# Complex analogues of the half-classical geometry

## TL;DR

This paper explores the complex analogues of the half-classical noncommutative geometry, focusing on the relations between variables and proposing a geometry that extends classical concepts into the complex noncommutative setting.

## Contribution

It introduces and studies the complex analogues of the half-classical geometry, particularly analyzing relations between variables to identify the most natural extension.

## Key findings

- Identification of the complex analogue of the half-classical geometry.
- Analysis of relations between variables $ ext{ab}^*$ and $ ext{a}^* ext{b}$.
- Proposal of the geometry arising from these relations as the 'correct' complex extension.

## Abstract

Under very strong axioms, there is precisely one real noncommutative geometry between the classical one and the free one, namely the half-classical one, coming from the relations $abc=cba$. We discuss here the complex analogues of this geometry, notably with a study of the geometry coming from the commutation relations between all the variables $\{ab^*,a^*b\}$, that we believe to be the "correct" one.