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

Synopsis

Usage:

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

Description

S must be a subset of vertices of the Dynkin diagram of g (as labelled by dynkinDiagram); or a matrix whose columns are the simple coroots of the subalgebra expanded in the basis of simple coroots of g.

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

i4 : h=𝔣_4; dynkinDiagram h

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

i6 : subLieAlgebra(h,matrix transpose{{1,0,0,0},{0,1,0,0},{0,0,1,0},-{2,3,2,1}}) -- simple coroots 1,2,3 and opposite of highest root

o6 = 𝔟
      4

o6 : simple LieAlgebra, subalgebra of h

Caveat

If S is a matrix, does not check if the map of root lattices leads to a valid Lie algebra embeddng.

Ways to use subLieAlgebra :

subLieAlgebra(LieAlgebra,List)
subLieAlgebra(LieAlgebra,Matrix)

For the programmer

The object subLieAlgebra is a method function.