## A study of vertex operator constructions for some infinite dimensional lie algebras

##### Abstract

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 and three we study the core of the smallest extended affine Lie algebra which is not of finite or affine type. Let T(S) be the Jordan algebra constructed from a semilattice S of Rν (ν ≥ 1). Let K(T(S)) be the Lie algebra obtained from the Jordan algebra T(S) by the Tits-Kantor-Koecher construction. The TKK algebra K(T(S)) is the universal central extension of the Lie algebra K(T(S)). When one specializes this construction to the (non-lattice) semilatticeS of R2, one obtains the core of the smallest extended affine Lie algebra which is not of finite or affine type. We present a complete description of this TKK algebra K(T(S)), which then allows us to give a faithful representation to this Lie algebra by vertex operators. It is interesting that in the construction of this TKK algebra the Clifford algebra enters the picture. The situation here is similar to, but more complicated than, that for the level 2 standard A(1)1-module and the level 1 standard B(1)1 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 vertex operator constructions for the toroidal Lie algebra of type Bl (l ≥ 3). The constructions are related to the folding of Dynkin diagram of D(1)l+1 and a two-cocycle necessary for the vertex operator constructions. From the construction it follows that the Fock space also affords a representation of the Clifford algebra W, which is spanned by the operators ωj(j in 2Z+1) with the relation ωiωj+ωj
ωi=-2 δi+j,0 (i,j in 2Z+1). Moreover, the construction also suggests a direct construction of the toroidal Lie algebra T(Bl) by vertex operators. In fact, the second construction generalizes the Lepowsky-Primc construction of the level one standard module of B(1)l to the toroidal case.