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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},{{{1,-1},{[b,c],[a,c]}},[a,b],
             {{1,-1},{[b,r4],[a,r4]}}},
         genWeights => {{1,0},{1,0},{2,0},{3,1},{4,1},{4,2}},
         genDiffs=>{[],[],[],{{-1},{[a,c]}},
             [a,a,c],{{1,1},{[r4],[a,r3]}}},genSigns=>{0,0,0,1,1,0})

o1 = L

o1 : LieAlgebra
i2 : dimTableLie 5

o2 = | 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 |
     | 0 0 0 0 0 |

              6        5
o2 : Matrix ZZ  <--- ZZ
i3 : eulerLie 5

o3 = {2, 1, 0, -1, -1}

o3 : List
i4 : homologyLie 5

o4 = | 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
o4 : Matrix ZZ  <--- ZZ

See also

Ways to use eulerLie :