Classification Of Irreducible Integrable Modules For A Family Of Lie Algebras

Representations theories play great roles in Lie algebras theories.The representations for Loop algebras, twisted Lie algebras,muti-loop Lie algebras and twisted muti-loop Lie algebras become one of the hotspots in Lie theories.Chari,Rao and Batra have done many studies on these aspects. Many good results have been turned out.In chapter two, we study the representations for the new Lie algebras we construct in our paper based on many relevant Lie algebras such as Loop algebras, twisted Lie algebras,muti-loop Lie algebras and twisted muti-loop Lie algebras. We construct Zⁿ—graded homomorphism φ| of U(L(η,μ))→ L₁ and make L₁ into a L|(η,μ)— module by defining the effect of L|(η,μ) on L₁ .Then we can restrict the effect of L|(η, μ) on L₁ to L₁(φ|). This way we also make L₁(φ|) into a module for Lie algebras L|(η,μ) .In chapter three,we mainly consider the following two induced modules ,induced L|(G,μ) — modulewhere B = U(L(G₊, μ))U(L|(η,μ)) and induced L(G,μ)—modulewhere B' = U(L(C₊, μ))U(L(η,μ)). We can get the only irreducible quotient of M(φ|) which we denote by V(φ|) and the only irreducible quotient of M(φ) which we denote by V(φ). At the same time,we give the definitions of graded and non-graded highest weight modules. We give an essential and sufficient condition on that L|(η,μ)—module L₁(φ|) is irreducible.In theorem 3.2.1 we present the description of irreducible L|(G,μ)—module V(φ) (?) L₁ and L(G,μ)—module V(φ|). In the last chapter, we define the integrable modules for Lie algebras L|(G,μ) and L(G, μ).At the same time, some good properties about them are given. Then we classify irreducible inte-grable modules with finite dimensional weight spaces for these Lie algebras. By Theorem 3.2.2, We reduce the problem to the classification of L|(G,μ)-module U(L|(G,μ))v(0|) which is easy to classify. According to Theorem 4.2.4, we complete the classification of irreducible integrable modules with finite dimensional weight spaces for the new algebras that we have constructed in this paper.