Derivation (differential algebra)

In mathematics, a derivation is a function on an algebra that 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:

[math]\displaystyle{ D(ab) = a D(b) + D(a) b. }[/math]

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. That is,

[math]\displaystyle{ [FG,N]=[F,N]G+F[G,N], }[/math]

where [math]\displaystyle{ [\cdot,N] }[/math] is the commutator with respect to [math]\displaystyle{ N }[/math]. 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

If A is a K-algebra, for K a ring, and D: A → A is a K-derivation, then

• If A has a unit 1, then D(1) = D(1²) = 2D(1), so that D(1) = 0. Thus by K-linearity, D(k) = 0 for all k ∈ K.
• If A is commutative, D(x²) = xD(x) + D(x)x = 2xD(x), and D(xⁿ) = nxⁿ⁻¹D(x), by the Leibniz rule.
• More generally, for any x₁, x₂, …, x_n ∈ A, it follows by induction that

  [math]\displaystyle{ D(x_1x_2\cdots x_n) = \sum_i x_1\cdots x_{i-1}D(x_i)x_{i+1}\cdots x_n }[/math]

which is [math]\displaystyle{ \sum_i D(x_i)\prod_{j\neq i}x_j }[/math] if for all i, D(x_i) commutes with [math]\displaystyle{ x_1,x_2,\ldots, x_{i-1} }[/math].

• For n > 1, Dⁿ is not a derivation, instead satisfying a higher-order Leibniz rule:

[math]\displaystyle{ D^n(uv) = \sum_{k=0}^n \binom{n}{k} \cdot D^{n-k}(u)\cdot D^k(v). }[/math]

Moreover, if M is an A-bimodule, write

[math]\displaystyle{ \operatorname{Der}_K(A,M) }[/math]

for the set of K-derivations from A to M.

• Der_K(A, M) is a module over K.
• Der_K(A) is a Lie algebra with Lie bracket defined by the commutator:

[math]\displaystyle{ [D_1,D_2] = D_1\circ D_2 - D_2\circ D_1. }[/math]

since it is readily verified that the commutator of two derivations is again a derivation.

• There is an A-module Ω_A/K (called the Kähler differentials) with a K-derivation d: A → Ω_A/K through which any derivation D: A → M factors. That is, for any derivation D there is a A-module map φ with

[math]\displaystyle{ D: A\stackrel{d}{\longrightarrow} \Omega_{A/K}\stackrel{\varphi}{\longrightarrow} M }[/math]

The correspondence [math]\displaystyle{ D\leftrightarrow \varphi }[/math] is an isomorphism of A-modules:

[math]\displaystyle{ \operatorname{Der}_K(A,M)\simeq \operatorname{Hom}_{A}(\Omega_{A/K},M) }[/math]

• If k ⊂ K is a subring, then A inherits a k-algebra structure, so there is an inclusion

[math]\displaystyle{ \operatorname{Der}_K(A,M)\subset \operatorname{Der}_k(A,M) , }[/math]

since any K-derivation is a fortiori a k-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

[math]\displaystyle{ {D(ab)=D(a)b+\varepsilon^{|a||D|}aD(b)} }[/math]

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

[math]\displaystyle{ {D(ab)=D(a)b+(-1)^{|a|}aD(b)} }[/math]

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.

Related notions

Hasse–Schmidt derivations are K-algebra homomorphisms

[math]\displaystyle{ A \to At. }[/math]

Composing further with the map which sends a formal power series [math]\displaystyle{ \sum a_n t^n }[/math] to the coefficient [math]\displaystyle{ a_1 }[/math] gives a derivation.

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, http://www.emis.de/monographs/KSM/index.html .

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