Irreducible modules over the universal central extension of the planar Galilean conformal algebra

TL;DR

This paper explores the representation theory of the universal central extension of the planar Galilean conformal algebra, constructing and classifying various irreducible modules, including Whittaker modules and tensor product modules, in a (2+1)-dimensional setting.

Contribution

It introduces new classes of irreducible modules over the universal central extension of the planar Galilean conformal algebra and provides criteria for their irreducibility and isomorphism classes.

Findings

Constructed a family of irreducible Whittaker modules over the algebra.

Established necessary and sufficient conditions for irreducibility of these modules.

Determined the isomorphism classes of certain tensor product modules.

Abstract

In this paper, we study the representation theory of the universal central extension $\mathcal{G}$ of the infinite-dimensional Galilean conformal algebra, introduced by Bagchi-Gopakumar, in $(2+1)$ dimensional space-time, which was named the planar Galilean conformal algebra by Aizawa. More precisely, we construct a family of Whittaker modules $W_{\psi_{m,n}}$ over $\mathcal{G}$ while the necessary and sufficient conditions for these modules to be irreducible are given when $m\in\mathbb{Z}_+, n\in\mathbb{N}$ and $m,n$ have the same parity. Moreover, the irreducible criteria of the tensor product modules $\Omega(\lambda,\eta,\sigma,0)\otimes R, \Omega (\lambda,\eta,0,\sigma)\otimes R$ over $\mathcal{G}$ are obtained, where $\Omega(\lambda,\eta,\sigma,0), \Omega (\lambda,\eta,0,\sigma)$ are $\mathcal{U}(\mathfrak{h})$-free modules of rank one over $\mathcal{G}$ and $R$ is an irreducible restricted module over $\mathcal{G}$. Also, the isomorphism classes of these tensor product modules are determined.