Derivation (abstract algebra)

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 DA → 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.

Properties

The Leibniz law itself has a number of immediate consequences. Firstly, if x1x2, … ,xn ∈ A, then it follows by mathematical induction that

$D(x_1x_2\cdots x_n) = \sum_i x_1\cdots x_{i-1}D(x_i)x_{i+1}\cdots x_n. \,$

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

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

since any K-derivation is a fortiori a k-derivation.

The set of k-derivations from A to M, Derk(A,M) is a module over k. Furthemore, the k-module Derk(A) forms a Lie algebra with Lie bracket defined by the commutator:

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

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 $\scriptstyle{D(ab)=D(a)b+\epsilon^{|a||D|}aD(b)}$, 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 $\scriptstyle{D(ab)=D(a)b+(-1)^{|a|}aD(b)}$, for odd |D|. They are called anti-derivations.

Examples of anti-derivations include the exterior derivative and the interior product acting on differential forms.

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 978-0387942698 .
• 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 .

Wikimedia Foundation. 2010.

Look at other dictionaries:

• Derivation — may refer to: Derivation (abstract algebra), a function on an algebra which generalizes certain features of the derivative operator Derivation (linguistics) Derivation in differential algebra, a unary function satisfying the Leibniz product law… …   Wikipedia

• Boolean algebra — This article discusses the subject referred to as Boolean algebra. For the mathematical objects, see Boolean algebra (structure). Boolean algebra, as developed in 1854 by George Boole in his book An Investigation of the Laws of Thought,[1] is a… …   Wikipedia

• Boolean algebra (introduction) — Boolean algebra, developed in 1854 by George Boole in his book An Investigation of the Laws of Thought , is a variant of ordinary algebra as taught in high school. Boolean algebra differs from ordinary algebra in three ways: in the values that… …   Wikipedia

• Exterior algebra — In mathematics, the exterior product or wedge product of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions. Like the cross product, and the scalar triple product, the exterior product of… …   Wikipedia

• Differential graded algebra — In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded algebra with an added chain complex structure that respects the algebra structure. Contents 1 Definition 2 Examples of DGAs 3 Other facts about …   Wikipedia

• Commutator — This article is about the mathematical concept. For the relation between canonical conjugate entities, see canonical commutation relation. For the type of electrical switch, see commutator (electric). In mathematics, the commutator gives an… …   Wikipedia

• List of mathematics articles (D) — NOTOC D D distribution D module D D Agostino s K squared test D Alembert Euler condition D Alembert operator D Alembert s formula D Alembert s paradox D Alembert s principle Dagger category Dagger compact category Dagger symmetric monoidal… …   Wikipedia

• Product rule — For Euler s chain rule relating partial derivatives of three independent variables, see Triple product rule. For the counting principle in combinatorics, see Rule of product. Topics in Calculus Fundamental theorem Limits of functions Continuity… …   Wikipedia

• Outline of algebraic structures — In universal algebra, a branch of pure mathematics, an algebraic structure is a variety or quasivariety. Abstract algebra is primarily the study of algebraic structures and their properties. Some axiomatic formal systems that are neither… …   Wikipedia

• List of algebraic structures — In universal algebra, a branch of pure mathematics, an algebraic structure is a variety or quasivariety. Abstract algebra is primarily the study of algebraic structures and their properties. Some axiomatic formal systems that are neither… …   Wikipedia