GradedLieAlgebras : Table of Contents

• GradedLieAlgebras -- a package for doing computations in graded Lie algebras
• - 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 -- make the subalgebra of boundaries
• center -- make the ideal of central elements
• coefficients(LieElement) -- get the coefficients and monomials of a Lie element
• computedDegree -- get the degree to which the computations have been performed
• 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(LieAlgebra) -- real description
• diff(LieAlgebra) -- get the differential of the generators
• differential -- make the derivation defined by the differential
• Differential Lie algebra tutorial
• 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 -- 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 -- the class of all Ext-algebras
• extAlgebra -- compute the Ext-algebra of a Lie algebra
• ExtElement -- the class of all Ext-algebra elements
• ExtElement + ExtElement -- addition of Ext-algebra elements
• ExtElement - ExtElement -- subtraction of Ext-algebra elements
• ExtElement ExtElement -- multiplication of Ext-algebra elements
• FGLieIdeal -- the class of all finitely generated Lie ideals
• FGLieSubAlgebra -- the class of all finitely generated Lie subalgebras
• Field -- name for an optional argument for lieAlgebra and holonomy
• First Lie algebra tutorial
• firstDegree -- get the degree of an element
• generators(ExtAlgebra) -- get the generators
• generators(LieAlgebra) -- get the generators
• generators(LieSubSpace) -- get the generators
• holonomy -- compute the holonomy Lie algebra associated to an arrangement or matroid
• Holonomy Lie algebras and symmetries
• holonomy(...,Field=>...) -- optional argument for holonomy
• holonomyLocal -- compute the Lie algebra for a local subalgebra of the holonomy Lie algebra
• Homomorphisms and derivations
• 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 -- get a Lie element in the polynomial ring representation
• 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 -- compute the Lie algebra whose enveloping algebra is the Koszul dual of a quadratic algebra
• LastWeightHomological -- name for an optional argument for lieAlgebra
• LieAlgebra -- the class of all Lie algebras
• 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
• lieAlgebra(...,Field=>...) -- optional argument for lieAlgebra
• lieAlgebra(...,LastWeightHomological=>...) -- optional argument for lieAlgebra
• lieAlgebra(...,Signs=>...) -- optional argument for lieAlgebra
• lieAlgebra(...,Weights=>...) -- optional argument for lieAlgebra
• LieAlgebraMap -- the class of all Lie algebra homomorphisms
• 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 -- the class of all Lie algebra derivations
• 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 -- the class of all Lie algebra elements
• 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 -- make the homology as a vector space
• LieIdeal -- the class of all Lie ideals
• lieIdeal -- make a Lie ideal
• lieRing -- get the internal ring for representation of Lie elements
• LieSubAlgebra -- the class of all Lie subalgebras
• lieSubAlgebra -- make a Lie subalgebra
• LieSubSpace -- the class of all Lie subspaces
• lieSubSpace -- make a Lie subspace
• LieSubSpace + LieSubSpace -- make the sum of two Lie subspaces
• LieSubSpace @ LieSubSpace -- make the intersection of two Lie subspaces
• listMultiply -- multiplication of lists
• 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
• mbRing -- a polynomial ring representation of the Lie algebra used for output
• member(LieElement,LieSubSpace) -- whether a Lie element belongs to a Lie subspace
• Minimal models, Ext-algebras and Koszul duals
• minimalModel -- compute the minimal model
• minimalPresentation(ZZ,LieAlgebra) -- compute a minimal presentation
• monomials(LieElement) -- get the monomials of a Lie element
• 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 Lie algebras and subspaces
• quotient(LieIdeal,FGLieSubAlgebra) -- make the quotient of a Lie ideal by a finitely generated Lie subalgebra
• 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
• Second Lie algebra tutorial
• sign -- get the sign of a homogeneous element
• Signs -- name for an optional argument for lieAlgebra
• source(LieAlgebraMap) -- get the source of a map
• source(LieDerivation) -- get the source of a map
• 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
• VectorSpace -- the class of all vector spaces
• 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
• ZZ _ ExtAlgebra -- get the zero element
• ZZ _ LieAlgebra -- get the zero element