# lieHomology -- make the homology as a vector space

## Description

The basis elements for the homology of a Lie algebra (which is of type VectorSpace) are represented as cycles in the Lie algebra.

## Synopsis

• Usage:
H=lieHomology(L)
• Inputs:
• Outputs:
• H, an instance of the type VectorSpace, the homology of the Lie algebra
 i1 : L=lieAlgebra({a,b},Signs=>{1,0},Weights=>{{2,0},{2,1}}, Field=>ZZ/3,LastWeightHomological=>true) o1 = L o1 : LieAlgebra i2 : D=differentialLieAlgebra({0_L,a}) o2 = D o2 : LieAlgebra i3 : H=lieHomology D o3 = H o3 : VectorSpace i4 : dims(10,H) o4 = | 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 1 0 0 0 0 | | 0 0 0 0 0 1 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 | 10 10 o4 : Matrix ZZ <--- ZZ i5 : basis(6,H) o5 = {(b a a), (b b a)} o5 : List