Quantum differential calculus

In quantum geometry or noncommutative geometry a quantum differential calculus or noncommutative differential structure on an algebra formula_1 over a field formula_2 means the specification of a space of differential forms over the algebra. The algebra formula_1 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 formula_1-formula_1-bimodule formula_6 over formula_1, i.e. one can multiply elements of formula_6 by elements of formula_1 in an associative way:
2. A linear map formula_11 obeying the Leibniz rule
3. formula_13
4. (optional connectedness condition) formula_14
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 formula_15 are constant functions.
An "exterior algebra" or "differential graded algebra" structure over formula_1 means a compatible extension of formula_6 to include analogues of higher order differential forms
obeying a graded-Leibniz rule with respect to an associative product on formula_19 and obeying formula_20. Here formula_21 and it is usually required that formula_19 is generated by formula_23. The product of differential forms is called the exterior or wedge product and often denoted formula_24. 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 formula_1 is commutative or functions on an actual space. This is because we do "not" demand that
since this would imply that formula_27, 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 formula_28 the algebra of polynomials in one variable the translation-covariant quantum differential calculi are parametrized by formula_29 and take the form
This shows how finite differences arise naturally in quantum geometry. Only the limit formula_31 has functions commuting with 1-forms, which is the special case of high school differential calculus.
2. For formula_32 the algebra of functions on an algebraic circle, the translation (i.e. circle-rotation)-covariant differential calculi are parametrized by formula_33 and take the form
This shows how formula_35-differentials arise naturally in quantum geometry.
3. For any algebra formula_1 one has a universal differential calculus defined by
where formula_38 is the algebra product. By axiom 3., any first order calculus is a quotient of this.