[Paper Review] A2-Planar Algebras II: Planar Modules

David Evans, Mathew Pugh|arXiv (Cornell University)|Jun 23, 2009

Algebraic structures and combinatorial models35 references4 citations

TL;DR

This paper introduces planar modules over $A_2$-planar algebras, generalizing Jones's planar algebra framework to encode the representation theory of quantum $SU(3)$. It constructs $A_2$-graph planar algebras from $SU(3)$ $\mathcal{ADE}$ graphs with cell systems and achieves a partial modular decomposition of these algebras into irreducible $A_2$-$PTL$-modules, particularly identifying components via eigenvalue analysis of key operators on Hilbert modules.

ABSTRACT

Generalizing Jones's notion of a planar algebra, we have previously introduced an A_2-planar algebra capturing the structure contained in the double complex pertaining to the subfactor for a finite SU(3) ADE graph with a flat cell system. We now introduce the notion of modules over an A_2-planar algebra, and describe certain irreducible Hilbert A_2-TL-modules. We construct an A_2-graph planar algebra associated to each pair (G,W) given by an SU(3) ADE graph G and a cell system W on G. A partial modular decomposition of these A_2-graph planar algebras is achieved.

Motivation & Objective

• To generalize Jones’s planar algebra framework to $A_2$-planar algebras, capturing the structure of $SU(3)$ subfactors and their associated modular invariants.
• To define and study planar modules over $A_2$-planar algebras, particularly focusing on irreducible Hilbert modules of lowest weight zero.
• To construct an $A_2$-graph planar algebra $P^{\mathcal{G}}$ for each pair $(\mathcal{G}, W)$, where $\mathcal{G}$ is an $SU(3)$ $\mathcal{ADE}$ graph and $W$ is a cell system on $\mathcal{G}$.
• To achieve a partial modular decomposition of $P^{\mathcal{G}}$ into irreducible $A_2$-$PTL$-modules using spectral analysis of operators like $\rho_{(2,2)}$, $U_1$, and $\widetilde{U}_1$.
• To determine the structure of $P^{\mathcal{G}}$ for specific graphs such as $\mathcal{E}^{(8)}$ and $\mathcal{D}^{(6)}$ by analyzing orthogonal complements and eigenvalue multiplicities.

Proposed method

• Generalize Jones’s planar algebra framework to $A_2$-planar algebras, which encode the representation theory of quantum $SU(3)$ via $A_2$-Temperley-Lieb algebras.
• Define planar modules over $A_2$-planar algebras, introducing the concept of lowest weight and constructing irreducible Hilbert $A_2$-$PTL$-modules of lowest weight zero.
• Construct the $A_2$-graph planar algebra $P^{\mathcal{G}}$ from an $SU(3)$ $\mathcal{ADE}$ graph $\mathcal{G}$ and a compatible cell system $W$, using trivalent vertex diagrams weighted by Ocneanu cells.
• Perform spectral analysis on key operators $\rho_{(2,2)}$, $U_1$, and $\widetilde{U}_1$ acting on graded components of $P^{\mathcal{G}}$, particularly on $P^{\mathcal{G}}_{+^{2}-^{2}}$.
• Use eigenvalue multiplicities and symmetry properties (e.g., conjugation invariance) to identify irreducible module components within $P^{\mathcal{G}}$, especially in $W^\perp \cap P^{\mathcal{G}}_{+^{2}-^{2}}$.
• Apply the Verlinde algebra and nimrep structure to relate the decomposition to modular invariants and fusion rules of $SU(3)$ conformal field theories.

Experimental results

Research questions

• RQ1 How can planar modules be defined and classified over $A_2$-planar algebras, particularly those of lowest weight zero?
• RQ2 What is the structure of the $A_2$-graph planar algebra $P^{\mathcal{G}}$ associated to an $SU(3)$ $\mathcal{ADE}$ graph $\mathcal{G}$ and a cell system $W$?
• RQ3 Which irreducible $A_2$-$PTL$-modules appear in the decomposition of $P^{\mathcal{G}}$, and how can they be identified via spectral data?
• RQ4 How does the modular decomposition of $P^{\mathcal{G}}$ reflect the underlying $SU(3)$ modular invariant and graph symmetry?
• RQ5 What constraints do eigenvalue multiplicities of $\rho_{(2,2)}$, $U_1$, and $\widetilde{U}_1$ impose on the possible irreducible components of $P^{\mathcal{G}}$?

Key findings

• The $A_2$-graph planar algebra $P^{\mathcal{E}^{(8)}}$ contains at least $H^{\beta_{(0,0)}}$, $H^{\beta_{(3,0)}}$, $H^{\beta_{(0,3)}}$, $H^{\beta_{(2,2)}}$, and two rank $(2,2)$ modules $H^{(2,2),\gamma_1,\varepsilon_2 i}$, $H^{(2,2),\gamma_2,\varepsilon_3 i}$ with $\omega = \pm i$, along with $H^{(3,0),\varepsilon_1}$ and $H^{(0,3),\varepsilon_1}$ for $\varepsilon_1 \in \{\pm 1\}$.
• For $\mathcal{D}^{(6)}$, the decomposition of $P^{\mathcal{D}^{(6)}}$ includes $H^{\beta_{(0,0)}}$, $H^{0,\overline{0}}$, and two rank $(2,2)$ modules $H^{(2,2),\gamma_1,\varepsilon_1}$, $H^{(2,2),\gamma_2,\varepsilon_2}$ with $\omega^2 = 1$, implying $\omega = \pm 1$.
• The dimension of $W^\perp \cap P^{\mathcal{E}^{(8)}}_{+^{2}-^{2}}$ is 6, and the eigenvalue $\omega^2 = -1$ for $\rho_{(2,2)}$ on $H^{(2,2),\gamma,\omega}_{+^{2}-^{2}}$ implies $\omega = \pm i$, restricting the possible module types.
• The dimension of $W^\perp \cap P^{\mathcal{D}^{(6)}}_{+^{2}-^{2}}$ is 2, and the eigenvalue 1 of $\rho_{(2,2)}$ with multiplicity two forces the presence of two rank $(2,2)$ modules rather than a single $H^{(3,0),\gamma}$ module.
• The action of $U_1$ and $\widetilde{U}_1$ on $H^{(3,0),\pm 1}_{+^{2}-^{2}}$ has eigenvalues $[4]\alpha\delta^{-2}$ and $0$ with multiplicity two, while on $H^{(0,3),\pm 1}_{+^{2}-^{2}}$ it has $[4]\alpha\delta^{-2}$ and $0$ with multiplicity two, indicating distinct spectral behavior.
• The decomposition of $P^{\mathcal{G}}$ is constrained by conjugation symmetry: if $\gamma_1$ is complex, then $\gamma_2 = \overline{\gamma}_1$, or both $\gamma_1, \gamma_2$ are real, ensuring $P^{\mathcal{G}}$ is closed under complex conjugation.