Derivation (differential algebra)

Short description: Algebraic generalization of the derivative

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(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,

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

where [⋅,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.

## Contents

* 1 Properties
* 2 Graded derivations
* 3 Related notions
* 4 See also
* 5 References

## 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(x1x2⋯xn)=∑ix1⋯xi−1D(xi)xi+1⋯xn

which is ∑iD(xi)∏j≠ixj if for all i, D(xi) commutes with x1,x2,…,xi−1.

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

Dn(uv)=∑k=0n(nk)⋅Dn−k(u)⋅Dk(v).
Moreover, if M is an A-bimodule, write

DerK(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:

[D1,D2]=D1∘D2−D2∘D1.
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⟶dΩA/K⟶φM
The correspondence D↔φ is an isomorphism of A-modules:

DerK(A,M)≃HomA(ΩA/K,M)

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

DerK(A,M)⊂Derk(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(ab)=D(a)b+ε|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(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

A->At.

Composing further with the map which sends a formal power series ∑antn to the coefficient a1 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 .

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Derivation (differential algebra). Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Derivation_(differential_algebra)&oldid=175491"