Quantum differential calculus

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(\omega b)=(a\omega )b,\ \forall a,b\in A,\ \omega \in \Omega ^{1}

2. A linear map ${{\rm {d}}}:A\to \Omega ^{1}$ obeying the Leibniz rule

{{\rm {d}}}(ab)=a({{\rm {d}}}b)+({{\rm {d}}}a)b,\ \forall a,b\in A

3. $\Omega ^{1}=\{a({{\rm {d}}}b)\ |\ a,b\in A\}$

4. (optional connectedness condition) $\ker \ {{\rm {d}}}=k1$

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 ${{\rm {d}}}$ 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

\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 $\Omega$ 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({{\rm {d}}}b)=({{\rm {db}}})a,\ \forall a,b\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 $A={{\mathbb C}}[x]$ the algebra of polynomials in one variable the translation-covariant quantum differential calculi are parametrized by $\lambda \in {\mathbb C}$ and take the form

\Omega ^{1}={{\mathbb C}}.{{\rm {d}}}x,\quad ({{\rm {d}}}x)f(x)=f(x+\lambda )({{\rm {d}}}x),\quad {{\rm {d}}}f={f(x+\lambda )-f(x) \over \lambda }{{\rm {d}}}x

This shows how finite differences arise naturally in quantum geometry. Only the limit $\lambda \to 0$ has functions commuting with 1-forms, which is the special case of high school differential calculus.

2. For $A={{\mathbb C}}[t,t^{{-1}}]$ the algebra of functions on an algebraic circle, the translation (i.e. circle-rotation)-covariant differential calculi are parametrized by $q\neq 0\in {\mathbb C}$ and take the form

\Omega ^{1}={{\mathbb C}}.{{\rm {d}}}t,\quad ({{\rm {d}}}t)f(t)=f(qt)({{\rm {d}}}t),\quad {{\rm {d}}}f={f(qt)-f(t) \over q(t-1)}\,{{\rm {dt}}}

This shows how $q$ -differentials arise naturally in quantum geometry.

3. For any algebra $A$ one has a universal differential calculus defined by

\Omega ^{1}=\ker(m:A\otimes A\to A),\quad {{\rm {d}}}a=1\otimes a-a\otimes 1,\quad \forall a\in A

where $m$ is the algebra product. By axiom 3., any first order calculus is a quotient of this.

Quantum differential calculus

Note

Examples

See also

Further reading