# LieSubSpace + LieSubSpace -- make the sum of two Lie subspaces

## Synopsis

• Operator: +
• Usage:
S = A + B
• Inputs:
• A, an instance of the type LieSubSpace, an instance of type LieSubSpace
• B, an instance of the type LieSubSpace, an instance of type LieSubSpace
• Outputs:
• S, an instance of the type LieSubSpace, an instance of type LieSubSpace, the sum of $A$ and $B$

## Description

If both $A$ and $B$ are instances of FGLieIdeal, then $S$ is of type FGLieIdeal. Otherwise, if both $A$ and $B$ are instances of LieIdeal, then $S$ is of type LieIdeal. If exactly one of $A$ and $B$ is an instance of LieIdeal, and the other is an instance of LieSubAlgebra, then $S$ is of type LieSubAlgebra. Otherwise, $S$ is of type LieSubSpace.

 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 : D=differentialLieAlgebra{0_F,0_F,0_F,a c,a a c,r4 - a r3} o2 = D o2 : LieAlgebra i3 : I=lieIdeal{b c - a c,a b,b r4 - a r4} o3 = I o3 : FGLieIdeal i4 : Q=D/I o4 = Q o4 : LieAlgebra i5 : f=map(Q,D) o5 = f o5 : LieAlgebraMap i6 : J=lieIdeal{a c} o6 = J o6 : FGLieIdeal i7 : K=inverse(f,J) o7 = K o7 : LieIdeal i8 : use D i9 : I+lieIdeal{a c} o9 = finitely generated ideal of D o9 : FGLieIdeal i10 : dims(6,oo) o10 = | 0 1 4 7 16 30 | | 0 0 0 0 2 9 | | 0 0 0 0 0 1 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | 6 6 o10 : Matrix ZZ <--- ZZ i11 : dims(6,K) o11 = | 0 1 4 7 16 30 | | 0 0 0 0 2 9 | | 0 0 0 0 0 1 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | 6 6 o11 : Matrix ZZ <--- ZZ