,b\in A\}$
4. (optional connectedness condition) $\ker \mathrm{d}=k1$

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 $\mathrm{d}$ are constant functions.

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

$$\mathrm{\Omega }={\oplus }_n{\mathrm{\Omega }}^n,\mathrm{d}:{\mathrm{\Omega }}^n\to {\mathrm{\Omega }}^{n+1}$$

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

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

since this would imply that $\mathrm{d}(ab-ba)=0,\mathrm{\forall }a,b\in A$, 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 $A=ℂ[x]$ the algebra of polynomials in one variable the translation-covariant quantum differential calculi are parametrized by $\lambda \in ℂ$ and take the form $${\mathrm{\Omega }}^1=ℂ.\mathrm{d}x,(\mathrm{d}x)f(x)=f(x+\lambda )(\mathrm{d}x),\mathrm{d}f=\frac{f(x+\lambda )-f(x)}{\lambda }\mathrm{d}x$$ 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 $A=ℂ[t,t^{-1}]$ the algebra of functions on an algebraic circle, the translation (i.e. circle-rotation)-covariant differential calculi are parametrized by $q\ne 0\in ℂ$ and take the form $${\mathrm{\Omega }}^1=ℂ.\mathrm{d}t,(\mathrm{d}t)f(t)=f(qt)(\mathrm{d}t),\mathrm{d}f=\frac{f(qt)-f(t)}{q(t-1)}\mathrm{d}\mathrm{t}$$ This shows how $q$-differentials arise naturally in quantum geometry.
3. For any algebra $A$ one has a universal differential calculus defined by $${\mathrm{\Omega }}^1=\ker (m:A\otimes A\to A),\mathrm{d}a=1\otimes a-a\otimes 1,\mathrm{\forall }a\in A$$ 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

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