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GradedLieAlgebras :: kernelBasisLie

kernelBasisLie -- a basis of the kernel of a Lie homomorphism or derivation in a specified degree

Synopsis

Description

i1 : L = lieAlgebra({a,b,c,r3,r4,r42},
         genWeights => {{1,0},{1,0},{2,0},{3,1},{4,1},{4,2}},
         genSigns=>{0,0,0,1,1,0},diffl=>true)/{b c - a c,a b,b r4 - a r4}

o1 = L

o1 : LieAlgebra
i2 : L.genDiffs = {L.zz,L.zz,L.zz,a c,a a c,r4 - a r3}

o2 = {0, 0, 0, (b c), (b b c), r4 - (a r3)}

o2 : List
i3 : M = minmodelLie 3

o3 = M

o3 : LieAlgebra
i4 : f = M.modelmap

o4 = f

o4 : MapLie
i5 : peekLie f

o5 = MapLie{fr_0 => a     }
            fr_1 => b
            fr_2 => c
            fr_3 => 0
            fr_4 => r3
            fr_5 => r3
            sourceLie => M
            targetLie => L
i6 : kernelTableLie(3,f)

o6 = | 0 1 3 |
     | 0 1 3 |
     | 0 0 0 |

              3        3
o6 : Matrix ZZ  <--- ZZ
i7 : kernelBasisLie(3,f)

o7 = {(fr_0 fr_1 fr_0), (fr_1 fr_1 fr_0),  - (fr_0 fr_2) + (fr_1 fr_2), (fr_0
     ------------------------------------------------------------------------
     fr_3), (fr_1 fr_3),  - fr_4 + fr_5}

o7 : List
i8 : kernelBasisLie(3,1,f)

o8 = {(fr_0 fr_3), (fr_1 fr_3),  - fr_4 + fr_5}

o8 : List
i9 : d = diffLie()

o9 = d

o9 : DerLie
i10 : kernelBasisLie(5,1,d)

o10 = { - (a a r3) + (b a r3),  - (a a r3) + (b b r3),  - (a a r3) + (b r4)}

o10 : List
i11 : cyclesBasisLie(5,1)

o11 = { - (a a r3) + (b a r3),  - (a a r3) + (b b r3),  - (a a r3) + (b r4)}

o11 : List

See also

Ways to use kernelBasisLie :