Module Structures On U(η) For Kac-Moody Algebras And Their Applications

Let g be a Kac-Moody algebra with Cartan subalgebra h.In this thesis,we first study the U(h)-free modules over the Kac-Moody algebras,say the full subcategory of g-Mod: H(g)={M∈g-Mod|Reu(h)u(g)M|(?)u(h)U(h))}. Using extension of Dynkin diagram,we show that:H(g) is nonempty if and only if g is of type Al(l≥1)or Cl(l≥2).We also describe the module structures when H is not empty.Next we use modules in H(g)to consider the category H for the basic Lie super-algebras and show that only for the basic Lie superalgebras of type B(0,l)(l≥1)the category H is nonempty.Meanwhile,we use the U(h)-free g-modules to construct a class of new modules over the related(full)toroidal Lie(super)algebras L(g),i.e. H1={M,∈L(g)-Md|Resu(h1)u(L(g))M(?)u(h1)U(h1)}, where h1=h+Cd1+…+Cdn.Finally,we determine the simplicities of generalized Verma modules over sll+2(C) induced from u(h)-free sll+1(C)-modules.