Laurent polynomial

A Laurent polynomial (after Pierre Alphonse Laurent ) is a generalization of the term polynomial in mathematics . Negative exponents are also allowed for the Laurent polynomial.

definition

A Laurent polynomial over a commutative ring is an expression of shape ${\displaystyle R}$

${\displaystyle p(X)=\sum _{k\in \mathbb {Z} }a_{k}X^{k},\quad a_{k}\in R}$,

in which only finitely many ring elements are different from 0. A Laurent polynomial can therefore be viewed as a Laurent series with only a finite number of coefficients different from 0. ${\displaystyle a_{k}}$

The ring of Laurent polynomials

With Laurent polynomials one calculates formally as follows:

Addition: , ${\displaystyle {}\quad \quad \sum _{i\in \mathbb {Z} }a_{i}X^{i}\,+\,\sum _{i\in \mathbb {Z} }b_{i}X^{i}=\sum _{i\in \mathbb {Z} }(a_{i}+b_{i})X^{i}}$

Multiplication: . ${\displaystyle \sum _{i\in \mathbb {Z} }a_{i}X^{i}\,\cdot \,\sum _{j\in \mathbb {Z} }b_{j}X^{j}=\sum _{k\in \mathbb {Z} }\left(\sum _{i,j:i+j=k}a_{i}b_{j}\right)X^{k}}$

These operations make the amount in a ring, called Laurent ring over . It is even an R-module if one defines the multiplication with elements in an obvious way as follows: ${\displaystyle R[X,X^{-1}]}$ ${\displaystyle R}$ ${\displaystyle a\in R}$

Scalar multiplication . ${\displaystyle a\cdot \sum _{i\in \mathbb {Z} }a_{i}X^{i}\,=\,\sum _{i\in \mathbb {Z} }(aa_{i})\,X^{i}}$

In many applications there is a field , then there is an - algebra . ${\displaystyle R}$ ${\displaystyle R[X,X^{-1}]}$ ${\displaystyle R}$

properties

• Obtained from the polynomial ring by the Indeterminate inverted . The Laurent ring above is the localization of after the semigroup produced by the positive potencies of . ${\displaystyle R[X,X^{-1}]}$ ${\displaystyle R[X]}$ ${\displaystyle X}$ ${\displaystyle R}$ ${\displaystyle R[X]}$ ${\displaystyle X}$
• The units of are of the form , where one unit is and . ${\displaystyle R[X,X^{-1}]}$ ${\displaystyle aX^{i}}$ ${\displaystyle a\in R}$ ${\displaystyle i\in \mathbb {Z} }$
• The Laurent-ring over is isomorphic to the group ring of about . ${\displaystyle R}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle R}$

Derivatives of the Laurent ring

It is a body. Then is the set of derivatives on a Lie algebra . The formal derivation ${\displaystyle R}$ ${\displaystyle R[X,X^{-1}]}$

${\displaystyle {\frac {\partial }{\partial X}}:\,\sum _{i\in \mathbb {Z} }a_{i}X^{i}\mapsto \sum _{i\in \mathbb {Z} }i\cdot a_{i}X^{i-1}}$

is such a derivation. Therefore, a derivation is also given for each by the definition and one can prove that this is the most general derivation . If there is such a derivation, then it is and one can show. ${\displaystyle p(X)\in R[X,X^{-1}]}$ ${\displaystyle \textstyle T_{p(X)}:=p(X){\frac {\partial }{\partial X}}}$ ${\displaystyle R[X,X^{-1}]}$ ${\displaystyle T}$ ${\displaystyle p(X):=T(1\cdot X)\in R[X,X^{-1}]}$ ${\displaystyle T=T_{p(X)}}$

The derivatives therefore form a basis. A short calculation confirms the commutator relations ${\displaystyle \textstyle d_{n}:=T_{-X^{n+1}}=-X^{n+1}{\frac {\partial }{\partial X}},\,n\in \mathbb {Z} }$

• ${\displaystyle [d_{m},d_{n}]\,=\,(m-n)d_{m+n}}$for everyone . ${\displaystyle m,n\in \mathbb {Z} }$

(see Witt algebra ). Further applies

• ${\displaystyle d_{0}(X^{n})=-n\cdot X^{n}}$for everyone . ${\displaystyle n\in \mathbb {Z} }$

This is why it is also called the degree derivation . ${\displaystyle -d_{0}\,}$

Individual evidence

1. ^ Igor Frenkel , James Lepowsky , Arne Meurman : Vertex Operator Algebras and the Monster , Academic Press, New York (1988) ISBN 0-12-267065-5 , sentence 1.9.1