# LieAlgebra * LieAlgebra -- free product of Lie algebras

## Synopsis

• Operator: *
• Usage:
M = L1*L2
• Inputs:
• Outputs:
• M, an instance of the type LieAlgebra, the free product of $L1$ and $L2$

## Description

 i1 : F1 = lieAlgebra({a,b},Signs=>{0,1},Weights=>{{2,0},{2,1}}, LastWeightHomological=>true) o1 = F1 o1 : LieAlgebra i2 : L1 = differentialLieAlgebra{0_F1,a} o2 = L1 o2 : LieAlgebra i3 : F2 = lieAlgebra({a,b,c},Weights=>{{1,0},{2,1},{3,2}}, Signs=>{1,1,1},LastWeightHomological=>true) o3 = F2 o3 : LieAlgebra i4 : L2 = differentialLieAlgebra{0_F2,a a,a b}/{b b+4 a c} o4 = L2 o4 : LieAlgebra i5 : M = L1*L2 o5 = M o5 : LieAlgebra i6 : describe(M) o6 = generators => {pr , pr , pr , pr , pr } 0 1 2 3 4 Weights => {{2, 0}, {2, 1}, {1, 0}, {2, 1}, {3, 2}} Signs => {0, 1, 1, 1, 1} ideal => { - (pr_2 pr_2 pr_2), (pr_3 pr_3) + 4 (pr_2 pr_4), (pr_2 pr_2 pr_3) + (pr_2 pr_2 pr_3) - (pr_3 pr_2 pr_2) - 4 (pr_2 pr_2 pr_3)} ambient => LieAlgebra{...10...} diff => {0, pr_0, 0, (pr_2 pr_2), (pr_2 pr_3)} Field => QQ computedDegree => 0 i7 : normalForm\ideal(M) o7 = {0, (pr_3 pr_3) + 4 (pr_2 pr_4), 0} o7 : List i8 : d = differential M o8 = d o8 : LieDerivation i9 : d (pr_1 pr_3) o9 = - (pr_3 pr_0) + 2 (pr_2 pr_2 pr_1) o9 : M