| The current study aims to explore the integral forms of the lattice vertex operator algebras.The essence of the integral forms in a general algebra is to choose a basis such that the multiplication of any two elements in the basis is a linear combination of integral coefficients of the basis.For example,the structure constants of Chevalley generators of simple Lie algebras are integers,and the integrality of Jacobi identity in vertex operator algebras,this makes it is possible to discuss integral forms on simple Lie algebras and vertex operator algebras.Kostant further gave the properties of integral forms of universal enveloping algebras on finite dimensional Lie algebras,the base elements are generated by the product of divided powers of the Chevalley generators.In the 1980s,Garland extended the Kostant Z-form to the enveloping algebras of affine algebras,and proved that the integral forms are preserved by the divided powers of generators.The work of Kostant and Garland on Z-form have played a significant role in the development of the theory of finite and infinite dimensional Lie algebras respectively.In this paper,we will extend their work and prove that the Schur Z-form on the lattice vertex algebras are also preserved by the divided powers of general vertex operators,which provides a possible way to further study the corresponding groups of vertex algebras.In this dissertation we study the action of the divided powers of the general vertex operators of even lattice vertex operator algebras VL on their integral forms(VL)Z,and finally come to the conclusion that the integral forms of vertex operator algebras associated to even lattices are preserved by the divided powers of Y(v,x).The vertex operator algebras associated to even lattices consists of two parts:group algebras C{L} and symmetric algebras S((?)+).In chapter 3,We study the action of the divided powers of vertex operators corresponding to these two parts on integral forms(VL)Z of even lattice vertex operator algebras,and conclude that the integral forms of vertex operator algebras associated to even lattices are preserved by the divided powers of these vertex operators.In chapter 4,we further investigate the action of the divided powers of vertex operators on the integral forms of lattice vertex operator algebras and its irreducible modules,and conclude that the integral forms of vertex operator algebras associated to even lattices and its irreducible module are preserved by the divided powers of vertex operators. |
