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
Ω
1
over
A
, i.e. one can multiply elements of
Ω
1
by elements of
A
in an associative way:
a
(
ω
b
)
=
(
a
ω
)
b
,
∀
a
,
b
∈
A
,
ω
∈
Ω
1
.
2. A linear map
d
:
A
→
Ω
1
obeying the Leibniz rule
d
(
a
b
)
=
a
(
d
b
)
+
(
d
a
)
b
,
∀
a
,
b
∈
A
3.
Ω
1
=
{
a
(
d
b
)
|
a
,
b
∈
A
}
4. (optional connectedness condition)
ker
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
d
are constant functions.
An exterior algebra or differential graded algebra structure over
A
means a compatible extension of
Ω
1
to include analogues of higher order differential forms
Ω
=
⊕
n
Ω
n
,
d
:
Ω
n
→
Ω
n
+
1
obeying a graded-Leibniz rule with respect to an associative product on
Ω
and obeying
d
2
=
0
. Here
Ω
0
=
A
and it is usually required that
Ω
is generated by
A
,
Ω
1
. 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
a
(
d
b
)
=
(
d
b
)
a
,
∀
a
,
b
∈
A
since this would imply that
d
(
a
b
−
b
a
)
=
0
,
∀
a
,
b
∈
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).
1. For
A
=
C
[
x
]
the algebra of polynomials in one variable the translation-covariant quantum differential calculi are parametrized by
λ
∈
C
and take the form
Ω
1
=
C
.
d
x
,
(
d
x
)
f
(
x
)
=
f
(
x
+
λ
)
(
d
x
)
,
d
f
=
f
(
x
+
λ
)
−
f
(
x
)
λ
d
x
This shows how finite differences arise naturally in quantum geometry. Only the limit
λ
→
0
has functions commuting with 1-forms, which is the special case of high school differential calculus.
2. 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
≠
0
∈
C
and take the form
Ω
1
=
C
.
d
t
,
(
d
t
)
f
(
t
)
=
f
(
q
t
)
(
d
t
)
,
d
f
=
f
(
q
t
)
−
f
(
t
)
q
(
t
−
1
)
d
t
This shows how
q
-differentials arise naturally in quantum geometry.
3. For any algebra
A
one has a universal differential calculus defined by
Ω
1
=
ker
(
m
:
A
⊗
A
→
A
)
,
d
a
=
1
⊗
a
−
a
⊗
1
,
∀
a
∈
A
where
m
is the algebra product. By axiom 3., any first order calculus is a quotient of this.
