[Paper Review] On the classification of easy quantum groups

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

Advanced Operator Algebra Research16 references4 citations

TL;DR

This paper completes the classification of easy quantum groups by identifying exactly seven free easy quantum groups—$O_n^+$, $S_n^+$, $H_n^+$, $B_n^+$, ${S_n'}^+$, ${B_n'}^+$, and $B_n^{ ext{#}+}$—and establishes a full classification in the half-liberated and non-hyperoctahedral cases. It introduces a new quantum group $B_n^{ ext{#}+}$, explains its $C^*$-algebraic structure via tensor and free products with $C^*(\mathbb{Z}_2)$, and proves that none of the full $C^*$-algebras for these groups are exact when $n \geq 5$. The work extends the combinatorial framework of noncrossing partitions to unify and classify these quantum groups systematically.

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 complete the classification of easy quantum groups in the free and half-liberated cases.
• To identify and characterize the missing quantum group $B_n^{\# +}$, which arises from the splitting of $B_n'$ in the free setting.
• To establish a correspondence between categories of noncrossing partitions and quantum groups via generating partitions.
• To analyze the $C^*$-algebraic structure of these quantum groups using tensor and free products with $C^*(\mathbb{Z}_2)$.
• To prove non-exactness of the full $C^*$-algebras for the seven free easy quantum groups when $n \geq 5$.

Proposed method

• Use of generating partitions to describe categories of noncrossing partitions associated with easy quantum groups.
• Application of category operations (tensor product, composition, involution) to classify closed categories of noncrossing partitions.
• Construction of $C^*$-algebras for ${B_n'}^+$ and $B_n^{ ext{#}+}$ via tensor and free products with $C^*(\mathbb{Z}_2)$, respectively.
• Computation of $K$-theory groups for $B_n^+$, ${B_n'}^+$, and $B_n^{ ext{#}+}$ using Voigt's result on $O_n^+$.
• Inductive proof that any noncrossing partition with even-sized singleton blocks can be built from basic partitions, establishing category closure.
• Use of the noncrossing rule on $\oplus$ and $\ominus$ signs to verify virtual connections between singletons in noncrossing configurations.

Experimental results

Research questions

• RQ1 How many free easy quantum groups exist between $S_n^+$ and $O_n^+$, and what distinguishes them combinatorially and algebraically?
• RQ2 Why does the classical group $B_n'$ split into two distinct free quantum groups, ${B_n'}^+$ and $B_n^{ ext{#}+}$, while others remain in one-to-one correspondence?
• RQ3 What is the $C^*$-algebraic structure of the new quantum group $B_n^{ ext{#}+}$, and how does it relate to $B_n^+$?
• RQ4 Are the full $C^*$-algebras of the seven free easy quantum groups exact, and what does this imply for their representation theory?
• RQ5 Can the half-liberated case be fully classified, and how do $O_n^*$, $H_n^*$, and $B_n^{ ext{#}*}$ fit into this framework?

Key findings

• There are exactly seven free easy quantum groups: $O_n^+$, $S_n^+$, $H_n^+$, $B_n^+$, ${S_n'}^+$, ${B_n'}^+$, and $B_n^{ ext{#}+}$, completing the classification in this setting.
• The group $B_n'$ splits into two distinct free quantum groups due to different $C^*$-algebraic structures: ${B_n'}^+$ arises from a tensor product with $C^*(\mathbb{Z}_2)$, while $B_n^{ ext{#}+}$ arises from a free product.
• The $C^*$-algebra of $B_n^{ ext{#}+}$ is isomorphic to the free product of $C^*(B_n^+)$ and $C^*(\mathbb{Z}_2)$, providing a conceptual explanation for the splitting.
• The $K$-theory of $B_n^+$, ${B_n'}^+$, and $B_n^{ ext{#}+}$ is computed using Voigt’s result on $O_n^+$, showing non-trivial $K$-groups.
• For $n \geq 5$, the full $C^*$-algebras of all seven free easy quantum groups are not exact, indicating non-trivial operator algebraic complexity.
• The half-liberated case is fully classified: there are exactly three half-liberated easy quantum groups besides the hyperoctahedral series $H_n^{(s)}$, including the new $B_n^{ ext{#}*}$.