matrix2LieAlg -- Converts from Matrix form to the LieAlgebra (with the standard basis)

Description

Let's build a 7-graded algebra and see the bases of each piece getting converted to the symbolic ements in the Lie algebra.

i1 : ea = exteriorExtension(3, 7, QQ);

i2 : ea.bases#0#0

o2 = | 1 0  0 0 0 0 0 |
     | 0 -1 0 0 0 0 0 |
     | 0 0  0 0 0 0 0 |
     | 0 0  0 0 0 0 0 |
     | 0 0  0 0 0 0 0 |
     | 0 0  0 0 0 0 0 |
     | 0 0  0 0 0 0 0 |

              7       7
o2 : Matrix QQ  <-- QQ

i3 : sln = ea.LieAlgebra

o3 = sln

o3 : PolynomialRing

i4 : ea.matrix2LieAlg (ea.bases#0#0)

o4 = h
      1

o4 : sln

i5 : ea.bases#0#8

o5 = | 0 0 0 1 0 0 0 |
     | 0 0 0 0 0 0 0 |
     | 0 0 0 0 0 0 0 |
     | 0 0 0 0 0 0 0 |
     | 0 0 0 0 0 0 0 |
     | 0 0 0 0 0 0 0 |
     | 0 0 0 0 0 0 0 |

              7       7
o5 : Matrix ZZ  <-- ZZ

i6 : ea.matrix2LieAlg (ea.bases#0#8)

o6 = E
      {0, 3}

o6 : sln

If you don't name the Lie algebra something it has an annoying printout.

For the programmer

The object matrix2LieAlg is a symbol.

The source of this document is in ExteriorExtensions.m2:839:0.