Structure And Representation For A Class Of Infinite-dimensional Lie Algebras

In Chapter one of the thesis we study the isomorphic classes and the derivation algebra of the Lie algebra L(E₁,E₂,E₃) := G (?) A with the Lie bracket given by (1) and A = C[t₁^±1,t₂^±1,…,t_v^±1] when L(E₁,E₂,E₃) is perfect. We prove that if the Lie algebra L(E₁,E₂,E₃) is perfect, then L(E₁,E₂,E₃) is either isomorphic to L(1,1,1), or isomorphic to L(t^s₁,t^s₂,1) for some s₁,s₂ ∈ Z₂^v with s₁,s₂ ≠ 0 and s₁≠s₂. We also prove that L(t^s₁,t^s₂,1) is finitely generated. Finally, we give the derivation algebra of the Lie algebra L(t^s₁,t^s₂,1). In Chapter two we give a perfect central extension L := L(?)K of L with Lie bracket given by (16), and prove that L is the uniersal central extension of L. In Chapter three we construct a level two homogeneous construction of the toroidal Lie algebra of type A₁. The Fock module obtained here is compoletely reducible over the the extended toroidal Lie algebra. In the last Chapter of the thesis we study the existence of the vertex operator representation for the uniersal central extension of the perfect Lie algebra L(E₁,E₂,E₃) with v = 2. We give a vertex operator representation for the Lie algbra L(t₁,t₂,1). Therefore, for every perfect Lie algebra L(E₁,E₂,E₃) with v = 2, its uniersal central extension has the vertex operator represention.