Galilean $W_3$ algebra

TL;DR

This paper constructs and analyzes the Galilean $W_3$ algebra, proving its simplicity, establishing reducibility criteria, and providing free field realizations within a lattice VOA framework.

Contribution

It introduces the Galilean $W_3$ algebra as a universal enveloping vertex algebra and proves its simplicity, offering new insights into its structure and representations.

Findings

Proved the simplicity of $ ext{Galilean } W_3$ algebra.

Established reducibility criteria for Verma modules.

Constructed free field realizations within a rank 4 lattice VOA.

Abstract

Galilean $W_3$ vertex operator algebra $\mathcal GW_3(c_L,c_M)$ is constructed as a universal enveloping vertex algebra of certain non-linear Lie conformal algebra. It is proved that this algebra is simple by using determinant formula of the vacuum module. Reducibility criterion for Verma modules is given, and the existence of subsingular vectors demonstrated. Free field realisation of $\mathcal GW_3(c_L,c_M)$ and its highest weight modules is obtained within a rank 4 lattice VOA.