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Derivation (differential algebra)

Algebraic generalization of the derivative

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) = a D(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 [\cdot,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.

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(x**n) = nx**n−1D(x), by the Leibniz rule.
  • More generally, for any x1, x2, …, x**n ∈ A, it follows by 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
    : which is \sum_i D(x_i)\prod_{j\neq i}x_j if for all i, D(xi) commutes with x_1,x_2,\ldots, x_{i-1}.
  • For n 1, D**n is not a derivation, instead satisfying a higher-order Leibniz rule: :: D^n(uv) = \sum_{k=0}^n \binom{n}{k} \cdot D^{n-k}(u)\cdot D^k(v). : Moreover, if M is an A-bimodule, write :: \operatorname{Der}_K(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: :: [D_1,D_2] = D_1\circ D_2 - D_2\circ D_1. : 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\stackrel{d}{\longrightarrow} \Omega_{A/K}\stackrel{\varphi}{\longrightarrow} M : The correspondence D\leftrightarrow \varphi is an isomorphism of A-modules: :: \operatorname{Der}K(A,M)\simeq \operatorname{Hom}{A}(\Omega_{A/K},M)
  • If k ⊂ K is a subring, then A inherits a k-algebra structure, so 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.

Graded derivations

Homogeneous derivation

Given a graded algebra A and a homogeneous linear map D of grade on A, D is a homogeneous derivation if :{D(ab)=D(a)b+\varepsilon^{|a||D|}aD(b)} for every homogeneous element a and every element b of A for a commutator factor . A graded derivation is sum of homogeneous derivations with the same ε.

If , this definition reduces to the usual case. If , however, then :{D(ab)=D(a)b+(-1)^{|a||D|}aD(b)} for odd , 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.

References

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