[GAP Forum] representations of Lie algebras

[degraaf at science.unitn.it]

Dear R. N. Tsai,

You asked:

> I didn't find a way to get explicit matrices for representation.
> Is this possible?


For this you can use the function "MatrixOfAction", for example

gap> L:= SimpleLieAlgebra( "A", 2, Rationals );
<Lie algebra of dimension 8 over Rationals>
gap> V:= HighestWeightModule( L, [1,1] );
<8-dimensional left-module over <Lie algebra of dimension 8 over Rationals>>
gap> x:= Basis(L)[1];
v.1
gap> MatrixOfAction( Basis(V), x );
[ [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ],
  [ 0, 0, 0, 2, 1, 0, 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, 1 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ] ]


or if you would like to have a function of Lie algebra elements alone:

gap> rep:= function( x ) return MatrixOfAction( Basis(V), x ); end;
function( x ) ... end
gap> rep( x );
[ [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ],
  [ 0, 0, 0, 2, 1, 0, 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, 1 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ] ]


I hope this answers your question. If not, or if you have any other
questions, then please ask again.

Best wishes,

Willem de Graaf