LieAlgebraModule @ LieAlgebraModule -- Take the tensor product of modules over different Lie algebras

Operator: @

Description

Produces a module over the direct sum of the Lie algebras of the two modules.

i1 : LL_(1,2,3,4) (simpleLieAlgebra("D",4)) @ LL_(5,6) (simpleLieAlgebra("G",2))

o1 = LL           (𝔡  ++ 𝔤 )
       1,2,3,4,5,6  4     2

o1 : irreducible LieAlgebraModule over 𝔡  ++ 𝔤
                                        4     2

A complicated way to define usual tensor product LieAlgebraModule ** LieAlgebraModule would be using the diagonal embedding:

i2 : g := simpleLieAlgebra("A",1)

o2 = 𝔞
      1

o2 : simple LieAlgebra

i3 : h := g ++ g

o3 = 𝔞  ++ 𝔞
      1     1

o3 : LieAlgebra

i4 : gdiag := subLieAlgebra(h,matrix {{1},{1}})

o4 = 𝔞
      1

o4 : simple LieAlgebra, subalgebra of 𝔞  ++ 𝔞
                                       1     1

i5 : M = LL_5 (g); M' = LL_2 (g);

i7 : M @ M'

o7 = LL   (𝔞  ++ 𝔞 )
       5,2  1     1

o7 : irreducible LieAlgebraModule over 𝔞  ++ 𝔞
                                        1     1

i8 : branchingRule(oo,gdiag)

o8 = LL (𝔞 ) ++ LL (𝔞 ) ++ LL (𝔞 )
       3  1       5  1       7  1

o8 : LieAlgebraModule over 𝔞
                            1

i9 : M ** M'

o9 = LL (𝔞 ) ++ LL (𝔞 ) ++ LL (𝔞 )
       3  1       5  1       7  1

o9 : LieAlgebraModule over 𝔞
                            1

Ways to use this method:

LieAlgebraModule @ LieAlgebraModule -- Take the tensor product of modules over different Lie algebras

The source of this document is in LieTypes.m2:2622:0.