# Quantum groups under very strong axioms

## TL;DR

This paper classifies intermediate quantum groups between $H_N$ and $U_N^+$ under very strong axioms, showing only eight basic solutions exist when assuming easiness, uniformity, and geometric coherence.

## Contribution

It proves that under the strongest axioms, the known basic quantum groups are the only solutions in the intermediate range.

## Key findings

- Only eight solutions under strong axioms.
- Classification of intermediate quantum groups.
- Use of intersection and generation operations.

## Abstract

We study the intermediate quantum groups $H_N\subset G\subset U_N^+$. The basic examples are $H_N,K_N,O_N,U_N,H_N^+,K_N^+,O_N^+,U_N^+$, which form a cube. Any other example $G$ sits inside the cube, and by using standard operations, namely intersection $\cap$ and generation $<\,,>$, can be projected on the faces and edges. We prove that under the strongest possible axioms, namely (1) easiness, (2) uniformity, and (3) geometric coherence of the various projection operations, the 8 basic solutions are the only ones.