Derivation (differential algebra)
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 : A → A that satisfies Leibniz's law:
More generally, if M is an A-bimodule, a K-linear map D : A → M 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 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**n ∈ A, it follows by induction that
D**n is not a derivation, instead satisfying a higher-order Leibniz rule:
- Moreover, if M is an A-bimodule, write
DerK(A, M) is a module over K.
DerK(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.](https://wikimedia.org/api/rest_v1/media/math/render/svg/9196a962e980d7f5237277994ccc82fe82b3c205)
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
If k ⊂ K is a subring, then A inherits a k-algebra structure, so there is an inclusion
Graded derivations
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.
Graded derivations of superalgebras (i.e. Z2-graded algebras) are often called superderivations.
Related notions
Hasse–Schmidt derivations are K-algebra homomorphisms
![{\displaystyle A\to A[[t]].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21d5019fc824697626911b9b24eff8d080bc8fe9)
See also
In differential geometry derivations are tangent vectors
Kähler differential
Hasse derivative
p-derivation
Wirtinger derivatives
Derivative of the exponential map