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GradedLieAlgebras :: LieAlgebra / List

LieAlgebra / List -- make a quotient Lie algebra

Synopsis

Description

Consider first the case where $L$ has zero differential, and where $L$ is finitely presented as a quotient of a free Lie algebra $F$. In this case, the output $Q$ is also finitely presented as a quotient of $F$.

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

o1 = F

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

o2 = L

o2 : LieAlgebra
i3 : Q = L/{a c}

o3 = Q

o3 : LieAlgebra
i4 : describe Q

o4 = generators => {a, b, c}
     Weights => {{1, 0}, {1, 0}, {1, 0}}
     Signs => {0, 0, 0}
     ideal => { - (b a),  - (c a)}
     ambient => F
     diff => {}
     Field => QQ
     computedDegree => 0
i5 : class\Q#ideal

o5 = {F, F}

o5 : List
i6 : F/Q#ideal==Q

o6 = true

In case $L$ has a non-zero differential, the program adds relations depending on the fact that the ideal should be invariant under the differential. These extra (non-normalized) relations may be looked upon using describe(LieAlgebra). Observe that $D$ is not free in this example, see differentialLieAlgebra.

i7 : F = lieAlgebra({a,b,c2,c3},Weights=>{{1,0},{1,0},{2,1},{3,2}},
          Signs=>{1,1,1,1},LastWeightHomological=>true)

o7 = F

o7 : LieAlgebra
i8 : D = differentialLieAlgebra{0_F,0_F,a a,b c2}

o8 = D

o8 : LieAlgebra
i9 : L = D/{a c2}

o9 = L

o9 : LieAlgebra
i10 : Q = L/{b c3}

o10 = Q

o10 : LieAlgebra
i11 : describe D

o11 = generators => {a, b, c2, c3}
      Weights => {{1, 0}, {1, 0}, {2, 1}, {3, 2}}
      Signs => {1, 1, 1, 1}
      ideal => { - (b a a)}
      ambient => F
      diff => {0, 0, (a a), (b c2)}
      Field => QQ
      computedDegree => 3
i12 : describe Q

o12 = generators => {a, b, c2, c3}
      Weights => {{1, 0}, {1, 0}, {2, 1}, {3, 2}}
      Signs => {1, 1, 1, 1}
      ideal => { - (b a a), (a c2),  - (a a a), (b c3),  - (b b c2)}
      ambient => F
      diff => {0, 0, (a a), (b c2)}
      Field => QQ
      computedDegree => 0
i13 : class\ideal(Q)

o13 = {F, F, F, F, F}

o13 : List
i14 : class\diff(Q)

o14 = {F, F, F, F}

o14 : List

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 a quotient of a finitely presented Lie algebra $N$ by an ideal $I$, where $N$ is given as $M$ modulo a lifting of the input list $x$ to $M$, and $I$ is the image of the natural map from $M$ to $N$ applied to $J$, see image(LieAlgebraMap,LieSubSpace).

i15 : F = lieAlgebra{a,b,c}

o15 = F

o15 : LieAlgebra
i16 : M = F/{a b}

o16 = M

o16 : LieAlgebra
i17 : f=map(M,M,{0_M,b,c})
warning: the map might not be well defined, 
         use isWellDefined

o17 = f

o17 : LieAlgebraMap
i18 : J=kernel f

o18 = J

o18 : LieIdeal
i19 : L = M/J

o19 = L

o19 : LieAlgebra
i20 : Q=L/{b c}

o20 = Q

o20 : LieAlgebra
i21 : N=ambient Q

o21 = N

o21 : LieAlgebra
i22 : describe Q

o22 = generators => {a, b, c}
      Weights => {{1, 0}, {1, 0}, {1, 0}}
      Signs => {0, 0, 0}
      ideal => ideal of N
      ambient => N
      diff => {}
      Field => QQ
      computedDegree => 0
i23 : use M
i24 : N==M/{b c}

o24 = true
i25 : ideal(Q)===new LieIdeal from image(map(N,M),J)

o25 = true

See also