Subfactors and Hecke groups

TL;DR

This paper explores the connection between Hecke groups and subfactor indices in von Neumann algebras, addressing a problem posed by V. F. R. Jones, through the application of cluster C*-algebras.

Contribution

It introduces a novel approach linking Hecke groups and subfactor indices using cluster C*-algebras, solving a longstanding problem.

Findings

Established a relationship between Hecke groups and subfactor indices.

Applied cluster C*-algebras to analyze subfactors.

Provided new insights into the structure of von Neumann algebra subfactors.

Abstract

We study a relation between the Hecke groups and the index of subfactors in a von Neumann algebra. Such a problem was raised by V. F. R. Jones. We solve the problem using the notion of a cluster C*-algebra.

Equations

$$\mathcal{D}\cong \mathbf{C}{P}^1/A^{\mathbf{Z}},$$

$$t{r}^2(A)\in [4,\infty )\bigcup \{4{\cos }^2\left(\frac{\pi }{n}\right)|n\ge 3\}.$$

$$\lambda \in [2,\infty )\bigcup \{2\cos \left(\frac{\pi }{n}\right)|n\ge 3\}.$$

$$x_k{x}_k^{\prime }$$

$$b_{ij}^{\prime }$$

$$K_0(\mathbb{A}(\mathbf{x},B))\cong \mathcal{A}(\mathbf{x},B),$$

$$z⟼\mathbf{k}z,wherez\in \mathbf{C}{P}^{1}and|\mathbf{k}|\ne 1.$$

$$A=\left(\begin{array}{cc}\sqrt{\mathbf{k}}& 0\\ 0& \frac{1}{\sqrt{\mathbf{k}}}\end{array}\right)\in PS{L}_2(\mathbf{C}):=S{L}_2(\mathbf{C})/\pm I.$$

$$t{r}^2(A)=\frac{(\mathbf{k}+1{)}^2}{\mathbf{k}}\in \mathbf{C}\backslash [0,4].$$

$$\mathcal{D}\cong \mathbf{C}{P}^1/A^{\mathbf{Z}}.$$

$$T_{\mathcal{D}}=\left\{t=\frac{R}{r}|t>1\right\}.$$

$$[4,\infty )\bigcup \{4{\cos }^2\left(\frac{\pi }{n}\right)|n\ge 3\},$$

$$t{r}^2(A)\in [4,\infty )\bigcup \{4{\cos }^2\left(\frac{\pi }{n}\right)|n\ge 3\}.$$

$$\mathbf{p}:\mathbf{C}{P}^1\to \mathbf{C}{P}^1$$

$$\chi (\mathbf{C}{P}^1)=2\chi (\mathbf{C}{P}^1)-\sum _{i=1}^2(e_i-1),$$

$$\mathcal{H}:=\mathbf{C}{P}^1\backslash \{D_0,\infty \},$$

$$\rho :\mathcal{D}\to \mathcal{H}.$$

$$\mathcal{D}\cong \mathbf{C}{P}^1\backslash \{D_0,D_{\infty }\},$$

$$\rho :\mathcal{D}\to \mathbb{H}/G_{\lambda }.$$

$${\lambda }^2=\frac{1}{\pi }\left|{\rho }^{-1}(\partial D(\lambda ))\right|$$

$$\lambda =tr(A)\in [2,\infty )\bigcup \{2\cos \left(\frac{\pi }{n}\right)|n\ge 3\}.$$

Full text

1. Introduction

The following problem can be found in [Jones 1991] [5, p.24]:

*“Consider the subgroup $G_{\lambda}$ of $SL_{2}(\mathbf{R})$ generated by $\left(\begin{smallmatrix}1&\lambda\cr 0&1\end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}0&1\cr-1&0\end{smallmatrix}\right)$. For what values of $\lambda>0$ is it discrete? Answer: $\lambda=2\cos\left({\pi\over n}\right),~{}n=3,4,\dots$ or $\lambda\geq 2$. (…) We have been unable to find any direct connection between this result and Theorem 3.1 (Jones Index Theorem). It is a tantalizing situation.” *

The aim of our note is to solve the problem in terms of the cluster $C^{*}$-algebras [6, Section 4.4.3]. To give an idea, let $\mathscr{D}=\{z=x+iy\in\mathbf{C}~{}|~{}r\leq|z|\leq R\}$ be an annulus in the complex plane. Consider the Schottky uniformization of $\mathscr{D}$, i.e.

