3-dimensional $\Lambda$-BMS Symmetry and its Deformations

TL;DR

This paper classifies and constructs quantum group deformations of the infinite-dimensional $ ext{Lambda}$-BMS symmetry algebra in 3D asymptotically AdS spacetimes, exploring their mathematical structures and potential physical implications.

Contribution

It provides a complete classification of Lie bialgebra structures and explicit Hopf algebra constructions for the $ ext{Lambda}$-BMS algebra, extending known finite-dimensional results to the infinite-dimensional case.

Findings

All possible Lie bialgebra structures are classified.

Explicit Hopf algebra deformations are constructed for selected cases.

Some $ ext{kappa}$-Poincaré deformations cannot be extended to the infinite-dimensional algebra.

Abstract

In this paper we study quantum group deformations of the infinite dimensional symmetry algebra of asymptotically AdS spacetimes in three dimensions. Building on previous results in the finite dimensional subalgebras we classify all possible Lie bialgebra structures and for selected examples, we explicitly construct the related Hopf algebras. Using cohomological arguments we show that this construction can always be performed by a so-called twist deformation. The resulting structures can be compared to the well-known $\kappa$-Poincar\'e Hopf algebras constructed on the finite dimensional Poincar\'e or (anti) de Sitter algebra. The dual $\kappa$ Minkowski spacetime is supposed to describe a specific non-commutative geometry. Importantly, we find that some incarnations of the $\kappa$-Poincar\'e can not be extended consistently to the infinite dimensional algebras. Furthermore, certain deformations can have potential physical applications if subalgebras are considered. Since the conserved charges associated with asymptotic symmetries in 3-dimensional form a centrally extended algebra we also discuss briefly deformations of such algebras. The presence of the full symmetry algebra might have observable consequences that could be used to rule out these deformations. }