Factorization of simple modules for certain restricted two-parameter quantum groups

We study the representations of the restricted two-parameter quantum groups of types B and G . For these restricted two-parameter quantum groups, we give some explicit conditions which guarantee that a simple module can be factored as the tensor product of a one-dimensional module with a module that is naturally a module for the quotient by central group-like elements. That is, given θ a primitive ℓ th root of unity, the factorization of simple 𝔲_θ ^y ,θ ^z (𝔤) -modules is possible, if and only if (2( y − z ), ℓ ) = 1 for 𝔤 = 𝔰𝔬_2n + 1 ; (3( y − z ), ℓ ) = 1 for 𝔤 = G 2 .