skew_algebra - Maple Help

Ore_algebra

 skew_algebra
 declare an Ore algebra

 Calling Sequence skew_algebra(t_1=l_1,..., t_n=l_n, options)

Parameters

 t_i - types of commutation l_i - lists of indeterminates whose lengths are determined by the corresponding t_i options - (optional) options described below

Description

 • The skew_algebra command declares an Ore algebra and returns a table that can be used by other functions of the Ore_algebra package.
 • An Ore algebra is an algebra used to represent linear operators that apply to functions or sequences, such as linear differentiation or recurrence operators.  It is an algebra of skew polynomials in the indeterminates ${x}_{1},...,{x}_{p},{d}_{1},...,{d}_{q}$ ruled by the following commutation relations: for any polynomial P in ${x}_{1},...,{x}_{p}$ and any i in $1,...,q$.

${d}_{i}P={\mathrm{\sigma }}_{i}\left(P\right){d}_{i}+{\mathrm{\delta }}_{i}\left(P\right)$

 Any pair ${x}_{i},{x}_{j}$ or ${d}_{i},{d}_{j}$ commute.  The sigma_is are algebra endomorphisms and the delta_is are additive functions that moreover satisfy the following skew Leibniz rule:

${\mathrm{\delta }}_{i}\left(PQ\right)={\mathrm{\sigma }}_{i}\left(P\right){\mathrm{\delta }}_{i}\left(Q\right)+{\mathrm{\delta }}_{i}\left(P\right)Q$

 • Weyl algebras are a special case of Ore algebras, obtained when all operators are differentiation operators. For more information, see Ore_algebra[Weyl_algebra].
 • The lists l_i involve the x_is and d_is, where the names x_i and the d_i may not be assigned.  Each list l_i consists of a pseudo-differential indeterminate d_i followed by one or more of the x_js.
 • The string t_i represents the type of the pseudo-derivative d_i.  It is either a predefined type or a user-defined type.  Possible commutations are described in Ore_algebra[commutation_rules].
 • Though Ore algebras are noncommutative algebras, they are represented with the standard commutative Maple product.  Every Ore_algebra function dealing with elements of an Ore algebra uses its normal form where all d_i appear on the right of the corresponding x_i.  A monomial ${x}^{a}{d}^{b}$, meant as a normal form, can therefore be printed either ${x}^{a}{d}^{b}$ or ${x}^{a}{d}^{b}$.
 • The sum in Ore 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).
 • 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

The following call declares an Ore algebra built on a differential operator Dx and on a shift operator Sn.  It also prepares the use of a function $\mathrm{\eta }\left(n\right)$ in the coefficients of the polynomials.

 > $\mathrm{with}\left(\mathrm{Ore_algebra}\right):$
 > $A≔\mathrm{skew_algebra}\left(\mathrm{diff}=\left[\mathrm{Dx},x\right],\mathrm{shift}=\left[\mathrm{Sn},n\right],\mathrm{func}=\mathrm{\eta }\right)$
 ${A}{≔}{\mathrm{Ore_algebra}}$ (1)

This is the name of a table.  Products in the algebra are performed using skew_product.

 > $\mathrm{skew_product}\left(\mathrm{Dx},x,A\right),\mathrm{skew_product}\left(\mathrm{Sn},n,A\right)$
 ${\mathrm{Dx}}{}{x}{+}{1}{,}\left({n}{+}{1}\right){}{\mathrm{Sn}}$ (2)
 > $\mathrm{skew_product}\left(\mathrm{Dx}\mathrm{Sn},xn,A\right)$
 $\left({x}{}{n}{+}{x}\right){}{\mathrm{Dx}}{}{\mathrm{Sn}}{+}\left({n}{+}{1}\right){}{\mathrm{Sn}}$ (3)
 > $\mathrm{skew_product}\left(\mathrm{Dx},\frac{1}{x},A\right)$
 $\frac{{\mathrm{Dx}}}{{x}}{-}\frac{{1}}{{{x}}^{{2}}}$ (4)
 > $\mathrm{skew_product}\left(\mathrm{Sn},\mathrm{\eta }\left(n\right),A\right)$
 ${\mathrm{\eta }}{}\left({n}{+}{1}\right){}{\mathrm{Sn}}$ (5)

The following declaration, however, is forbidden.

 > $\mathrm{skew_algebra}\left(\mathrm{diff}=\left[\mathrm{Dx},x\right],\mathrm{shift}=\left[\mathrm{Sx},x\right]\right)$