Irreducible modules over the Virasoro algebra
The paper under review contributes with two new classes of examples of simple modules over the Virasoro Lie algebra \(\mathrm{Vir}\). This Lie algebras has a basis indexed by integers (together with an additional central element). The first class consists of simple Whittaker modules for the Whittaker pair \((\mathrm{Vir},\mathrm{Vir}_{\geq m})\) (in the sense of \textit{P. Batra} and \textit{V. Mazorchuk}, J. Pure Appl. Algebra 215, No. 7, 1552--1568 (2011; Zbl 1228.17008)]). Here \(\mathrm{Vir}_{\geq m}\), \(m\in\mathbb{N}\), is a subalgebra generated by all basis vectors corresponding to indices \(\geq m\). The authors show that in most case the (centrally reduced) universal Whittaker module is simple. An explicit criterion for simplicity of the latter module is given. The second class is obtained using the following trick: the authors realize the Virasoro algebra as a subalgebra of the Heisenberg-Virasoro Lie algebra. The latter algebra has a natural family of modules of the intermediate series (weight modules with one-dimensional weight spaces). Furthermore, the Heisenberg-Virasoro algebra has many more endomorphisms than the Virasoro algebra. So, the authors take a simple module of the intermediate series over the Heisenberg-Virasoro algebra, twist it by an endomorphism, and then restrict back to the Virasoro algebra. In most cases this gives a simple Virasoro module and the authors determine isomorphism classes of the modules which can be obtained in this way.
scientific article; zbMATH DE number 5987892
scientific article