In mathematics, a derivation is a function on an algebra that generalizes certain features of the derivative operator. Specifically, given an algebra <math>A</math> over a ring or a field <math>K</math>, a <math>K</math>-derivation is a <math>K</math>-linear map <math>D:A\to A</math> that satisfies Leibniz's law:

:<math> D(ab) = a D(b) + D(a) b.</math>

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

Derivations occur in many different contexts in diverse areas of mathematics. The partial derivative with respect to a variable is an <math>\mathbb{R}</math>-derivation on the algebra of real-valued differentiable functions on <math>\mathbb{R}^n</math>. The Lie derivative with respect to a vector field is an <math>\mathbb{R}</math>-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 <math>A</math> is noncommutative, then the commutator with respect to an element of the algebra <math>A</math> defines a linear endomorphism of <math>A</math> to itself, which is a derivation over <math>K</math>. That is,

:<math>[FG,N]=[F,N]G+F[G,N]\, ,</math>

where <math>[\cdot,N]</math> is the commutator with respect to <math>N</math>. An algebra <math>A</math> equipped with a distinguished derivation <math>d</math> forms a differential algebra, and is itself a significant object of study in areas such as differential Galois theory.

Properties

If <math>A</math> is a <math>K</math>-algebra, for <math>K</math> a ring, and is a <math>K</math>-derivation, then

  • If <math>A</math> has a unit 1, then <math>D(1) = D(1^2) = 2D(1)</math>, so that <math>D(1) = 0</math>. Thus by <math>K</math>-linearity, <math>D(k) = 0</math> for all <math>k\in K</math>.
  • If <math>A</math> is commutative, then <math>D(x^2) = xD(x) + D(x)x = 2xD(x)</math>, and <math>D(x^n) = nx^{n-1}D(x)</math>, by the Leibniz rule.
  • More generally, for any <math>x_1, x_2, \ldots, x_n \in A</math>, it follows by induction that
  • : <math>D(x_1x_2\cdots x_n) = \sum_i x_1\cdots x_{i-1}D(x_i)x_{i+1}\cdots x_n </math>

: which is <math>\textstyle \sum_i D(x_i)\prod_{j\neq i}x_j</math> if for all <math>i</math>, <math>D(x_i)</math> commutes with <math>x_1,x_2,\ldots, x_{i-1}</math>.

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

:: <math>D^n(uv) = \sum_{k=0}^n \binom{n}{k} \cdot D^{n-k}(u)\cdot D^k(v).</math>

: Moreover, if <math>M</math> is an <math>A</math>-bimodule, write

:: <math> \operatorname{Der}_K(A,M)</math>

:for the set of <math>K</math>-derivations from <math>A</math> to <math>M</math>.

  • <math>\mathrm{Der}_K(A,M)</math> is a module over <math>K</math>.
  • <math>\mathrm{Der}_K(A)</math> is a Lie algebra with Lie bracket defined by the commutator:

:: <math>[D_1,D_2] = D_1\circ D_2 - D_2\circ D_1.</math>

: since it is readily verified that the commutator of two derivations is again a derivation.

  • There is an <math>A</math>-module <math>\Omega_{A/K}</math> (called the Kähler differentials) with a <math>K</math>-derivation <math>d:A\to\Omega_{A/K}</math> through which any derivation <math>D:A\to M</math> factors. That is, for any derivation <math>D</math>' there is a <math>A</math>-module map <math>\varphi</math> with

:: <math> D: A\stackrel{d}{\longrightarrow} \Omega_{A/K}\stackrel{\varphi}{\longrightarrow} M </math>

: The correspondence <math> D\leftrightarrow \varphi</math> is an isomorphism of <math>A</math>-modules:

:: <math> \operatorname{Der}_K(A,M)\simeq \operatorname{Hom}_{A}(\Omega_{A/K},M)</math>

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

:: <math>\operatorname{Der}_K(A,M)\subset \operatorname{Der}_k(A,M) ,</math>

: since any <math>K</math>-derivation is a fortiori a <math>k</math>-derivation.

Graded derivations

Given a graded algebra <math>A</math> and a homogeneous linear map <math>D</math> of grade <math>|D|</math> on <math>A</math>, <math>D</math> is a homogeneous derivation if

:<math>{D(ab)=D(a)b+\varepsilon^{|a||D|}aD(b)}</math>

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

If <math>\varepsilon = 1</math>, this definition reduces to the usual case. If <math>\varepsilon = -1</math>, however, then

:<math>{D(ab)=D(a)b+(-1)^{|a||D|}aD(b)}</math>

for odd <math>|D|</math>, and <math>D</math> 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., <math>\mathbb{Z}_2</math>-graded algebras) are often called superderivations.

Hasse–Schmidt derivations are <math>K</math>-algebra homomorphisms

:<math>A \to At.</math>

Composing further with the map that sends a formal power series <math>\sum a_n t^n</math> to the coefficient <math>a_1</math> 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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