Level-1/2 Vertex Representation Of Quantum N-toroidal Algebra Of Type C_n

Quantum toroidal algebra(also named double affine quantum algebra)is the affinization of quantum affine algebra.In 1995,Ginzburg-Kapranov-Vasserot([12])introduced the quantum toroidal algebra of type A and gave a geometric representation of it simultaneously.Subse-quently,Varagnolo and Vasserot[36]obtained the Schur-Weyl duality between representations of the quantum toroidal algebras Uq(gtor)and elliptic Cherednik algebras in type A.Later,more and more mathematicians paid attention to the structure and the representation theory of quantum toroidal algebras and took correlative research.In this paper we also named quan-tum toroidal algebras as quantum 2-toroidal algebras.In the recent joint paper[11],quantum N-toroidal algebra was constructed as a natural generalization of quantum 2-toroidal algebra Uq(gtor)analogous to the relations of N-toroidal Lie algebra and 2-toroidal Lie algebra.We aim to realize a level-1/2 vertex representation of quantum N-toroidal algebra of type C.Of course,we will get a level-1/2 vertex representation of quantum 2-toroidal algebra as a special case and this construction could be recognized as a generalization of a level-1/2 vertex repre-sentation of quantum affinc algebra of type C[23].A sketch of the structure of this paper goes as below.In the first chapter,it is the introduction detailing the background of the concepts related to the paper,such as Lie algebra,Kac-Moody algebra,affine Lie algebra,quantum group,quantum affine algebra,quantum toroidal algebra,etc.In the second chapter,a recall of prerequisite knowledge correlated with the article will be needed.It includes the definition of Lie algebra,the representation and module of Lie alge-bra,the root system,Cartan matrix,central extension,tensor product.We will also narrate the definition and structure of affine Kac-Moody algebra,quantum group,quantum affine algebra,quantum toroidal algebra.For the later purpose,we detail the root systems of simple Lie alge-bra,affine Kac-Moody algebra and the relevant N-toroidal Lie algebra of type C.The structure of quantum affine algebra of type C will also be shown.Lastly,the symbol descriptions will be listed.In the third chapter,we review the definition of the quantum N-toroidal algebra of type C.Subsequently,We show the explicit process of constructing the basic vertex operators and the Fock space.Then we introduce the chief vertex operators,thus the main result of the paper will be given:a level-1/2 vertex representation of the quantum N-toroidal algebra of type C.As a special case,we gain a level-1/2 vertex representation of quantum toroidal algebra of type C when N=2In the final chapter,we prove the main theorem of this paper in detail.For the need of the proof,we demonstrate the operational relations of the main vertex operators.Subsequently we verify the construction in chaper three satisfies all the generating relations of the target algebra.