8.8 Weyl algebra

So far we have explained Groebner basis computation in commutative polynomial rings. However Groebner basis can be considered in more general non-commutative rings. Weyl algebra is one of such rings and Risa/Asir implements fundamental operations in Weyl algebra and Groebner basis computation in Weyl algebra.

The n dimensional Weyl algebra over a field K, D=K<x1,…,xn,D1,…,Dn> is a non-commutative algebra which has the following fundamental relations:

xi*xj-xj*xi=0, Di*Dj-Dj*Di=0, Di*xj-xj*Di=0 (i!=j), Di*xi-xi*Di=1

D is the ring of differential operators whose coefficients are polynomials in K[x1,…,xn] and Di denotes the differentiation with respect to xi. According to the commutation relation, elements of D can be represented as a K-linear combination of monomials x1^i1*…*xn^in*D1^j1*…*Dn^jn. In Risa/Asir, this type of monomial is represented by <<i1,…,in,j1,…,jn>> as in the case of commutative polynomial. That is, elements of D are represented by distributed polynomials. Addition and subtraction can be done by +, -, but multiplication is done by calling dp_weyl_mul() because of the non-commutativity of D.

[0] A=<<1,2,2,1>>;
(1)*<<1,2,2,1>>
[1] B=<<2,1,1,2>>;
(1)*<<2,1,1,2>>
[2] A*B;
(1)*<<3,3,3,3>>
[3] dp_weyl_mul(A,B);
(1)*<<3,3,3,3>>+(1)*<<3,2,3,2>>+(4)*<<2,3,2,3>>+(4)*<<2,2,2,2>>
+(2)*<<1,3,1,3>>+(2)*<<1,2,1,2>>

The following functions are available for Groebner basis computation in Weyl algebra: dp_weyl_gr_main(), dp_weyl_gr_mod_main(), dp_weyl_gr_f_main(), dp_weyl_f4_main(), dp_weyl_f4_mod_main(). Computation of the global b function is implemented as an application.

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