create an algebra of linear differential operators
diff_algebra(l_1, ..., l_n)
list Di,xi or a list comm,ai
indeterminates (variable names)
indeterminates (parameter names)
indeterminates (differential operator names)
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 Di,xi, 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 comm,ai, 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 xa⁢Db can therefore be printed either xa⁢Db or xa⁢Db.
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].
A ≔ diff_algebra⁡Dx,x,Dy,y
A ≔ Ore_algebra
The following calls are equivalent. The first syntax is more convenient to input numerous commutative parameters.
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