In quantum geometry or noncommutative geometry a quantum differential calculus or noncommtuative 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 `coordinate algebra' 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 for an actual space of a differentiable structure. In ordinary differential geometry one can multiply differential 1-forms by functions form the left and the right and has an exterior derivative. Correspondingly, a first order quantum differential calculus means at least the following:
1. An A − A-bimodule Ω1 over A, i.e. one can multiply elements elements of Ω1 by elements of A in an associative way:
2. A linear map obeying the Leibniz rule
4. (optional connectedness condition)
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 are constant functions.
An exterior algebra or differential graded algebra structure over A means a compatible extension of Ω1 to include analogues of higher order differential forms
obeying a graded-Leibniz rule with respect to an associative product on Ω and obeying d2 = 0. Here Ω0 = A and it is usually required that Ω is generated by A,Ω1. The product of differential forms is called the exterior or `wedge' product and often denoted . 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.
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
since this would imply that , 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).
1. For the algebra of polynomials in one variable the translation-covariant quantum differential calculi are parametrized by and take the form
This shows how finite differences arise naturally in quantum geometry. Only the limit has functions commuting with 1-forms, which is the special case of high school differential calculus.
2. For the algebra of functions on an algebraic circle, the translation (i.e. circle-rotation)-covariant differential calculi are parametrized by and take the form
This shows how q-differentials arise naturally in quantum geometry.
3. For any algebra A one has a universal differential calculus defined by
where m is the algebra product. By axiom 3., any first order calculus us a quotient of this.