Difference between revisions of "Category:ApCoCoA-1:Package weyl"

The package `Weyl' is designed to perform various computations in the Weyl algebras. In particular, one can compute a left Groebner basis of the left ideals in Weyl algebras.

Recall that the ${\displaystyle n}$ dimensional Weyl algebra ${\displaystyle A_{n}=K<x_{1},...,x_{n},y_{1},...,y_{n}>}$ over a field ${\displaystyle K}$, is a non-commutative free associative algebra modulo the following commutation relations:

${\displaystyle [x_{i},x_{j}]=0,[y_{i},y_{j}]=0,[y_{i},x_{j}]=0}$ for ${\displaystyle i\neq j,}$

and

${\displaystyle [y_{i},x_{i}]=1,}$

where ${\displaystyle [a,b]=ab-ba}$. From the last relation we have ${\displaystyle y_{i}\cdot x_{i}=x_{i}\cdot y_{i}+1}$, i.e. ${\displaystyle A_{n}(K)}$ is non-commutative. If ${\displaystyle K}$ is a field of characteristic 0 then ${\displaystyle A_{n}(K)}$ is simple. For the positive characteristic there exist non-trivial two-sided ideals of ${\displaystyle A_{n}(K)}$. The package Weyl also provides a function for computing the reduced two-sided Groebner basis of these two-sided ideals.

${\displaystyle A_{n}(K)}$ may be considered as a ring of differential operators with polynomial coefficients in ${\displaystyle K[x_{1},..,x_{n}]}$ and ${\displaystyle y_{i}}$ denotes the differentiation with respect to ${\displaystyle x_{i}}$. According to the commutation relation, elements of ${\displaystyle A_{n}(K)}$ can be represented as a ${\displaystyle K}$-linear combination of monomials ${\displaystyle x_{1}^{i_{1}}\cdot ...\cdot x_{n}^{i_{n}}\cdot y_{1}^{j_{1}}\cdot ...y_{n}^{j_{n}}}$. Such a representation of an element ${\displaystyle f}$ of ${\displaystyle A_{n}(K)}$ is called the Standard Form of ${\displaystyle f}$.

In Weyl package, we represent such monomial as x[1]^i[1]*...*x[n]^i[n]*y[1]^j[1]*...*y[n]^j[n] like in the case of commutative polynomial. That is, Weyl Polynomials of ${\displaystyle A_{n}(K)}$ that are in the Standard Form are given as polynomials in ApCoCoA. However, Weyl polynomials that are not in their standard form should have to be first converted into the standard form using the function Weyl.WStandardForm(L). Where L is a list of Lists. See below for the description of the use of this function.

Addition and subtraction can be done by +, -, but because of the non-commutativity of ${\displaystyle A_{n}(K),}$ the multiplication is performed by calling the function Weyl.WMul(). Hence in Weyl package, one can work with the Weyl algebra by using the ApCoCoA ring environment for the commutative polynomial rings in 2n indeterminates over the field ${\displaystyle \mathbb {Q} }$ or ${\displaystyle \mathbb {Z} /p}$ where ${\displaystyle p}$ is a prime number.

Note:

1) Due to commutativity of ApCoCoA, we have yx=xy. Therefore, for every function in the package, it is not possible to check if the input polynomial F is in the standard form or not. Thus if any of the monomial in F is of the form y^A*x^B then 1st use WStandardForm(L) to convert F into its standard form (here L is a list of lists representing monomials in F).

2) To make all functions in Weyl package different from functions in other packages of ApCoCoA, some of the function names in the Weyl package starts with the letter 'W'. For example, WStandardlForm(), WGB(), etc.