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 {{nowrap|D : A → A}} that satisfies Leibniz's law:

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

More generally, if M is an A-bimodule, a K-linear map {{nowrap|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 {{nowrap|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,

: $[FG,N]=[F,N]G+F[G,N],$

where $[\cdot,N]$ is the commutator with respect to $N$ . 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 {{math|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 {{math|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 {{math|x₁, x₂, …, x_n ∈ A}}, it follows by induction that
: $D(x_1x_2\cdots x_n) = \sum_i x_1\cdots x_{i-1}D(x_i)x_{i+1}\cdots x_n$

: which is $\sum_i D(x_i)\prod_{j\neq i}x_j$ if for all {{mvar|i}}, {{math|D(x_i)}} commutes with $x_1,x_2,\ldots, x_{i-1}$ .

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

:: $D^n(uv) = \sum_{k=0}^n \binom{n}{k} \cdot D^{n-k}(u)\cdot D^k(v).$

: Moreover, if M is an A-bimodule, write

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

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

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

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

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

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

:: $D: A\stackrel{d}{\longrightarrow} \Omega_{A/K}\stackrel{\varphi}{\longrightarrow} M$

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

:: $\operatorname{Der}_K(A,M)\simeq \operatorname{Hom}_{A}(\Omega_{A/K},M)$

If {{math|k ⊂ K}} is a subring, then A inherits a k-algebra structure, so 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.

Graded derivations

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

: ${D(ab)=D(a)b+\varepsilon^$

aD(b)}

for every homogeneous element a and every element b of A for a commutator factor {{nowrap|1=ε = ±1}}. A graded derivation is sum of homogeneous derivations with the same ε.

If {{nowrap|1=ε = 1}}, this definition reduces to the usual case. If {{nowrap|1=ε = −1}}, however, then

: ${D(ab)=D(a)b+(-1)^$

aD(b)}

for odd {{abs|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

: $A \to A t .$

Composing further with the map that sends a formal power series $\sum a_n t^n$ to the coefficient $a_1$ gives a derivation.

References

{{Citation|first=Nicolas|last=Bourbaki|authorlink=Nicolas Bourbaki|title=Algebra I|year=1989|publisher=Springer-Verlag|isbn=3-540-64243-9|series=Elements of mathematics}}.
{{citation|first=David|authorlink=David Eisenbud|last=Eisenbud|title=Commutative algebra with a view toward algebraic geometry|isbn=978-0-387-94269-8|publisher=Springer-Verlag|year=1999|edition=3rd.}}.
{{citation|first=Hideyuki|last=Matsumura|title=Commutative algebra|publisher=W. A. Benjamin|year=1970|series=Mathematics lecture note series|isbn=978-0-8053-7025-6}}.
{{citation|title=Natural operations in differential geometry|first1=Ivan|last1=Kolař|first2=Jan|last2=Slovák|first3=Peter W.|last3=Michor|year=1993|publisher=Springer-Verlag|url=http://www.emis.de/monographs/KSM/index.html}}.

Category:Differential algebra

Derivation (differential algebra)

Properties

Graded derivations

Related notions

See also

References