Derivation (abstract algebra)

In abstract algebra, 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) = (Da)b + a(Db).

More generally, a K-linear map D of A into an A-module M, satisfying 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. 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

$D(x_1x_2\cdots x_n) = \sum_i x_1\cdots x_{i-1}D(x_i)x_{i+1}\cdots x_n. \,$

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). If A is unital, then D(1) = 0 since D(1) = D(1·1) = D(1) + D(1). Thus, since D is K linear, it follows that D(x) = 0 for all x ∈ K.

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. Furthemore, 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

If we have a graded algebra A, and D is an homogeneous linear map of grade d = |D| on A then D is an homogeneous derivation if $\scriptstyle{D(ab)=D(a)b+\epsilon^{|a||D|}aD(b)}$, 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 $\scriptstyle{D(ab)=D(a)b+(-1)^{|a|}aD(b)}$, for odd |D|. They are called anti-derivations.

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

Graded derivations of superalgebras (i.e. Z₂-graded algebras) are often called superderivations.

See also

• In elemental differential geometry derivations are tangent vectors
• Kähler differential
• p-derivation

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 978-0387942698 .
• Matsumura, Hideyuki (1970), Commutative algebra, Mathematics lecture note series, W. A. Benjamin, ISBN 978-0805370256 .
• Kolař, Ivan; Slovák, Jan; Michor, Peter W. (1993), Natural operations in differential geometry, Springer-Verlag, http://www.emis.de/monographs/KSM/index.html .