# Quantum Klein Space and Superspace

## TL;DR

This paper develops an algebraic quantum group-based framework for quantizing complex Minkowski space and its real forms, including the Kleinian signature, and extends these constructions to superspaces.

## Contribution

It introduces a novel algebraic quantization of Minkowski space and its real forms within the quantum groups framework, including supersymmetric extensions.

## Key findings

- Constructed quantum metrics for various signatures.
- Provided explicit presentations of quantized coordinate algebras.
- Extended quantizations to $
abla=1$ superspaces.

## Abstract

We give an algebraic quantization, in the sense of quantum groups, of the complex Minkowski space, and we examine the real forms corresponding to the signatures $(3,1)$, $(2,2)$, $(4,0)$, constructing the corresponding quantum metrics and providing an explicit presentation of the quantized coordinate algebras. In particular, we focus on the Kleinian signature $(2,2)$. The quantizations of the complex and real spaces come together with a coaction of the quantizations of the respective symmetry groups. We also extend such quantizations to the $\mathcal{N}=1$ supersetting.