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:

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

(the last equality holds if, for all commutes with ).

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 c ∈ K, D(c) = D(c⋅1) = c⋅D(1) = 0.

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

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:

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

Higher order

If D is a derivation, higher order derivation follows a law similar to the binomial identity:

Graded derivations

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.

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 .