Laurent polynomial

A Laurent polynomial ( after Pierre Alphonse Laurent ) is a generalization of the concept of polynomial mathematics. When Laurent polynomial and negative exponents are allowed.

Definition

Laurent a polynomial over a commutative ring is an expression of the form

Wherein only many ring members are different from 0 at. A Laurent polynomial can therefore be considered a finite number different from 0 coefficient as a Laurent series with only.

The ring of Laurent polynomials

With Laurent polynomials is expected formally as follows:

Addition:

Multiplication.

These operations make the deal about a ring, called the Laurent ring. There even is an R- module if we define multiplication by elements in an obvious way as follows:

Scalar multiplication: .

In many applications a body which is then algebra.

Properties

• Obtained from the polynomial ring, by inverting the Indeterminate. The Laurent- ring over it the localization after the semigroup generated by the positive powers of.
• The units are of the form, wherein a unit and.
• The Laurent- ring over is isomorphic to the group ring of about.

Derivations of Laurent- ring

It is a body. Then the amount of the derivative to a ion Lie algebra. The formal derivation

Such a derivation. Therefore, a derivation is also given to each by the definition and one can prove that this is the most common Derivation on. ( If such a derivation, it is and it can be shown. )

The derivations, therefore form a basis. By a short calculation confirms to the commutation relations

• For everyone.

(see Witt algebra ). Next apply

• For everyone.

Therefore, it is also called the degree derivation.

Swell

Igor Frenkel, James Lepowsky, Arne Meurman: Vertex Operator Algebras and the Monster, Academic Press, New York ( 1988) ISBN 0-12-267065-5

• Commutative Algebra
• Theory of Lie groups