Modules over the planar Galilean conformal algebra arising from free modules of rank one

Jin Cheng, Dongfang Gao and Ziting Zeng

Vol. 326 (2023), No. 2, 227–249

DOI: 10.2140/pjm.2023.326.227

Abstract

The planar Galilean conformal algebra $\mathcal{𝒢}$ introduced by Bagchi-Gopakumar and Aizawa is an infinite-dimensional extension of the finite-dimensional Galilean conformal algebra in $(2+1)$-dimensional space-time. In this paper, we give a complete classification of $\mathcal{𝒰}(ℂL_0)$-free modules of rank 1 and $\mathcal{𝒰}(\mathfrak{𝔥})$-free modules of rank 1 over $\mathcal{𝒢}$, where $\mathfrak{𝔥}$ is the Cartan subalgebra (a nilpotent self-normalizing subalgebra) of $\mathcal{𝒢}$, $ℂL_0$ is a subalgebra of $\mathfrak{𝔥}$. Also, we determine the necessary and sufficient conditions for these modules to be irreducible, and find the maximal proper submodules when these modules are not irreducible.

Keywords

planar Galilean conformal algebra, free module, irreducible module, isomorphism

Mathematical Subject Classification

Primary: 17B10, 17B65, 17B66, 17B68

Milestones

Received: 26 March 2023

Revised: 21 November 2023

Accepted: 5 December 2023

Published: 9 January 2024