Derivation (differential algebra)

2025-01-21T18:16:22Z

137.204.150.25: /* Graded derivations */

{{Short description|Algebraic generalization of the derivative}}
In [[mathematics]], a '''derivation''' is a function on an [[algebra over a field|algebra]] that generalizes certain features of the [[derivative]] operator. Specifically, given an algebra ''A'' over a [[ring (mathematics)|ring]] or a [[field (mathematics)|field]] ''K'', a ''K''-derivation is a ''K''-[[linear map]] {{nowrap|''D'' : ''A'' → ''A''}} that satisfies [[Product rule|Leibniz's law]]:

:<math> D(ab) = a D(b) + D(a) b.</math>

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 Der''K''(''A''). The collection of ''K''-derivations of ''A'' into an ''A''-module ''M'' is denoted by {{nowrap|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 '''R'''''n''. 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,
:<math>[FG,N]=[F,N]G+F[G,N],</math>
where <math>[\cdot,N]</math> is the commutator with respect to <math>N</math>. 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 {{math|''D'': ''A'' → ''A''}} is a ''K''-derivation, then
* If ''A'' has a unit 1, then ''D''(1) = ''D''(12) = 2''D''(1), so that ''D''(1) = 0. Thus by ''K''-linearity, ''D''(''k'') = 0 for all {{math|''k'' ∈ ''K''}}.
* If ''A'' is commutative, ''D''(''x''2) = ''xD''(''x'') + ''D''(''x'')''x'' = 2''xD''(''x''), and ''D''(''x''''n'') = ''nx''''n''−1''D''(''x''), by the Leibniz rule.
* More generally, for any {{math|''x''1, ''x''2, …, ''x''''n'' ∈ ''A''}}, it follows by [[mathematical induction|induction]] that
*: <math>D(x_1x_2\cdots x_n) = \sum_i x_1\cdots x_{i-1}D(x_i)x_{i+1}\cdots x_n </math>
: which is <math display="inline">\sum_i D(x_i)\prod_{j\neq i}x_j</math> if for all {{mvar|i}}, {{math|''D''(''xi'')}} commutes with <math>x_1,x_2,\ldots, x_{i-1}</math>.
* For ''n'' > 1, ''D''''n'' is not a derivation, instead satisfying a higher-order Leibniz rule:
:: <math>D^n(uv) = \sum_{k=0}^n \binom{n}{k} \cdot D^{n-k}(u)\cdot D^k(v).</math>
: Moreover, if ''M'' is an ''A''-bimodule, write
:: <math> \operatorname{Der}_K(A,M)</math>
:for the set of ''K''-derivations from ''A'' to ''M''.
* {{nowrap|Der''K''(''A'', ''M'')}} is a [[module (mathematics)|module]] over ''K''.
* Der''K''(''A'') is a [[Lie algebra]] with Lie bracket defined by the [[commutator]]:
:: <math>[D_1,D_2] = D_1\circ D_2 - D_2\circ D_1.</math>
: since it is readily verified that the commutator of two derivations is again a derivation.
* There is an ''A''-module {{math|Ω''A''/''K''}} (called the [[Kähler differentials]]) with a ''K''-derivation {{math|''d'': ''A'' → Ω''A''/''K''}} through which any derivation {{math|''D'': ''A'' → ''M''}} factors. That is, for any derivation ''D'' there is a ''A''-module map {{mvar|φ}} with
:: <math> D: A\stackrel{d}{\longrightarrow} \Omega_{A/K}\stackrel{\varphi}{\longrightarrow} M </math>
: The correspondence <math> D\leftrightarrow \varphi</math> is an isomorphism of ''A''-modules:
:: <math> \operatorname{Der}_K(A,M)\simeq \operatorname{Hom}_{A}(\Omega_{A/K},M)</math>
* If {{math|''k'' ⊂ ''K''}} is a [[subring]], then ''A'' inherits a ''k''-algebra structure, so there is an inclusion
:: <math>\operatorname{Der}_K(A,M)\subset \operatorname{Der}_k(A,M) ,</math>
: 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
:<math>{D(ab)=D(a)b+\varepsilon^{|a||D|}aD(b)}</math>
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
:<math>{D(ab)=D(a)b+(-1)^{|a||D|}aD(b)}</math>
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 form]]s.

Graded derivations of [[superalgebra]]s (i.e. '''Z'''2-graded algebras) are often called '''superderivations'''.

==Related notions==

[[Hasse–Schmidt derivation]]s are ''K''-algebra homomorphisms

:<math>A \to A[[t]].</math>

Composing further with the map that sends a [[formal power series]] <math>\sum a_n t^n</math> to the coefficient <math>a_1</math> gives a derivation.

==See also==
*In [[differential geometry]] derivations are [[Tangent space#Definition via derivations|tangent vector]]s
*[[Kähler differential]]
*[[Hasse derivative]]
*[[p-derivation|''p''-derivation]]
*[[Wirtinger derivatives]]
*[[Derivative of the exponential map]]

==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=Kolař|first2=Jan|last2=Slovák|first3=Peter W.|last3=Michor|year=1993|publisher=Springer-Verlag|url=http://www.emis.de/monographs/KSM/index.html}}.

[[Category:Differential algebra]]

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Derivation (differential algebra)