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 Template:Nowrap that satisfies Leibniz's law:

More generally, if M is an A-bimodule, a K-linear map Template:Nowrap that satisfies the Leibniz law is also called a derivation. The collection of all K-derivations of A to itself is denoted by Der_K(A). The collection of K-derivations of A into an A-module M is denoted by Template:Nowrap.

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,

where <math>[\cdot,N]</math> is the commutator with respect to <math>N</math>. 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.

PropertiesEdit

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(1²) = 2D(1), so that D(1) = 0. Thus by K-linearity, D(k) = 0 for all Template:Math.
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 Template: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 display="inline">\sum_i D(x_i)\prod_{j\neq i}x_j</math> if for all Template:Mvar, Template:Math commutes with <math>x_1,x_2,\ldots, x_{i-1}</math>.

For n > 1, Dⁿ 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 M is an A-bimodule, write

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

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

Template:Nowrap is a module over K.
Der_K(A) is a Lie algebra with Lie bracket defined by the commutator:

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

There is an A-module Template:Math (called the Kähler differentials) with a K-derivation Template:Math through which any derivation Template:Math factors. That is, for any derivation D there is a A-module map Template:Mvar 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 A-modules:

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

If Template:Math is a subring, then A inherits a k-algebra structure, so there is an inclusion

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

since any K-derivation is a fortiori a k-derivation.

Graded derivationsEdit

Template:Anchor

Given a graded algebra A and a homogeneous linear map D of grade Template:Abs on A, D is a homogeneous derivation if

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

for every homogeneous element a and every element b of A for a commutator factor Template:Nowrap. A graded derivation is sum of homogeneous derivations with the same ε.

If Template:Nowrap, this definition reduces to the usual case. If Template:Nowrap, however, then

for odd Template:Abs, 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 notionsEdit

Hasse–Schmidt derivations are K-algebra homomorphisms

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.

Derivation (differential algebra)

Contents

PropertiesEdit

Graded derivationsEdit

Related notionsEdit

See alsoEdit

ReferencesEdit