Tensor Product Module Of The Quantum Group U_q（sl₂）

Let g be a finite-dimensional semi-simple Lie algebra,and q ∈IF,q ≠2,q2≠1.The quantum group Uq(g)is the quantization of the universal enveloping algebra of Lie algebras,and its representation theory has important applications in mathematics,physics.and other branches of mathematicsWhen g =sI27 some scholars constructed the module structure of Uq(sI2)on the Laurent polynomial ring C[K±1],and determined its irreducibility.The tensor product structures of these irreducible modules and finite-dimensional Uq(sI2)-modules are given,as well as their submodules and quotient modulesIn this paper,we study the structure of the tensor product between the free modules of the quantum group Uq(sI2),and then we determined the irreducibility of these modules The submodules and quotients and their properties in the reducible case are given.In addition,the structures of the tensor products of these modules and infinite-dimensional Ug(sI2)-modules are also studied,their irreducibility is determined,and submodules and their properties in the case of reducibility are given.