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Derivation (differential algebra)

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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:

D ( a b ) = D ( a ) b + a D ( b ) .

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

The Leibniz law itself has a number of immediate consequences. Firstly, if x1, x2, ..., xnA, then it follows by mathematical induction that

D ( x 1 x 2 x n ) = i x 1 x i 1 D ( x i ) x i + 1 x n = i D ( x i ) j i x j

(the last equality holds if, for all i ,   D ( x i ) commutes with x 1 , x 2 , , x i 1 ).

In particular, if A is commutative and x1 = x2 = ... = xn, then this formula simplifies to the familiar power rule D(xn) = nxn−1D(x). Secondly, if A has a unit element 1, then D(1) = 0 since D(1) = D(1·1) = D(1) + D(1). Moreover, because D is K-linear, it follows that "the derivative of any constant function is zero"; more precisely, for any xK, D(x) = D(x·1) = x·D(1) = 0.

If kK is a subring, and A is a k-algebra, then there is an inclusion

Der K ( A , M ) Der k ( A , M ) ,

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

The set of k-derivations from A to M, Derk(A, M) is a module over k. Furthermore, the k-module Derk(A) forms a Lie algebra with Lie bracket defined by the commutator:

[ D 1 , D 2 ] = D 1 D 2 D 2 D 1 .

It is readily verified that the Lie bracket of two derivations is again a derivation.

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

D ( a b ) = D ( a ) b + ε | a | | D | a D ( b )

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

D ( a b ) = D ( a ) b + ( 1 ) | a | a D ( b )

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.

References

Derivation (differential algebra) Wikipedia


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