Quantum differential calculus explained

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 \Omega ^{1} over A , i.e. one can multiply elements of \Omega ^{1} by elements of A in an associative way: $$a(\backslash omega\; b)=(a\backslash omega)b,\backslash \; \backslash forall\; a,b\backslash in\; A,\backslash \; \backslash omega\backslash in\backslash Omega^1\; .$$

1. A linear map

{\rm d}:A\to\Omega ^{1} obeying the Leibniz rule $$(ab)=a(b)\; +\; (a)b,\backslash \; \backslash forall\; a,b\backslash in\; A$$\Omega ^{1=\{a({\rm} d}b) | a,b\in A\}

1. (optional connectedness condition)

\ker {\rm d} = k 1

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

{\rm d} are constant functions.

An exterior algebra or differential graded algebra structure over

A means a compatible extension of \Omega ^{1} to include analogues of higher order differential forms

$$\backslash Omega=\backslash oplus\_n\backslash Omega^n,\backslash \; :\backslash Omega^n\backslash to\backslash Omega^$$

obeying a graded-Leibniz rule with respect to an associative product on

\Omega and obeying {\rm d} ^{2=0} . Here \Omega ^{0=A} and it is usually required that \Omega is generated by A,\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

$$a(b)\; =\; (b)a,\backslash \; \backslash forall\; a,b\backslash in\; A$$

since this would imply that

{\rm d}(ab - ba) =0 , \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 λ \in C and take the form $$\backslash Omega^1=.x,\backslash quad\; (x)f(x)=f(x+\backslash lambda)(x),\backslash quad\; f=x$$ This shows how finite differences arise naturally in quantum geometry. Only the limit λ \to 0 has functions commuting with 1-forms, which is the special case of high school differential calculus.

1. 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 $$\backslash Omega^1=.t,\backslash quad\; (t)f(t)=f(qt)(t),\backslash quad\; f=\backslash ,$$ This shows how q -differentials arise naturally in quantum geometry.

1. For any algebra

A one has a universal differential calculus defined by $$\backslash Omega^1=\backslash ker(m:A\backslash otimes\; A\backslash to\; A),\backslash quad\; a=1\backslash otimes\; a-a\backslash otimes\; 1,\backslash quad\backslash forall\; a\backslash in\; A$$ 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