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GradedLieAlgebras : Index
A
B
C
D
E
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I
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T
U
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X
Y
Z
- 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
boundaries(LieAlgebra)
-- make the subalgebra of boundaries
center
-- make the ideal of central elements
center(LieAlgebra)
-- 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
computedDegree(ExtAlgebra)
-- get the degree to which the computations have been performed
computedDegree(LieAlgebra)
-- get the degree to which the computations have been performed
cycles
-- make the subalgebra of cycles
cycles(LieAlgebra)
-- 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)
-- real description
describe(LieAlgebra)
-- real description
describe(LieAlgebraMap)
-- real description
describe(LieDerivation)
-- real description
describe(VectorSpace)
-- 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
differentialLieAlgebra(List)
-- 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
dims(ZZ,ExtAlgebra)
-- compute the dimensions of a Lie algebra, Ext-algebra or vector space
dims(ZZ,LieAlgebra)
-- compute the dimensions of a Lie algebra, Ext-algebra or vector space
dims(ZZ,VectorSpace)
-- compute the dimensions of a Lie algebra, Ext-algebra or vector space
dims(ZZ,ZZ,ExtAlgebra)
-- compute the dimensions of a Lie algebra, Ext-algebra or vector space
dims(ZZ,ZZ,LieAlgebra)
-- compute the dimensions of a Lie algebra, Ext-algebra or vector space
dims(ZZ,ZZ,VectorSpace)
-- 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
extAlgebra(ZZ,LieAlgebra)
-- 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
firstDegree(ExtElement)
-- get the degree of an element
firstDegree(LieDerivation)
-- get the degree of an element
firstDegree(LieElement)
-- get the degree of an element
generators(ExtAlgebra)
-- get the generators
generators(LieAlgebra)
-- get the generators
generators(LieSubSpace)
-- get the generators
GradedLieAlgebras
-- a package for doing computations in graded Lie algebras
holonomy
-- compute the holonomy Lie algebra associated to an arrangement or matroid
Holonomy Lie algebras and symmetries
holonomy(...,Field=>...)
-- optional argument for holonomy
holonomy(List)
-- compute the holonomy Lie algebra associated to an arrangement or matroid
holonomy(List,List)
-- 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
holonomyLocal(ZZ,LieAlgebra)
-- 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
indexForm(LieElement)
-- get a Lie element in the polynomial ring representation
innerDerivation
-- make the derivation defined by right Lie multiplication by a Lie element
innerDerivation(LieElement)
-- 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
isMember(LieElement,LieSubSpace)
-- whether a Lie element belongs to a Lie subspace
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
koszulDual(PolynomialRing)
-- compute the Lie algebra whose enveloping algebra is the Koszul dual of a quadratic algebra
koszulDual(QuotientRing)
-- 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
lieAlgebra(List)
-- make a free Lie algebra
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
lieDerivation(LieAlgebraMap,List)
-- make a graded derivation
lieDerivation(List)
-- make a graded 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
lieHomology(LieAlgebra)
-- make the homology as a vector space
LieIdeal
-- the class of all Lie ideals
lieIdeal
-- make a Lie ideal
lieIdeal(LieSubSpace)
-- make a Lie ideal
lieIdeal(List)
-- 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
lieSubAlgebra(List)
-- 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
lieSubSpace(List)
-- make a Lie subspace
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
Minimal models, Ext-algebras and Koszul duals
minimalModel
-- compute the minimal model
minimalModel(ZZ,LieAlgebra)
-- 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
normalForm(LieElement)
-- 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(List,LieAlgebra)
-- get a random element of a Lie algebra
random(ZZ,LieAlgebra)
-- get a random element of a Lie algebra
random(ZZ,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
sign(ExtElement)
-- get the sign of a homogeneous element
sign(LieDerivation)
-- get the sign of a homogeneous element
sign(LieElement)
-- 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(List,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
VectorSpace
-- the class of all vector spaces
weight
-- get the weight of a homogeneous element
weight(ExtElement)
-- get the weight of a homogeneous element
weight(LieDerivation)
-- get the weight of a homogeneous element
weight(LieElement)
-- get the weight of a homogeneous element
zeroDerivation
-- make a derivation from the zero map
zeroDerivation(LieAlgebra)
-- make a derivation from the zero map
zeroIdeal
-- make the zero ideal
zeroIdeal(LieAlgebra)
-- make the zero ideal
zeroMap
-- make the zero map
zeroMap(LieAlgebra,LieAlgebra)
-- make the zero map
ZZ _ ExtAlgebra
-- get the zero element
ZZ _ LieAlgebra
-- get the zero element