Ore_algebra - Maple Programming Help

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Ore_algebra

 diff_algebra
 create an algebra of linear differential operators

 Calling Sequence diff_algebra(l_1, ..., l_n)

Parameters

 l_i - list $\left[{\mathrm{D}}_{i},{x}_{i}\right]$ or a list $\left[\mathrm{comm},{a}_{i}\right]$ x_i - indeterminates (variable names) a_i - indeterminates (parameter names) D_i - indeterminates (differential operator names)

Description

 • 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:

$\mathrm{D}\left[i\right]\cdot x\left[i\right]=x\left[i\right]\cdot \mathrm{D}\left[i\right]+1,\mathbf{for}i=1,...,n$

 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].

Examples

 > $\mathrm{with}\left(\mathrm{Ore_algebra}\right):$
 > $A≔\mathrm{diff_algebra}\left(\left[\mathrm{Dx},x\right],\left[\mathrm{Dy},y\right]\right)$
 ${A}{≔}{\mathrm{Ore_algebra}}$ (1)
 > $\mathrm{skew_product}\left(\mathrm{Dx},x,A\right),\mathrm{skew_product}\left(\mathrm{Dy},y,A\right)$
 ${\mathrm{Dx}}{}{x}{+}{1}{,}{\mathrm{Dy}}{}{y}{+}{1}$ (2)
 > $\mathrm{skew_product}\left(\mathrm{Dx}\mathrm{Dy},xy,A\right)$
 ${\mathrm{Dx}}{}{\mathrm{Dy}}{}{x}{}{y}{+}{\mathrm{Dx}}{}{x}{+}{\mathrm{Dy}}{}{y}{+}{1}$ (3)
 > $\mathrm{skew_product}\left(\mathrm{Dx},{x}^{10},A\right)$
 ${\mathrm{Dx}}{}{{x}}^{{10}}{+}{10}{}{{x}}^{{9}}$ (4)

The following calls are equivalent.  The first syntax is more convenient to input numerous commutative parameters.

 > $\mathrm{skew_algebra}\left(\mathrm{comm}=\left\{a,b,c,d,e,f,g,h\right\},\mathrm{diff}=\left[\mathrm{Dx},x\right]\right)$
 ${\mathrm{Ore_algebra}}$ (5)
 > $\mathrm{diff_algebra}\left(\left[\mathrm{comm},a\right],\left[\mathrm{comm},b\right],\left[\mathrm{comm},c\right],\left[\mathrm{comm},d\right],\left[\mathrm{comm},e\right],\left[\mathrm{comm},f\right],\left[\mathrm{comm},g\right],\left[\mathrm{comm},h\right],\left[\mathrm{Dx},x\right]\right)$
 ${\mathrm{Ore_algebra}}$ (6)
 > $\mathrm{evalb}\left(=\right)$
 ${\mathrm{true}}$ (7)