---
abstract: 'We define new noncommutative spheres with partial commutation relations for the coordinates. We investigate the quantum groups acting maximally on them, which yields new quantum versions of the orthogonal group: They are partially commutative in a way such that they do *not* interpolate between the classical and the free quantum versions of the orthogonal group. Likewise we define non-interpolating, partially commutative quantum versions of the symmetric group recovering Bichon’s quantum automorphism groups of graphs. They fit with the mixture of classical and free independence as recently defined by Speicher and Wysoczanski (rediscovering $\Lambda$-freeness of Mlotkowski), due to some weakened version of a de Finetti theorem.'
address: 'Saarland University, Fachbereich Mathematik, Postfach 151150, 66041 Saarbrücken, Germany'
author:
- Roland Speicher
- Moritz Weber
bibliography:
- 'BibFileMixedQG.bib'
nocite: '[@*]'
title: Quantum groups with partial commutation relations
---

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Introduction
============

Motivated by the recent preprint on mixtures of classical and free indepence by Wysoczanski and the first author [@SpW] – where the notion of and results on $\Lambda$-freeness of Mlotkowski [@Mlo] were rediscovered – we ask for the corresponding quantum symmetries. The mixture of independences goes as follows. Let ${\varepsilon}=({\varepsilon}_{ij})_{i,j\in \{1,\ldots,n\}}$ 