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Ore_algebra

  

diff_algebra

  

create an algebra of linear differential operators

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

diff_algebra(l_1, ..., l_n)

Parameters

l_i

-

list Di,xi or a list comm,ai

x_i

-

indeterminates (variable names)

a_i

-

indeterminates (parameter names)

D_i

-

indeterminates (differential operator names)

Description

• 

The diff_algebra command declares an Ore algebra and returns a table that can be used by other functions of the Ore_algebra package.

• 

A Weyl algebra is an algebra of noncommutative polynomials in the indeterminates x_1,..., x_n, D_1,..., D_n ruled by the following commutation relations:

Dixi=xiDi+1,fori=1,...,n

  

Any other pair of indeterminates commute.

• 

Note that Weyl algebras are a special case of Ore algebras.  For more information, see Ore_algebra.

• 

The name x_i may not be assigned.

• 

The name D_i may not be assigned.  It is used to denote the differential indeterminate D_i associated to the base indeterminate x_i, that is, the operator of differentiation with respect to x_i.

• 

When the list l_i is of the form Di,xi, the names x_i and D_i may not be assigned.  Both indeterminates commute with any other indeterminate of the algebra.

• 

When the list l_i is of the form comm,ai, the name a_i may not be assigned.  It denotes a parameter that commutes with any other indeterminate of the algebra.

• 

Though Weyl algebras are noncommutative algebras, their elements are represented with the standard commutative Maple product. Every Ore_algebra function dealing with elements of a Weyl algebra uses its normal form where all D_i appear on the right of the corresponding x_i.  A monomial xaDb can therefore be printed either xaDb or xaDb.

• 

The sum in Weyl algebras is performed by using the `+` operator, while the product is performed by the Ore_algebra function skew_product (see the Examples section below).

• 

It is also possible to declare a Weyl algebra by using Ore_algebra[skew_algebra].

• 

Options are available to control the ground ring of the algebra and the action of the operators on Maple objects.  See Ore_algebra[declaration_options].

Examples

withOre_algebra:

Adiff_algebraDx,x,Dy,y

A:=Ore_algebra

(1)

skew_productDx,x,A,skew_productDy,y,A

Dxx+1,Dyy+1

(2)

skew_productDxDy,xy,A

DxDyxy+Dxx+Dyy+1

(3)

skew_productDx,x10,A

Dxx10+10x9

(4)

The following calls are equivalent.  The first syntax is more convenient to input numerous commutative parameters.

skew_algebracomm=a,b,c,d,e,f,g,h,diff=Dx,x

Ore_algebra

(5)

diff_algebracomm,a,comm,b,comm,c,comm,d,comm,e,comm,f,comm,g,comm,h,Dx,x

Ore_algebra

(6)

evalb=

true

(7)

See Also

Ore_algebra

Ore_algebra/skew_algebra

Ore_algebra/skew_product

Ore_algebra/Weyl_algebra

 


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