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 , 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 , 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  can therefore be printed either  or .

The sum in Weyl algebras is performed by using the `+` of Maple, 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].