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:

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 $\left[{\mathrm{D}}_i\,x_i\right]$, 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 $\left[\mathrm{comm}\,a_i\right]$, 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 $x^a{\mathrm{D}}^b$ can therefore be printed either $x^a{\mathrm{D}}^b$ or $x^a{\mathrm{D}}^b$.

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].

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

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

$A≔\mathrm{Ore\_algebra}$

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

$\mathrm{Dx}x+1,\mathrm{Dy}y+1$

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

$\mathrm{Dx}\mathrm{Dy}xy+\mathrm{Dx}x+\mathrm{Dy}y+1$

$\mathrm{skew\_product}\left(\mathrm{Dx}\,x^{10}\,A\right)$

$\mathrm{Dx}{x}^{10}+10x^9$

$\mathrm{skew\_algebra}\left(\mathrm{comm}=\left\{a\,b\,c\,d\,e\,f\,g\,h\right\}\,\mathrm{diff}=\left[\mathrm{Dx}\,x\right]\right)$

$\mathrm{Ore\_algebra}$

$\mathrm{diff\_algebra}\left(\left[\mathrm{comm}\,a\right]\,\left[\mathrm{comm}\,b\right]\,\left[\mathrm{comm}\,c\right]\,\left[\mathrm{comm}\,d\right]\,\left[\mathrm{comm}\,e\right]\,\left[\mathrm{comm}\,f\right]\,\left[\mathrm{comm}\,g\right]\,\left[\mathrm{comm}\,h\right]\,\left[\mathrm{Dx}\,x\right]\right)$

$\mathrm{Ore\_algebra}$

$\mathrm{evalb}\left(=\right)$

$\mathrm{true}$