LAUSR

We prove that the tensor product of a simple and a finite dimensional $${\mathfrak {sl}}_n$$ sl n -module has finite type socle. This is applied to reduce classification of simple $${\mathfrak {q}}(n)$$ q ( n ) -supermodules to that of simple $${\mathfrak {sl}}_n$$ sl n -modules. Rough structure of simple $${\mathfrak {q}}(n)$$ q ( n ) -supermodules, considered as $${\mathfrak {sl}}_n$$ sl n -modules, is described in terms of the combinatorics of category $${\mathcal {O}}$$ O .