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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 $D:A\to A$ that satisfies Leibniz’s law:

D(ab)=aD(b)+D(a)b.

More generally, if $M$ is an $A$ –bimodule, a $K$ -linear map $D:A\to M$ that satisfies the Leibniz law is also called a derivation. The collection of all $K$ -derivations of $A$ to itself is denoted by $\mathrm {Der} _{K}(A)$ . The collection of $K$ -derivations of $A$ into an $A$ -module $M$ is denoted by $\mathrm {Der} _{K}(A,M)$ .

Derivations occur in many different contexts in diverse areas of mathematics. The partial derivative with respect to a variable is an $\mathbb {R}$ -derivation on the algebra of real-valued differentiable functions on $\mathbb {R} ^{n}$ . The Lie derivative with respect to a vector field is an $\mathbb {R}$ -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,

[FG,N]=[F,N]G+F[G,N]\,,

where $[\cdot ,N]$ is the commutator with respect to $N$ . 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 $A$ is a $K$ -derivation, then

If $A$ has a unit 1, then $D(1)=D(1^{2})=2D(1)$ , so that $D(1)=0$ . Thus by $K$ -linearity, $D(k)=0$ for all $k\in K$ .
If $A$ is commutative, then $D(x^{2})=xD(x)+D(x)x=2xD(x)$ , and $D(x^{n})=nx^{n-1}D(x)$ , by the Leibniz rule.
More generally, for any $x_{1},x_{2},\ldots ,x_{n}\in A$ , it follows by induction that
$D(x_{1}x_{2}\cdots x_{n})=\sum _{i}x_{1}\cdots x_{i-1}D(x_{i})x_{i+1}\cdots x_{n}$

which is

\textstyle \sum _{i}D(x_{i})\prod _{j\neq i}x_{j}

if for all

i

,

D(x_{i})

commutes with

x_{1},x_{2},\ldots ,x_{i-1}

.

For $n>1$ , $D^{n}$ is not a derivation, instead satisfying a higher-order Leibniz rule:

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

\operatorname {Der} _{K}(A,M)

for the set of

K

-derivations from

A

to

M

.

$\mathrm {Der} _{K}(A,M)$ is a module over $K$ .
$\mathrm {Der} _{K}(A)$ is a Lie algebra with Lie bracket defined by the commutator:

[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 $\Omega _{A/K}$ (called the Kähler differentials) with a $K$ -derivation $d:A\to \Omega _{A/K}$ through which any derivation $D:A\to M$ factors. That is, for any derivation $D$ ‘ there is a $A$ -module map $\varphi$ with

D:A{\stackrel {d}{\longrightarrow }}\Omega _{A/K}{\stackrel {\varphi }{\longrightarrow }}M

The correspondence

D\leftrightarrow \varphi

is an isomorphism of

A

-modules:

\operatorname {Der} _{K}(A,M)\simeq \operatorname {Hom} _{A}(\Omega _{A/K},M)

If $k\subset K$ is a subring, then $A$ inherits a $k$ -algebra structure, so there is an inclusion

\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(ab)=D(a)b+\varepsilon ^{|a||D|}aD(b)}

for every homogeneous element $A$ and every element $b$ of $A$ for a commutator factor $\varepsilon =\pm 1$ . A graded derivation is sum of homogeneous derivations with the same $\varepsilon$ .

If $\varepsilon =1$ , this definition reduces to the usual case. If $\varepsilon =-1$ , however, then

{D(ab)=D(a)b+(-1)^{|a||D|}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., $\mathbb {Z} _{2}$ -graded algebras) are often called superderivations.

Related notions

Hasse–Schmidt derivations are $K$ -algebra homomorphisms

A\to A[[t]].

Composing further with the map that sends a formal power series $\sum a_{n}t^{n}$ to the coefficient $a_{1}$ gives a derivation.

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.

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Properties

Graded derivations

Related notions

See also

References