YuanHuai Bai
Let M be the ideal in Z[v] generated by v-1 and a odd prime p, U be a quantum algebra over A-Z[V]M with a symmetric Cartan matrix. Let k be algebraically closed field of characteristic zero. Consider a ring homomorphism A→k (v→ζ) and let Uk = U A k, Uk be infinitesimal quantum algebra of Uk,ζ be a primitive p-th root of 1. Let V and M be finite dimensional integrable Uk-modules. In this paper we show that Uk module isomorphisms V M ≌ M V when at least one of V or M is injective, or least one of V or M is trivial Uk module. Moreover, if V is indecomposable Uk module, M is indecomposable Uk module and M restricts to an indecomposable Uk module, then we show that V M is indecomposable Uk module. Finally, if V and M is Uk module isomorphisms and with unique simple submodule, we prove that the V and M also is Uk module isomorphisms, this shows Uk structure on indecomposable injective Uk module is unique.