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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

Menu

Authors

Version

This documentation describes version 1.4.1.1 of WeylAlgebras.

Citation

If you have used this package in your research, please cite it as follows:

@misc{WeylAlgebrasSource,
  title = {{WeylAlgebras: D-modules. Version~1.4.1.1}},
  author = {Anton Leykin and Harrison Tsai},
  howpublished = {A \emph{Macaulay2} package available at
    "http://people.math.gatech.edu/~aleykin3/Dmodules"}
}

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
    • factorWeylAlgebra -- factor a Weyl algebra element
    • factorWeylAlgebra1 -- give one factorisation of a Weyl algebra element
    • 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
    • factorWeylAlgebra(RingElement) -- see factorWeylAlgebra -- factor a Weyl algebra element
    • 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
    • StopAfter (missing documentation)

For the programmer

The object WeylAlgebras is a package, defined in WeylAlgebras.m2, with auxiliary files in WeylAlgebras/.

The source of this document is in WeylAlgebras/DOC/main.m2:55:0.