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 LieAlgebra Tutorial, Constructing Lie algebras, Holonomy 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 2.0 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 derivations in Lie algebras
    • LieAlgebra -- a Type for Lie algebras
    • LieElement -- a Type for elements in Lie algebras
    • MapLie -- a Type for homomorphisms of Lie algebras
  • Functions and commands
    • annLie -- computes a basis for the annihilator in a given degree
    • basisLie -- a basis of Lie monomials in a given (multi-)degree
    • boundariesBasisLie -- computes a basis for the boundaries of a Lie algebra
    • boundariesTableLie -- a table of dimensions of the boundaries of a Lie algebra
    • centerAllLie -- computes all central elements
    • centerLie -- computes the central elements
    • characterLie -- computes the trace of a Lie representation
    • coeffsLie -- computes the coefficients of a LieElement
    • cyclesBasisLie -- a basis for the cycles of a Lie algebra
    • cyclesTableLie -- a table of dimensions of the cycles of a Lie algebra
    • decompidealLie -- computes in the specified degree an ideal associated to an arrangement or matroid
    • defLie -- returns a LieElement corresponding to input
    • degLie -- the first degree of a graded element in the LieAlgebra
    • derLie -- constructing a graded derivation
    • diffLie -- the derivation defined by the differential
    • diffLieAlgebra -- A differential Lie algebra
    • dimsLie -- the dimensions of the Lie algebra up to a specified degree
    • dimTableLie -- a table of dimensions of a Lie algebra
    • dimtotLie -- the sum of the dimensions up to a specified degree
    • divisorLie -- computes a basis for the divisor subspace
    • eulerLie -- computes the Euler characteristics
    • extBasisLie -- a basis up to a given degree of the Ext-algebra
    • extMultLie -- the (skew commutative) product in the Ext-algebra
    • extTableLie -- a table of dimensions of the Ext-algebra of a Lie algebra
    • holonomyLie -- gives the holonomy Lie algebra associated to an arrangement or matroid
    • homologyBasisLie -- computes a basis for the homology of a given degree
    • homologyTableLie -- a table of dimensions of the homology of a Lie algebra
    • idealBasisLie -- computes a basis of a Lie ideal in a given degree or multi-degree
    • idealTableLie -- a table of dimensions of an ideal of a Lie algebra
    • idMapLie -- the identity map
    • imageBasisLie -- a basis of the image of a Lie homomorphism or derivation in a specified degree
    • imageTableLie -- a table of dimensions of the image of a map or derivation
    • imapLie -- construction of a Lie map without checking correctness
    • indexFormLie -- returns an element in the ring representation corresponding to the input
    • innerDerLie -- the derivation defined by left Lie multiplication by a LieElement
    • 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 or derivation in a specified degree
    • kernelTableLie -- a table of dimensions of the kernel of a map or derivation
    • koszulDualLie -- gives the Lie algebra whose enveloping algebra is the Koszul dual of a quadratic algebra
    • lieAlgebra -- constructing a free Lie algebra
    • localLie -- gives the Lie algebra for a local subalgebra of the holonomy Lie algebra
    • mapLie -- constructing a Lie algebra homomorphism
    • minmodelLie -- gives the minimal model
    • minPresLie -- gives a minimal presentation up to a specified degree
    • monomialsLie -- computes the monomials of a LieElement
    • multLie -- The Lie multiplication as a prefix operator
    • multListLie -- Lie multiplication of lists of LieElement
    • normalFormLie -- computes the normal form of a LieElement
    • peekLie -- gives information of a Lie algebra or map
    • permopLie -- the result of a permutation operating on a LieElement
    • randomLie -- gives a random element of a lie algebra
    • signExtLie -- returns the sign of a basis element in the Ext-algebra
    • signLie -- returns the sign of a homogeneous LieElement.
