9 Vertex Operator Algebra Analogue of Embedding D 8 into E 8

Let LD8(1, 0) and LE8(1, 0) be the simple vertex operator algebras associated to untwisted affine Lie algebra ĝD8 and ĝE8 with level 1 respectively. In the 1980s by I. Frenkel, Lepowsky and Meurman as one of the many important preliminary steps toward their construction of the moonshine module vertex operator algebra, they use roots lattice showing that LD8(1, 0) can embed into LE8(1, 0) as a vertex operator subalgebra([5, 6, 8]). Their construct is a base of vertex operator theory. But the embedding they gave using the fact Lg(1, 0) is isomorphic to its root lattice vertex operator algebra VL. In this paper, we give an explicitly construction of the embedding and show that as an LD8(1, 0)-module, LE8(1, 0) is isomorphic to the extension of LD8(1, 0) by its simple module LD8(1, ω8). It may be convenient to be used for conformal field theory.