Representations of the Orbifold VOAS Lsl2^(k,0)K and the Commutant VOAS CLsom^(1,0)⊗3(Lsom^(3,0))

For the Klein group K , k ∈ Z ⩾ 1 and m ∈ Z ⩾ 4 , we study the representations of the orbifold vertex operator algebra L sl 2 ^ ( k , 0 ) K and the commutant vertex operator algebra of L so m ^ ( 3 , 0 ) in L so m ^ ( 1 , 0 ) ⊗ 3 which can be realized as the orbifold vertex operator subalgebra L sl 2 ^ ( 2 m , 0 ) K or its extension. All the irreducible modules for L sl 2 ^ ( k , 0 ) K and C L so m ^ ( 1 , 0 ) ⊗ 3 ( L so m ^ ( 3 , 0 ) ) are classified and constructed explicitly.