On quantum deformations of (anti-)de Sitter algebras in (2+1) dimensions

TL;DR

This paper reviews quantum deformations of (2+1) and (3+1) dimensional (A)dS algebras, classifies their Lie bialgebras, and introduces generalized ppa-deformations with multiple parameters, connecting these structures to quantum spacetimes.

Contribution

It provides a comprehensive classification of (2+1) (A)dS Lie bialgebras and introduces multi-parameter ppa-deformations, extending previous models to higher dimensions.

Findings

Classification of (2+1) (A)dS Lie bialgebras

Introduction of three-parameter ppa-deformation for (2+1) (A)dS

Two-parameter ppa-deformation preserving isotropy in (3+1) dimensions

Abstract

Quantum deformations of (anti-)de Sitter algebras in (2+1) dimensions are revisited, and several features of these quantum structures are reviewed. In particular, the classification problem of (2+1) (A)dS Lie bialgebras is presented and the associated noncommutative quantum (A)dS spaces are also analysed. Moreover, the flat limit (or vanishing cosmological constant) of all these structures leading to (2+1) quantum Poincar\'e algebras and groups is simultaneously given by considering the cosmological constant as an explicit Lie algebra parameter in the (A)dS algebras. By making use of this classification, a three-parameter generalization of the \kappa-deformation for the (2+1) (A)dS algebras and quantum spacetimes is given. Finally, the same problem is studied in (3+1) dimensions, where a two-parameter generalization of the \kappa-(A)dS deformation that preserves the space isotropy is found.