Derivation (abstract algebra)

In Differential algebra, an area of mathematics devoted to the algebraic study of differential equations, 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 DA → A that satisfies Leibniz's law:

${\displaystyle D(ab)=(Da)b+a(Db).}$

More generally, if M is an A-module, a K-linear map D:AM which 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. 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 x1x2, … ,xn ∈ A, then it follows by mathematical induction that

${\displaystyle D(x_{1}x_{2}\cdots x_{n})=\sum _{i}x_{1}\cdots x_{i-1}D(x_{i})x_{i+1}\cdots x_{n}.\,}$

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 x ∈ K, D(x) = D(x·1) = x·D(1) = 0.

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

${\displaystyle \operatorname {Der} _{K}(A,M)\subset \operatorname {Der} _{k}(A,M),\,}$

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:

${\displaystyle [D_{1},D_{2}]=D_{1}\circ D_{2}-D_{2}\circ D_{1}.}$

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

If we have a graded algebra A, and D is a homogeneous linear map of grade d = |D| on A then D is a homogeneous derivation if ${\displaystyle \scriptstyle {D(ab)=D(a)b+\epsilon ^{|a||D|}aD(b)}}$, ε = ±1 acting on homogeneous elements of A. A graded derivation is sum of homogeneous derivations with the same ε.

If the commutator factor ε = 1, this definition reduces to the usual case. If ε = −1, however, then ${\displaystyle \scriptstyle {D(ab)=D(a)b+(-1)^{|a|}aD(b)}}$, for odd |D|. They are called anti-derivations.

Examples of anti-derivations include the exterior derivative and the interior product acting on differential forms.

References

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