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WeylAlgebras -- algorithms for D-modules

Description

To begin, read the D-modules tutorial.

How to make Weyl algebras

Basic commands

  • gbw -- compute a Gröbner basis with respect to a weight vector
  • inw -- get the initial term or ideal with respect to a weight vector
  • Fourier -- Fourier transform for Weyl algebra
  • Dtransposition -- standard transposition for Weyl algebra
  • stafford -- computes 2 generators for a given ideal in the Weyl algebra
  • makeCyclic -- finds a cyclic generator of a D-module
  • Dprune -- prunes a D-module

Basic invariants of D-modules

Programming aids

  • createDpairs -- pairs coordinate and derivation variables in a Weyl algebra
  • extractDiffsAlgebra -- underlying polynomial ring in the differentials of a Weyl algebra
  • extractVarsAlgebra -- underlying polynomial ring in the ordinary variables of a Weyl algebra
  • Dtrace -- set or get the depth of comments made by D-module routines

Authors

Version

This documentation describes version 1.4.1.1 of WeylAlgebras.

Source code

The source code from which this documentation is derived is in the file WeylAlgebras.m2. The auxiliary files accompanying it are in the directory WeylAlgebras/.

Exports

  • Functions and commands
    • characteristicIdeal -- characteristic ideal of a D-module
    • createDpairs -- pairs coordinate and derivation variables in a Weyl algebra
    • Ddim -- dimension of a D-module
    • Dprune -- prunes a D-module
    • DsingularLocus -- singular locus of a D-module
    • Dtrace -- set or get the depth of comments made by D-module routines
    • Dtransposition -- standard transposition for Weyl algebra
    • extractDiffsAlgebra -- underlying polynomial ring in the differentials of a Weyl algebra
    • extractVarsAlgebra -- underlying polynomial ring in the ordinary variables of a Weyl algebra
    • Fourier -- Fourier transform for Weyl algebra
    • FourierInverse -- see Fourier -- Fourier transform for Weyl algebra
    • gbw -- compute a Gröbner basis with respect to a weight vector
    • holonomicRank -- holonomic rank of a D-module
    • inw -- get the initial term or ideal with respect to a weight vector
    • isHolonomic -- determines whether a D-module (or ideal in Weyl algebra) is holonomic
    • makeCyclic -- finds a cyclic generator of a D-module
    • makeWeylAlgebra -- Weyl algebra corresponding to a polynomial ring
    • getHomSwitch -- see setHomSwitch -- toggle the use of homogeneous Weyl algebra
    • setHomSwitch -- toggle the use of homogeneous Weyl algebra
    • stafford -- computes 2 generators for a given ideal in the Weyl algebra
  • Methods
    • characteristicIdeal(Ideal) -- see characteristicIdeal -- characteristic ideal of a D-module
    • characteristicIdeal(Module) -- see characteristicIdeal -- characteristic ideal of a D-module
    • createDpairs(PolynomialRing) -- see createDpairs -- pairs coordinate and derivation variables in a Weyl algebra
    • Ddim(Ideal) -- see Ddim -- dimension of a D-module
    • Ddim(Module) -- see Ddim -- dimension of a D-module
    • Dprune(Matrix) -- see Dprune -- prunes a D-module
    • Dprune(Module) -- see Dprune -- prunes a D-module
    • DsingularLocus(Ideal) -- see DsingularLocus -- singular locus of a D-module
    • DsingularLocus(Module) -- see DsingularLocus -- singular locus of a D-module
    • Dtrace(Sequence) -- see Dtrace -- set or get the depth of comments made by D-module routines
    • Dtrace(ZZ) -- see Dtrace -- set or get the depth of comments made by D-module routines
    • Dtransposition(ChainComplex) -- see Dtransposition -- standard transposition for Weyl algebra
    • Dtransposition(Ideal) -- see Dtransposition -- standard transposition for Weyl algebra
    • Dtransposition(Matrix) -- see Dtransposition -- standard transposition for Weyl algebra
    • Dtransposition(RingElement) -- see Dtransposition -- standard transposition for Weyl algebra
    • extractDiffsAlgebra(PolynomialRing) -- see extractDiffsAlgebra -- underlying polynomial ring in the differentials of a Weyl algebra
    • extractVarsAlgebra(PolynomialRing) -- see extractVarsAlgebra -- underlying polynomial ring in the ordinary variables of a Weyl algebra
    • Fourier(ChainComplex) -- see Fourier -- Fourier transform for Weyl algebra
    • Fourier(Ideal) -- see Fourier -- Fourier transform for Weyl algebra
    • Fourier(Matrix) -- see Fourier -- Fourier transform for Weyl algebra
    • Fourier(Module) -- see Fourier -- Fourier transform for Weyl algebra
    • Fourier(RingElement) -- see Fourier -- Fourier transform for Weyl algebra
    • FourierInverse(ChainComplex) -- see Fourier -- Fourier transform for Weyl algebra
    • FourierInverse(Ideal) -- see Fourier -- Fourier transform for Weyl algebra
    • FourierInverse(Matrix) -- see Fourier -- Fourier transform for Weyl algebra
    • FourierInverse(Module) -- see Fourier -- Fourier transform for Weyl algebra
    • FourierInverse(RingElement) -- see Fourier -- Fourier transform for Weyl algebra
    • gbw(Ideal,List) -- see gbw -- compute a Gröbner basis with respect to a weight vector
    • gbw(Matrix,List) -- see gbw -- compute a Gröbner basis with respect to a weight vector
    • holonomicRank(Ideal) -- see holonomicRank -- holonomic rank of a D-module
    • holonomicRank(Module) -- see holonomicRank -- holonomic rank of a D-module
    • inw(Ideal,List) -- see inw -- get the initial term or ideal with respect to a weight vector
    • inw(Matrix,List) -- see inw -- get the initial term or ideal with respect to a weight vector
    • inw(RingElement,List) -- see inw -- get the initial term or ideal with respect to a weight vector
    • isHolonomic(Ideal) -- see isHolonomic -- determines whether a D-module (or ideal in Weyl algebra) is holonomic
    • isHolonomic(Module) -- see isHolonomic -- determines whether a D-module (or ideal in Weyl algebra) is holonomic
    • makeCyclic(Matrix) -- see makeCyclic -- finds a cyclic generator of a D-module
    • makeWeylAlgebra(PolynomialRing) -- see makeWeylAlgebra -- Weyl algebra corresponding to a polynomial ring
    • setHomSwitch(Boolean) -- see setHomSwitch -- toggle the use of homogeneous Weyl algebra
    • stafford(Ideal) -- see stafford -- computes 2 generators for a given ideal in the Weyl algebra
  • Symbols
    • dpairInds -- see createDpairs -- pairs coordinate and derivation variables in a Weyl algebra
    • dpairVars -- see createDpairs -- pairs coordinate and derivation variables in a Weyl algebra
    • AnnG -- see makeCyclic -- finds a cyclic generator of a D-module
    • Generator -- see makeCyclic -- finds a cyclic generator of a D-module
    • SetVariables -- see makeWeylAlgebra -- Weyl algebra corresponding to a polynomial ring

For the programmer

The object WeylAlgebras is a package.

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