Derivation (differential algebra)

Template:Short description 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 ${\mathrm{D}\mathrm{e}\mathrm{r}}_K(A)$. The collection of $K$-derivations of $A$ into an $A$-module $M$ is denoted by ${\mathrm{D}\mathrm{e}\mathrm{r}}_K(A,M)$.

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 ${ℝ}^n$. 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,

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

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 Template:Math 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 $D(x^n)=n{x}^{n-1}D(x)$, by the Leibniz rule.
• More generally, for any $x_1,x_2,\dots ,x_n\in A$, 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 ${\sum }_{i}D(x_i){\prod }_{j\ne i}{x}_j$ if for all $i$, $D(x_i)$ commutes with $x_1,x_2,\dots ,x_{i-1}$.

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

$D^n(uv)=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})\cdot D^{n-k}(u)\cdot D^k(v).$

Moreover, if $M$ is an $A$-bimodule, write

${\mathrm{Der}}_K(A,M)$

for the set of $K$-derivations from $A$ to $M$.

• ${\mathrm{D}\mathrm{e}\mathrm{r}}_K(A,M)$ is a module over $K$.
• ${\mathrm{D}\mathrm{e}\mathrm{r}}_K(A)$ 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 ${\mathrm{\Omega }}_{A/K}$ (called the Kähler differentials) with a $K$-derivation $d:A\to {\mathrm{\Omega }}_{A/K}$ through which any derivation $D:A\to M$ factors. That is, for any derivation $D$' there is a $A$-module map $\phi$ with

$D:A\stackrel{d}{⟶}{\mathrm{\Omega }}_{A/K}\stackrel{\phi }{⟶}M$

The correspondence $D\leftrightarrow \phi$ is an isomorphism of $A$-modules:

${\mathrm{Der}}_K(A,M)\simeq {\mathrm{Hom}}_A({\mathrm{\Omega }}_{A/K},M)$

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

${\mathrm{Der}}_K(A,M)\subset {\mathrm{Der}}_k(A,M),$

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

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

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

If $\epsilon =1$, this definition reduces to the usual case. If $\epsilon =-1$, 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.

See also

• In differential geometry derivations are tangent vectors
• Kähler differential
• Hasse derivative
• p-derivation
• Wirtinger derivatives
• Derivative of the exponential map

References

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