Options Available When Declaring Ore Algebras
The option characteristic=p, where p is 0 or any positive integer different from 1, is used to declare the characteristic of the algebra.
The option alg_relations=s, where s is an equation of a list or set of polynomial equations, introduces algebraic relations between commutative parameters; a polynomial p is meant as the equation p=0.
The option comm=s, where s is a name or a list or set of names, introduces commutative parameters; in case of a commutative algebra of polynomials, use Ore_algebra[poly_algebra] instead of Ore_algebra[skew_algebra].
The option polynom=s, where s is a name or a list or set of names, introduces indeterminates that are to be viewed as polynomial indeterminates (that is, may not appear rationally).
The option func=s, where s is a name or a list or set of names, introduces names of functions that are allowed to appear in the coefficients of the elements of the algebra.
The option action=s, where s is a set or list of equations of the form
proc(f,n) ... end proc
overloads the default actions of the operators on Maple objects. u is any of the indeterminates of the algebra that was declared in a commutation, and the right-hand side is a procedure that implements the action of the operator u on Maple objects. More specifically, a call to this procedure with an expression f and a non-negative integer n as arguments returns the (u@@n)(f) (see the Examples section below).
Changing the Ground Field
Here is an example of operators over a finite field.
A ≔ skew_algebra⁡diff=Dx,x:
A ≔ skew_algebra⁡diff=Dx,x,characteristic=5:
Here are Ore algebras on a polynomial ring and on a rational function field. The types of coefficients allowed differ accordingly. In particular, generic functions are allowed in the rational case only, and have to be explicitly declared.
A ≔ skew_algebra⁡diff=Dx,x,polynom=x:
On the other hand, both following inputs are illegal:
Error, (in Ore_algebra:-skew_product) skew polynomials must be members of the algebra
This is an error:
A ≔ skew_algebra⁡diff=Dx,x,func=η:
A ≔ skew_algebra⁡diff=Dx,x,comm=i,alg_relations=i2+1:
This is not:
A ≔ skew_algebra⁡diff=Dx,x,comm=i,alg_relations=i2+1,func=η:
Action on Maple Objects
Each commutation type has its default action on Maple objects. For instance, the diff commutation acts on functions f(x) and not on sequences u(n):
By changing the action, you can view the previous Weyl algebra as acting on sequences u(n) rather than on functions f(x).
Dx=proc(u,order) local res; global n;
res:=u; to order do res:=subs(n=n+1,n*res) end do; res
x=proc(u,order) global n;
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