Overview of Weyl Algebras - Maple Help

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Overview of Weyl Algebras

Description

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Weyl algebras are algebras of linear differential operators with polynomial coefficients.  They are particular cases of Ore algebras.

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A Weyl algebra is an algebra of noncommutative polynomials in the indeterminates x1,...,xn,D1,...,Dn ruled by the following commutation relations:

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

  

Any other pair of indeterminates commute.

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In the previous equation, x_i and D_i represent multiplication by x_i and differentiation with respect to x_i respectively.  The (noncommutative) inner product in the Ore algebra represents the composition of operators. Therefore, the identity reduces to the Leibniz rule:

diffxifx1,...,xn,xi=xidifffx1,...,xn,xi+fx1,...,xn

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Since Weyl algebras are particular cases of Ore algebras, you can use most commands of the Ore_algebra package on Weyl algebras without knowing the definition of Ore algebras. For details, see Ore_algebra.

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More specifically, Weyl algebras are defined as operators with polynomial coefficients.

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The commands available for Weyl algebras are most of those of the Ore_algebra package, namely the following.

  

Building an algebra

diff_algebra

dual_algebra

reverse_algebra

skew_algebra

  

Calculations in an algebra

annihilators

dual_polynomial

rand_skew_poly

reverse_polynomial

skew_elim

skew_gcdex

skew_pdiv

skew_power

skew_prem

skew_product

 

 

  

Action on Maple objects

applyopr

 

 

 

  

Converters

Ore_to_DESol

Ore_to_diff

 

 

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The skew_algebra and diff_algebra commands declare new algebras to work with.  They return a table needed by other Ore_algebra procedures.  The diff_algebra command creates a Weyl algebra.  The skew_algebra command creates a general Ore algebra, but can also be used to create a Weyl algebra. (The latter alternative is in fact more convenient in the case of Weyl algebras with numerous commutative parameters.)

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The skew_product and skew_power commands implement the arithmetic of Weyl algebras.  Skew polynomials in a Weyl algebra are represented by commutative polynomials of Maple.  The sum of skew polynomials is performed using the Maple `+` command. Their product, however, is performed using the skew_product command. Correspondingly, powers of skew polynomials are computed using the skew_power command.

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The rand_skew_poly command generates a random element of a Weyl algebra.

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The applyopr command applies an operator of a Weyl algebra to a function.

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The annihilators, skew_pdiv, skew_prem, skew_gcdex, and skew_elim commands implement a skew Euclidean algorithm in Weyl algebras and provide with related functionalities, such as computing remainders, gcds, (limited) elimination.  The annihilators command makes it possible to compute a lcm of two skew polynomials.  The skew_pdiv command computes pseudo-divisions in a Weyl algebra, while skew_prem simply computes corresponding pseudo-remainders.  The skew_gcdex command performs extended gcd computation in a Weyl algebra. When possible, the skew_elim command eliminates an indeterminate between two skew polynomials.

Examples

withOre_algebra:

A:=diff_algebraDx,x,Dy,y,Dz,z:

skew_productDx,x,A

Dxx+1

(1)

skew_productDy,y,A

Dyy+1

(2)

skew_productDz,z,A

Dzz+1

(3)

skew_productDxDyDz,xyz,A

DxDyDzxyz+DxDyxy+DxDzxz+DyDzyz+Dxx+Dyy+Dzz+1

(4)

skew_productDx3,x5,A

Dx3x5+15Dx2x4+60Dxx3+60x2

(5)

See Also

Ore_algebra, UsingPackages, with


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