Drinfel’d Realizations Of Quantum N-loop Algebras

Quantum algebra is one of the most important research directions in quantum group theory,and it is an important generaliztion of Kac-Moody algebra.Its structure and representation theory are very rich and profound.Recently,Gao-Jing-Xia-Zhang defined the general structure of quantum N-toroidal algebras[12].This work is a natural generalization of classical quantum toroidal algebras,just like the relation between 2-toroidal Lie algebras and Ntoroidal Lie algebras.At the same time,Gao-Jing-Xia-Zhang also discovered the relationship between quantum N-toroidal algebra and generalized quantum GIM algebra.In their followup research work,the vertex representation theory of quantum N-toroidal algebra also was constructed.Based on these works,this article first gives the definition of quantum N-Loop algebra,which is a subalgebra of decentralized quantum N-toroidal algebra,and can also be regarded as the N-fold affineization of quantum groups.By using quantum a set of minimal generators and generating relations of quantum N-Loop algebra,we define an associative algebra u0.Meanwhile,we discuss the relationship between the associative algebra u0 and the quantum N-loop algebra(?)·It provides a basis for subsequent research on the finitedimensional representation of quantum N-toroidal algebra.The specific work arrangements are as follows:The first chapter introduces the research background of this article and the mathematical symbols needed in the article.The second chapter is mainly to provide the relevant knowledge base for the full text,the definition and basic properties of simple Lie algebra,Kac-Moody algebra,quantum group,quantum toroidal algebra and quantum N-loop algebra are considered.The third chapter first gives the definition of quantum N-loop algebra(?)and then discusses the properties of(?).Since that the quantum N-loop algebra(?)has infinite generating elements and infinite generating relations,which means that its structure is very complicated and that brings certain difficulties to the follow-up research.This paper gives a set of finite generators {x±(0),x±((?)es)(s∈J),K±1} of(?)and the finitely generated relationships that relate only to these generators.That is,associative algebra u0 are subsets of the generators and relations of the quantum N-loop algebra(?),respectively.The main content of this paper discusses the relation between the associative algebra up and the quantum N-1oop algebra(?).Chapter 4 first naturally defines a natural mappingσ from u0 to(?).At the same time,it is also proved that the mapping is an algebraic epimorphism from u0 to(?).Therefore,the quantum N-loop algebra(?)is isomorphic to a quotient algebra of the associative algebra u0,namely(?)/ker σ.In the fifth Chapter,the ideal ker σ corresponding to u0 will be further discussed.Considering the complexity of the ideal ker σ,some elements of ker σ have been found in the paper,which provides the basis for the subsequent research.