In mathematics, a derivation is a function on an algebra that generalizes certain features of the derivative operator. Specifically, given an algebra over a ring or a field , a -derivation is a -linear map that satisfies Leibniz's law:
The same definition generalizes word-for-word whenever is an -bimodule. The collection of all -derivations of to itself is denoted by , and the collection of -derivations of into an -module is denoted by .
Derivations occur in many diverse areas of mathematics. The partial derivative with respect to a variable is an -derivation on the algebra of real-valued differentiable functions on . The Lie derivative with respect to a vector field is an -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. An algebra equipped with a distinguished derivation forms a differential algebra, and is itself a significant object of study in areas such as differential Galois theory.
Inner derivations The commutator with respect to a fixed element defines a linear endomorphism of to itself: . That map is a derivation over , since the algebraic identity
is precisely the Leibniz rule for . (The map is trivial — constant 0 — if is commmutative.) Such derivations are called inner derivations. In general, not all elements of need be inner; the next term in the short exact sequence of Lie algebras is the level-1 Hochschild cohomology:
Properties
If is a -algebra, for a ring, and is a -derivation, then If has a unit 1, then , so that . Thus by -linearity, for all . If is commutative, then , and , by the Leibniz rule. More generally, for any , it follows by induction that
which is if for all , commutes with . For , is not a derivation, instead satisfying a higher-order Leibniz rule:
Moreover, if is an -bimodule, write
for the set of -derivations from to . is a module over . 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 -module (called the Kähler differentials) with a -derivation through which any derivation factors. That is, for any derivation ' there is a -module map with
The correspondence is an isomorphism of -modules:
If is a subring, then inherits a -algebra structure, so there is an inclusion
since any -derivation is a fortiori a -derivation.
Graded derivations
Given a graded algebra and a homogeneous linear map of grade on , is a homogeneous derivation if
for every homogeneous element and every element of for a commutator factor . A graded derivation is sum of homogeneous derivations with the same .
If , this definition reduces to the usual case. If , however, then
for odd , and 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., -graded algebras) are often called superderivations.
Related notions
Hasse–Schmidt derivations are -algebra homomorphisms
Composing further with the map that sends a formal power series to the coefficient gives a derivation.
See also In differential geometry derivations are tangent vectors Kähler differential Hasse derivative p-derivation Wirtinger derivatives Derivative of the exponential map
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