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# Derivation (differential algebra)

In mathematics, a derivation is a function on an algebra which 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 : AA that satisfies Leibniz's law:

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

• If A is commutative, D(x2) = xD(x) + D(x)x = 2xD(x), and D(x**n) = nx**n−1D(x), by the Leibniz rule.

• More generally, for any x1, x2, ..., x**nA, it follows by induction that

which isif for allcommutes with.
• D**n is not a derivation, instead satisfying a higher-order Leibniz rule:

Moreover, if M is an A-bimodule, write
for the set of K-derivations from A to M.
• DerK(A, M) is a module over K.

• DerK(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 (called the Kähler differentials) with a K-derivation through which any derivation factors. That is, for any derivation D there is a A-module map with

The correspondenceis an isomorphism of A-modules:
• If kK is a subring, then A inherits a k-algebra structure, so there is an inclusion

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

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

for every homogeneous element a and every element b of A for a commutator factor ε = ±1. A graded derivation is sum of homogeneous derivations with the same ε.

If ε = 1, this definition reduces to the usual case. If ε = −1, however, then

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.

Hasse–Schmidt derivations are K-algebra homomorphisms

Composing further with the map which sends aformal power seriesto the coefficientgives a derivation.

• In differential geometry derivations are tangent vectors

• Kähler differential

• Hasse derivative

• p-derivation

• Wirtinger derivatives

• Derivative of the exponential map

## References

[1]
Citation Linkwww.emis.deNatural operations in differential geometry
Sep 24, 2019, 5:40 PM
[2]
Citation Linkwww.emis.deNatural operations in differential geometry
Sep 24, 2019, 5:40 PM
[3]
Citation Linken.wikipedia.orgThe original version of this page is from Wikipedia, you can edit the page right here on Everipedia.Text is available under the Creative Commons Attribution-ShareAlike License.Additional terms may apply.See everipedia.org/everipedia-termsfor further details.Images/media credited individually (click the icon for details).
Sep 24, 2019, 5:40 PM