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Derivation (differential algebra)

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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 {{nowrap|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 {{nowrap|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 {{nowrap|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 Dcolon Ato 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 kin 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 {{nowrap|x1, x2, ..., xn ∈ A}}, it follows by induction that


D(x_1x_2cdots x_n) = sum_i x_1cdots x_{i-1}D(x_i)x_{i+1}cdots x_n
which is sum_i D(x_i)prod_{jneq 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 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.


[D_1,D_2] = D_1circ D_2 - D_2circ 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:Ato Omega_{A/K} through which any derivation D:Ato M factors. That is, for any derivation D there is a A-module map varphi with


D: Astackrel{d}{longrightarrow} Omega_{A/K}stackrel{varphi}{longrightarrow} M
The correspondence Dleftrightarrow varphi is an isomorphism of A-modules:
operatorname{Der}_K(A,M)simeq operatorname{Hom}_{A}(Omega_{A/K},M)
  • If {{nowrap|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

{{Anchor|Homogeneous derivation|Graded derivation}}Given a graded algebra A and a homogeneous linear map D of grade {{abs|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 {{nowrap|1=ε = ±1}}. A graded derivation is sum of homogeneous derivations with the same ε.If {{nowrap|1=ε = 1}}, this definition reduces to the usual case. If {{nowrap|1=ε = −1}}, however, then
{D(ab)=D(a)b+(-1)^{|a|}aD(b)}
for odd {{abs|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

  • {{Citation|first=Nicolas|last=Bourbaki|authorlink=Nicolas Bourbaki|title=Algebra I|year=1989|publisher=Springer-Verlag|isbn=3-540-64243-9|series=Elements of mathematics}}.
  • {{citation|first=David|authorlink=David Eisenbud|last=Eisenbud|title=Commutative algebra with a view toward algebraic geometry|isbn=978-0-387-94269-8|publisher=Springer-Verlag|year=1999|edition=3rd.}}.
  • {{citation|first=Hideyuki|last=Matsumura|title=Commutative algebra|publisher=W. A. Benjamin|year=1970|series=Mathematics lecture note series|isbn=978-0-8053-7025-6}}.
  • {{citation|title=Natural operations in differential geometry|first1=Ivan|last1=KolaÅ™|first2=Jan|last2=Slovák|first3=Peter W.|last3=Michor|year=1993|publisher=Springer-Verlag|url=http://www.emis.de/monographs/KSM/index.html}}.


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