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 that satisfies Leibniz's law:
More generally, if "M" is an "A"-bimodule, a "K"-linear map 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 .
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". 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 is a "K"-derivation, then
Graded derivations.
Given a graded algebra "A" and a homogeneous linear map "D" of grade on "A", "D" is a homogeneous derivation if 
for every homogeneous element "a" and every element "b" of "A" for a commutator factor . A graded derivation is sum of homogeneous derivations with the same "ε".
If , this definition reduces to the usual case. If , however, then
for odd , 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
Composing further with the map which sends a formal power series formula_15 to the coefficient formula_16 gives a derivation.