- Source: Derivation (differential algebra)
In mathematics, a derivation is a function on an algebra that generalizes certain features of the derivative operator. Specifically, given an algebra A over a ring or a field K, a K-derivation is a K-linear map D : A → A that satisfies Leibniz's law:
D
(
a
b
)
=
a
D
(
b
)
+
D
(
a
)
b
.
{\displaystyle D(ab)=aD(b)+D(a)b.}
More generally, if M is an A-bimodule, a K-linear map D : A → M that satisfies the Leibniz law is also called a derivation. The collection of all K-derivations of A to itself is denoted by DerK(A). The collection of K-derivations of A into an A-module M is denoted by DerK(A, M).
Derivations occur in many different contexts in diverse areas of mathematics. The partial derivative with respect to a variable is an R-derivation on the algebra of real-valued differentiable functions on Rn. The Lie derivative with respect to a vector field is an R-derivation on the algebra of differentiable functions on a differentiable manifold; more generally it is a derivation on the tensor algebra of a manifold. It follows that the adjoint representation of a Lie algebra is a derivation on that algebra. The Pincherle derivative is an example of a derivation in abstract algebra. If the algebra A is noncommutative, then the commutator with respect to an element of the algebra A defines a linear endomorphism of A to itself, which is a derivation over K. That is,
[
F
G
,
N
]
=
[
F
,
N
]
G
+
F
[
G
,
N
]
,
{\displaystyle [FG,N]=[F,N]G+F[G,N],}
where
[
⋅
,
N
]
{\displaystyle [\cdot ,N]}
is the commutator with respect to
N
{\displaystyle N}
. An algebra A equipped with a distinguished derivation d forms a differential algebra, and is itself a significant object of study in areas such as differential Galois theory.
Properties
If A is a K-algebra, for K a ring, and D: A → A is a K-derivation, then
If A has a unit 1, then D(1) = D(12) = 2D(1), so that D(1) = 0. Thus by K-linearity, D(k) = 0 for all k ∈ K.
If A is commutative, D(x2) = xD(x) + D(x)x = 2xD(x), and D(xn) = nxn−1D(x), by the Leibniz rule.
More generally, for any x1, x2, …, xn ∈ A, it follows by induction that
D
(
x
1
x
2
⋯
x
n
)
=
∑
i
x
1
⋯
x
i
−
1
D
(
x
i
)
x
i
+
1
⋯
x
n
{\displaystyle D(x_{1}x_{2}\cdots x_{n})=\sum _{i}x_{1}\cdots x_{i-1}D(x_{i})x_{i+1}\cdots x_{n}}
which is
∑
i
D
(
x
i
)
∏
j
≠
i
x
j
{\textstyle \sum _{i}D(x_{i})\prod _{j\neq i}x_{j}}
if for all i, D(xi) commutes with
x
1
,
x
2
,
…
,
x
i
−
1
{\displaystyle x_{1},x_{2},\ldots ,x_{i-1}}
.
For n > 1, Dn is not a derivation, instead satisfying a higher-order Leibniz rule:
D
n
(
u
v
)
=
∑
k
=
0
n
(
n
k
)
⋅
D
n
−
k
(
u
)
⋅
D
k
(
v
)
.
{\displaystyle D^{n}(uv)=\sum _{k=0}^{n}{\binom {n}{k}}\cdot D^{n-k}(u)\cdot D^{k}(v).}
Moreover, if M is an A-bimodule, write
Der
K
(
A
,
M
)
{\displaystyle \operatorname {Der} _{K}(A,M)}
for the set of K-derivations from A to M.
DerK(A, M) is a module over K.
DerK(A) is a Lie algebra with Lie bracket defined by the commutator:
[
D
1
,
D
2
]
=
D
1
∘
D
2
−
D
2
∘
D
1
.
{\displaystyle [D_{1},D_{2}]=D_{1}\circ D_{2}-D_{2}\circ D_{1}.}
since it is readily verified that the commutator of two derivations is again a derivation.
There is an A-module ΩA/K (called the Kähler differentials) with a K-derivation d: A → ΩA/K through which any derivation D: A → M factors. That is, for any derivation D there is a A-module map φ with
D
:
A
⟶
d
Ω
A
/
K
⟶
φ
M
{\displaystyle D:A{\stackrel {d}{\longrightarrow }}\Omega _{A/K}{\stackrel {\varphi }{\longrightarrow }}M}
The correspondence
D
↔
φ
{\displaystyle D\leftrightarrow \varphi }
is an isomorphism of A-modules:
Der
K
(
A
,
M
)
≃
Hom
A
(
Ω
A
/
K
,
M
)
{\displaystyle \operatorname {Der} _{K}(A,M)\simeq \operatorname {Hom} _{A}(\Omega _{A/K},M)}
If k ⊂ K is a subring, then A inherits a k-algebra structure, so there is an inclusion
Der
K
(
A
,
M
)
⊂
Der
k
(
A
,
M
)
,
{\displaystyle \operatorname {Der} _{K}(A,M)\subset \operatorname {Der} _{k}(A,M),}
since any K-derivation is a fortiori a k-derivation.
Graded derivations
Given a graded algebra A and a homogeneous linear map D of grade |D| on A, D is a homogeneous derivation if
D
(
a
b
)
=
D
(
a
)
b
+
ε
|
a
|
|
D
|
a
D
(
b
)
{\displaystyle {D(ab)=D(a)b+\varepsilon ^{|a||D|}aD(b)}}
for every homogeneous element a and every element b of A for a commutator factor ε = ±1. A graded derivation is sum of homogeneous derivations with the same ε.
If ε = 1, this definition reduces to the usual case. If ε = −1, however, then
D
(
a
b
)
=
D
(
a
)
b
+
(
−
1
)
|
a
|
a
D
(
b
)
{\displaystyle {D(ab)=D(a)b+(-1)^{|a|}aD(b)}}
for odd |D|, and D is called an anti-derivation.
Examples of anti-derivations include the exterior derivative and the interior product acting on differential forms.
Graded derivations of superalgebras (i.e. Z2-graded algebras) are often called superderivations.
Related notions
Hasse–Schmidt derivations are K-algebra homomorphisms
A
→
A
[
[
t
]
]
.
{\displaystyle A\to A[[t]].}
Composing further with the map which sends a formal power series
∑
a
n
t
n
{\displaystyle \sum a_{n}t^{n}}
to the coefficient
a
1
{\displaystyle a_{1}}
gives a derivation.
See also
In differential geometry derivations are tangent vectors
Kähler differential
Hasse derivative
p-derivation
Wirtinger derivatives
Derivative of the exponential map
References
Bourbaki, Nicolas (1989), Algebra I, Elements of mathematics, Springer-Verlag, ISBN 3-540-64243-9.
Eisenbud, David (1999), Commutative algebra with a view toward algebraic geometry (3rd. ed.), Springer-Verlag, ISBN 978-0-387-94269-8.
Matsumura, Hideyuki (1970), Commutative algebra, Mathematics lecture note series, W. A. Benjamin, ISBN 978-0-8053-7025-6.
Kolař, Ivan; Slovák, Jan; Michor, Peter W. (1993), Natural operations in differential geometry, Springer-Verlag.
Kata Kunci Pencarian:
- Derivation (differential algebra)
- Differential algebra
- Derivation
- Elementary function
- Differential-algebraic system of equations
- Differential graded algebra
- Differential (mathematics)
- Kähler differential
- Exterior algebra
- Liouville's theorem (differential algebra)
