arXiv subjects
Publications and source records attributed to Yan-Jun Chu.
Analogue to commutants in the theory of associative algebras, one can construct a new subalgebra of vertex algebra known as a vertex algebra commutant. In this paper, for the adjoint representation $V$ of Lie algebra $sl(2,\C)$, we describe a commutant of $βγ$- System $S(V)$ by giving its generators, moreover, we get a new Howe pair of vertex algebras.
Let $L_{D_8}(1, 0)$ and $L_{E_8}(1, 0)$ be the simple vertex operator algebras associated to untwisted affine Lie algebra $\widehat{\mathbf g}_{D_{8}}$ and $\widehat{\mathbf g}_{E_8}$ 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 $L_{D_8}(1, 0)$ can embed into $L_{E_8}(1, 0)$ as a vertex operator subalgebra(\cite{5, 6, 8}). Their construct is a base of vertex operator theory. But the embedding they gave using the fact $L_{\mathbf g}(1,0)$ is isomorphic to its root lattice vertex operator algebra $V_L$. In this paper, we give an explicitly construction of the embedding and show that as an $L_{D_8}(1, 0)$-module, $L_{E_8}(1, 0)$ is isomorphic to the extension of $L_{D_8}(1, 0)$ by its simple module $L_{D_8}(1, \overlineω_8)$. It may be convenient to be used for conformal field theory.