[논문 리뷰] A planar algebra construction of the Haagerup subfactor

Emily Peters|ArXiv.org|2009. 02. 08.

Algebraic structures and combinatorial models참고 문헌 12인용 수 6

한 줄 요약

이 논문은 평면 대수 기법을 사용하여 Haagerup 하위인자를 구성하며, 특히 $TL_{12}$ 대수에서의 특정 토글 불변량과 내적 관계가 필요한 구조를 유도함을 보여준다. 핵심 결과는 쌍대 기저 벡터와 노름 관계의 명시적 계산으로, 이는 $TL_{12}$ 대수에서의 평면 대수적 관계와 내적 항등식을 통해 하위인자의 존재를 확인한다. 구성은 Jones-Wenzl 아이소템포텐셜과 $\sqrt{13}$을 포함하는 스펙트럼 계산에 기반하여 하위인자의 지수와 쌍대성 성질을 확립한다.

ABSTRACT

Most known examples of subfactors occur in families, coming from algebraic objects such as groups, quantum groups and rational conformal field theories. The Haagerup subfactor is the smallest index finite-depth subfactor which does not occur in one of these families. In this paper we construct the planar algebra associated to the Haagerup subfactor, which provides a new proof of the existence of the Haagerup subfactor. Our technique is to find the Haagerup planar algebra as a singly generated subfactor planar algebra, contained inside of a graph planar algebra.

연구 동기 및 목표

• To construct the Haagerup subfactor using planar algebra methods rather than traditional von Neumann algebra techniques.
• To compute inner products of tangles in $TL_{12}$ involving Jones-Wenzl idempotents and specific basis elements to verify duality and norm conditions.
• To establish that the dual basis vectors $\hat{w}_{i,i+1}(T)$ and $\hat{w}_{i,i+6}(T)$ are scalar multiples of $\alpha_{i,i+1}(T)$ and $\alpha_{i,i+6}(T)$, respectively, under the condition $\rho(T) = -T$.
• To verify the consistency of the planar algebra structure by computing key inner products such as $\langle \hat{w}_{4,5}(T), \hat{w}_{10,11}(T) \rangle = \frac{17 - 5\sqrt{13}}{144}$ and $\|N\|^2 = \frac{16}{351}(13 + 11\sqrt{13})$.

제안 방법

• The construction uses the Temperley-Lieb algebra $TL_{12}$ with quantum integers $[n]$ and the Jones-Wenzl idempotent $f^{(12)}$ to define tangle invariants and inner products.
• Inner products are computed via summation over basis elements $\beta \in B(TL_n)$, weighted by coefficients from $f^{(n)}$, with contributions evaluated diagrammatically.
• Dual basis vectors $\hat{w}_{i,i+1}(T)$ and $\hat{w}_{i,i+6}(T)$ are defined via orthogonal projection and normalization, using the relation $\hat{w}_{i,i+6}(T) = \frac{11 + \sqrt{13}}{36} \alpha_{i,i+6}(T)$.
• The norm of $N = \text{Join}_2(T,T) - \text{Proj}_{TL_6}(\text{Join}_2(T,T)) - \text{Proj}_{ATL_6(T)}(\text{Join}_2(T,T))$ is computed using orthogonal decomposition and known norms of $f^{(6)}$ and $\hat{w}_{6,12}$.
• Sphericality and rotation rules are applied to simplify inner products involving starred tangles, reducing them to evaluations of $Z(T^3)$ and $Z((\rho^{1/2}T)^3)$.
• The paper uses the identity $f^{(n)} \cdot f^{(n)} = f^{(n)}$ to simplify inner products and confirms that certain inner products vanish due to coefficient sums being zero.

실험 결과

연구 질문

• RQ1 Can the Haagerup subfactor be constructed purely within the framework of planar algebras using $TL_{12}$ and Jones-Wenzl idempotents?
• RQ2 What is the precise norm of the vector $N = (1 - \text{Proj}_{TL_6} - \text{Proj}_{ATL_6(T)}) \text{Join}_2(T,T)$, and how does it relate to the subfactor index?
• RQ3 Why is $\hat{w}_{i,i+6}(T)$ a scalar multiple of $\alpha_{i,i+6}(T)$, and what is the exact scalar in terms of $\sqrt{13}$?
• RQ4 What is the value of the inner product $\langle \hat{w}_{4,5}(T), \hat{w}_{10,11}(T) \rangle$, and how is it computed via basis contributions in $f^{(12)}$?
• RQ5 How do the coefficients of basis elements in $f^{(6)}$ and $f^{(12)}$ determine the non-vanishing contributions to the inner products in the planar algebra?

주요 결과

• The inner product $\langle \alpha_{i,i+1}(T), w_{i,i+1}(T) \rangle = 76 - 20\sqrt{13}$, which is used to normalize $\hat{w}_{i,i+1}(T)$.
• The dual vector $\hat{w}_{i,i+6}(T)$ is exactly $\frac{11 + \sqrt{13}}{36} \alpha_{i,i+6}(T)$, confirming it is a scalar multiple due to vanishing cross-terms.
• The inner product $\langle \hat{w}_{4,5}(T), \hat{w}_{10,11}(T) \rangle = \frac{17 - 5\sqrt{13}}{144}$, computed via summing contributions from five specific $TL_6$ basis elements.
• The norm of $N$ is $\|N\|^2 = \frac{16}{351}(13 + 11\sqrt{13})$, derived from orthogonal decomposition and known norms of $f^{(6)}$ and $\hat{w}_{6,12}$.
• The norm of the projection of $\text{Join}_3(T,T)$ onto $ATL_5(T)$ is $\left\| \text{Proj}_{ATL_5(T)}(\text{Join}_3(T,T)) \right\|^2 = \frac{[5]^2}{[6]}$, with $[5] = \frac{\sin(5\pi/13)}{\sin(\pi/13)}$.
• The inner product $\langle \alpha_{i,i+6}(T), w_{i-1,i}(T) \rangle = 0$ and $\langle \alpha_{i,i+6}(T), w_{i+5,i+6}(T) \rangle = 0$, due to coefficient sums in $f^{(6)}$ being zero over contributing basis elements.