[GAP Forum] semsimple algebras and subalgebras

[Willem de Graaf]

Dear R. N. Tsai,

> Also getting the highest weight of the
> adjoint rep works for simple algebras; but I think it has trouble with
> direct sum of simple algebras :

In that case the module is not irreducible, so there is no single highest
weight.

All the best,

Willem de Graaf


On Mon, Feb 17, 2014 at 9:32 PM, R.N. Tsai <r_n_tsai at yahoo.com> wrote:

> Dear Willem and forum,
>
> Thanks for the quick response. The function you created "DecomposeAdRep" is
> exactly what I was looking for. Also getting the highest weight of the
> adjoint rep works for simple algebras; but I think it has trouble with
> direct sum of simple algebras :
>
> # A1+A1 -> A1 (or D2->D1 or O4->O3 ) branching rule for adjoint rep :
> gap> r:=LieAlgebraAndSubalgebras("A1 A1");;
> gap> L:= r.liealg;;
> gap> K:= r.subalgs[1];;
> # this gives the correct decomposition : 6->3+1+1+1;
> gap> dec:=DecomposeAdRep( L, K );
> [ <vector space of dimension 3 over Rationals>, <vector space of dimension
> 1 over Rationals>,
>   <vector space of dimension 1 over Rationals>, <vector space of dimension
> 1 over Rationals> ]
>
> # however this has a problem; dim should be 6.
> gap> wt:= PositiveRootsAsWeights( RootSystem(L) );;
> gap> hw:= wt[ Length(wt) ];
> [ 0, 2 ]
> gap> dim:=DimensionOfHighestWeightModule(L,hw);
> 3
>
> Regards,
> R.N.
>
>
>   On Monday, February 17, 2014 12:39 AM, Willem de Graaf <
> degraaf at science.unitn.it> wrote:
>  Dear R.N. Tsai,
>
>
> You asked:
>
> > I'd like to identify
> > the subalgebra irreps with concrete subspaces of the main algebra. Is
> there
> > a (hopefully simple) way to extract this information?
>
> Not directly, however it is not so difficult to write some code for that.
> At the bottom of this message please find a GAP function that does that.
> It is followed by an example.
>
> One remark: in your piece of code you use the command
>
>       Dimension(HighestWeightModule(sub,bra[1][k]))
>
> It is much more efficient to use
>
>       DimensionOfHighestWeughtModule( sub, bra[1][k] );
>
> as this avoids constructing the module.
>
> You also asked:
>
> > is there a general way to get the weight of the adjoint rep for an
> > arbitrary semisimple algebra?
>
> If the Lie algebra is denoted L, then you can do
>
> wt:= PositiveRootsAsWeights( RootSystem(L) );;
> hw:= wt[ Length(wt) ];
>
> I am glad that the sla package is of use to you.
>
> Best wishes,
>
> Willem de Graaf
>
> # function:
>
> DecomposeAdRep:= function( L, K )
>
>         # K is a subalgebra of L, both semisimple in char 0;
>         # we return the decomposition of L as K module.
>
>     local cg, e, x, ad, i, spaces, spaces0, h, ww, sp, mat, es, hwv;
>
>     cg:= CanonicalGenerators( RootSystem(K) );
>     e:= List( Basis(L), x -> [ ] );
>     for x in cg[1] do
>         ad:= TransposedMat( AdjointMatrix(Basis(L),x) );
>         for i in [1..Length(ad)] do
>             Append( e[i], ad[i] );
>         od;
>     od;
>     spaces:= [ List( NullspaceMat(e), u -> u*Basis(L) ) ];
>
>     for h in cg[3] do
>         spaces0:= [ ];
>         for ww in spaces do
>             sp:= Basis( Subspace(L,ww), ww );
>             mat:= List( ww, u-> Coefficients(sp,h*u) );
>             es:= Eigenspaces( LeftActingDomain(L), mat );
>             for i in [1..Length(es)] do
>                 Add( spaces0, List( Basis(es[i]), x -> x*ww ) );
>             od;
>         od;
>         spaces:= spaces0;
>     od;
>
>     hwv:= Concatenation( spaces );
>     spaces:= [ ];
>     for i in [1..Length(hwv)] do
>         Add( spaces, MutableBasisOfClosureUnderAction( LeftActingDomain(L),
>                  cg[2], "left", [ hwv[i] ], \*, Zero(L), Dimension(L) ) );
>     od;
>     return List( spaces, u -> Subspace(L,BasisVectors(u)) );
>
> end;
>
> # example:
>
> gap> r:= LieAlgebraAndSubalgebras("G2");;
> gap> L:= r.liealg;;
> gap> K:= r.subalgs[5];;
> gap> DecomposeAdRep( L, K );
> [ <vector space over Rationals, with 8 generators>,
>   <vector space over Rationals, with 3 generators>,
>   <vector space over Rationals, with 3 generators> ]
>
>
>
>
>