type for algebras of commutative polynomials - Maple 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+xy+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 q1.  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!

type1,OreAlgebra

false

(1)

A commutative algebra of polynomials.

withOre_algebra:

A:=poly_algebraa,b,c:

typeA,CommAlgebra,typeA,OreAlgebra

true,true

(2)

typea2+b2+c21,SkewPolynomialA

true

(3)

Skew algebras of linear differential operators.

A:=diff_algebraDx,x:

typeA,CommAlgebra,typeA,SkewAlgebra,typeA,SkewParamAlgebra

false,true,false

(4)

typexDx+1,SkewPolynomialA,typeDx+1x,SkewPolynomialA

true,true

(5)

A:=diff_algebraDx,x,polynom=x:

typeA,CommAlgebra,typeA,SkewAlgebra,typeA,SkewParamAlgebra

false,true,false

(6)

typexDx+1,SkewPolynomialA,typeDx+1x,SkewPolynomialA

true,false

(7)

Skew algebras of linear q-recurrence operators.

A:=qshift_algebraSn,qn:

typeA,CommAlgebra,typeA,SkewAlgebra,typeA,SkewParamAlgebra

false,false,true

(8)

typeqnSn1qn+1,SkewPolynomialA

true

(9)

See Also

Ore_algebra, Ore_algebra/diff_algebra, Ore_algebra/poly_algebra, Ore_algebra/shift_algebra, Ore_algebra/skew_algebra, type


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