# On the representation theory of the Bondi-Metzner-Sachs group and its variants in three space-time dimensions

## TL;DR

This paper develops an extended representation theory for the BMS group and its variants in three-dimensional space-time, crucial for understanding asymptotic symmetries in General Relativity.

## Contribution

It introduces an extended Wigner-Mackey theory to classify irreducible unitary representations of BMS-like groups in 3D, including non-locally compact cases.

## Key findings

- Irreducible representations are induced from compact, cyclic little groups of even order.
- The extended theory applies to groups in any space-time dimension d ≥ 3.
- The classification aids in understanding asymptotic symmetries in gravitational theories.

## Abstract

The original Bondi-Metzner-Sachs (BMS) group B is the common asymptotic symmetry group of all asymptotically flat Lorentzian radiating 4-dim space-times. As such, B is the best candidate for the universal symmetry group of General Relativity (G.R.). In 1973, with this motivation, P. J. McCarthy classified all relativistic B-invariant-systems in terms of strongly continuous irreducible unitary repesentations (IRS) of B. Here we introduce the analogue B(2,1) of the BMS group B in 3 space-time dimensions. B(2,1) itself admits thirty-four analogues both real in all signatures and in complex space-times. In order to find the IRS of both B(2,1) and its analogues we need to extend Wigner-Mackey's theory of induced representations. The necessary extension is described and is reduced to the solution of three problems. These problems are solved in the case where B(2,1) and its analogues are equipped with the Hilbert topology. The extended theory is necessary in order to construct the IRS of both B and its analogues in any number d of space-time dimensions, d is greater or equal to 3, and also in order to construct the IRS of their supersymmetric counterparts. We use the extended theory to obtain the necessary data in order to construct the IRS of B(2,1): The main results of the representation theory are: The IRS are induced from little groups which are compact. The finite little groups are cyclic groups of even order. The inducing construction is exhaustive notwithstanding the fact that B(2,1) is not locally compact in the employed Hilbert topology.