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LieAlgebraMap -- the class of all Lie algebra homomorphisms


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

See also

Methods that use an object of class LieAlgebraMap :

For the programmer

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