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

for some illustrations of ways to use this package.

Caveat

Computations with squares in characteristic 2 is not supported in the current version.

Authors

Clas Löfwall <clas.lofwall@gmail.com>
Samuel Lundqvist <samuel@math.su.se>

Certification

Version 3.0 of this package was accepted for publication in volume 11 of The Journal of Software for Algebra and Geometry on 28 August 2020, in the article Software for doing computations in graded Lie algebras. That version can be obtained from the journal or from the Macaulay2 source code repository.

Version

This documentation describes version 3.0 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
- ExtAlgebra -- the class of all Ext-algebras
- ExtElement -- the class of all Ext-algebra elements
- FGLieIdeal -- the class of all finitely generated Lie ideals
- FGLieSubAlgebra -- the class of all finitely generated Lie subalgebras
- LieAlgebra -- the class of all Lie algebras
- LieAlgebraMap -- the class of all Lie algebra homomorphisms
- LieDerivation -- the class of all Lie algebra derivations
- LieElement -- the class of all Lie algebra elements
- LieIdeal -- the class of all Lie ideals
- LieSubAlgebra -- the class of all Lie subalgebras
- LieSubSpace -- the class of all Lie subspaces
- VectorSpace -- the class of all vector spaces
Functions and commands
- boundaries -- make the subalgebra of boundaries
- center -- make the ideal of central elements
- computedDegree -- get the degree to which the computations have been performed
- cycles -- make the subalgebra of cycles
- differential -- make the derivation defined by the differential
- differentialLieAlgebra -- make a differential Lie algebra
- dims -- compute the dimensions of a Lie algebra, Ext-algebra or vector space
- extAlgebra -- compute the Ext-algebra of a Lie algebra
- firstDegree -- get the degree of an element
- holonomy -- compute the holonomy Lie algebra associated to an arrangement or matroid
- holonomyLocal -- compute the Lie algebra for a local subalgebra of the holonomy Lie algebra
- indexForm -- get a Lie element in the polynomial ring representation
- innerDerivation -- make the derivation defined by right Lie multiplication by a Lie element
- koszulDual -- compute the Lie algebra whose enveloping algebra is the Koszul dual of a quadratic algebra
- lieAlgebra -- make a free Lie algebra
- lieDerivation -- make a graded derivation
- lieHomology -- make the homology as a vector space
- lieIdeal -- make a Lie ideal
- lieSubAlgebra -- make a Lie subalgebra
- lieSubSpace -- make a Lie subspace
- listMultiply -- multiplication of lists
- minimalModel -- compute the minimal model
- normalForm -- compute the normal form of a LieElement
- sign -- get the sign of a homogeneous element
- weight -- get the weight of a homogeneous element
- zeroDerivation -- make a derivation from the zero map
- zeroIdeal -- make the zero ideal
- zeroMap -- make the zero map
Methods
- - ExtElement -- unary negation
- - LieAlgebraMap -- unary negation
- - LieDerivation -- unary negation
- - LieElement -- unary negation
- ambient(LieAlgebra) -- get the ambient Lie algebra
- annihilator(FGLieSubAlgebra) -- make the annihilator Lie subalgebra
- basis(List,ExtAlgebra) -- compute a basis
- basis(List,LieAlgebra) -- compute a basis
- basis(List,VectorSpace) -- compute a basis
- basis(ZZ,ExtAlgebra) -- compute a basis
- basis(ZZ,LieAlgebra) -- compute a basis
- basis(ZZ,VectorSpace) -- compute a basis
- basis(ZZ,ZZ,ExtAlgebra) -- compute a basis
- basis(ZZ,ZZ,LieAlgebra) -- compute a basis
- basis(ZZ,ZZ,VectorSpace) -- compute a basis
- "boundaries(LieAlgebra)" -- see boundaries -- make the subalgebra of boundaries
