A study of vertex operator constructions for some infinite dimensional lie algebras
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In Chapter one of the thesis we construct a module for the toroidal Lie algebra and the extended toroidal Lie algebra of type A1. The Fock module representation obtained here is faithful and completely reducible over the extended toroidal Lie algebra. We also study the level two vertex operator representation of the toroidal Lie algebra of type A1. This generalizes the Lepowsky-Wilson study of the principal level two standard module for A(1)a . In Chapter two we present a complete description of the TKK algebra K(T(S)), which allows us in Chapter three to give a faithful representation to this Lie algebra by vertex operators. In the construction of this TKK algebra by vertex operators the Clifford algebra enters the picture. The situation here is similar to but more complicated than that for the level 2 standard A(1)a -module and the level 1 standard B(1)ɩ - module, where the Lie algebras of operators act on a vector space of mixed boson-fermion states. In the last chapter of the thesis we give two constructions for the toroidal Lie algebra of type B1 (ɩ ≥ ɜ) by vertex operators. The first construction is related to the folding of Dynkin diagram of D(1)ɩ +1 and a two-cocycle necessary for the vertex operator construction. This construction also suggests a direct construction of the toroidal Lie algebra of type B1 by vertex operators. In fact, the second construction generalizes the Lepowsky-Primc construction of the level one standard module of B(1)ɩ to the toroidal case.