Unitary representations for the Schr\"{o}dinger-Virasoro Lie algebra

TL;DR

This paper classifies conjugate-linear anti-involutions and characterizes unitary Harish-Chandra modules over the Schrödinger-Virasoro algebra, showing they correspond to modules over the Virasoro algebra.

Contribution

It provides a complete classification of conjugate-linear anti-involutions and establishes that unitary Harish-Chandra modules over the Schrödinger-Virasoro algebra are equivalent to those over the Virasoro algebra.

Findings

Only two classes of conjugate-linear anti-involutions exist.

Unitary Harish-Chandra modules over Schrödinger-Virasoro are the same as those over Virasoro.

The structure of unitary modules is fully characterized.

Abstract

In this paper, conjugate-linear anti-involutions and unitary Harish-Chandra modules over the Schr\"{o}dinger-Virasoro algebra are studied. It is proved that there are only two classes conjugate-linear anti-involutions over the Schr\"{o}dinger-Virasoro algebra. The main result of this paper is that a unitary Harish-Chandra module over the Schr\"{o}dinger-Virasoro algebra is simply a unitary Harish-Chandra module over the Virasoro algebra.