Automorphisms and Verma modules for Generalized Schr\"{o}dinger-Virasoro algebras

TL;DR

This paper studies the automorphisms and Verma modules of a class of infinite-dimensional Lie algebras called generalized Schr"{o}dinger-Virasoro algebras, providing a complete classification of their automorphism groups and module irreducibility.

Contribution

It completely determines the automorphism groups and the irreducibility conditions of Verma modules for generalized Schr"{o}dinger-Virasoro algebras, a new class of infinite-dimensional Lie algebras.

Findings

Automorphism groups of the algebras are fully characterized.

Irreducibility conditions for Verma modules are explicitly described.

The structure of generalized Schr"{o}dinger-Virasoro algebras is clarified.

Abstract

Let $\mathbb{F}$ be a field of characteristic 0, $G$ an additive subgroup of $\mathbb{F}$, $\alpha\in \mathbb{F}$ satisfying $\alpha\notin G, 2\alpha\in G$. We define a class of infinite-dimensional Lie algebras which are called generalized Schr\"{o}dinger-Virasoro algebras and use $\mathfrak{gsv}[G,\alpha]$ to denote the one corresponding to $G$ and $\alpha$. In this paper the automorphism group and irreducibility of Verma modules for $\mathfrak{gsv}[G,\alpha]$ are completely determined.