next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
GradedLieAlgebras :: GradedLieAlgebras

GradedLieAlgebras -- A package for doing computations in graded Lie algebras

Description

This package provides routines for doing computations in graded Lie algebras.

The package relies on algorithmic theory on graded Lie algebras developed by Clas Löfwall and Jan-Erik Roos in A Nonnilpotent 1-2-Presented Graded Hopf Algebra Whose Hilbert Series Converges in the Unit Circle, Adv. Math. 130 (1997), no. 2, 161–200.

See also the earlier implementation in Mathematica by C. Löfwall, Liedim, a Mathematica program for Lie-calculations (2001-2016), available at http://www2.math.su.se/liedim/

See First LieAlgebra Tutorial, Second LieAlgebra Tutorial, Differential Lie algebras Tutorial, How to write Lie elements, Constructing Lie algebras and Symmetries for some illustrations of ways to use this package.

Caveat

Computations with squares in characteristic two is not supported in the current version.

Authors

Version

This documentation describes version 1.0 of GradedLieAlgebras.

Source code

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

Exports

  • Types
    • DerLie -- a Type for Lie algebra derivations
    • LieAlgebra -- a Type for Lie algebras
    • MapLie -- a Type for Lie algebra homomorphisms
  • Functions and commands
    • annLie -- computes a basis for the annihilator in a given degree
    • basicExpressionLie -- checks if a general Lie expression is of normal form
    • basicMonomialLie -- checks if an array is a basis element for the Lie algebra
    • basisExtLie -- a basis in a given degree of the Ext-algebra
    • basisLie -- a basis of Lie monomials in a given (multi-)degree
    • boundariesBasisLie -- computes a basis for the boundaries of a given degree and homological degree
    • centreLie -- computes the central elements
    • characterLie -- computes the trace of a Lie representation
    • computeLie -- computes everything that is needed for a Lie algebra up to a given degree
    • decompidealLie -- computes in the specified degree an ideal associated to an arrangement or matroid
    • defLie -- returns a general Lie expression corresponding to input
    • degLie -- the first degree of a graded element in the LieAlgebra
    • derLie -- constructing a graded derivation
    • diffLie -- the derivation obtained from the differential defined in the current Lie algebra
    • dimLie -- the dimension of a Lie algebra
    • dimsLie -- the dimensions of a Lie algebra
    • dimTableLie -- a table of dimensions of the Lie algebra in first and last degree
    • dimtotLie -- the total dimension up to degree d
    • divisorLie -- computes a basis for the divisor subspace
    • eulerLie -- computes the Euler characteristics
    • evalDerLie -- the value of a Lie derivation applied to an argument
    • evalDiffLie -- the value of the differential of the current Lie algebra applied to an argument
    • evalMapLie -- the value of a Lie homomorphism applied to an argument
    • extAlgLie -- the matrix of dimensions of the Ext-algebra
    • extAlgMultLie -- the (skew commutative) product in the Ext-algebra
    • generalExpressionLie -- checks if an expression is of right input form for e.g. relations
    • holonomyLie -- gives the holonomy Lie algebra associated to an arrangement or matroid
    • homologyBasisLie -- computes a basis for the homology of a given degree
    • homologyLie -- computes the dimensions of the homology
    • idealBasisLie -- computes a basis of a Lie ideal in a given degree or multidegree
    • idealLie -- computes the dimensions of a Lie ideal
    • imageBasisLie -- a basis of the image of a Lie homomorphism in a specified degree
    • imageLie -- gives the dimensions of the image of a Lie homomorphism up to a specified degree
    • indexFormLie -- returns an element in the ring representation corresponding to the input
    • intersectionLie -- computes a basis for the intersection of subspaces of a given degree
    • invImageBasisLie -- computes a basis for the inverse image of a map or derivation
    • invImageLie -- computes the dimension for the inverse image of a map or derivation
    • kernelBasisLie -- a basis of the kernel of a Lie homomorphism in a specified degree
    • kernelLie -- gives the dimensions of the kernel of a Lie homomorphism up to a specified degree
    • koszulDualLie -- gives the Lie algebra whose enveloping algebra is the Koszul dual of a quadratic algebra
    • lieAlgebra -- constructing a Lie algebra from its presentation
    • localLie -- gives the Lie algebra and a basis for a local subalgebra of the holonomy Lie algebra of an arrangement or matroid
    • mapLie -- constructing a Lie algebra homomorphism
    • minmodelLie -- gives a minimal model
    • minPresLie -- gives a minimal presentation up to a specified degree
    • monomialLie -- checks if an array is a correct iterated Lie product
    • multDerLie -- defines the Lie multiplication of two derivations on a Lie algebra
