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GradedLieAlgebras :: LieAlgebra / LieIdeal

LieAlgebra / LieIdeal -- make a quotient Lie algebra

Synopsis

Description

Any object of type LieAlgebra is a finitely presented (differential) Lie algebra modulo an ideal, which is an object of type LieIdeal (and which might be zero). If the input Lie algebra $L$ is finitely presented, then the output Lie algebra $Q$ is simply presented as a quotient of $L$ by the input ideal $I$. (Observe that each time L/I is executed, a new different copy of L/I is produced.)

i1 : F = lieAlgebra{a,b,c}

o1 = F

o1 : LieAlgebra
i2 : L = F/{a b}

o2 = L

o2 : LieAlgebra
i3 : f=map(L,L,{0_L,b,c})
warning: the map might not be well defined, 
         use isWellDefined

o3 = f

o3 : LieAlgebraMap
i4 : I=kernel f

o4 = I

o4 : LieIdeal
i5 : Q = L/I

o5 = Q

o5 : LieAlgebra
i6 : describe Q

o6 = generators => {a, b, c}
     Weights => {{1, 0}, {1, 0}, {1, 0}}
     Signs => {0, 0, 0}
     ideal => I
     ambient => L
     diff => {}
     Field => QQ
     computedDegree => 0
i7 : Q===L/I

o7 = false
i8 : Q==L/I

o8 = true

If the input Lie algebra $L$ is given as a finitely presented Lie algebra $M$ modulo an ideal $J$ that is not (known to be) finitely generated (e.g., the kernel of a homomorphism ), then the output Lie algebra $Q$ is presented as $M$ modulo the ideal that is the inverse image of the natural map from $M$ to $L$ applied to the input ideal $I$.

i9 : F = lieAlgebra{a,b,c}

o9 = F

o9 : LieAlgebra
i10 : M = F/{a b}

o10 = M

o10 : LieAlgebra
i11 : f=map(M,M,{0_M,b,c})
warning: the map might not be well defined, 
         use isWellDefined

o11 = f

o11 : LieAlgebraMap
i12 : J=kernel f

o12 = J

o12 : LieIdeal
i13 : L = M/J

o13 = L

o13 : LieAlgebra
i14 : X=lieAlgebra{x}

o14 = X

o14 : LieAlgebra
i15 : g=map(X,L,{0_X,x,x})
warning: the map might not be well defined, 
         use isWellDefined

o15 = g

o15 : LieAlgebraMap
i16 : I=kernel g

o16 = I

o16 : LieIdeal
i17 : Q=L/I

o17 = Q

o17 : LieAlgebra
i18 : ambient Q===M

o18 = true
i19 : ideal(Q)===inverse(map(L,M),I)

o19 = true

See also