# The $\kappa$-(A)dS noncommutative spacetime

## TL;DR

This paper constructs the $
abla$-(A)dS noncommutative spacetime by quantizing its semiclassical Poisson structure, revealing a quantum sphere structure and extending $
abla$-Minkowski spacetime to include a cosmological constant.

## Contribution

It provides the first explicit construction of the $
abla$-(A)dS noncommutative spacetime, generalizing $
abla$-Minkowski to non-zero cosmological constant with a unique quantization approach.

## Key findings

- The $
abla$-(A)dS noncommutative spacetime is the only such generalization with a primitive time translation generator.
- It features a quadratic subalgebra of spatial coordinates forming a quantum sphere.
- The spacetime can be represented as a noncommutative pseudosphere in ambient coordinates.

## Abstract

The (3+1)-dimensional $\kappa$-(A)dS noncommutative spacetime is explicitly constructed by quantizing its semiclassical counterpart, which is the $\kappa$-(A)dS Poisson homogeneous space. This turns out to be the only possible generalization of the well-known $\kappa$-Minkowski spacetime to the case of non-vanishing cosmological constant, under the condition that the time translation generator of the corresponding quantum (A)dS algebra is primitive. Moreover, the $\kappa$-(A)dS noncommutative spacetime is shown to have a quadratic subalgebra of local spatial coordinates whose first-order brackets in terms of the cosmological constant parameter define a quantum sphere, while the commutators between time and space coordinates preserve the same structure of the $\kappa$-Minkowski spacetime. When expressed in ambient coordinates, the quantum $\kappa$-(A)dS spacetime is shown to be defined as a noncommutative pseudosphere.