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# subLieAlgebra -- Define a sub-Lie algebra of an existing one

- Usage:
    `subLieAlgebra(g,S)`
- Inputs:
  - g, an instance of the type LieAlgebra,
  - S, a list, a matrix, or a string,
- Outputs:
  - h, an instance of the type LieAlgebra,

## Description

For the purposes of this function, a sub-Lie algebra means an embedding of a Lie algebra into another up to linear equivalence. According to Dynkin's theory, this means that it is determined by the restriction of the embedding to the Cartan subalgebra, and that is the data provided by S. Specifically, S must be either a subset of vertices of the Dynkin diagram of g (as labelled by dynkinDiagram):

    i1 : g=𝔢_8; dynkinDiagram g

    o2 =         o 2
                 |
         o---o---o---o---o---o---o
         1   3   4   5   6   7   8

    i3 : subLieAlgebra(g,{1,2,3,4,5,8})

    o3 = 𝔡  ++ 𝔞
          5     1

    o3 : LieAlgebra, subalgebra of g

The vertices are labelled from 1 to the rank of g; because we frequently want to consider the lowest root, it is labelled 0:

    i4 : h=𝔣_4; dynkinDiagram h

    o5 = o---o=>=o---o
         1   2   3   4

    i6 : subLieAlgebra(h,{0,1,2,3})

    o6 = 𝔟
          4

    o6 : simple LieAlgebra, subalgebra of h

Or S must be a matrix whose columns are the simple coroots of the subalgebra expanded in the basis of simple coroots of g:

    i7 : g=𝔢_6

    o7 = g

    o7 : simple LieAlgebra

    i8 : h=subLieAlgebra(g,{2,4,{0,0,1,0,1,0},{1,0,0,0,0,1}}); describe h

    o9 = 𝔣
          4

    i10 : branchingRule(adjointModule g,h)

    o10 = LL       (h) ++ LL       (h)
            0,0,0,1         1,0,0,0

    o10 : LieAlgebraModule over h

Or S is the string principal, which is currently the only predefined subalgebra:

    i11 : g=𝔞_2; h=subLieAlgebra(g,"principal"); describe h

    o13 = 𝔞
           1

    i14 : V=LL_(2,4) g; qdim V

           12     10     8     6     4     2         -2     -4     -6     -8
    o15 = q   + 2q   + 4q  + 5q  + 7q  + 7q  + 8 + 7q   + 7q   + 5q   + 4q   +
          -----------------------------------------------------------------------
            -10    -12
          2q    + q

    o15 : ZZ[q]

    i16 : W=branchingRule(V,h); describe W

                            2                      2
    o17 = LL (h) ++ (LL (h))  ++ LL (h) ++ (LL (h))  ++ LL  (h) ++ LL  (h)
            0          4           6          8           10         12

    i18 : character W

           12     10     8     6     4     2         -2     -4     -6     -8
    o18 = x   + 2x   + 4x  + 5x  + 7x  + 7x  + 8 + 7x   + 7x   + 5x   + 4x   +
           1      1      1     1     1     1         1      1      1      1
          -----------------------------------------------------------------------
            -10    -12
          2x    + x
            1      1

    o18 : ZZ[x ]
              1

In simply laced types, principal specialisation (character of principal subalgebra) and q-dimension agree.

## Caveat

If S is a matrix, does not check if the map of Cartan subalgebras leads to a valid Lie algebra embedding.

## Ways to use subLieAlgebra:

- `subLieAlgebra(LieAlgebra,List)`
- `subLieAlgebra(LieAlgebra,Matrix)`
- `subLieAlgebra(LieAlgebra,String)`

## For the programmer

The object subLieAlgebra is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/LieAlgebraRepresentations/documentation.m2:332:0.