In quantum geometry or noncommutative geometry a quantum differential calculus or noncommutative differential structure on an algebra A {\displaystyle A} over a field k {\displaystyle k} means the specification of a space of differential forms over the algebra. The algebra A {\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:
An A {\displaystyle A} - A {\displaystyle A} -bimodule Ω 1 {\displaystyle \Omega ^{1}} over A {\displaystyle A} , i.e. one can multiply elements of Ω 1 {\displaystyle \Omega ^{1}} by elements of A {\displaystyle A} in an associative way: a ( ω b ) = ( a ω ) b , ∀ a , b ∈ A , ω ∈ Ω 1 . {\displaystyle a(\omega b)=(a\omega )b,\ \forall a,b\in A,\ \omega \in \Omega ^{1}.}
A linear map d : A → Ω 1 {\displaystyle {\rm {d}}:A\to \Omega ^{1}} obeying the Leibniz rule d ( a b ) = a ( d b ) + ( d a ) b , ∀ a , b ∈ A {\displaystyle {\rm {d}}(ab)=a({\rm {d}}b)+({\rm {d}}a)b,\ \forall a,b\in A}
Ω 1 = { a ( d b ) | a , b ∈ A } {\displaystyle \Omega ^{1}=\{a({\rm {d}}b)\ |\ a,b\in A\}}
(optional connectedness condition) ker d = k 1 {\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 d {\displaystyle {\rm {d}}} are constant functions. An exterior algebra or differential graded algebra structure over A {\displaystyle A} means a compatible extension of Ω 1 {\displaystyle \Omega ^{1}} to include analogues of higher order differential forms
Ω = ⊕ n Ω n , d : Ω n → Ω n + 1 {\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 Ω {\displaystyle \Omega } and obeying d 2 = 0 {\displaystyle {\rm {d}}^{2}=0} . Here Ω 0 = A {\displaystyle \Omega ^{0}=A} and it is usually required that Ω {\displaystyle \Omega } is generated by A , Ω 1 {\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.
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