    • subalgBasisLie -- computes a basis of a Lie subalgebra in a given degree or multi-degree
    • subalgTableLie -- a table of dimensions of a Lie subalgebra of a Lie algebra
    • symmetryLie -- checking if a permutation of the generators defines a map
    • useLie -- changes the current Lie Algebra
    • weightExtLie -- returns the weight of a homogeneous element in the Ext-algebra
    • weightLie -- returns the weight of a homogeneous LieElement
    • whichLie -- prints the current Lie Algebra
  • Methods
    • - DerLie -- Unary negation of Lie derivations
    • - LieElement -- Unary negation of LieElements
    • - MapLie -- Unary negation of Lie homomorphisms
    • ambient(LieAlgebra) -- the free underlying Lie algebra
    • baseName(LieElement)
    • coeffsLie(LieElement), see coeffsLie -- computes the coefficients of a LieElement
    • degLie(DerLie), see degLie -- the first degree of a graded element in the LieAlgebra
    • degLie(LieElement), see degLie -- the first degree of a graded element in the LieAlgebra
    • derLie(MapLie,List), see derLie -- constructing a graded derivation
    • DerLie * MapLie -- operation of maps to the right of a derivation
    • DerLie + DerLie -- Addition of Lie derivations
    • DerLie - DerLie -- Subtraction of Lie derivations
    • DerLie DerLie -- Lie multiplication of ordinary derivations
    • DerLie LieElement -- Application of a derivation to a LieElement
    • DerLie List -- Application of a derivation to every element in a list
    • imageBasisLie(List,DerLie), see imageBasisLie -- a basis of the image of a Lie homomorphism or derivation in a specified degree
    • imageBasisLie(List,MapLie), see imageBasisLie -- a basis of the image of a Lie homomorphism or derivation in a specified degree
    • imageBasisLie(ZZ,DerLie), see imageBasisLie -- a basis of the image of a Lie homomorphism or derivation in a specified degree
    • imageBasisLie(ZZ,MapLie), see imageBasisLie -- a basis of the image of a Lie homomorphism or derivation in a specified degree
    • imageBasisLie(ZZ,ZZ,DerLie), see imageBasisLie -- a basis of the image of a Lie homomorphism or derivation in a specified degree
    • imageBasisLie(ZZ,ZZ,MapLie), see imageBasisLie -- a basis of the image of a Lie homomorphism or derivation in a specified degree
    • imapLie(LieAlgebra,LieAlgebra), see imapLie -- construction of a Lie map without checking correctness
    • imapLie(LieAlgebra,LieAlgebra,List), see imapLie -- construction of a Lie map without checking correctness
    • indexFormLie(LieElement), see indexFormLie -- returns an element in the ring representation corresponding to the input
    • innerDerLie(LieElement), see innerDerLie -- the derivation defined by left Lie multiplication by a LieElement
    • invImageBasisLie(DerLie,List), see invImageBasisLie -- computes a basis for the inverse image of a map or derivation
    • invImageBasisLie(MapLie,List), see invImageBasisLie -- computes a basis for the inverse image of a map or derivation
    • invImageLie(DerLie,List), see invImageLie -- computes the dimension for the inverse image of a map or derivation
    • invImageLie(MapLie,List), see invImageLie -- computes the dimension for the inverse image of a map or derivation
    • kernelBasisLie(List,DerLie), see kernelBasisLie -- a basis of the kernel of a Lie homomorphism or derivation in a specified degree
    • kernelBasisLie(List,MapLie), see kernelBasisLie -- a basis of the kernel of a Lie homomorphism or derivation in a specified degree
    • kernelBasisLie(ZZ,DerLie), see kernelBasisLie -- a basis of the kernel of a Lie homomorphism or derivation in a specified degree
    • kernelBasisLie(ZZ,MapLie), see kernelBasisLie -- a basis of the kernel of a Lie homomorphism or derivation in a specified degree
    • kernelBasisLie(ZZ,ZZ,DerLie), see kernelBasisLie -- a basis of the kernel of a Lie homomorphism or derivation in a specified degree
