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## 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`.

``` A=<<1,2,2,1>>;
(1)*<<1,2,2,1>>
 B=<<2,1,1,2>>;
(1)*<<2,1,1,2>>
 A*B;
(1)*<<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 avilable 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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