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

- 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

## See also

- LieSubSpace @ LieSubSpace -- make the intersection of two Lie subspaces

## Ways to use this method:

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

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/GradedLieAlgebras/doc.m2:1597:0.