type/CommAlgebra - Maple Programming Help

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type/CommAlgebra

type for algebras of commutative polynomials

type/OreAlgebra

type for all commutative and skew algebras

type/SkewAlgebra

type for simple skew algebras

type/SkewParamAlgebra

type for other skew algebras

type/SkewPolynomial

type for skew polynomials

 Calling Sequence type(A, CommAlgebra) type(A, OreAlgebra) type(A, SkewAlgebra) type(A, SkewParamAlgebra) type(P, SkewPolynomial(A))

Parameters

 A - table that denotes an algebra P - polynomial in such an algebra

Description

 • The type CommAlgebra checks if the algebra A is an algebra of commutative polynomials, as declared by Ore_algebra[poly_algebra] (or Ore_algebra[skew_algebra] with no commutation and commutative parameters only).
 • The type SkewAlgebra checks if the algebra A is built by using Ore_algebra[skew_algebra] with commutations of the form

$yx=py+sx+yx+r$

 for constants p, r, and s only.  This is the case for the commutation types delta, diff, euler, shift, and their dual forms.
 • The type SkewParamAlgebra checks if the algebra A is built by using Ore_algebra[skew_algebra] with commutations of the form

$yx=qxy+py+sx+r$

 for constants p, q, r, and s with at least one commutation with $q\ne 1$.  This is the case for the commutation types qdelta, qdiff, qdilat, qshift, shift+qshift, and their dual forms.
 • The type OreAlgebra checks if the algebra A is any of the above.
 • The type SkewPolynomial checks if the membership of the polynomial P in the algebra A.  When this algebra allows rational function coefficients, a polynomial with rational function coefficients is a member of the algebra.

Examples

Not an algebra!

 > $\mathrm{type}\left(1,\mathrm{OreAlgebra}\right)$
 ${\mathrm{false}}$ (1)

A commutative algebra of polynomials.

 > $\mathrm{with}\left(\mathrm{Ore_algebra}\right):$
 > $A≔\mathrm{poly_algebra}\left(a,b,c\right):$
 > $\mathrm{type}\left(A,\mathrm{CommAlgebra}\right),\mathrm{type}\left(A,\mathrm{OreAlgebra}\right)$
 ${\mathrm{true}}{,}{\mathrm{true}}$ (2)
 > $\mathrm{type}\left({a}^{2}+{b}^{2}+{c}^{2}-1,\mathrm{SkewPolynomial}\left(A\right)\right)$
 ${\mathrm{true}}$ (3)

Skew algebras of linear differential operators.

 > $A≔\mathrm{diff_algebra}\left(\left[\mathrm{Dx},x\right]\right):$
 > $\mathrm{type}\left(A,\mathrm{CommAlgebra}\right),\mathrm{type}\left(A,\mathrm{SkewAlgebra}\right),\mathrm{type}\left(A,\mathrm{SkewParamAlgebra}\right)$
 ${\mathrm{false}}{,}{\mathrm{true}}{,}{\mathrm{false}}$ (4)
 > $\mathrm{type}\left(x\mathrm{Dx}+1,\mathrm{SkewPolynomial}\left(A\right)\right),\mathrm{type}\left(\mathrm{Dx}+\frac{1}{x},\mathrm{SkewPolynomial}\left(A\right)\right)$
 ${\mathrm{true}}{,}{\mathrm{true}}$ (5)
 > $A≔\mathrm{diff_algebra}\left(\left[\mathrm{Dx},x\right],\mathrm{polynom}=x\right):$
 > $\mathrm{type}\left(A,\mathrm{CommAlgebra}\right),\mathrm{type}\left(A,\mathrm{SkewAlgebra}\right),\mathrm{type}\left(A,\mathrm{SkewParamAlgebra}\right)$
 ${\mathrm{false}}{,}{\mathrm{true}}{,}{\mathrm{false}}$ (6)
 > $\mathrm{type}\left(x\mathrm{Dx}+1,\mathrm{SkewPolynomial}\left(A\right)\right),\mathrm{type}\left(\mathrm{Dx}+\frac{1}{x},\mathrm{SkewPolynomial}\left(A\right)\right)$
 ${\mathrm{true}}{,}{\mathrm{false}}$ (7)

Skew algebras of linear q-recurrence operators.

 > $A≔\mathrm{qshift_algebra}\left(\left[\mathrm{Sn},{q}^{n}\right]\right):$
 > $\mathrm{type}\left(A,\mathrm{CommAlgebra}\right),\mathrm{type}\left(A,\mathrm{SkewAlgebra}\right),\mathrm{type}\left(A,\mathrm{SkewParamAlgebra}\right)$
 ${\mathrm{false}}{,}{\mathrm{false}}{,}{\mathrm{true}}$ (8)
 > $\mathrm{type}\left(\frac{{q}^{n}}{1-{q}^{n}}\mathrm{Sn}+1,\mathrm{SkewPolynomial}\left(A\right)\right)$
 ${\mathrm{true}}$ (9)