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 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. That is,
where is the commutator with respect to . 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.
If A is a K-algebra, for K a ring, and D: A → A is a K-derivation, then
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.
Hasse–Schmidt derivations are K-algebra homomorphisms
Composing further with the map which sends a formal power series to the coefficient gives a derivation.
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 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. That is,
where is the commutator with respect to . 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.
If A is a K-algebra, for K a ring, and D: A → A is a K-derivation, then
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.
Hasse–Schmidt derivations are K-algebra homomorphisms
Composing further with the map which sends a formal power series to the coefficient gives a derivation.