Integral Basis For Vertex Algebras And The Universal Enveloping Algebras Of Some Lie Algebras

Vertex algebras are introduced by Borcherds.They are very important in some areas of Mathematics,such as the representation theory of infinite-dimensional Lie algebras,al-gebraic geometry,the theory of finite groups,integrable systems,and modular functions,etc.The theory of vertex algebras also serves as the rigorous mathematical foundation for two-dimensional conformal field theory and string theory,extensively studied by physi-cists.While they are usually defined over any field of characteristic zero,the coefficients of the most important Jacobi identity make sense in any commutative ring,so it is natural to consider the vertex algebras over Z.Similar with using Chevalley bases to create Lie algebras over Z,one can use integral basis to create the integral form of vertex algebras.Given an integral form of a vertex algebra,one can also construct vertex algebras over fields of prime characteristic p.In the theory of vertex algebras,the most important vertex algebras are Virasoro vertex algebras,affine vertex algebras and lattice vertex algebras.In this paper,we are going to construct the integral forms for them when the level and central charge are some special values.Prom the construction of these vertex algebras,we can see that their structures have a lot to do with the structure of the universal enveloping algebras of the corresponding Lie algebras.So to get their integral form,we need to consider the integral form of these universal enveloping algebras.For the simplest vertex operator algebra-Virasoro vertex algebra VVir(2k,0).We take the level as 2k to create a Chevalley-type basis for Virasoro algebra,and then construct the integral basis for Virasoro vertex algebra VVir(2k,0).For affine vertex algebra Vk(sl2).There are two ways to get the integral form for U(sl2):One is given by the normal ordered basis;The other is given by generators.Accordingly,when k is integral,the integral form of Vk(sl2)can also be created by two ways.Since the integral forms for VVir(2k,0)and Vk(sI2)are provided,we can provide integral form for their tensor product,i.e.,tensor product of their integral form.Besides,we consider the integral form for the universal enveloping algebra U(C)of affine-Virasoro algebra L,which is the algebraic combination of affine algebra sl2 and Virasoro Lie algebra L.By the generating relations of L,we can easily get its Chevalley-type basis,then we construct the integral basis for U(L).Finally,we take lattice vertex algebra VZ? into consideration.It is constructed based on Heisenberg vertex algebra M(1).When the level is 1,we first find an integral basis for M(1)as a vector space,and then use its relationship with VZ? to construct an integral basis for VZ? and a generating set of the integral form in Vza.