# Classification of irreducible Harish-Chandra modules over gap-$p$ Virasoro algebras

## TL;DR

This paper classifies irreducible Harish-Chandra modules over gap-$p$ Virasoro algebras, showing they are either highest weight, lowest weight, or intermediate series modules, extending understanding of these algebraic structures.

## Contribution

It provides a complete classification of irreducible Harish-Chandra modules for gap-$p$ Virasoro algebras, a new class related to quantum tori and Heisenberg-Virasoro algebra.

## Key findings

- All irreducible Harish-Chandra modules are highest weight, lowest weight, or intermediate series.
- The classification extends known results from classical Virasoro algebra to gap-$p$ versions.
- The work links these modules to subalgebras isomorphic to Vir, graded by $p\,\mathbb Z$.

## Abstract

We prove that any irreducible Harish-Chandra modules for a class of Lie algebras, which we call gap-$p$ Virasoro algebras, must be a highest weight module, a lowest weight module, or a module of intermediate series.These algebras are closely related to the Heisenberg-Virasoro algebra and the algebra of derivations over a quantum torus. They also contain subalgebras which are isomorphic to the Virasoro algebra $Vir$, but graded by $p\mathbb Z$(unlike $Vir$ by $\mathbb Z$).