The Generalized Vertex Algebras From Affine Lie Algebra With Short Roots

Frankel and Kac[1,9,10]and Segal[11] had constructed the level-one representationsof a?ne Kac-Moody algebras A(n1),D_n⁽¹⁾,E₆⁽¹⁾,E7_<sup>(1),E8_<sup>(1)by means of vertex operators in1981. Goddard-Nahm-Olive-Schwimmer[3] gave the same representations for the casesB_n⁽¹⁾,C_n⁽¹⁾,G₂⁽¹⁾,F₄⁽¹⁾in 1986. In addition, Xu and Jiang [4] have introduced another setof vertex operators in 1990 which are constructed for the level-one representations of thecases B_n⁽¹⁾and G₂⁽¹⁾.Xu[5] gave the level-one representations of the a?ne Lie algebras withfirst kind in 1991. Kac[12] gave the lattice vertex algebra by the representations of a?neLie algebras and solved the case of A_n⁽¹⁾,D_n⁽¹⁾,E₆⁽¹⁾,E₇⁽¹⁾,E₈⁽¹⁾by lattice vertex algebra ,i.e, there are only long roots(p=1).In this paper we consider the case of short roots. We want to use the vertex operatorsof a?ne Lie Algebras with first kind to construct vertex algebras .We find that we can'tdirectly construct a vertex algebra by means of vertex operators of Xu[5],so we modifythe vertex operators a little.In the end we construct a kind of generalized vertex algebrawith the modified vertex operators .