Vertex Representations Of Quantum N-toroidal Algebras For Types F₄ And A₁

Recently,Gao-Jing-Xia-Zhang[13]defined the structures of quantum N-toroidal alge-bras uniformally,which are a kind of natural generalizations of the classical quantum toroidal algebras,just like the relation between 2-toroidal Lie algebras and N-toroidal Lie algebras Based on this work,the paper will construct a level-one vertex representation of quantum N-toroidal algebras for type F4.In particular,quantum affine algebras and quantum toroidal al-gebras both are special subalgebras of quantum N-toroidal algebras.Thus,we will also obtain level-one vertex representations of them.Besides,we will also construct a higher level vertex representation of quantum N-toroidal algebras for type Al.The dissertation is organized as follows:In Chapter 1,we will mainly introduce the background of Lie algebras,the mathematical symbols needed in this paper and their corresponding meanings.In Chapter 2,we will emphat-ically review the relevant knowledge of Lie algebras to better understand this paper,including the basic concepts of Lie algebras,Kac-Moody algebras,toroidal Lie algebras,quantum group-s,quantum affine algebras and so on.For requirements of studying further,we will recall the related knowledge of simple Lie algebras g,affine Lie algebras g and quantum affine algebras Uq(g)of type F4 in Chapter 2In the first part of Chapter 3,we will introduce the concept of quantum N-toroidal alge-bras firstly.Then we will construct the Fock space,introduce "normal order",a special class of quasi-cocycles and several important vertex operators.Thus we will obtain the theorem of a level-one vertex representation of quantum N-toroidal algebras for type F4.In the second part,we will check the above main theorem in detailsIn Chapter 4,we will construct a level-two vertex representation of quantum N-toroidal algebras for Al.Similarly,we give the Fock space,vertex operators,etc.At last,we will check the above construction is the vertex representation of quantum N-toroidal algebras for Al.As a special case,we will also get a level-two vertex representation of quantum toroidal algebras of A1,which is a natural generalization of quantum affine algebras of A1.