Derivation (differential algebra) explained

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 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 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 .

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 [ ⋅ ,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 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 .
• If A is commutative, D(x²) = xD(x) + D(x)x = 2xD(x), and D(xⁿ) = nxⁿ⁻¹D(x), by the Leibniz rule.
• More generally, for any, it follows by induction that

D(x _{1x} _{2 …} x _{n)} = \sum _{i} x _{1 …} x _{i-1} D(x _{i)x} _{i+1} … x _{n}

which is $\backslash sum\_i\; D(x\_i)\backslash prod\_x\_j$ if for all, commutes with

x _{1,x} _{2,\ldots,} x _{i-1} .

• For n > 1, Dⁿ is not a derivation, instead satisfying a higher-order Leibniz rule:

D ^{n(uv)} = \sum ^{n} _{k =0} \binom{n}{k} ⋅ D ^{n-k} (u) ⋅ 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.

• is a module over K.
• Der_K(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 (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

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 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

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|} 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.

Related notions

Hasse–Schmidt derivations are K-algebra homomorphisms

A \to A[[t]]. \sum a _{n} t ^{n} to the coefficient a _{1} gives a derivation.

See also

• In differential geometry derivations are tangent vectors
• Kähler differential
• Hasse derivative
• p-derivation
• Wirtinger derivatives
• Derivative of the exponential map

References

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