derivation (abstract 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 that satisfies Leibniz's law:

:$D(ab) = a D(b) + D(a) b.$

More generally, if ''M'' is an ''A''-bimodule, a ''K''-linear map 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 .

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,
: G,N ,N+F ,N/math>
where cdot,N/math> is the commutator with respect to $N$. 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) = 2''D''(1), so that ''D''(1) = 0. Thus by ''K''-linearity, ''D''(''k'') = 0 for all .
* 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 , it follows by induction that
::$D(x_1x_2\cdots x_n) = \sum_i x_1\cdots x_D(x_i)x_\cdots x_n$
: which is $\sum_i D(x_i)\prod_x_j$ if for all , commutes with $x_1,x_2,\ldots, x_$.
* For ''n''>1, ''D''^''n'' is not a derivation, instead satisfying a higher-order Leibniz rule:
::$D^n(uv) = \sum_^n \binom \cdot D^(u)\cdot D^k(v).$
:Moreover, if ''M'' is an ''A''-bimodule, write
:: $\operatorname_K(A,M)$
:for the set of ''K''-derivations from ''A'' to ''M''.
* is a module over ''K''.
* Der_''K''(''A'') is a Lie algebra with Lie bracket defined by the commutator:
:: _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 (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
:: $D: A\stackrel \Omega_\stackrel M$
: The correspondence $D\leftrightarrow \varphi$ is an isomorphism of ''A''-modules:
:: $\operatorname_K(A,M)\simeq \operatorname_(\Omega_,M)$
*If is a subring, then ''A'' inherits a ''k''-algebra structure, so there is an inclusion
::$\operatorname_K(A,M)\subset \operatorname_k(A,M) ,$
: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 on ''A'', ''D'' is a homogeneous derivation if
:$$
for every homogeneous element ''a'' and every element ''b'' of ''A'' for a commutator factor . A graded derivation is sum of homogeneous derivations with the same ''ε''.

If , this definition reduces to the usual case. If , however, then
:$$
for odd , 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.

Related notions

Hasse–Schmidt derivations are ''K''-algebra homomorphisms

:A \to A t.

Composing further with the map which sends a formal power series $\sum a_n t^n$ to the coefficient $a_1$ gives a derivation.

See also

*In differential geometry derivations are tangent vectors
*Kähler differential
*Hasse derivative
* p-derivation
* Wirtinger derivatives
* Derivative of the exponential map

References

* .
* .
* .
* {{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.

Differential algebra