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GradedLieAlgebras :: eulerLie

eulerLie -- computes the Euler characteristics

Synopsis

Description

For each first degree d, where d goes from 1 to n, the alternating sum of the dimensions of the Lie algebra in homological degree 0 to d-1 is computed. As we know, the same numbers are obtained using the homology of the Lie algebra instead.

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)

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, (a c), (a a c), r4 - (a r3)}

o2 : List
i3 : Q=L/{b c - a c,a b,b r4 - a r4}

o3 = Q

o3 : LieAlgebra
i4 : dimTableLie 5

o4 = | 2 1 1 1 2 |
     | 0 0 1 3 5 |
     | 0 0 0 1 2 |
     | 0 0 0 0 0 |
     | 0 0 0 0 0 |

              5        5
o4 : Matrix ZZ  <--- ZZ
i5 : eulerLie 5

o5 = {2, 1, 0, -1, -1}

o5 : List
i6 : homologyTableLie 5

o6 = | 2 1 0 0 0 |
     | 0 0 0 1 1 |
     | 0 0 0 0 0 |
     | 0 0 0 0 0 |
     | 0 0 0 0 0 |

              5        5
o6 : Matrix ZZ  <--- ZZ

See also

Ways to use eulerLie :