Derivation (mathematics)

In various sub-areas of mathematics , images are called derivatives if they meet Leibniz's rule . The concept of derivations is a generalization of the derivation known from school mathematics .

definition

Let it be a commutative ring with one , for example a body like or . Besides, be a - algebra . A ( -linear) derivation (also -derivation) of is a -linear mapping that ${\displaystyle R}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle A}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle A}$ ${\displaystyle R}$ ${\displaystyle D\colon A\to A}$

${\displaystyle D(a_{1}a_{2})=D(a_{1})a_{2}+a_{1}D(a_{2})}$ for all ${\displaystyle a_{1},a_{2}\in A}$

Fulfills. The -linear property says that for all and the equations ${\displaystyle R}$ ${\displaystyle a_{1},a_{2}\in A}$ ${\displaystyle r\in R}$

${\displaystyle D(a_{1}+a_{2})=D(a_{1})+D(a_{2})}$

and

${\displaystyle D(ra_{1})=rD(a_{1})}$

be valid. The definition includes rings by considering them as -algebras. If it is mapped into a module or bimodule , the definition can be given analogously. ${\displaystyle A}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle D}$

General properties

• Is an algebra with identity element , the following applies . This also applies to everyone . ${\displaystyle A}$ ${\displaystyle 1_{A}}$ ${\displaystyle D(1_{A})=0}$ ${\displaystyle D(r)=0}$ ${\displaystyle r\in R}$
• The core of a derivation is a sub-algebra.
• The set of derivations of with values ​​in forms a Lie algebra with the commutator : are and derivations, so too ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle D_{1}}$ ${\displaystyle D_{2}}$

${\displaystyle [D_{1},D_{2}]=D_{1}\circ D_{2}-D_{2}\circ D_{1}.}$

• For an item is , a derivation. Derivations of this type are called internal derivatives . The Hochschild cohomology is the quotient of the module of the derivatives after the sub-module of the inner derivatives. ${\displaystyle b\in A}$ ${\displaystyle D_{b}\colon A\to A}$ ${\displaystyle D_{b}(a)=ba-ab}$ ${\displaystyle H^{1}(A,A)}$
• In commutative algebra holds true for all and all non-negative integers . ${\displaystyle A}$ ${\displaystyle D(a^{n})=na^{n-1}D(a)}$ ${\displaystyle a\in A}$ ${\displaystyle n}$

Examples

• The derivation of real functions is a derivation. This is what the product rule says . ${\displaystyle f\colon D\subseteq \mathbb {R} \to \mathbb {R} }$
• For is the formal derivation ${\displaystyle A=R[X]}$

${\displaystyle \sum a_{i}X^{i}\mapsto \sum ia_{i}X^{i-1}}$

a -linear derivation of with values ​​in . ${\displaystyle R}$ ${\displaystyle A}$ ${\displaystyle A}$

• Be a manifold. Then the is Cartan deriving a -linear Derivation of values in the space of 1-forms on . ${\displaystyle X}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle C^{\infty }(X)}$ ${\displaystyle A^{1}(X)}$ ${\displaystyle X}$
• One of the reformulations of the Jacobi identity for Lie algebras says that the adjoint representation operates through derivatives:

${\displaystyle [X,[A,B]]=[[X,A],B]+[A,[X,B]].}$

Derivatives and Kähler differentials

By definition -linear derivatives of a commutative algebra are classified by the module of Kähler differentials , i.e. That is, there is a natural bijection between the -linear derivatives of with values ​​in a module and the -linear mappings . Every derivation arises as a concatenation of the universal derivation with a linear mapping . ${\displaystyle R}$ ${\displaystyle A}$ ${\displaystyle \Omega _{A/R}}$ ${\displaystyle R}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle M}$ ${\displaystyle A}$ ${\displaystyle \Omega _{A/R}\to M}$ ${\displaystyle D\colon A\to M}$ ${\displaystyle \mathrm {d} \colon A\to \Omega _{A/R}}$ ${\displaystyle A}$ ${\displaystyle \Omega _{A/R}\to M}$

Anti-derivatives

definition

If a - or - graduated -algebra, then a -linear graduated mapping is called an antiderivation , if ${\displaystyle A}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle D\colon A\to A}$

${\displaystyle D(a_{1}a_{2})=D(a_{1})a_{2}+(-1)^{|a_{1}|}\cdot a_{1}D(a_{2})}$

holds for all homogeneous elements ; where denotes the degree of . ${\displaystyle a_{1},a_{2}\in A}$ ${\displaystyle |a_{1}|}$ ${\displaystyle a_{1}}$

Examples

• The external derivation of differential forms is an anti-derivation:

${\displaystyle \mathrm {d} (\omega \wedge \eta )=\mathrm {d} \omega \wedge \eta +(-1)^{|\omega |}\cdot \omega \wedge \mathrm {d} \eta .}$

literature

• Siegfried Bosch : Algebra , 7th edition 2009, Springer-Verlag, ISBN 3-540-40388-4 , doi: 10.1007 / 978-3-540-92812-6 .