new LieAlgebra from Matrix -- Define a Lie algebra from its Cartan matrix

Description

new LieAlgebra from M

If M is a valid Cartan matrix, it will reorder if needed the rows/columns of M to a standard form and then output the corresponding Lie algebra g.

i1 : M = matrix {{2, 0, -3, 0}, {0, 2, 0, -1}, {-1, 0, 2, 0}, {0, -1, 0, 2}}

o1 = | 2  0  -3 0  |
     | 0  2  0  -1 |
     | -1 0  2  0  |
     | 0  -1 0  2  |

              4       4
o1 : Matrix ZZ  <-- ZZ

i2 : h := new LieAlgebra from M

o2 = 𝔤  ++ 𝔞
      2     2

o2 : LieAlgebra

i3 : cartanMatrix h

o3 = | 2  -1 0  0  |
     | -3 2  0  0  |
     | 0  0  2  -1 |
     | 0  0  -1 2  |

              4       4
o3 : Matrix ZZ  <-- ZZ

Ways to use this method:

new LieAlgebra from Matrix -- Define a Lie algebra from its Cartan matrix

The source of this document is in LieTypes.m2:2650:0.