Galilean quantum gravity with cosmological constant and the extended q-Heisenberg algebra

TL;DR

This paper develops a Galilean gravity theory in 2+1 dimensions with a cosmological constant, utilizing Chern-Simons gauge theory and quantum algebra structures, advancing the understanding of non-relativistic quantum gravity models.

Contribution

It introduces a novel Galilean gravity framework with a cosmological constant based on Chern-Simons theory and explores its quantum algebraic structure, extending previous classical and quantum studies.

Findings

Identifies an r-matrix compatible with the Chern-Simons action.

Shows the bi-algebra structure as the classical double of the extended Heisenberg algebra.

Quantum theory is largely determined by the quantum double of the extended q-deformed Heisenberg algebra.

Abstract

We define a theory of Galilean gravity in 2+1 dimensions with cosmological constant as a Chern-Simons gauge theory of the doubly-extended Newton-Hooke group, extending our previous study of classical and quantum gravity in 2+1 dimensions in the Galilean limit. We exhibit an r-matrix which is compatible with our Chern-Simons action (in a sense to be defined) and show that the associated bi-algebra structure of the Newton-Hooke Lie algebra is that of the classical double of the extended Heisenberg algebra. We deduce that, in the quantisation of the theory according to the combinatorial quantisation programme, much of the quantum theory is determined by the quantum double of the extended q-deformed Heisenberg algebra.