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

## Synopsis

• Function: eulers
• Usage:
l = eulers(n,L)
• Inputs:
• 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