- "center(LieAlgebra)" -- see center -- make the ideal of central elements
- coefficients(LieElement) -- get the coefficients and monomials of a Lie element
- "computedDegree(ExtAlgebra)" -- see computedDegree -- get the degree to which the computations have been performed
- "computedDegree(LieAlgebra)" -- see computedDegree -- get the degree to which the computations have been performed
- "cycles(LieAlgebra)" -- see cycles -- make the subalgebra of cycles
- decompose(LieAlgebra) -- compute the ideal associated to an arrangement or matroid
- degreeLength(LieAlgebra) -- get the length of the weight of a generator
- "describe(ExtAlgebra)" -- see describe(LieAlgebra) -- real description
- describe(LieAlgebra) -- real description
- "describe(LieAlgebraMap)" -- see describe(LieAlgebra) -- real description
- "describe(LieDerivation)" -- see describe(LieAlgebra) -- real description
- "describe(VectorSpace)" -- see describe(LieAlgebra) -- real description
- diff(LieAlgebra) -- get the differential of the generators
- "differentialLieAlgebra(List)" -- see differentialLieAlgebra -- make a differential Lie algebra
- dim(List,ExtAlgebra) -- compute the dimension
- dim(List,LieAlgebra) -- compute the dimension
- dim(List,VectorSpace) -- compute the dimension
- dim(ZZ,ExtAlgebra) -- compute the dimension
- dim(ZZ,LieAlgebra) -- compute the dimension
- dim(ZZ,VectorSpace) -- compute the dimension
- dim(ZZ,ZZ,ExtAlgebra) -- compute the dimension
- dim(ZZ,ZZ,LieAlgebra) -- compute the dimension
- dim(ZZ,ZZ,VectorSpace) -- compute the dimension
- "dims(ZZ,ExtAlgebra)" -- see dims -- compute the dimensions of a Lie algebra, Ext-algebra or vector space
- "dims(ZZ,LieAlgebra)" -- see dims -- compute the dimensions of a Lie algebra, Ext-algebra or vector space
- "dims(ZZ,VectorSpace)" -- see dims -- compute the dimensions of a Lie algebra, Ext-algebra or vector space
- "dims(ZZ,ZZ,ExtAlgebra)" -- see dims -- compute the dimensions of a Lie algebra, Ext-algebra or vector space
- "dims(ZZ,ZZ,LieAlgebra)" -- see dims -- compute the dimensions of a Lie algebra, Ext-algebra or vector space
- "dims(ZZ,ZZ,VectorSpace)" -- see dims -- compute the dimensions of a Lie algebra, Ext-algebra or vector space
- euler(LieAlgebra) -- compute the Euler derivation
- eulers(ZZ,LieAlgebra) -- compute the list of Euler characteristics
- "extAlgebra(ZZ,LieAlgebra)" -- see extAlgebra -- compute the Ext-algebra of a Lie algebra
- ExtElement + ExtElement -- addition of Ext-algebra elements
- ExtElement - ExtElement -- subtraction of Ext-algebra elements
- ExtElement ExtElement -- multiplication of Ext-algebra elements
- "firstDegree(ExtElement)" -- see firstDegree -- get the degree of an element
- "firstDegree(LieDerivation)" -- see firstDegree -- get the degree of an element
- "firstDegree(LieElement)" -- see firstDegree -- get the degree of an element
- generators(ExtAlgebra) -- get the generators
- generators(LieAlgebra) -- get the generators
- generators(LieSubSpace) -- get the generators
- "holonomy(List)" -- see holonomy -- compute the holonomy Lie algebra associated to an arrangement or matroid
- "holonomy(List,List)" -- see holonomy -- compute the holonomy Lie algebra associated to an arrangement or matroid
- "holonomyLocal(ZZ,LieAlgebra)" -- see holonomyLocal -- compute the Lie algebra for a local subalgebra of the holonomy Lie algebra
- ideal(LieAlgebra) -- get the relations in a Lie algebra
- image(LieAlgebraMap) -- make the image of a Lie algebra map
- image(LieAlgebraMap,LieSubSpace) -- make the image of a Lie subspace under a Lie algebra map
- image(LieDerivation) -- make the image of a Lie derivation
- image(LieDerivation,LieSubSpace) -- make the image of a Lie subspace under a Lie derivation
