Weyl Algebras - Maple Help

Overview of Weyl Algebras

Description

 • Weyl algebras are algebras of linear differential operators with polynomial coefficients.  They are particular cases of Ore algebras.
 • A Weyl algebra is an algebra of noncommutative polynomials in the indeterminates ${x}_{1},...,{x}_{n},{\mathrm{D}}_{1},...,{\mathrm{D}}_{n}$ ruled by the following commutation relations:

${\mathrm{D}}_{i}{x}_{i}={x}_{i}{\mathrm{D}}_{i}+1,\mathrm{for}i=1,...,n$

 Any other pair of indeterminates commute.
 • 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:

$\mathrm{diff}\left({x}_{i}f\left({x}_{1},\mathrm{...},{x}_{n}\right),{x}_{i}\right)={x}_{i}\mathrm{diff}\left(f\left({x}_{1},\mathrm{...},{x}_{n}\right),{x}_{i}\right)+f\left({x}_{1},\mathrm{...},{x}_{n}\right)$

 • 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.
 • More specifically, Weyl algebras are defined as operators with polynomial coefficients.
 • The commands available for Weyl algebras are most of those of the Ore_algebra package, namely the following.
 Building an algebra

 Calculations in an algebra

 Action on Maple objects

 Converters

 • 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.)
 • 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.
 • The rand_skew_poly command generates a random element of a Weyl algebra.
 • The applyopr command applies an operator of a Weyl algebra to a function.
 • 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

 > $\mathrm{with}\left(\mathrm{Ore_algebra}\right):$
 > $A≔\mathrm{diff_algebra}\left(\left[\mathrm{Dx},x\right],\left[\mathrm{Dy},y\right],\left[\mathrm{Dz},z\right]\right):$
 > $\mathrm{skew_product}\left(\mathrm{Dx},x,A\right)$
 ${\mathrm{Dx}}{}{x}{+}{1}$ (1)
 > $\mathrm{skew_product}\left(\mathrm{Dy},y,A\right)$
 ${\mathrm{Dy}}{}{y}{+}{1}$ (2)
 > $\mathrm{skew_product}\left(\mathrm{Dz},z,A\right)$
 ${\mathrm{Dz}}{}{z}{+}{1}$ (3)
 > $\mathrm{skew_product}\left(\mathrm{Dx}\mathrm{Dy}\mathrm{Dz},xyz,A\right)$
 ${\mathrm{Dx}}{}{\mathrm{Dy}}{}{\mathrm{Dz}}{}{x}{}{y}{}{z}{+}{\mathrm{Dx}}{}{\mathrm{Dy}}{}{x}{}{y}{+}{\mathrm{Dx}}{}{\mathrm{Dz}}{}{x}{}{z}{+}{\mathrm{Dy}}{}{\mathrm{Dz}}{}{y}{}{z}{+}{\mathrm{Dx}}{}{x}{+}{\mathrm{Dy}}{}{y}{+}{\mathrm{Dz}}{}{z}{+}{1}$ (4)
 > $\mathrm{skew_product}\left({\mathrm{Dx}}^{3},{x}^{5},A\right)$
 ${{\mathrm{Dx}}}^{{3}}{}{{x}}^{{5}}{+}{15}{}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{4}}{+}{60}{}{\mathrm{Dx}}{}{{x}}^{{3}}{+}{60}{}{{x}}^{{2}}$ (5)