Derivation (differential algebra)

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Function on an algebra which generalizes certain features of derivative operator

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:

D ( a b ) = a D ( b ) + D ( a ) b . {\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 DerK(A). The collection of K-derivations of A into an A-module M is denoted by DerK(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,

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

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

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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, …, xn ∈ A, it follows by induction that D ( x 1 x 2 ⋯ x n ) = ∑ i x 1 ⋯ x i − 1 D ( x i ) x i + 1 ⋯ x n {\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 ∑ i D ( x i ) ∏ j ≠ i x j {\textstyle \sum _{i}D(x_{i})\prod _{j\neq i}x_{j}} if for all i, D(xi) commutes with x 1 , x 2 , … , x i − 1 {\displaystyle 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 ( u v ) = ∑ k = 0 n ( n k ) ⋅ D n − k ( u ) ⋅ D k ( v ) . {\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 Der K ⁡ ( A , M ) {\displaystyle \operatorname {Der} _{K}(A,M)} for the set of K-derivations from A to M.

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

[ D 1 , D 2 ] = D 1 ∘ D 2 − D 2 ∘ D 1 . {\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 ΩA/K (called the Kähler differentials) with a K-derivation d: A → ΩA/K through which any derivation D: A → M factors. That is, for any derivation D there is a A-module map φ with

D : A ⟶ d Ω A / K ⟶ φ M {\displaystyle D:A{\stackrel {d}{\longrightarrow }}\Omega _{A/K}{\stackrel {\varphi }{\longrightarrow }}M} The correspondence D ↔ φ {\displaystyle D\leftrightarrow \varphi } is an isomorphism of A-modules: Der K ⁡ ( A , M ) ≃ Hom A ⁡ ( Ω A / K , M ) {\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

Der K ⁡ ( A , M ) ⊂ Der k ⁡ ( A , M ) , {\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

D ( a b ) = D ( a ) b + ε | a | | D | a D ( b ) {\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

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

Hasse–Schmidt derivations are K-algebra homomorphisms

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

Composing further with the map that sends a formal power series ∑ a n t n {\displaystyle \sum a_{n}t^{n}} to the coefficient a 1 {\displaystyle a_{1}} gives a derivation.

Properties

Graded derivations

Related notions

See also