Derivation (differential algebra)

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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 : AA that satisfies Leibniz's law:

More generally, if M is an A-bimodule, a K-linear map D : AM 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. 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 x1, x2, ..., xnA, it follows by induction that
which is if for all commutes with .
  • Dn 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.

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 kK 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. Z2-graded algebras) are often called superderivations.

See also

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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