Derivation

A **derivation** on an algebra *A* over a ring or a field *k* is a linear map

D
:
A
⟶
A

{\displaystyle \scriptstyle{ D: A \longrightarrow A}}

 that satisfies Leibniz' law:

D
(
a
b
# )
D
(
a
)
⋅
b
a
⋅
D
(
b
)

{\displaystyle \scriptstyle{ D(ab) = D(a) \cdot b + a \cdot D(b)}}

If

x
1

,

x
2

,
⋯
,

x
n

∈
A

{\displaystyle \scriptstyle{ x\_1, x\_2, \cdots , x\_n \in A}}

 the above formula generalize to

D
(

x
1

⋅

x
2

⋯

x
n

# )

∑
i

x
1

⋅

x

i
−
1

D
(

x
i

)

x

i
1

⋅

x
n

{\displaystyle \scriptstyle{ D(x\_1 \cdot x\_2 \cdots x\_n) = \sum\_i {x\_1 \cdot x\_{i-1} D(x\_i) x\_{i+1} \cdot x\_n } }}

.
If

A

{\displaystyle \scriptstyle A}

 is unital, then

D
(
1
# )
0

{\displaystyle \scriptstyle{ D(1) = 0}}

since

D
(
1
)

=
D
(
1
⋅
1
)

=
D
(
1
)
⋅
1
1
⋅
D
(
1
)

=
D
(
1
)
D
(
1
)

{\displaystyle \scriptstyle{
\begin{align}
D(1) &=
D(1 \cdot 1)\\ &=
D(1) \cdot 1 + 1 \cdot D(1) \\ &=
D(1) + D(1)
\end{align}}}

If

A

{\displaystyle \scriptstyle A}

 is commutative, then for any

x
∈
A

{\displaystyle \scriptstyle{ x \in A}}

,

D
(

x
2

# )
2
x
D
(
x
)

{\displaystyle \scriptstyle{ D(x^2) = 2xD(x)}}

and

D
(

x
n

# )
n

x

n
−
1

D
(
x
)

{\displaystyle \scriptstyle{ D(x^n) = nx^{n-1}D(x)}}

.
Examples of derivations are:

* Derivatives

d

d
x

(
u
⋅
v
# )

d
u

d
x

⋅
v
u
⋅

d
v

d
x

{\displaystyle \dfrac{d}{dx}(u\cdot v) = \dfrac{du}{dx} \cdot v + u\cdot \dfrac{dv}{dx}}

* Partial derivatives

∂

∂
x

(
u
⋅
v
# )

∂
u

∂
x

⋅
v
u
⋅

∂
v

∂
x

{\displaystyle \dfrac{\partial}{\partial x}(u \cdot v) = \dfrac{\partial u}{\partial x} \cdot v + u\cdot \dfrac{\partial v}{\partial x} }

* Lie derivatives

L

X

(
f
g
# )
(

L

X

f
)
g
f

L

X

g

{\displaystyle \mathcal{L}\_X(fg)=(\mathcal{L}\_Xf) g + f\mathcal{L}\_Xg}

* Pincherle derivative

(
T
S

)
′

=

T
′

S
T

S
′

{\displaystyle \scriptstyle{ (TS)^\prime = T^\prime\!S + TS^\prime }}

* Commutator

[
A
,
B
C
]

=
[
A
,
B
]
C
B
[
A
,
C
]

=
A
B
C
−
B
A
C
B
A
C
−
B
C
A

=
A
(
B
C
)
−
(
B
C
)
A

{\displaystyle
\begin{align}
[A, BC] &= [A, B]C + B[A, C] \\
 &= ABC - BAC + BAC - BCA \\
 &= A(BC) - (BC)A
\end{align}
}

All these examples are tightly related, with the concept of derivation as the major unifying theme.

## Graded derivations

If we have a graded algebra

A

{\displaystyle \scriptstyle{A}}

, and

D

{\displaystyle \scriptstyle{ D }}

 is an homogeneous linear map of grade

# d

D

{\displaystyle \scriptstyle{d = \left| D \right|}}

 on

A

{\displaystyle \scriptstyle{A}}

 then

D

{\displaystyle \scriptstyle{ D }}

 is an **homogeneous derivation** if

D
(
a
b
# )
D
(
a
)
b

ϵ

a

D

a
D
(
b
)

,

# ϵ
±
1

{\displaystyle \scriptstyle{ \epsilon = \pm1 }}

acting on homogeneous elements of

A

{\displaystyle \scriptstyle{A}}

.

A **graded derivation** is sum of homogeneous derivations with the same

ϵ

{\displaystyle \scriptstyle{ \epsilon }}

.

If the commutator factor

# ϵ
1

{\displaystyle \scriptstyle{ \epsilon = 1 }}

, this definition reduces to the usual case.

If

# ϵ
−
1

{\displaystyle \scriptstyle{ \epsilon = -1 }}

, however, we have

D
(
a
b
# )
D
(
a
)
b
(
−
1

)

a

a
D
(
b
)

{\displaystyle \scriptstyle{ D(ab)=D(a)b+(-1)^{|a|}aD(b) }}

, for odd

D

{\displaystyle \scriptstyle{\left| D \right|}}

. They are called **antiderivations**.

Examples of anti-derivations include the exterior derivative and the interior product acting on differential forms.

Graded derivations of superalgebras (i.e.

Z

2

{\displaystyle \scriptstyle\mathbb{Z}\_2 }

-graded algebras) are often called **superderivations**.

## See also

* In elemental differential geometry derivations are tangent vectors