# $\kappa$-de Sitter and $\kappa$-Poincar\'e symmetries emerging from Chern-Simons (2+1)D gravity with a cosmological constant

## TL;DR

This paper demonstrates how $ppa$-de Sitter and $ppa$-Poincare9 symmetries emerge as quantum deformations in (2+1)D gravity with a cosmological constant, revealing associated non-commutative spacetimes.

## Contribution

It introduces a new r-matrix compatible with the Chern-Simons action, deriving deformed symmetries and non-commutative spacetime structures in three-dimensional gravity.

## Key findings

- Deformed $ppa$-de Sitter and $ppa$-Poincare9 symmetries derived from Chern-Simons gravity.
- Explicit form of the non-commutative spacetime associated with these symmetries.
- Symmetries reduce to $ppa$-Poincare9 in the zero cosmological constant limit.

## Abstract

Defining a new r-matrix compatible with the scalar product at the basis of the Chern-Simons action for a particle coupled to (2+1) Lorentzian gravity with cosmological constant, I show how deformed symmetries of $\kappa$-de Sitter and, in the vanishing cosmological limit, of $\kappa$-Poincar\'e kind, arise naturally as quantum-deformation of three dimensional gravity. I obtain moreover the non-commutative spacetime associated to these kinds of symmetries.