Centralizers of discrete Temperley-Lieb-Jones subfactors

TL;DR

The paper addresses the realization problem for discrete TLJ-type standard invariants by computing the centralizer invariant $N\subseteq M^{\phi}$ with $\phi=\tau\circ E$ and translating it into a combinatorial invariant via a tracial fair and balanced $\delta$-graph $\Gamma_{tr}$. It introduces a universal-cover–like construction: $\Gamma_{tr}$ is derived from the original graph $\Gamma$ by a path-weight quotient, and its structure is controlled by weights along loops, making $T(\mathbf{M})$ tractable through $T_{0}(\mathbf{M})$. A key result is that for tracial TLJ-type invariants, $T_{0}(\Gamma)=W^{\times}(\Gamma)$, providing an obstruction to realization because $T_{0}(\mathbf{M})\subseteq\mathcal{F}(M)$ must lie in the fundamental group of any realizable $\mathrm{II}_1$ factor; nontrivial $T_{0}$ hence obstructs realization. The framework yields concrete obstructions, e.g., the $A^{\delta}_{-\infty,\infty}$ graph can realize only if its $T_{0}$-group embeds into $\mathcal{F}(M)$, with broader implications for the uniqueness of interpolated free group factors and related realization questions.

Abstract

Discrete, unimodular inclusions of factors $(N\subseteq M, E)$ with $N$ of type $\rm{II}_{1}$ have a natural notion of standard invariant, generalizing the finite index case. When the unitary tensor category of $N$-$N$ bimodules generated by $_{N}L^{2}(M, τ\circ E)_{N}$ is equivalent to the Temperley-Lieb-Jones category $\text{TLJ}(δ)$, the associated discrete standard invariants are classified in terms of fair and balanced $δ$-graphs. Many examples of these subfactors naturally arise in the context of the Guionnet-Jones-Shlyakhtenko (GJS) construction for graphs. In this paper, we compute the discrete standard invariant of the centralizer subfactor $N\subseteq M^φ$ for the canonical state $φ=τ\circ E$, which is again a discrete subfactor of $\text{TLJ}(δ)$-type. We show that the associated fair and balanced $δ$-graph behaves analogously to a universal covering space of the original fair and balanced $δ$-graph. As an application, we obtain an obstruction to the realization of discrete tracial TLJ-type standard invariants by subfactors of a $\rm{II}_{1}$ factor $M$ in terms of the fundamental group of M.

Theorems & Definitions (43)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Definition 2.1
• Remark 2.2
• Remark 2.3
• Definition 2.4
• Definition 2.6
• Proposition 2.7