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:

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

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

If A is a K-algebra, for K a ring, and ${\displaystyle D\colon A\to A}$ 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 ${\displaystyle k\in K.}$
• 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

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

which is ${\displaystyle \sum _{i}D(x_{i})\prod _{j\neq i}x_{j}}$ if for all ${\displaystyle i,\ D(x_{i})}$ commutes with ${\displaystyle x_{1},x_{2},\cdots ,x_{i-1}}$.

• Dⁿ is not a derivation, instead satisfying a higher-order Leibniz rule:

${\displaystyle D^{n}(uv)=\sum _{k=0}^{n}{\binom {n}{k}}\cdot D^{n-k}(u)\cdot D^{k}(v).}$

Moreover, if M is an A-bimodule, write

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

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:

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

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

• There is an A-module ${\displaystyle \Omega _{A/K}}$ (called the Kähler differentials) with a K-derivation ${\displaystyle d:A\to \Omega _{A/K}}$ through which any derivation ${\displaystyle D:A\to M}$ factors. That is, for any derivation D there is a A-module map ${\displaystyle \varphi }$ with

${\displaystyle D:A{\stackrel {d}{\longrightarrow }}\Omega _{A/K}{\stackrel {\varphi }{\longrightarrow }}M}$

The correspondence ${\displaystyle D\leftrightarrow \varphi }$ is an isomorphism of A-modules:

${\displaystyle \operatorname {Der} _{K}(A,M)\simeq \operatorname {Hom} _{A}(\Omega _{A/K},M)}$

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

Graded derivations

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

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

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

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

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.

Related notions

Hasse–Schmidt derivations are K-algebra homomorphisms

${\displaystyle A\to A[[t]].}$

Composing further with the map which sends a formal power series ${\displaystyle \sum a_{n}t^{n}}$ to the coefficient ${\displaystyle a_{1}}$ 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.