Ore Algebra Options - Maple Help

Options Available When Declaring Ore Algebras

Description

 • 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

 u = 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).

Examples

Changing the Ground Field

 Here is an example of operators over a finite field.
 > $\mathrm{with}\left(\mathrm{Ore_algebra}\right):$
 > $A≔\mathrm{skew_algebra}\left(\mathrm{diff}=\left[\mathrm{Dx},x\right]\right):$
 > $\mathrm{skew_product}\left(\mathrm{Dx},{x}^{5},A\right)$
 ${\mathrm{Dx}}{}{{x}}^{{5}}{+}{5}{}{{x}}^{{4}}$ (1)
 > $A≔\mathrm{skew_algebra}\left(\mathrm{diff}=\left[\mathrm{Dx},x\right],\mathrm{characteristic}=5\right):$
 > $\mathrm{skew_product}\left(\mathrm{Dx},{x}^{5},A\right)$
 ${\mathrm{Dx}}{}{{x}}^{{5}}$ (2)
 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.
 > $A≔\mathrm{skew_algebra}\left(\mathrm{diff}=\left[\mathrm{Dx},x\right],\mathrm{polynom}=x\right):$
 > $\mathrm{skew_product}\left(\mathrm{Dx},x,A\right)$
 ${\mathrm{Dx}}{}{x}{+}{1}$ (3)
 On the other hand, both following inputs are illegal:
 > $\mathrm{skew_product}\left(\mathrm{Dx},\frac{x}{x-1},A\right)$
 > $\mathrm{skew_product}\left(\mathrm{Dx},\mathrm{\eta }\left(x\right),A\right)$
 > $A≔\mathrm{skew_algebra}\left(\mathrm{diff}=\left[\mathrm{Dx},x\right]\right):$
 > $\mathrm{skew_product}\left(\mathrm{Dx},x,A\right)$
 ${\mathrm{Dx}}{}{x}{+}{1}$ (4)
 > $\mathrm{skew_product}\left(\mathrm{Dx},\frac{x}{x-1},A\right)$
 ${-}\frac{{1}}{{\left({x}{-}{1}\right)}^{{2}}}{+}\frac{{\mathrm{Dx}}{}{x}}{{x}{-}{1}}$ (5)
 This is an error:
 > $\mathrm{skew_product}\left(\mathrm{Dx},\mathrm{\eta }\left(x\right),A\right)$
 > $A≔\mathrm{skew_algebra}\left(\mathrm{diff}=\left[\mathrm{Dx},x\right],\mathrm{func}=\mathrm{\eta }\right):$
 > $\mathrm{skew_product}\left(\mathrm{Dx},x,A\right)$
 ${\mathrm{Dx}}{}{x}{+}{1}$ (6)
 > $\mathrm{skew_product}\left(\mathrm{Dx},\frac{x}{x-1},A\right)$
 ${-}\frac{{1}}{{\left({x}{-}{1}\right)}^{{2}}}{+}\frac{{\mathrm{Dx}}{}{x}}{{x}{-}{1}}$ (7)
 > $\mathrm{skew_product}\left(\mathrm{Dx},\mathrm{\eta }\left(x\right),A\right)$
 ${\mathrm{Dx}}{}{\mathrm{\eta }}{}\left({x}\right){+}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}\right)$ (8)
 This is an error:
 > $A≔\mathrm{skew_algebra}\left(\mathrm{diff}=\left[\mathrm{Dx},x\right],\mathrm{comm}=i,\mathrm{alg_relations}={i}^{2}+1\right):$
 > $\mathrm{skew_product}\left(i\mathrm{Dx},-i\mathrm{\eta }\left(x\right),A\right)$
 This is not:
 > $A≔\mathrm{skew_algebra}\left(\mathrm{diff}=\left[\mathrm{Dx},x\right],\mathrm{comm}=i,\mathrm{alg_relations}={i}^{2}+1,\mathrm{func}=\mathrm{\eta }\right):$
 > $\mathrm{skew_product}\left(i\mathrm{Dx},-i\mathrm{\eta }\left(x\right),A\right)$
 ${\mathrm{Dx}}{}{\mathrm{\eta }}{}\left({x}\right){+}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}\right)$ (9)

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):
 > $A≔\mathrm{skew_algebra}\left(\mathrm{diff}=\left[\mathrm{Dx},x\right]\right):$
 > $\mathrm{applyopr}\left(x,f\left(x\right),A\right)$
 ${x}{}{f}{}\left({x}\right)$ (10)
 > $\mathrm{applyopr}\left(\mathrm{Dx},f\left(x\right),A\right)$
 $\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)$ (11)
 > $\mathrm{applyopr}\left(x,u\left(n\right),A\right)$
 ${x}{}{u}{}\left({n}\right)$ (12)
 > $\mathrm{applyopr}\left(\mathrm{Dx},u\left(n\right),A\right)$
 ${0}$ (13)
 By changing the action, you can view the previous Weyl algebra as acting on sequences u(n) rather than on functions f(x).
 > A:=skew_algebra(diff=[Dx,x],polynom=x,action={     Dx=proc(u,order) local res; global n;         res:=u; to order do res:=subs(n=n+1,n*res) end do; res     end proc,     x=proc(u,order) global n;         subs(n=n-order,u)     end proc}):
 > $\mathrm{applyopr}\left(x,f\left(x\right),A\right)$
 ${f}{}\left({x}\right)$ (14)
 > $\mathrm{applyopr}\left(\mathrm{Dx},f\left(x\right),A\right)$
 $\left({n}{+}{1}\right){}{f}{}\left({x}\right)$ (15)
 > $\mathrm{applyopr}\left(x,u\left(n\right),A\right)$
 ${u}{}\left({n}{-}{1}\right)$ (16)
 > $\mathrm{applyopr}\left(\mathrm{Dx},u\left(n\right),A\right)$
 $\left({n}{+}{1}\right){}{u}{}\left({n}{+}{1}\right)$ (17)