[TABLE]

where $\mathbf{C}P^{1}:=\mathbf{C}\cup\{\infty\}$ is the Riemann sphere and $A\in SL_{2}(\mathbf{C})/\pm I$ is a matrix acting on the $\mathbf{C}P^{1}$ by the Möbius transformation. It follows from [Glubokov & Nikolaev 2018] [3] and Section 2.2, that the index of subfactors in a von Neumann algebra coincides with the square of trace of matrix $A$, i.e.

[TABLE]

To solve the Jones Problem, we prove in Section 3 that $\mathscr{D}$ is a ramified double cover of the orbifold $\mathbb{H}/G_{\lambda}$, where $\mathbb{H}:=\{x+iy\in\mathbf{C}~{}|~{}y>0\}$ is the Lobachevsky half-plane and the group $G_{\lambda}$ acts on $\mathbb{H}$ by the linear fractional trasformations. Since such a cover takes the square root of the moduli parameter $tr^{2}~{}(A)$ of $\mathscr{D}$, we conclude that $\lambda=tr~{}(A)$. In other words, the Jones Index Theorem (1.2) is equivalent to the following well-known result:

Theorem 1.1**.**

([Hecke 1936] [4, Satz 1,2 & 6])* The $G_{\lambda}$ is a discrete subgroup of $SL_{2}(\mathbf{R})$ if and only if*

[TABLE]

Remark 1.2*.*

The group $G_{\lambda}$ appears in the study of the Riemann zeta function [Hecke 1936] [4]. In particular, Hecke’s Theorem says that the space of automorphic functions corresponding to $G_{\lambda}$ is (i) infinite-dimensional, if $\lambda>2$ or (ii) finite-dimensional, if $\lambda\in\{2\cos\left({\pi\over n}\right)~{}|~{}n\geq 3\}$. The proof of this fact is purely analytic [Hecke 1936] [4]. On the other hand, cases (i) and (ii) follow from the Sherman-Zelevinsky Theorem for the cluster $C^{*}$-algebras of rank 2, see [Glubokov & Nikolaev 2018] [3].

The article is organized as follows. Section 2 contains a brief review of preliminary results. Theorem 1.1 is proved in Section 3.

2. Preliminaries

The cluster $C^{*}$-algebras and their $K$-theory are covered in [6, Section 4.4.3]. A correspondence between the cluster $C^{*}$-algebra of an annulus $\mathscr{D}$ and the Jones Index Theorem was established in [Glubokov & Nikolaev 2018] [3]. The Hecke groups were introduced in [Hecke 1936] [4].

2.1. Cluster $C^{*}$-algebras

A cluster algebra $\mathscr{A}(\mathbf{x},B)$ of rank $n$ is a subring of the field of rational functions in $n$ variables depending on a cluster of variables $\mathbf{x}=(x_{1},\dots,x_{n})$ and a skew-symmetric matrix $B=(b_{ij})\in M_{n}(\mathbf{Z})$; the pair $(\mathbf{x},B)$ is called a seed. A new cluster $\mathbf{x}^{\prime}=(x_{1},\dots,x_{k}^{\prime},\dots,x_{n})$ and a new skew-symmetric matrix $B^{\prime}=(b_{ij}^{\prime})$ is obtained from $(\mathbf{x},B)$ by the exchange relations:

[TABLE]

