[Paper Review] On the classification of easy quantum groups - The nonhyperoctahedral and the half-liberated case

Moritz Weber|arXiv (Cornell University)|Jan 23, 2012

Algebraic structures and combinatorial models9 citations

TL;DR

This paper extends the classification of easy quantum groups by introducing new examples and providing a complete classification in the half-liberated case. Building on Banica and Speicher's framework, it shows that the six known easy groups split into seven free easy quantum groups, with a full characterization of the half-liberated case through partition-induced intertwiner spaces.

ABSTRACT

In 2009, Banica and Speicher began to study the compact quantum subgroups of the free orthogonal quantum group containing the symmetric group S_n. They focused on those whose intertwiner spaces are induced by some partitions. These so-called easy quantum groups have a deep connection to combinatorics. We continue their work on classifying these objects introducing some new examples of easy quantum groups. In particular, we show that the six easy groups O_n, S_n, H_n, B_n, S_n' and B_n' split into seven cases on the side of free easy quantum groups. Also, we give a complete classification in the half-liberated case.

Motivation & Objective

• To extend the classification of easy quantum groups beyond the known cases.
• To identify and construct new examples of easy quantum groups.
• To provide a complete classification of easy quantum groups in the half-liberated setting.
• To clarify the structural relationship between the six known groups and their free/half-liberated counterparts.

Proposed method

• Using partition categories to model intertwiner spaces of compact quantum groups.
• Applying combinatorial techniques to classify quantum groups based on partition rules.
• Extending the framework of Banica and Speicher to include half-liberated quantum groups.
• Analyzing the structure of partition categories to determine when they define quantum groups.
• Establishing a correspondence between partition classes and quantum group families.
• Verifying that the resulting quantum groups are indeed compact and easy via representation-theoretic constraints.

Experimental results

Research questions

• RQ1 How do the six known easy quantum groups—O_n, S_n, H_n, B_n, S_n', and B_n'—relate to the free easy quantum groups?
• RQ2 What new examples of easy quantum groups can be constructed using partition-based intertwiners?
• RQ3 How does the half-liberated case differ from the fully free and classical cases in terms of classification?
• RQ4 Can a complete classification be achieved for the half-liberated quantum groups?
• RQ5 What structural properties distinguish the seven resulting free easy quantum groups from one another?

Key findings

• The six known easy quantum groups O_n, S_n, H_n, B_n, S_n', and B_n' are shown to split into seven distinct free easy quantum groups.
• A complete classification is established for the half-liberated case of easy quantum groups.
• New examples of easy quantum groups are constructed through partition-induced intertwiner spaces.
• The classification reveals a refined structural hierarchy among the quantum groups based on their partition categories.
• The half-liberated case is fully characterized, showing that it forms a distinct and complete class within the easy quantum group framework.
• The results confirm and extend the combinatorial classification program initiated by Banica and Speicher.