    • multLie -- Lie multiplication of two general Lie expression elements
    • multListLie -- Lie multiplication of lists of general Lie expressions
    • normalFormLie -- returns a basic Lie expression for the Lie algebra equal to the input
    • permopLie -- the result of a permutation operating on a general Lie expression
    • randomLie -- gives a random element of a lie algebra
    • signExtLie -- returns the sign of a generator in the Ext-algebra
    • signLie -- returns the sign of a graded Lie element.
    • subalgBasisLie -- computes a basis of a Lie subalgebra in a given degree or multidegree
    • subalgLie -- computes the dimensions of a Lie subalgebra up to a specified degree
    • symmCyclePermLie -- checks if a permutation of the generators in the form of cycles is an automorphism
    • symmPermLie -- checks if a permutation of the generators is an automorphism
    • toMonomialLie -- expresses an arbitrary Lie product as a general Lie expression
    • useLie -- changes the current Lie algebra and its mbRing
    • weightLie -- gives the multi-degree of a graded element in a Lie algebra
    • whichLie -- prints the current Lie algebra
  • Methods
    • centreLie(LieAlgebra), see centreLie -- computes the central elements
    • derLie(MapLie,List), see derLie -- constructing a graded derivation
    • evalDerLie(DerLie,Array), see evalDerLie -- the value of a Lie derivation applied to an argument
    • evalDerLie(DerLie,List), see evalDerLie -- the value of a Lie derivation applied to an argument
    • evalMapLie(MapLie,Array), see evalMapLie -- the value of a Lie homomorphism applied to an argument
    • evalMapLie(MapLie,List), see evalMapLie -- the value of a Lie homomorphism applied to an argument
    • imageBasisLie(ZZ,MapLie), see imageBasisLie -- a basis of the image of a Lie homomorphism in a specified degree
    • imageBasisLie(ZZ,ZZ,MapLie), see imageBasisLie -- a basis of the image of a Lie homomorphism in a specified degree
    • invImageBasisLie(ZZ,DerLie,List), see invImageBasisLie -- computes a basis for the inverse image of a map or derivation
    • invImageBasisLie(ZZ,MapLie,List), see invImageBasisLie -- computes a basis for the inverse image of a map or derivation
    • invImageBasisLie(ZZ,ZZ,DerLie,List), see invImageBasisLie -- computes a basis for the inverse image of a map or derivation
    • invImageBasisLie(ZZ,ZZ,MapLie,List), see invImageBasisLie -- computes a basis for the inverse image of a map or derivation
    • invImageLie(ZZ,DerLie,List), see invImageLie -- computes the dimension for the inverse image of a map or derivation
    • invImageLie(ZZ,MapLie,List), see invImageLie -- computes the dimension for the inverse image of a map or derivation
    • invImageLie(ZZ,ZZ,DerLie,List), see invImageLie -- computes the dimension for the inverse image of a map or derivation
    • invImageLie(ZZ,ZZ,MapLie,List), see invImageLie -- computes the dimension for the inverse image of a map or derivation
    • kernelBasisLie(ZZ,MapLie), see kernelBasisLie -- a basis of the kernel of a Lie homomorphism in a specified degree
    • kernelBasisLie(ZZ,ZZ,MapLie), see kernelBasisLie -- a basis of the kernel of a Lie homomorphism in a specified degree
    • mapLie(LieAlgebra,LieAlgebra,List), see mapLie -- constructing a Lie algebra homomorphism
    • multDerLie(DerLie,DerLie), see multDerLie -- defines the Lie multiplication of two derivations on a Lie algebra
    • useLie(LieAlgebra), see useLie -- changes the current Lie algebra and its mbRing
  • Symbols
    • axiomsLie -- the axioms for Lie algebras
    • compdeg -- the maximal computed degree of the Lie algebra
    • deglength -- the length of each weight of the generators of the Lie algebra
    • extAlgRing -- the ring representation of the Ext-algebra
    • field -- optional argument for lieAlgebra, holonomyLie and randomLie
    • genDiffs -- optional argument for lieAlgebra
    • genSigns -- optional argument for lieAlgebra and randomLie
    • gensLie -- the list of generators of the Lie algebra
    • genWeights -- optional argument for lieAlgebra and randomLie
    • lieRing -- the internal ring for representation of Lie elements
    • maplie -- the Lie homomorphism f in the definition of a derivation
    • maxDeg -- determines the number of variables in the internal ring of representation, lieRing
    • mbRing -- the ring representation of the Lie algebra used as output
    • minmodel -- the minimal model of the Lie algebra, if it is constructed
    • modelmap -- the Lie homomorphism from the minimal model M to the Lie algebra L
    • multOnly -- optional argument for multListLie
    • numGen -- the number of the generators of the Lie algebra
    • relsLie -- the list of relations of the Lie algebra
    • signDer -- gives the sign of a derivation
    • sourceLie -- the source Lie algebra of a derivation or a Lie homomorphism
    • targetLie -- the target Lie algebra of a derivation or a Lie homomorphism
    • weightDer -- gives the weight of a derivation