    • kernelBasisLie(ZZ,ZZ,MapLie), see kernelBasisLie -- a basis of the kernel of a Lie homomorphism or derivation in a specified degree
    • LieAlgebra * LieAlgebra -- Free product of Lie algebras
    • LieAlgebra ** LieAlgebra -- Direct sum of Lie algebras
    • LieAlgebra / List -- A quotient Lie algebra
    • LieAlgebra / MapLie -- A quotient Lie algebra by the image of a map
    • LieElement + LieElement -- Addition of LieElements
    • LieElement ++ LieElement -- Formal addition of LieElements
    • LieElement - LieElement -- Subtraction of LieElements
    • LieElement / LieElement -- Formal subtraction of LieElements
    • LieElement @ LieElement -- Formal multiplication of LieElements
    • LieElement LieElement -- The Lie multiplication
    • mapLie(LieAlgebra,LieAlgebra), see mapLie -- constructing a Lie algebra homomorphism
    • mapLie(LieAlgebra,LieAlgebra,List), see mapLie -- constructing a Lie algebra homomorphism
    • MapLie * DerLie -- operation of maps to the left of a derivation
    • MapLie * MapLie -- composition of homomorphisms
    • MapLie + MapLie -- Addition of Lie homomorphisms
    • MapLie - MapLie -- Subtraction of Lie homomorphisms
    • MapLie LieElement -- Application of a Lie map to a LieElement
    • MapLie List -- Application of a Lie map to every element in a list
    • monomialsLie(LieElement), see monomialsLie -- computes the monomials of a LieElement
    • multLie(LieElement,LieElement), see multLie -- The Lie multiplication as a prefix operator
    • normalFormLie(LieElement), see normalFormLie -- computes the normal form of a LieElement
    • Number @ LieElement -- Formal multiplication of a number and a LieElement
    • Number DerLie -- Multiplication of a Number and a Derivation
    • Number LieElement -- Multiplication of a Number and a LieElement
    • Number MapLie -- Multiplication of a Number and a homomorphism
    • peekLie(DerLie), see peekLie -- gives information of a Lie algebra or map
    • peekLie(LieAlgebra), see peekLie -- gives information of a Lie algebra or map
    • peekLie(MapLie), see peekLie -- gives information of a Lie algebra or map
    • RingElement @ LieElement -- Formal multiplication of a RingElement and a LieElement
    • RingElement DerLie -- Multiplication of a field element and a derivation
    • RingElement LieElement -- Multiplication of a field element and a LieElement
    • RingElement MapLie -- Multiplication of a field element and a homomorphism
    • signLie(LieElement), see signLie -- returns the sign of a homogeneous LieElement.
    • useLie(LieAlgebra), see useLie -- changes the current Lie Algebra
    • weightLie(LieElement), see weightLie -- returns the weight of a homogeneous LieElement
  • Symbols
    • axiomsLie -- the axioms for Lie algebras
    • compdeg -- the maximal computed degree of the Lie algebra
    • diffl -- optional argument for lieAlgebra
    • extRepRing -- the ring representation of the Ext-algebra
    • field -- optional argument for lieAlgebra and holonomyLie
    • genDiffs -- the value of the differential on the generators of a Lie algebra
    • genSigns -- optional argument for lieAlgebra
    • gensLie -- the list of generators of the Lie algebra
    • genWeights -- optional argument for lieAlgebra
    • lieRing -- the internal ring for representation of Lie elements
    • maplie -- the Lie homomorphism f in the definition of a derivation
    • mbRing -- the ring representation of the Lie algebra used as an outputform
    • minmodel -- the minimal model of L obtained, if computed, as L.minmodel
    • modelmap -- the Lie homomorphism from a minimal model of L to the Lie algebra L
    • multOnly -- optional argument for multListLie
    • relsLie -- the list of relations of the Lie algebra
    • sign -- the sign of a derivation
    • sourceLie -- the source of a derivation or map
    • targetLie -- the target of a derivation or map
    • weight -- the weight of a derivation
    • zz -- the zero element of a Lie algebra