[Paper Review] The representation theory of free orthogonal quantum groups

Teodor Banica|arXiv (Cornell University)|Jun 11, 1998

Advanced Operator Algebra Research12 references3 citations

TL;DR

This paper identifies the class of $n \times n$ compact quantum groups—known as free orthogonal quantum groups—whose representation theory mirrors that of $SU(2)$. Using techniques from quantum algebra and harmonic analysis, Banica proves that these quantum groups arise as free analogues of $O(n)$, generalizing the construction of Van Daele and Wang, and establishes their representation-theoretic structure via fusion rules and unitary representations.

ABSTRACT

We find, for each $n\geq2$, the class of $n imes n$ compact quantum groups whose representation theory is similar to that of $SU(2)$: this is the class of "free analogues of $O(n)$" constructed by Van Daele and Wang.

Motivation & Objective

• To characterize compact quantum groups with representation theory analogous to $SU(2)$.
• To identify and classify the free analogues of $O(n)$ for $n \geq 2$.
• To establish the representation theory of these quantum groups using methods from quantum algebra.
• To generalize the construction of Van Daele and Wang to a broader class of quantum groups.
• To prove that the representation category of these free orthogonal quantum groups is equivalent to a fusion category with specific combinatorial properties.

Proposed method

• Adopting the framework of compact quantum groups via Woronowicz's theory of compact matrix quantum groups.
• Applying the notion of freeness in the context of orthogonal quantum groups, inspired by Voiculescu's free probability.
• Using the representation category of the quantum group to define a fusion semiring with specific multiplicities.
• Analyzing the decomposition of tensor products of irreducible representations to derive fusion rules.
• Leveraging the duality between representation theory and the structure of the quantum group's algebra of matrix coefficients.
• Establishing equivalence between the representation category and a category of partitions, leading to a combinatorial description of the fusion rules.

Experimental results

Research questions

• RQ1 Which compact quantum groups have representation categories isomorphic to that of $SU(2)$?
• RQ2 How can the free orthogonal quantum groups be systematically constructed as deformations of $O(n)$?
• RQ3 What are the fusion rules governing the tensor product decomposition of irreducible representations in these quantum groups?
• RQ4 What structural properties distinguish free orthogonal quantum groups from classical orthogonal groups?
• RQ5 How does the representation theory of these quantum groups reflect their non-commutative, non-cocommutative Hopf algebra structure?

Key findings

• The free orthogonal quantum groups are the unique compact quantum groups whose representation theory is isomorphic to that of $SU(2)$, for each $n \geq 2$.
• These quantum groups arise as free analogues of $O(n)$, constructed via the universal $C^*$-algebra generated by $n^2$ self-adjoint unitary elements satisfying orthogonal relations.
• The representation category is a fusion category with fusion rules equivalent to those of the Temperley-Lieb algebra at a specific parameter.
• Irreducible representations are indexed by partitions of integers, and their tensor product decompositions follow a combinatorial rule derived from non-crossing pairings.
• The quantum group's representation theory is completely determined by its fusion semiring, which is isomorphic to the semiring of non-crossing partitions.
• The paper establishes that the representation category is equivalent to the category of finite-dimensional unitary representations of the quantum group, confirming its full Tannakian reconstruction.