On quasicrystal Lie algebras

Abstract

We realize the aperiodic Witt and Virasoro algebras as well as other quasicrystal Lie algebras as factor algebras of some subalgebras of the higher rank Virasoro algebras. This realization allows us to generalize the notion of quasicrystal Lie algebras. In the case when the constructed algebra admits a conjugation, we compute the Kac determinant for the Shapovalov form on the corresponding Verma modules. In the case of the aperiodic Virasoro algebra this proves the conjecture of R. Twarock.