LieAlgebraMap -- the class of all Lie algebra homomorphisms

Description

This type represents homomorphisms from $M$ to $L$, where $M$ and $L$ are Lie algebras defined over the same field. Use the constructor map(LieAlgebra,LieAlgebra,List) to define homomorphisms. A homomorphism is given by the value of the generators and it should preserve weight and sign. Observe that the zero element is considered to have any weight and sign. The zero element in L is denoted by 0_L. An object of type LieAlgebraMap need not be well defined as a map and need not commute with the differentials. Use isWellDefined(ZZ,LieAlgebraMap) to check if this is true.

i1 : M = lieAlgebra({a,b,c},Weights=>{2,1,1})

o1 = M

o1 : LieAlgebra

i2 : L = lieAlgebra{a,b}

o2 = L

o2 : LieAlgebra

i3 : f = map(L,M,{a b,a,0_L})

o3 = f

o3 : LieAlgebraMap

i4 : isWellDefined(3,f)
the map is well defined for all degrees
the map commutes with the differential for all degrees

o4 = true

i5 : describe f

o5 = a =>  - (b a)
     b => a
     c => 0
     source => M
     target => L

i6 : use M

i7 : f (a c - a b)

o7 =  - (a b a)

o7 : L

Methods that use an object of class LieAlgebraMap :

- LieAlgebraMap -- unary negation
describe(LieAlgebraMap) -- see describe(LieAlgebra) -- real description
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
inverse(LieAlgebraMap,LieSubSpace) -- make the inverse image of a Lie subspace under a Lie algebra map
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
kernel(LieAlgebraMap) -- make the kernel of a map
LieAlgebra / LieAlgebraMap -- make a quotient Lie algebra
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 * LieAlgebraMap -- composition of a derivation and a homomorphism
Number LieAlgebraMap -- multiplication of a number and a homomorphism
RingElement LieAlgebraMap -- multiplication of a field element and a homomorphism
source(LieAlgebraMap) -- get the source of a map
target(LieAlgebraMap) -- get the target of a map
trace(ZZ,LieSubSpace,LieAlgebraMap) -- compute the trace of a Lie algebra map acting on a Lie subspace

For the programmer

The object LieAlgebraMap is a type, with ancestor classes HashTable < Thing.

LieAlgebraMap -- the class of all Lie algebra homomorphisms

Description

See also

Methods that use an object of class LieAlgebraMap :

For the programmer