- "indexForm(LieElement)" -- see indexForm -- get a Lie element in the polynomial ring representation
- "innerDerivation(LieElement)" -- see innerDerivation -- make the derivation defined by right Lie multiplication by a Lie element
- inverse(LieAlgebraMap,LieSubSpace) -- make the inverse image of a Lie subspace under a Lie algebra map
- inverse(LieDerivation,LieSubSpace) -- make the inverse image of a Lie subspace under a Lie derivation
- isIsomorphism(LieAlgebraMap) -- whether a Lie map is an isomorphism
- isSurjective(LieAlgebraMap) -- whether a Lie map is surjective
- isWellDefined(ZZ,LieAlgebraMap) -- whether a Lie map is well defined
- isWellDefined(ZZ,LieDerivation) -- whether a Lie derivation is well defined
- kernel(LieAlgebraMap) -- make the kernel of a map
- kernel(LieDerivation) -- make the kernel of a map
- "koszulDual(PolynomialRing)" -- see koszulDual -- compute the Lie algebra whose enveloping algebra is the Koszul dual of a quadratic algebra
- "koszulDual(QuotientRing)" -- see koszulDual -- compute the Lie algebra whose enveloping algebra is the Koszul dual of a quadratic algebra
- "lieAlgebra(List)" -- see lieAlgebra -- make a free Lie algebra
- LieAlgebra * LieAlgebra -- free product of Lie algebras
- LieAlgebra ++ LieAlgebra -- direct sum of Lie algebras
- LieAlgebra / LieAlgebraMap -- make a quotient Lie algebra
- LieAlgebra / LieIdeal -- make a quotient Lie algebra
- LieAlgebra / List -- make a quotient Lie algebra
- LieAlgebra == LieAlgebra -- whether two Lie algebras are defined in the same way
- LieAlgebraMap * LieAlgebraMap -- composition of homomorphisms
- LieAlgebraMap * LieDerivation -- composition of a homomorphism and a derivation
- LieAlgebraMap + LieAlgebraMap -- addition of Lie homomorphisms
- LieAlgebraMap - LieAlgebraMap -- subtraction of Lie homomorphisms
- LieAlgebraMap == LieAlgebraMap -- whether two Lie algebra homomorphisms are defined in the same way
- LieAlgebraMap @ LieElement -- formal application of a Lie map to a Lie element
- LieAlgebraMap \ List -- apply a Lie homomorphism to every element in a list
- LieAlgebraMap \\ List -- formal application of a Lie map to every element in a list
- LieAlgebraMap LieElement -- apply a Lie homomorphism
- "lieDerivation(LieAlgebraMap,List)" -- see lieDerivation -- make a graded derivation
- "lieDerivation(List)" -- see lieDerivation -- make a graded derivation
- LieDerivation * LieAlgebraMap -- composition of a derivation and a homomorphism
- LieDerivation + LieDerivation -- addition of Lie derivations
- LieDerivation - LieDerivation -- subtraction of Lie derivations
- LieDerivation @ LieElement -- formal application of a derivation to a Lie element
- LieDerivation \ List -- apply a derivation to every element in a list
- LieDerivation \\ List -- formal application of a derivation to every element in a list
- LieDerivation LieDerivation -- Lie multiplication of ordinary derivations
- LieDerivation LieElement -- apply a derivation
- LieElement + LieElement -- addition of Lie elements
- LieElement ++ LieElement -- formal addition of Lie elements
- LieElement - LieElement -- subtraction of Lie elements
- LieElement / LieElement -- formal subtraction of Lie elements
- LieElement @ LieElement -- formal multiplication of Lie elements
- LieElement LieElement -- multiplication of Lie elements
- "lieHomology(LieAlgebra)" -- see lieHomology -- make the homology as a vector space
- "lieIdeal(LieSubSpace)" -- see lieIdeal -- make a Lie ideal
- "lieIdeal(List)" -- see lieIdeal -- make a Lie ideal
- "lieSubAlgebra(List)" -- see lieSubAlgebra -- make a Lie subalgebra
- "lieSubSpace(List)" -- see lieSubSpace -- make a Lie subspace
- LieSubSpace + LieSubSpace -- make the sum of two Lie subspaces
