Derivation (differential algebra)
In various areas of mathematics is called illustrations as derivations, if they satisfy the Leibniz rule. The concept of derivations is a generalization of the familiar from school mathematics dissipation.
Definition
It is a commutative ring with unit, such as a body or as. Furthermore, it is an algebra. A ( linear ) Derivation of a linear map, the
Met. The property - linear states that for all and the equations
And
. apply The definition includes rings by being perceives as algebras. Forms in a module or bimodule from, so you can specify the definition analog.
Main Features
- Is an algebra with identity, then applies. This also applies to all.
- The core of a derivation is a subalgebra.
- The set of derivations of forms with values in the commutator a Lie algebra: Are and derivations, as well as
- For an element, a derivation. Derivations of this type are called inner derivations. The Hochschild cohomology is the quotient of the derivations module according to the sub-module of inner derivations.
- In a commutative algebra applies to any and all non-negative integers.
Examples
- The derivation of real functions is a derivation. This means the product rule.
- For the formal derivation
- Be a manifold. Then the Cartan derivative is a linear derivation of with values in the space of 1-forms.
- One of the reformulations of the Jacobi identity for Lie algebras states that the adjoint representation operated by derivations:
Derivations and Kähler differentials
By definition derivations of a commutative algebra by the module of Kähler differentials - linear be classified, ie there is a natural bijection between the linear derivations of with values in a module and the - linear maps. Each derivation arises as a concatenation of the universal derivation with a - linear mapping.
Anti derivations
Definition
Is a - or - graduate algebra, this means a linear graduated Figure a Antiderivation when
Valid for all homogeneous elements; it refers to the degree of.
Examples
- The exterior derivative of differential forms is a Antiderivation:
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