On the classification of easy quantum groups

2012/01/23 by Moritz Weber, Weber, Moritz · 4 citations
Mathematics · #05E10 #16T30 #17B37 #20G42 (Secondary) #46L65 (Primary) 46L54 #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA) #Quantum Algebra (math.QA) #Random Matrices and Applications #math.OA #math.QA #msc:05E10 #msc:16T30 #msc:17B37 #msc:20G42 #msc:46L54 #msc:46L65

paper · pdf · doi:10.48550/arxiv.1201.4723

39 pages; appeared in Advances in Mathematics, Vol. 245, pages 500-533, 2013

openalex publication_date 2012/01/23 · arxiv created 2013/12/05 · arxiv updated 2013/12/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Full classification of de Finetti type theorems for *-random variables in classical and free probability
• Tubular partitions and representations of the quantum automorphism group of a homogeneous rooted tree
• Finite de Finetti theorems for free easy quantum groups
• A Connection between Easy Quantum Groups, Varieties of Groups and Reflection Groups

Related

• Unitary Easy Quantum Groups: the free case and the group case
• The full classification of orthogonal easy quantum groups
• The combinatorics of an algebraic class of easy quantum groups
• Easy quantum groups
• Quantum groups based on spatial partitions