The seed $(\mathbf{x}^{\prime},B^{\prime})$ is said to be a mutation of $(\mathbf{x},B)$ in direction $k$, where $1\leq k\leq n$; the algebra $\mathscr{A}(\mathbf{x},B)$ is generated by cluster variables $\{x_{i}\}_{i=1}^{\infty}$ obtained from the initial seed $(\mathbf{x},B)$ by the iteration of mutations in all possible directions $k$. The Laurent phenomenon says that $\mathscr{A}(\mathbf{x},B)\subset\mathbf{Z}[\mathbf{x}^{\pm 1}]$, where $\mathbf{Z}[\mathbf{x}^{\pm 1}]$ is the ring of the Laurent polynomials in variables $\mathbf{x}=(x_{1},\dots,x_{n})$ depending on an initial seed $(\mathbf{x},B)$. The $\mathscr{A}(\mathbf{x},B)$ is a commutative algebra with an additive abelian semigroup consisting of the Laurent polynomials with positive coefficients. Thus the algebra $\mathscr{A}(\mathbf{x},B)$ is a countable abelian group with an order satisfying the Riesz interpolation property, i.e. a dimension group. A cluster $C^{*}$-algebra $\mathbb{A}(\mathbf{x},B)$ is an AF-algebra, such that

[TABLE]

where $\cong$ is an isomorphism of the dimension groups [6, Section 4.4.3].

2.2. Schottky uniformization of $\mathscr{D}$

Consider the Riemann surface $\mathscr{D}$ (an annulus) defined by the formula (1.1). We shall use the Schottky uniformization of $\mathscr{D}$ by the loxodromic transformations. Namely, let $\mathbf{C}P^{1}:=\mathbf{C}~{}\cup~{}\{\infty\}$ be the Riemann sphere and consider the Möbius transformation of $\mathbf{C}P^{1}$ given by the formula:

[TABLE]

It is easy to see, that (2.4) can be written in the matrix form:

[TABLE]

It is well known, that (2.4) is a loxodromic transformation if and only if

[TABLE]

The Schottky uniformization of $\mathscr{D}$ is given by the formula:

[TABLE]

2.3. Admissible values of $tr^{2}~{}(A)$

Recall that the moduli space of the annulus $\mathscr{D}$ is given by the formula:

[TABLE]

We consider a cluster $C^{*}$-algebra $\mathbb{A}(\mathscr{D})$ associated to a canonical triangulation of $\mathscr{D}$ [Fomin, Shapiro & Thurston 2008] [1, Example 4.4]. It follows from the Sherman-Zelevinsky Theorem for the algebra $\mathbb{A}(\mathscr{D})$, that the admissible values of the “index” ${(t+1)^{2}\over t}$ must belong to the set:

[TABLE]

see [Glubokov & Nikolaev 2018] [3] for the proof.

We set $\mathbf{k}=t$ in the formulas (2.4) - (2.6). Comparing (2.6) and (2.9), one gets

[TABLE]

Remark 2.1*.*

It follows from (2.6) that $A$ is a loxodromic transformation if and only if $tr^{2}~{}(A)\in(4,\infty)$. The case $tr^{2}~{}(A)\in\{4\cos^{2}\left({\pi\over n}\right)~{}|~{}n\geq 3\}$ corresponds to an elliptic transformation $A$ of order $n$. Finally, that case $tr^{2}~{}(A)=4$ gives a parabolic transformation $A$. Note that for the elliptic and parabolic transformations, the values of parameter $t$ in (2.8) are the $n$-th roots of unity.

3. Proof of theorem 1.1

We split the proof in a series of lemmas.

Lemma 3.1**.**

Let $p_{1},p_{2}\in\mathbf{C}P^{1}$ be two points on the Riemann sphere, such that $p_{1}\neq p_{2}$. Then there exists a double covering map

[TABLE]

ramified over the points $p_{1}$ and $p_{2}$.

Proof.

(i) Recall that the necessary condition for the existence of $\mathbf{p}$ is given by the Riemann-Hurwitz formula:

[TABLE]

where $\chi(\mathbf{C}P^{1})$ is the Euler characteristic and $e_{i}$ is the degree of the map $z\mapsto z^{e_{i}}$ in the ramification point $p_{i}$. Since $\chi(\mathbf{C}P^{1})=e_{1}=e_{2}=2$, we conclude that the condition (3.2) is satisfied.

