Researches Of The Representations Of The TKK Algebra

The extended affine Lie algebras (EALA's for short) were first introduced in the paper [H-KT] as a natural generalization of affine Kac-Moody algebras. Subsequently, in [BGK] and [AABGP], the authors classified for EALA's and found that EALA's allow not only the multiple variables Laurent polynomial rings as their coordinate algebras but also certain alternative and Jordan algebras and the quantum torus as their coordinate algebras.The representations for EALA's coordinatized by the multiple variables Laurent polynomial rings, i.e. toroidal Lie algebras, have been constructed in [EM], [MRY], [Bi1], [BB], etc. In [BS], [G3, 4, 5], [BGT] and [GZ], the authors studied the representations for EALA's over quantum tori. In [BGT], a general vertex operator construction based on the Fock space for affine Lie albebras of type A has been given, which allows the authors to give a unified treatment for both the homogeneous and principle realizations of the affine Lie algebras (gl)|^_N as well as for some EALA's coordinatized by certain quantum tori.In [AABGP], the authors introduce the concept of semilattice to descibe the extended affine root systems of EALA's, and define a Jordan algebra from a semilattice, then construct an EALA's of type A\ which is coordinatized by the Jordan algebra, i.e. TKK algebra. Let S be a set in a v—dimensional Euclidean space R^v(v≥ 1). A discrete subset S of R^v is called a semilattice (or lattice) in R^v if S spans R^v , and satisfies the conditions 0 ∈ S, -S = S,S+2S (?) S(or S+S (?) S). Let J = J(S) be the Jordan algebra constructed from a semilattice S of R^v(v ≥ 1) . A Lie algebra (?)(J(S)) can be obtained from the Jordan algebra by the Tits-kantor-koecher constuction([J]), i.e. (?)(J(S)):={sl₂(C) (?)J)(?) Inder(J), where Inder(J) = [L_J,L_J], and L_J is the set of multiplication operators for J. The TKK algebra (?)(J(S)) is defined to be the universal central extension of the Lie algebra (?)(J(S)). If v = 2, we call this algebra (?)(J(S)), obtained from the smallest nonlattic semilattic, the baby-TKK algebra.In [T2] the author gave a vertex representation of the baby-TKK algebrain terms of the bosonic and fermionic fields. It is known that, up to isomorphism, there are only one lattice and one nonlattice smeilattice in R2. A lattice is a semilattice. In chapter I of this thesis, we first study the structure of the TKK algebra Q(J(S)) with the lattice S = Z2 of R2. Next, by using results from [BGT], we construct a representation of this TKK algebra with only bosonic fields. Moreover, through the vertex operators decompositon and restricting the Fock space given in §1.3, we obtain two classes of representations of the baby-TKK algebra by bosonic and fermionic fields. One of the representations recovers the construction given in [T2].The free fields construction was first given by Wakimoto ([W2]) for the affine Lie algebra si 2 and in a great generality by Feigin and Frenkel ([FF]) for the affine Lie algebras sln. In [GZ], the authors gave a class of highest weight representations of the EALA's gl2(Cq) by the method of the free fields. In chapter II of this paper, by using the free fields construction, we obtain a vertex representation for the Tits-Kantor-Koecher Lie algebra determined by the lattic of R2.