Category:ApCoCoA-1:Package weyl

In the previous version of the ApCoCoA, one can use Groebner basis computation in commutative polynomial rings. However, Groebner basis can also be computed in non-commutative algebras. One such computation for Groebner basis in Weyl algebras was implemented in CoCoALib. Weyl package is designed to use this implementation and other operations in Weyl algebra.

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 ${\displaystyle [a,b]=ab-ba}$. Thus ${\displaystyle y_{i}\cdot x_{i}=x_{i}\cdot y_{i}+1}$ in ${\displaystyle A_{n}(K)}$.

${\displaystyle A_{n}(K)}$ 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 ${\displaystyle x_{1}^{i_{1}}\cdot ...\cdot x_{n}^{i_{n}}\cdot y_{1}^{j_{1}}\cdot ...y_{n}^{j_{n}}}$. The element in this form is then called in Normal Form.

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 Normal Form are given as polynomials in ApCoCoA. However, Weyl polynomials that are not in Normal form should be first converted into Normal form using the function Weyl.WNormalForm(L). Where L is a list of Lists. See below for description of the use of this function.

Addition and subtraction can be done by +, -, but because of the non-commutativity of ${\displaystyle A_{n}(K),}$ multiplication is performed by calling the function Weyl.WMul(). Hence in Weyl package, one can work with Weyl algebra using 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, yx=xy. So for every function in the package, it is not possible to check if the input polynomial F is in normal form or not. Thus if any of the monomial in F is of the form y^A*x^B then 1st use WNormalForm(L) to convert F into its normal 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, each function name in Weyl package starts with letter 'W'. For example, WNormalForm(), WGB(), etc.