Inquantum geometry ornoncommutative geometry aquantum differential calculus ornoncommutative differential structure on an algebra${\displaystyle A}$ over a field${\displaystyle k}$ means the specification of a space ofdifferential forms over the algebra. The algebra${\displaystyle A}$ here is regarded as acoordinate 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${\displaystyle {\mathrm{\Omega }}^1}$ over${\displaystyle A}$, i.e. one can multiply elements of${\displaystyle {\mathrm{\Omega }}^1}$ by elements of${\displaystyle A}$ in an associative way:$${\displaystyle a(\omega b)=(a\omega )b,\mathrm{\forall }a,b\in A,\omega \in {\mathrm{\Omega }}^1.}$$
2. A linear map${\displaystyle \mathrm{d}:A\to {\mathrm{\Omega }}^1}$ obeying the Leibniz rule$${\displaystyle \mathrm{d}(ab)=a(\mathrm{d}b)+(\mathrm{d}a)b,\mathrm{\forall }a,b\in A}$$
3. ${\displaystyle {\mathrm{\Omega }}^1=\{a(\mathrm{d}b)|a,b\in A\}}$
4. (optional connectedness condition)${\displaystyle \ker \mathrm{d}=k1}$

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

Anexterior algebra ordifferentialgraded algebra structure over${\displaystyle A}$ means a compatible extension of${\displaystyle {\mathrm{\Omega }}^1}$ to include analogues of higher order differential forms

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

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

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 donot demand that

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

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

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$${\displaystyle {\mathrm{\Omega }}^1=\mathbb{C}.\mathrm{d}x,(\mathrm{d}x)f(x)=f(x+\lambda )(\mathrm{d}x),\mathrm{d}f=\frac{f(x+\lambda )-f(x)}{\lambda }\mathrm{d}x}$$ This shows how finite differences arise naturally in quantum geometry. Only the limit${\displaystyle \lambda \to 0}$ 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${\displaystyle q\ne 0\in \mathbb{C}}$ and take the form$${\displaystyle {\mathrm{\Omega }}^1=\mathbb{C}.\mathrm{d}t,(\mathrm{d}t)f(t)=f(qt)(\mathrm{d}t),\mathrm{d}f=\frac{f(qt)-f(t)}{q(t-1)}\mathrm{d}\mathrm{t}}$$ This shows how${\displaystyle q}$-differentials arise naturally in quantum geometry.
3. For any algebra${\displaystyle A}$ one has auniversal differential calculus defined by$${\displaystyle {\mathrm{\Omega }}^1=\ker (m:A\otimes A\to A),\mathrm{d}a=1\otimes a-a\otimes 1,\mathrm{\forall }a\in A}$$ where${\displaystyle m}$ is the algebra product. By axiom 3., any first order calculus is a quotient of this.

• Quantum geometry
• Noncommutative geometry
• Quantum calculus
• Quantum group
• Quantum spacetime

• Connes, A. (1994),Noncommutative geometry,Academic Press,ISBN 0-12-185860-X
• Majid, S. (2002),A quantum groups primer, London Mathematical Society Lecture Note Series, vol. 292,Cambridge University Press,doi:10.1017/CBO9780511549892,ISBN 978-0-521-01041-2,MR 1904789

• Noncommutative geometry
• Algebraic structures