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Derivation (abstract algebra)
In abstract algebra, a derivation on an algebra A over a field k is a linear map
- D : A → A
that satisfies Leibniz' law:
- D(ab) = (Da)b + a(Db).
As a consequence, if A is unital,
then
- D(1) = 0 since
D1 = D(1·1) = D1 + D1.
Examples of derivations are partial derivatives, Lie derivatives, the Pincherle derivative, and the commutator with respect to an element of the algebra. All these examples are tightly related, with the concept of derivation as the major unifying theme.
See also: Kähler differential
If we have a Z2 graded algebra A, D is an antiderivation if
- D(ab) = (Da)b + (−1)deg(a)a(Db).
The same proof showing D(1)=0 applies, if A is unital.
10-26-2009 08:16:03
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The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details