Realization of level two integrable highest weight representations of the affine Lie algebra A(,7)('(2))

The representation theory of affine Lie algebras has been a very important area of research during the last two decades because of its interactions with several other areas of mathematics and physics. One of the crucial features of affine Lie algebra representation theory is the existence of explicit constructions of certain level one representations in terms of vertex operators. The level of the representation is the number by which the canonical central element of the affine Lie algebra acts. One way to give explicit constructions of higher level representations is to realize it inside the tensor product of known level one representations to obtain the desired representations. It is known that the principal character of the integrable highest weight representations of an affine Lie algebra has an infinite product of Rogers-Ramanujan type. In 1982, using their vertex operator realization of level one representations, Lepowsky and Wilson gave a Lie theoretic proof of the famous Rogers-Ramanujan identities. Since then, the combinatorial identities of Rogers-Ramanujan type have been used by several researchers to give explicit realizations of certain integrable highest weight representations of affine Lie algebras. In most cases the realizations are given in terms of the {dollar}{lcub}cal Z{rcub}{dollar}-algebras. It was shown by Bos that the product side of the Rogers-Ramanujan identities only occur in certain low level representations of the affine Lie algebras {dollar}Asbsp{lcub}1{rcub}{lcub}(1){rcub}, Asbsp{lcub}2{rcub}{lcub}(1){rcub}, Asbsp{lcub}2{rcub}{lcub}(2){rcub}, Csbsp{lcub}3{rcub}{lcub}(1){rcub}, Fsbsp{lcub}4{rcub}{lcub}(1){rcub}, Gsbsp{lcub}2{rcub}{lcub}(1){rcub},{dollar} and {dollar}Asbsp{lcub}7{rcub}{lcub}(2){rcub}.{dollar} These representations have already been constructed. The last of these to be constructed were the level two representations {dollar}L(Lambdasb0 + Lambdasb1){dollar} and {dollar}L(Lambdasb3){dollar} for {dollar}Asbsp{lcub}7{rcub}{lcub}(2){rcub}.{dollar}; In this thesis we give the realizations of the remaining level two representations, {dollar}L(2Lambdasb0), L(Lambdasb2),{dollar} and {dollar}L(Lambdasb4),{dollar} for {dollar}Asbsp{lcub}7{rcub}{lcub}(2){rcub}{dollar} whose principal characters contain the product sides of the Rogers-Ramanujan identities. Here due to the differences in the character formulas, we use a "two column" construction to obtain explicit realizations of these three modules, which are in contrast to the constructions of the representations listed above which use a "single column" construction.