On The Category Of Modules For A Vertex Operator Algebra

Vertex algebra, or vertex operator algebra (VOA,brie?y), is an important algebraicstructure in conformal field theory and statistical mechanics. and it is also the structureof Quant Fields Theories in strict. Researching the modules for a VOA is the usualmethod of studying the properties of VOA, besides that the homological algebra is thenew tool to study the algebraic structure of VOA. In this thesis, the category of modulesfor a VOA have been discussed by using homological algebra, and some results have beengot.In chapter 3, it is shown that all modules for a VOA V forms a category at first,its objects are the V - modules, and its morphisms are the V - module homomorphisms.Secondly, it is shown that there exists direct sum module for 2 given V - modules, allthe morphisms for the given 2 V - modules forms an Abelian group as well as that thecomposition satisfy the bidistribution law; thus it is an additive category. Finally, it isshown that there exist kernel and cokernel for any V - module homomorphism, and thatevery monomorphism is the kernel of its cokernel, every epimorphism is the cokernel ofits kernel, therefore, it is an Abelian category. In addition, it is given the condition tojudge if the V - module is projective object and it is shown that the category of modulesfor a VOA is projective enough.In chapter 4, the concept of homological group (module) of a VOA with the V -module coe?cients is introduced, and the low order (0,1) homological group of a VOAassociated with the 1-rank positive definite lattice is calculated out as an example. theprojective resolution of trivial module C is introduced by using the results of chapter 3 aswell as the derived functor of Hom functor, hence, the concept of the homological group ofa VOA with the coe?cients in a given module is introduced. Moreover, the homologicalgroup of a VOA is defined, and it is shown that the homological groups of this VOA areisomorphism to the homological groups of a Lie algebra respectively by using the resultsgiven by E. Frankel in [49]. Finally, the Lie bracket relations of the Lie algebra attachedto the VOA associated with a 1-rank even positive definite lattice are calculated out, andtherefore the low-order (0,1) homological groups of the above VOA are calculated out.