量子代数模的张量积与不可分解模

Tensor Product of Modules and Indecomposables Module for Quantum Algebras

令M是Z[v]的由v-1和奇素数p生成的理想,U是A=Z[v]M上相伴于对称Cartan矩阵的量子代数.k是特征为零的代数闭域,A→k(v(?)ξ)是环同态.U_k=U(?)_Ak,u_k是U_k的无穷小量子代数.令ξ是1的p次本原根.本文证明了:若有限维可积U_k模M,V中至少有一个是内射模,或者M,V中有一个模作为u_k模是平凡的,则有U_k模同构M(?)V≌V(?)M.我们还证明了:若有限维可积U_k模V作为u_k模是不可分解的,有限维可积U_k模M是不可分解的,且M|_(uk)是平凡的,则V(?)M是不可分解U_k模.令V和M是有限维可积U_k模,作为u_k模是同构的且具有单基座,本文证明V和M作为U_k模也是同构的.由此得到:不可分解内射u_k模提升为U_k模是唯一的.

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.