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doc /// Key makeWeylAlgebra (makeWeylAlgebra,PolynomialRing) [makeWeylAlgebra,SetVariables] SetVariables Headline Weyl algebra corresponding to a polynomial ring Usage makeWeylAlgebra R makeWA R Inputs R:PolynomialRing the (commutative) polynomial ring SetVariables=>Boolean indicates whether the generators should be assigned to global variables Outputs :PolynomialRing the (non-commutative) Weyl algebra Description Text Given a polynomial ring @EM "R"@ with variables @EM "x_1,..,x_n"@, this routine returns a Weyl algebra with variables @EM "x_1,..,x_n"@ and @EM "dx_1,..,dx_n"@. Example R = QQ[x,y,z] D = makeWeylAlgebra R Text To skip naming the ring, use parentheses. Example makeWA(QQ[x,y,z]) Caveat The polynomial ring R must be commutative. SeeAlso [monoid, WeylAlgebra] Subnodes createDpairs extractDiffsAlgebra extractVarsAlgebra -- createThetaRing /// -* doc /// Key createThetaRing (createThetaRing, PolynomialRing) ThetaRing isGeneric WtoT Headline Usage Inputs Outputs Description Text Example n = 4 W = QQ[t_0..t_n, x_0..x_n, dx_0..dx_n, WeylAlgebra => {x_0..x_n => dx_0..dx_n}] createThetaRing W W.ThetaRing /// *- document { Key => {createDpairs, (createDpairs, PolynomialRing), dpairInds, dpairVars}, Headline => "pairs coordinate and derivation variables in a Weyl algebra", Usage => "createDpairs A", Inputs => { "A" => "the Weyl algebra" }, Consequences => { {"attaches to ", TT "A", " a pair of keys to help distinguish the coordinate variables from the derivation variables."} }, "Since the Weyl algebra has commutation rules, this routine attaches to the Weyl algebra two keys to organize the variables. The first key 'dpairVars' contains 3 lists: a list of the coordinate variables, a list of the derivative variables, and a list of the central variables. The second key 'dpairInds' also contains 3 lists of the corresponding indices to 'dpairVars'.", EXAMPLE lines /// W = QQ[x,y,Dx,Dy, WeylAlgebra => {x=>Dx,y=>Dy}] createDpairs W W.dpairVars W.dpairInds ///, SeeAlso => {[monoid, WeylAlgebra]} } doc /// Key extractVarsAlgebra (extractVarsAlgebra, PolynomialRing) Headline underlying polynomial ring in the ordinary variables of a Weyl algebra Usage extractVarsAlgebra D Inputs D:PolynomialRing a Weyl algebra Outputs :PolynomialRing Description Text Extracts from a Weyl algebra the polynomial ring in its ordinary variables. Example D = makeWA(QQ[x,y]) R = extractVarsAlgebra D describe R SeeAlso extractDiffsAlgebra /// doc /// Key extractDiffsAlgebra (extractDiffsAlgebra, PolynomialRing) Headline underlying polynomial ring in the differentials of a Weyl algebra Usage extractDiffsAlgebra D Inputs D:PolynomialRing a Weyl algebra Outputs :PolynomialRing Description Text Extracts from a Weyl algebra the polynomial ring in its differentials only. Example D = makeWA(QQ[x,y]) S = extractDiffsAlgebra D describe S SeeAlso extractVarsAlgebra ///
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