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:

D(ab)=D(a)b+aD(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 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 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. 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 $D:A\to A$ 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 $k\in K$ .
If A is commutative, D(x²)=xD(x)+D(x)x=2xD(x) and D(xⁿ)=nx^n-1D(x), by the Leibniz rule.
More generally, for any x₁, x₂, ..., x_n ∈ A, it follows by induction that

D(x_{1}x_{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(x_{i})

commutes with

x_1,x_2,\cdots, x_{i-1}

.

Dⁿ is not a derivation, instead satisfying a 'higher order' Leibniz rule:

D^{n}(uv)=\sum _{k=0}^{n}{n \choose k}\cdot D^{n-k}(u)\cdot D^{k}(v).

Moreover, if M is an A-bimodule, write

{\text{Der}}_{K}(A,M)

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:

[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 $\Omega _{A/K}$ (called the Kähler differentials) with a K-derivation $d:A\to \Omega _{A/K}$ through which any derivation $D:A\to M$ factors. That is, for any derivation D there is a A-module map $\varphi$ with

D:A{\stackrel {d}{\longrightarrow }}\Omega _{A/K}{\stackrel {\varphi }{\longrightarrow }}M

The correspondence

D\leftrightarrow \varphi

is an isomorphism of A-modules:

{\text{Der}}_{K}(A,M)\simeq {\text{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

Given a graded algebra A and a homogeneous linear map D of grade |D| 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 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

{D(ab)=D(a)b+(-1)^{|a|}aD(b)}

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. Z₂-graded algebras) are often called superderivations.

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 .

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Derivation (differential algebra)

Properties

Graded derivations

See also

References