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.
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.
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:
This is not:
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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