(ii) The sufficient condition for the existence of $\mathbf{p}$ can be verified directly using [Gersten 1987] [2, Theorem 1.5]. We leave it as an exercise to the reader.

Lemma 3.1 is proved. ∎

Remark 3.2*.*

We assume further that in lemma 3.1 we have $p_{1}=0$ and $p_{2}=\infty$. Our assumption is not restrictive, since any two points $p_{1},p_{2}\in\mathbf{C}P^{1}$ can be put into such a position by a Möbius transformation.

Lemma 3.3**.**

Consider a Riemann surface:

[TABLE]

where $D_{0}$ is a disk containing point $0\in\mathbf{C}P^{1}$. Then there exists a double covering map

[TABLE]

Proof.

Let $\mathbf{p}:\mathbf{C}P^{1}\to\mathbf{C}P^{1}$ be the double covering map ramified at [math] and $\infty$, see lemma 3.1 and remark 3.2. Recall that

[TABLE]

where $D_{\infty}$ is a disk containing point $\infty\in\mathbf{C}P^{1}$. We use a homotopy to contract $D_{\infty}$ to the point $\infty$, and set the map $\rho\equiv\mathbf{p}$. Comparing (3.3) and (3.5), we conclude that $\rho$ is the required double covering map. Lemma 3.3 is proved. ∎

Lemma 3.4**.**

$\mathscr{H}\cong\mathbb{H}/G_{\lambda}$, where $\lambda\in[2,\infty)~{}\bigcup~{}\{2\cos\left({\pi\over n}\right)~{}|~{}n\geq 3\}$.

Proof.

It is well known, that the Hecke orbifold $\{\mathbb{H}/G_{\lambda}~{}|~{}\lambda>2\}$ is a topological sphere $S^{2}$ with a hole $D(\lambda)$ of radius $\lambda/2$, one elliptic fixed point $e_{2}$ of order 2 and one puncture $c$, see e.g. [Schmidt & Sheingorn 1995] [7, p. 255]. The elliptic point $e_{2}=i$ is a fixed point of the matrix $\left(\begin{smallmatrix}0&1\cr-1&0\end{smallmatrix}\right)$ having order 2 in the group $SL_{2}(\mathbf{Z})/\pm I$.

Since $S^{2}\cong\mathbf{C}P^{1}$, we use (3.3) to identify $D(\lambda)\equiv D_{0}$ and $e_{2}\equiv\infty$. Thus one gets $\mathscr{H}\cong\mathbb{H}/G_{\lambda}$. Using lemma 3.3, we obtain a double covering map $\rho$ ramified in the points $e_{2}$ and $0\in D(\lambda)$, i.e.

[TABLE]

Remark 3.5*.*

Notice that $\rho^{-1}(D(\lambda))$ is a disk and $\rho^{-1}(e_{2})$ is a regular point of $\mathscr{D}$. The corresponding ramification points are shown in Figure 1.

To determine admissible values of the moduli parameter $\lambda$, observe that $\lambda={1\over\pi}|\partial D(\lambda)|$, where $\partial D(\lambda)$ is the boundary of the disk $D(\lambda)$. Observe that the local map at the point $0\in D(\lambda)$ is given by the formula $z\mapsto z^{2}$. Therefore using the polar coordinates, we conclude that:

[TABLE]

is a moduli parameter of the Riemann surface $\mathscr{D}$. But according to (1.2) any such a parameter must coincide with the $tr^{2}~{}(A)$, where $A$ is the matrix in the Schottky uniformization (1.1). Taking positive values of the square root, one gets from (1.2)

[TABLE]

Lemma 3.4 is proved. ∎

Remark 3.6*.*

For the sake of brevity, lemma 3.4 is proved for the continuous moduli $\lambda\in(2,\infty)$. The case of the discrete moduli is treated likewise, see remark 2.1.

Theorem 1.1 follows from lemma 3.4.