- 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 K-derivation is a K-linear map D: A → A that satisfies Leibniz's law:
- D(ab) = (Da)b + a(Db).
More generally, a K-linear map D of A into an A-module M, satisfying 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. 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.
The Leibniz law itself has a number of immediate consequences. Firstly, if x1, x2, … ,xn ∈ A, then it follows by mathematical induction that
In particular, if A is commutative and x1 = x2 = … = xn, then this formula simplifies to the familiar power rule D(xn) = nxn−1D(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 k-algebra, then there is an inclusion
since any K-derivation is a fortiori a k-derivation.
It is readily verified that the Lie bracket of two derivations is again a derivation.
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 anti-derivations.
Graded derivations of superalgebras (i.e. Z2-graded algebras) are often called superderivations.
- 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 978-0387942698 .
- Matsumura, Hideyuki (1970), Commutative algebra, Mathematics lecture note series, W. A. Benjamin, ISBN 978-0805370256 .
- Kolař, Ivan; Slovák, Jan; Michor, Peter W. (1993), Natural operations in differential geometry, Springer-Verlag, http://www.emis.de/monographs/KSM/index.html .
Wikimedia Foundation. 2010.