Free Inhomogeneous Wreath Product of Quantum Groups
Josse van Dobben de Bruyn, Amaury Freslon, Prem Nigam Kar, David E. Roberson, Peter Zeman
TL;DR
The paper defines the free inhomogeneous wreath product $(\mathbb{G}_1,\dots,\mathbb{G}_m)\wr\wr_* \mathbb{H}$ as a robust noncommutative analogue of classical inhomogeneous wreath products and proves it forms a compact quantum group. It then shows how quantum automorphism groups of connected graphs can be decomposed inductively along the block-tree, reducing their computation to quantum automorphism groups of blocks and their vertex stabilizers, provided certain isomorphism/quantum-isomorphism conditions hold. This leads to concrete algorithms for computing quantum automorphism groups of forests, outerplanar graphs, and block graphs, expressed via free products, free wreath products, and the FIWP. The results unify and extend known quantum symmetry descriptions for these graph classes and provide a practical framework for algorithmic computation of quantum automorphism groups in graph-theoretic settings with structured decompositions.
Abstract
We introduce the free inhomogeneous wreath product of compact matrix quantum groups, which generalizes the free wreath product (Bichon 2004). We use this to present a general technique to determine quantum automorphism groups of connected graphs in terms of their maximal biconnected subgraphs, provided that we have sufficient information about their quantum automorphism groups. We show that this requirement is met for forests, outerplanar graphs, and block graphs leading to algorithms to compute the quantum automorphism groups of these graphs.