Derivation (differential algebra)
In mathematics, 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:
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 Der_{K}(A). The collection of K-derivations of A into an A-module 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 R-derivation on the algebra of real-valued differentiable functions on R^{n}. 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. 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
If A is a K-algebra, for K a ring, and is a K-derivation, then
- If A has a unit 1, then D(1) = D(1^{2}) = 2D(1), so that D(1) = 0. Thus by K-linearity, D(k) = 0 for all
- If A is commutative, D(x^{2}) = xD(x) + D(x)x = 2xD(x), and D(x^{n}) = nx^{n−1}D(x), by the Leibniz rule.
- More generally, for any x_{1}, x_{2}, ..., x_{n} ∈ A, it follows by induction that
- which is if for all commutes with .
- D^{n} is not a derivation, instead satisfying a higher-order Leibniz rule:
- Moreover, if M is an A-bimodule, write
- for the set of K-derivations from A to M.
- Der_{K}(A, M) is a module over K.
- Der_{K}(A) is a Lie algebra with Lie bracket defined by the commutator:
- since it is readily verified that the commutator of two derivations is again a derivation.
- There is an A-module (called the Kähler differentials) with a K-derivation through which any derivation factors. That is, for any derivation D there is a A-module map with
- The correspondence is an isomorphism of A-modules:
- If k ⊂ K is a subring, then A inherits a k-algebra structure, so there is an inclusion
- 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
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
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. Z_{2}-graded algebras) are often called superderivations.
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.