- LieSubSpace @ LieSubSpace -- make the intersection of two Lie subspaces
- map(LieAlgebra) -- get the map of a minimal model
- map(LieAlgebra,LieAlgebra) -- make a natural Lie algebra homomorphism
- map(LieAlgebra,LieAlgebra,List) -- make a Lie algebra homomorphism
- map(LieDerivation) -- get the map in the definition of a Lie derivation
- member(LieElement,LieSubSpace) -- whether a Lie element belongs to a Lie subspace
- "minimalModel(ZZ,LieAlgebra)" -- see minimalModel -- compute the minimal model
- minimalPresentation(ZZ,LieAlgebra) -- compute a minimal presentation
- monomials(LieElement) -- get the monomials of a Lie element
- "normalForm(LieElement)" -- see normalForm -- compute the normal form of a LieElement
- Number @ LieElement -- formal multiplication of a number and a Lie element
- Number ExtElement -- multiplication of a number and an Ext-algebra element
- Number LieAlgebraMap -- multiplication of a number and a homomorphism
- Number LieDerivation -- multiplication of a number and a derivation
- Number LieElement -- multiplication of a number and a Lie element
- numgens(LieAlgebra) -- get the number of generators
- quotient(LieIdeal,FGLieSubAlgebra) -- make the quotient of a Lie ideal by a finitely generated Lie subalgebra
- "random(List,LieAlgebra)" -- see random(ZZ,LieAlgebra) -- get a random element of a Lie algebra
- random(ZZ,LieAlgebra) -- get a random element of a Lie algebra
- "random(ZZ,ZZ,LieAlgebra)" -- see random(ZZ,LieAlgebra) -- get a random element of a Lie algebra
- RingElement @ LieElement -- formal multiplication of a ring element and a Lie element
- RingElement ExtElement -- multiplication of a field element and a Ext-algebra element
- RingElement LieAlgebraMap -- multiplication of a field element and a homomorphism
- RingElement LieDerivation -- multiplication of a field element and a derivation
- RingElement LieElement -- multiplication of a field element and a Lie element
- ScriptedFunctor _ LieAlgebra -- get the identity homomorphism
- "sign(ExtElement)" -- see sign -- get the sign of a homogeneous element
- "sign(LieDerivation)" -- see sign -- get the sign of a homogeneous element
- "sign(LieElement)" -- see sign -- get the sign of a homogeneous element
- source(LieAlgebraMap) -- get the source of a map
- source(LieDerivation) -- get the source of a map
- "standardForm(List,LieAlgebra)" -- see standardForm(RingElement,LieAlgebra) -- get a Lie element in standard form
- standardForm(RingElement,LieAlgebra) -- get a Lie element in standard form
- target(LieAlgebraMap) -- get the target of a map
- target(LieDerivation) -- get the target of a map
- trace(ZZ,LieSubSpace,LieAlgebraMap) -- compute the trace of a Lie algebra map acting on a Lie subspace
- use(LieAlgebra) -- set the generators
- "weight(ExtElement)" -- see weight -- get the weight of a homogeneous element
- "weight(LieDerivation)" -- see weight -- get the weight of a homogeneous element
- "weight(LieElement)" -- see weight -- get the weight of a homogeneous element
- "zeroDerivation(LieAlgebra)" -- see zeroDerivation -- make a derivation from the zero map
- "zeroIdeal(LieAlgebra)" -- see zeroIdeal -- make the zero ideal
- "zeroMap(LieAlgebra,LieAlgebra)" -- see zeroMap -- make the zero map
- ZZ _ ExtAlgebra -- get the zero element
- ZZ _ LieAlgebra -- get the zero element
Symbols
- Field -- name for an optional argument for lieAlgebra and holonomy
- LastWeightHomological -- name for an optional argument for lieAlgebra
- lieRing -- get the internal ring for representation of Lie elements
- mbRing -- a polynomial ring representation of the Lie algebra used for output
- Signs -- name for an optional argument for lieAlgebra

For the programmer

The object GradedLieAlgebras is a package.