TensorProduct( list ) O
TensorProduct( V, W ) O
Here list must be a list of Lie algebra modules. This function returns the tensor product of the elements in the list (as a Lie algebra module). The modules must be defined over the same Lie algebras.
In the second form is short for TensorProduct( [ V, W ] ).
Elements of the tensor product V1ļÄVk are linear combinations of v1ļÄvk, where the vi are arbitrary basis elements of Vi. In GAP a tensor element like that is printed as
v_1<x> ... <x>v_kFurthermore, the zero of a tensor product is printed as
<0-tensor>This does not mean that all tensor products have the same zero element: zeros of different tensor products have different families.
gap> L:= SimpleLieAlgebra("G",2,Rationals);;
gap> V:= HighestWeightModule( L, [ 1, 0 ] );;
gap> W:= TensorProduct( [ V, V, V ] );
<343-dimensional left-module over <Lie algebra of dimension
14 over Rationals>>
gap> w:= Basis(W)[1];
1*(1*v0<x>1*v0<x>1*v0)
gap> Basis(L)[1]^w;
<0-tensor>
gap> Basis(L)[7]^w;
1*(1*v0<x>1*v0<x>y1*v0)+1*(1*v0<x>y1*v0<x>1*v0)+1*(y1*v0<x>1*v0<x>1*v0)
ExteriorPower( V, k ) O
Here V must be an algebra module. This function returns the k-th exterior power of V (as an algebra module).
Elements of the exterior power Ùk V are linear combinations of vi1Ù¼Ùvik, where the vij are basis elements of V, and 1 £ i1 < i2¼ < ik. In GAP a wedge element like that is printed as
v_1/\ ... /\v_kFurthermore, the zero of an exterior power is printed as
<0-wedge>This does not mean that all exterior powers have the same zero element: zeros of different exterior powers have different families.
gap> L:= SimpleLieAlgebra("G",2,Rationals);;
gap> V:= HighestWeightModule( L, [ 1, 0 ] );;
gap> W:= ExteriorPower( V, 3 );
<35-dimensional left-module over <Lie algebra of dimension 14 over Rationals>>
gap> w:= Basis(W)[1];
1*(1*v0/\y1*v0/\y3*v0)
gap> Basis(L)[2]^w;
-1*(1*v0/\y1*v0/\y1*v0)
gap> Basis(L)[7]^w;
1*(1*v0/\y1*v0/\y4*v0)+1*(y1*v0/\y1*v0/\y3*v0)
SymmetricPower( V, k ) O
Here V must be an algebra module. This function returns the k-th symmetric power of V (as an algebra module).
Elements of the symmetric power Symmk(V) are linear combinations of vi1¼vik, where the vij are basis elements of V, and 1 £ i1 £ i2¼ £ ik. In GAP a symmetric element like that is printed as
v_1.v_2 ... .v_kFurthermore, the zero of a symmetric power is printed as
<0-symmetric element>This does not mean that all symmetric powers have the same zero element: zeros of different symmetric powers have different families.
gap> L:= SimpleLieAlgebra("G",2,Rationals);;
gap> V:= HighestWeightModule( L, [ 1, 0 ] );;
gap> W:= SymmetricPower( V, 3 );
<84-dimensional left-module over <Lie algebra of dimension 14 over Rationals>>
gap> w:= Basis(W)[1];
1*(1*v0.1*v0.1*v0)
gap> Basis(L)[2]^w;
<0-symmetric element>
gap> Basis(L)[7]^w;
3*(1*v0.1*v0.y1*v0)
DirectSumOfAlgebraModules( list ) O
DirectSumOfAlgebraModules( V, W ) O
Here list must be a list of algebra modules. This function returns the direct sum of the elements in the list (as an algebra module). The modules must be defined over the same algebras.
In the second form is short for DirectSumOfAlgebraModules( [ V, W ] )
gap> L:= SimpleLieAlgebra( "C", 3, Rationals );; gap> V:= HighestWeightModule( L, [ 1, 1, 0 ] ); <64-dimensional left-module over <Lie algebra of dimension 21 over Rationals>> gap> W:= HighestWeightModule( L, [ 0, 0, 2 ] ); <84-dimensional left-module over <Lie algebra of dimension 21 over Rationals>> gap> U:= DirectSumOfAlgebraModules( V, W ); <148-dimensional left-module over <Lie algebra of dimension 21 over Rationals>>[Top] [Previous] [Up] [Index]
GAP 4 manual