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:

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:

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.

$\mathrm{with}\left(\mathrm{Ore\_algebra}\right)\:$

$A≔\mathrm{diff\_algebra}\left(\left[\mathrm{Dx}\,x\right]\,\left[\mathrm{Dy}\,y\right]\,\left[\mathrm{Dz}\,z\right]\right)\:$

$\mathrm{skew\_product}\left(\mathrm{Dx}\,x\,A\right)$

$\mathrm{Dx}x+1$

$\mathrm{skew\_product}\left(\mathrm{Dy}\,y\,A\right)$

$\mathrm{Dy}y+1$

$\mathrm{skew\_product}\left(\mathrm{Dz}\,z\,A\right)$

$\mathrm{Dz}z+1$

$\mathrm{skew\_product}\left(\mathrm{Dx}\mathrm{Dy}\mathrm{Dz}\,xyz\,A\right)$

$\mathrm{Dx}\mathrm{Dy}\mathrm{Dz}xyz+\mathrm{Dx}\mathrm{Dy}xy+\mathrm{Dx}\mathrm{Dz}xz+\mathrm{Dy}\mathrm{Dz}yz+\mathrm{Dx}x+\mathrm{Dy}y+\mathrm{Dz}z+1$

$\mathrm{skew\_product}\left({\mathrm{Dx}}^3\,x^5\,A\right)$

${\mathrm{Dx}}^3{x}^5+15{\mathrm{Dx}}^2{x}^4+60\mathrm{Dx}{x}^3+60x^2$