 Derivation (abstract algebra)

In abstract algebra, 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 Kderivation is a Klinear map D: A → A that satisfies Leibniz's law:
 D(ab) = (Da)b + a(Db).
More generally, a Klinear map D of A into an Amodule M, satisfying the Leibniz law is also called a derivation. The collection of all Kderivations of A to itself is denoted by Der_{K}(A). The collection of Kderivations of A into an Amodule 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 Rderivation on the algebra of realvalued differentiable functions on R^{n}. The Lie derivative with respect to a vector field is an Rderivation on the algebra of differentiable functions on a differentiable manifold; more generally it is a derivation on the tensor algebra of a manifold. 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.
Contents
Properties
The Leibniz law itself has a number of immediate consequences. Firstly, if x_{1}, x_{2}, … ,x_{n} ∈ A, then it follows by mathematical induction that
In particular, if A is commutative and x_{1} = x_{2} = … = x_{n}, then this formula simplifies to the familiar power rule D(x^{n}) = nx^{n−1}D(x). If A is unital, then D(1) = 0 since D(1) = D(1·1) = D(1) + D(1). Thus, since D is K linear, it follows that D(x) = 0 for all x ∈ K.
If k ⊂ K is a subring, and A is a kalgebra, then there is an inclusion
since any Kderivation is a fortiori a kderivation.
The set of kderivations from A to M, Der_{k}(A,M) is a module over k. Furthemore, the kmodule Der_{k}(A) forms a Lie algebra with Lie bracket defined by the commutator:
It is readily verified that the Lie bracket of two derivations is again a derivation.
Graded derivations
If we have a graded algebra A, and D is an homogeneous linear map of grade d = D on A then D is an homogeneous derivation if , ε = ±1 acting on homogeneous elements of A. A graded derivation is sum of homogeneous derivations with the same ε.
If the commutator factor ε = 1, this definition reduces to the usual case. If ε = −1, however, then , for odd D. They are called antiderivations.
Examples of antiderivations include the exterior derivative and the interior product acting on differential forms.
Graded derivations of superalgebras (i.e. Z_{2}graded algebras) are often called superderivations.
See also
 In elemental differential geometry derivations are tangent vectors
 Kähler differential
 pderivation
References
 Bourbaki, Nicolas (1989), Algebra I, Elements of mathematics, SpringerVerlag, ISBN 3540642439.
 Eisenbud, David (1999), Commutative algebra with a view toward algebraic geometry (3rd. ed.), SpringerVerlag, ISBN 9780387942698.
 Matsumura, Hideyuki (1970), Commutative algebra, Mathematics lecture note series, W. A. Benjamin, ISBN 9780805370256.
 Kolař, Ivan; Slovák, Jan; Michor, Peter W. (1993), Natural operations in differential geometry, SpringerVerlag, http://www.emis.de/monographs/KSM/index.html.
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