Extension of vertex operator algebras by a self-dual simple module
Let \(V\) be a regular vertex operator algebra and \((M, Y_M(\cdot,z))\) a \(V\)-module. By the use of a certain weight-one primary semisimple element \(h\in V\) one may change the vertex operators into \(Y_M(\Delta (h,z)\cdot,z)\) and obtain another \(V\)-module \(\tilde M\) [cf. \textit{H.-S. Li}, The physical superselection principle in vertex operator algebra theory, preprint]. In the paper under review the author shows that if \(\tilde V\) is self-dual, then \(\overline V=V\oplus\tilde V\) is a vertex operator algebra (or superalgebra) and that \(M\oplus \tilde M\) is a (possibly twisted) \(\overline V\)-module. The constructed vertex operator algebra \(\overline V\) is regular; this follows from a somewhat more general theorem ensuring that an extended vertex operator algebra \(\overline V=V^0\oplus V^1\) of a regular vertex operator algebra \(V^0\) by a simple current \(V^0\)-module \(V^1\) is regular. As an illustrative example the author considers the extension \(\overline V\) of the vertex operator algebra \(V=L(\ell,0)\), \(\ell\in 2\mathbb{N}\), associated to the affine Lie algebra \(A^{(1)}_1\).
scientific article; zbMATH DE number 1014041
scientific article