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 D : A → A that satisfies Leibniz's law:

$D(ab)=D(a)b+aD(b).$

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 Der_K(A). The collection of K-derivations of A into an A-module M is denoted by Der_K(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, ..., x_n ∈ A, then it follows by mathematical induction that

$D(x_1x_2\cdots x_n) = \sum_i x_1\cdots x_{i-1}D(x_i)x_{i+1}\cdots x_n = \sum_i D(x_i)\prod_{j\neq i}x_j$

(the last equality holds if, for all $i,\ D(x_{i})$ commutes with $x_1,x_2,\cdots, x_{i-1}$).

In particular, if A is commutative and x₁ = x₂ = ... = x_n, 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

$\operatorname{Der}_K(A,M)\subset \operatorname{Der}_k(A,M) ,$

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

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

$[D_1,D_2] = D_1\circ D_2 - D_2\circ D_1.$

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

Graded derivations

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

${D(ab)=D(a)b+\varepsilon^{|a||D|}aD(b)}$

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

${D(ab)=D(a)b+(-1)^{|a|}aD(b)}$

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.

See also

• In differential geometry derivations are tangent vectors
• Kähler differential
• Hasse derivative
• p-derivation
• Wirtinger derivatives
• Derivative of the exponential map

References

• Bourbaki, Nicolas (1989), Algebra I, Elements of mathematics, Springer-Verlag, ISBN 3-540-64243-9.
• Eisenbud, David (1999), Commutative algebra with a view toward algebraic geometry (3rd. ed.), Springer-Verlag, ISBN 978-0-387-94269-8.
• Matsumura, Hideyuki (1970), Commutative algebra, Mathematics lecture note series, W. A. Benjamin, ISBN 978-0-8053-7025-6.
• Kolař, Ivan; Slovák, Jan; Michor, Peter W. (1993), Natural operations in differential geometry, Springer-Verlag.