Induced modules for orbifold vertex operator algebras. (English) Zbl 1002.17015

Lam, Ching Hung

J. Math. Soc. Japan 53, No. 3, 541-557 (2001).

Let \(V\) be a simple vertex operator algebra and \(G\) a finite abelian subgroup of \(\text{Aut} V\). In [C. Dong and G. Mason, Duke Math. J. 86, 305-321 (1997; Zbl 0890.17031)] it is proved that the orbifold vertex operator algebra \(V^G\) is also simple, and that \(V^\xi\), the sum of all irreducible \(G\)-modules which afford the character \(\xi\in\text{Irr} G\), is an irreducible \(V^G\)-module. In the paper under review the author assumes that \(V^G\) is rational, and that for any irreducible \(V^G\)-module \(L\) and any \(\xi\in\text{Irr} G\) the tensor product \(V^\xi\widehat\otimes_{V^G}L\) is an irreducible \(V^G\)-module. Under these assumptions the author obtains a decomposition of an irreducible \(V\)-module into a direct sum of irreducible \(V^G\)-modules. The author also defines an induced module from \(V^G\) to \(V\) and shows that every irreducible \(V\)-module is a quotient module of some induced module. In addition, it is proved that \(V\) is rational in this case.

Reviewer: Mirko Primc (Zagreb)

MSC:

17B69

Vertex operators; vertex operator algebras and related structures

Keywords:

vertex operator algebras; orbifold; induced modules

Citations:

Zbl 0890.17031