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# LieAlgebra ++ LieAlgebra -- direct sum of Lie algebras

- Operator: ++
- Usage:
    `S = L++M`
- Inputs:
  - L, an instance of the type LieAlgebra,
  - M, an instance of the type LieAlgebra,
- Outputs:
  - S, an instance of the type LieAlgebra, the direct sum of $L$ and $M$

## 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 : T=L1++L2

    o5 = T

    o5 : LieAlgebra

    i6 : describe(T)

    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), (pr_0 pr_2), (pr_0 pr_3), (pr_0 pr_4), (pr_1 pr_2), (pr_1 pr_3), (pr_1 pr_4)}
         ambient => LieAlgebra{...10...}
         diff => {0, pr_0, 0, (pr_2 pr_2), (pr_2 pr_3)}
         Field => QQ
         computedDegree => 0

    i7 : normalForm\ideal(T)

    o7 = {0, (pr_3 pr_3) + 4 (pr_2 pr_4), 0,  - (pr_2 pr_0),  - (pr_3 pr_0),
         ------------------------------------------------------------------------
         (pr_0 pr_4), (pr_2 pr_1), (pr_3 pr_1), (pr_1 pr_4)}

    o7 : List

## Ways to use this method:

- LieAlgebra ++ LieAlgebra -- direct sum of Lie algebras

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/GradedLieAlgebras/doc.m2:2991:0.