eulers(ZZ,LieAlgebra) -- compute the list of Euler characteristics

Synopsis

Function: eulers
Usage:

l = eulers(n,L)
Inputs:
- n, an integer, the maximal degree
- L, an instance of the type LieAlgebra,
Outputs:
- l, a list, the list of Euler characteristics from degree $1$ to $n$

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 : F=lieAlgebra({a,b,c,r3,r4,r42},
          Weights => {{1,0},{1,0},{2,0},{3,1},{4,1},{4,2}},
          Signs=>{0,0,0,1,1,0},LastWeightHomological=>true)

o1 = F

o1 : LieAlgebra

i2 : L=differentialLieAlgebra{0_F,0_F,0_F,a c,a a c,r4 - a r3}

o2 = L

o2 : LieAlgebra

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

o3 = Q

o3 : LieAlgebra

i4 : dims(5,Q)

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 : eulers(5,Q)

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

o5 : List

i6 : H=lieHomology Q

o6 = H

o6 : VectorSpace

i7 : dims(5,H)

o7 = | 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
o7 : Matrix ZZ  <-- ZZ

Ways to use this method: