Isometry Theorem for Continuous Quiver of Type $\tilde{A}$

XIAOWEN GAO¹ and MINGHUI ZHAO²^∗ ¹ School of Science, Beijing Forestry University, Beijing 100083, P. R. China [email protected] (X.Gao) ² School of Science, Beijing Forestry University, Beijing 100083, P. R. China [email protected] (M.Zhao)

Abstract.

The Isometry Theorem for continuous quiver of type $A$ plays an important role in persistent homology. In this paper, we shall generalize Isometry Theorem to continuous quiver of type $\tilde{A}$ .

Key words and phrases:

Continuous quiver, Isometry Theorem

2010 Mathematics Subject Classification:

Primary 16G20; Secondary 55N31.

^∗ Corresponding author

1. Introduction

Representation theory of quivers is important in persistent homology and widely used in topological data analysis.

Gabriel gave the classification of indecomposable representations of quivers of finite type in [11]. In [10], Crawley-Boevey gave a classification of indecomposable representations of $\mathbb{R}$ . In [6], Botnan gave a classification of indecomposable representations of infinite zigzag. Igusa, Rock and Todorov introduced general continuous quivers of type $A$ and classified indecomposable representations in [14]. In [1, 2, 18, 19], Appel, Sala and Schiffmann introduced continuum quivers independently. By using these results, persistence diagrams corresponding to representations are defined.

The Isometry Theorem shows that the interleaving distance between two representations of a given quiver is equal to the bottleneck distance between corresponding persistence diagrams. The stability part of Isometry Theorem for continuous quiver $\mathbb{R}$ was given by Cohen-Steiner-Edelsbrunner-Harer in [9], Chazal-De Silva-Glisse-Oudot in [8] and Bauer-Lesnick in [3]. The converse stability part was given by Bubenik-Scott in [7] and Lesnick in [16]. In addition, Botnan studied the stability of zigzag persistence modules in [5]. For the history of Isometry Theorem, see [17].

In [13], Hanson and Rock introduced continuous quivers of type $\tilde{A}$ and gave a classification of indecomposable representations. In[15, 18, 19], Igusa-Todorov and Sala-Schiffmann also study continuous quivers of type $\tilde{A}$ and its persistence representations.

In this paper, we shall give an Isometry Theorem for continuous quivers of type $\tilde{A}$ . The proof of this theorem is based on the equivalence between the category of representations of continuous quivers of type $\tilde{A}$ and that of continuous quivers of type $A$ with automorphism.

In Section 2, we shall recall Isometry Theorem for continuous quivers of type $A$ . The Isometry Theorem for continuous quivers of type $\tilde{A}$ will be given in Section 3 and its proof will be given in Section 4.

2. Continuous quivers of type ${A}$

2.1. Continuous quivers of type ${A}$

Let $\mathbb{R}$ be the set of real numbers and $<$ be the normal order on $\mathbb{R}$ . Following notations in [14], ${A}_{\mathbb{R}}=(\mathbb{R},<)$ is called a continuous quiver of type $A$ .

Let $k$ be a fixed field. A representation of ${A}_{\mathbb{R}}$ over $k$ is given by $\mathbb{V}=(\mathbb{V}(x),\mathbb{V}(x,y))$ , where $\mathbb{V}(x)$ is a $k$ -vector space for any $x\in\mathbb{R}$ and $\mathbb{V}(x,y):\mathbb{V}(x)\rightarrow\mathbb{V}(y)$ is a $k$ -linear map for any $x<y\in\mathbb{R}$ satisfying that $\mathbb{V}(x,y)\circ\mathbb{V}(y,z)=\mathbb{V}(x,z)$ for any $x<y<z\in\mathbb{R}$ . Let $\mathbb{V}=(\mathbb{V}(x),\mathbb{V}(x,y))$ and $\mathbb{W}=(\mathbb{W}(x),\mathbb{W}(x,y))$ be two representations of ${A}_{\mathbb{R}}$ . A family of $k$ -linear maps $\varphi=(\varphi(x))_{x\in\mathbb{R}}$ is called a morphism from $\mathbb{V}$ to $\mathbb{W}$ , if $\varphi(y)\mathbb{V}(x,y)$ = $\mathbb{W}(x,y)\varphi(x)$ , for any $x<y\in\mathbb{R}$ .

Denoted by $\mathrm{Rep}_{k}({A}_{\mathbb{R}})$ the category of representations of ${A}_{\mathbb{R}}$ and by $\mathrm{Rep}^{pwf}_{k}({A}_{\mathbb{R}})$ the subcategory of pointwise finite-dimensional representations.

For each $a,b\in\mathbb{R}$ , we use the notation $|a,b|$ for one of $(a,b),[a,b),(a,b]$ and $[a,b]$ . In this paper, we allow $a=-\infty$ or $b=+\infty$ , that is the notation $|a,b|$ may mean $(a,+\infty)$ , $(-\infty,b)$ or $(-\infty,+\infty)$ , too.

Denote ${T}_{|a,b|}=({T}_{|a,b|}(x),{T}_{|a,b|}(x,y))$ as the following representation of ${A}_{\mathbb{R}}$ , where

{T}_{|a,b|}(x)=\left\{\begin{aligned} k,&\qquad x\in|a,b|,\\ 0,&\qquad\textrm{otherwise};\\ \end{aligned}\right.

and

{T}_{|a,b|}(x,y)=\left\{\begin{aligned} 1_{k},&\qquad x<y\,\,\textrm{and}\,\,x,y\in|a,b|,\\ 0,&\qquad\textrm{otherwise}.\\ \end{aligned}\right.

The representation ${T}_{|a,b|}$ is called an interval representation.

Let $\mathbb{V}$ be a pointwise finite-dimensional representation of ${{A}}_{\mathbb{R}}$ . In [10, 4, 14], it is proved that it can be decomposed into the direct sum of indecomposable interval representations

\mathbb{V}=\bigoplus_{i\in I}{T}_{|a_{i},b_{i}|}.

2.2. Isometry Theorem

In this section, we follow the notation in [17].

Let $\mathbb{V}=(\mathbb{V}(x),\mathbb{V}(x,y))$ and $\mathbb{W}=(\mathbb{W}(x),\mathbb{W}(x,y))$ be two representations of ${A}_{\mathbb{R}}$ . For $0\leq\varepsilon\in\mathbb{R}$ , a family of linear map $\alpha(x):\mathbb{V}(x)\rightarrow\mathbb{W}(x+\varepsilon)$ is called a morphism of degree $\varepsilon$ from $\mathbb{V}$ to $\mathbb{W}$ if the diagram

is commutative for any $x\leq y\in\mathbb{R}$ . Denoted by $\textrm{Hom}_{{A}_{\mathbb{R}}}^{\varepsilon}({\mathbb{V}},{\mathbb{W}})$ the set of morphisms of degree $\varepsilon$ from ${\mathbb{V}}$ to ${\mathbb{W}}$ .

An $\varepsilon$ -interleaving between $\mathbb{V}$ and $\mathbb{W}$ is two families of morphisms $\alpha\in\textrm{Hom}_{{A}_{\mathbb{R}}}^{\varepsilon}({\mathbb{V}},{\mathbb{W}})$ and $\beta\in\textrm{Hom}_{{A}_{\mathbb{R}}}^{\varepsilon}({\mathbb{W}},{\mathbb{V}})$ such that the diagrams

are commutative for any $x\in\mathbb{R}$ .

The interleaving distance of representations of ${A}_{\mathbb{R}}$ is defined as

d_{i,{A}_{\mathbb{R}}}(\mathbb{V},\mathbb{W})=\inf\{\varepsilon\geq 0\mid\textrm{there is an $\varepsilon$-interleaving between $\mathbb{V}$ and $\mathbb{W}$}\}.

For a pointwise finite-dimensional representation $\mathbb{V}=\bigoplus_{i\in I}{T}_{|a_{i},b_{i}|}$ of ${{A}}_{\mathbb{R}}$ . The set of intervals $|a_{i},b_{i}|$ for all $i\in I$ is called the persistence barcodes of $\mathbb{V}$ . The multiset $dgm(\mathbb{V})$ of points with coordinate $(a_{i},b_{i})$ is called the persistence diagram of $\mathbb{V}$ .

Let $A$ and $B$ be two multisets of points in the extended plane $\bar{\mathbb{R}}^{2}$ . A partial matching between $A$ and $B$ is a subset $P$ of $A\times B$ such that

(1)

there is at most one point $b\in B$ such that $(a,b)\in P$ for any $a\in A$ ;
(2)

there is at most one point $a\in A$ such that $(a,b)\in P$ for any $b\in B$ .

The bottleneck cost $c(P)$ of the partial matching $P$ is defined as

c(P)=\max\{\sup_{(a,b)\in P}||a-b||_{\infty},\sup_{s=(s_{x},s_{y})\in S}{\frac{s_{x}-s_{y}}{2}}\},

where $S$ is the set of unmatched points in $A\cup B$ . The bottleneck distance between $A$ and $B$ is defined as

d_{b,\mathbb{R}^{2}}(A,B)=\inf_{\textrm{partial matching }P}c(P).

Theorem 2.1 ([17]).

Let $\mathbb{V}$ and $\mathbb{W}$ be two pointwise finite-dimensional representations of continuous quiver ${{A}}_{\mathbb{R}}$ of type ${A}$ . Then,

d_{b,\mathbb{R}^{2}}(dgm(\mathbb{V}),dgm(\mathbb{W}))=d_{i,{A}_{\mathbb{R}}}(\mathbb{V},\mathbb{W}).

3. Continuous quivers of type $\tilde{A}$

3.1. Continuous quivers of type $\tilde{A}$

We define an equivalence ” $\sim$ ” on $\mathbb{R}$ , where $x\sim y$ if and only if $y-x\in\mathbb{Z}$ for any $x,y\in\mathbb{R}$ . Let $[x]$ be the equivalent class of $x$ for any $x\in\mathbb{R}$ and define $[x]<[y]$ if and only if $x<y$ and $|x-y|<\frac{1}{2}$ . Let $\tilde{{A}}_{\mathbb{R}}=(\mathbb{R}/{\sim},<)$ , which is called a continuous quiver of type $\tilde{A}$ .

A representation of $\tilde{{A}}_{\mathbb{R}}$ over $k$ is given by $\textbf{V}=(\textbf{V}([x]),\textbf{V}([x],[y]))$ , where $\textbf{V}([x])$ is a $k$ -vector space, and $\textbf{V}([x],[y]):\textbf{V}([x])\rightarrow\textbf{V}([y])$ is a $k$ -linear map for any $[x]<[y]\in\mathbb{R}/{\sim}$ satisfying that $\textbf{V}([x],[y])\circ\textbf{V}([y],[z])=\textbf{V}([x],[z])$ for any $[x]<[y]<[z]\in\mathbb{R}/{\sim}$ . Let $\textbf{V}=(\textbf{V}([x]),\textbf{V}([x],[y]))$ and $\textbf{W}=(\textbf{W}([x]),\textbf{W}([x],[y]))$ be representations of $\tilde{{A}}_{\mathbb{R}}$ . A family of $k$ -linear maps $f([x]):\textbf{V}([x])\rightarrow\textbf{W}([x])$ is called a morphism from V to W, if it satisfies $f([y])\textbf{V}([x],[y])$ = $\textbf{W}([x],[y])f([x])$ , for any $[x]<[y]$ .

Let $\textbf{V}=(\textbf{V}([x]),\textbf{V}([x],[y]))$ be a representation of $\tilde{{A}}_{\mathbb{R}}$ over $k$ . The representation V are called nilpotent, provided that the linear map $\textbf{V}([x_{n}],[x])\circ\textbf{V}([x_{n-1}],[x_{n}])\circ\cdots\circ\textbf{V}([x_{1}],[x_{2}])\circ\textbf{V}([x],[x_{1}])$ is nilpotent for any $[x]<[x_{1}]<[x_{2}]<\cdots<[x_{n-1}]<[x_{n}]<[x]$ .

Denoted by $\mathrm{Rep}^{nil}_{k}(\tilde{{A}}_{\mathbb{R}})$ the category of nilpotent representations of $\tilde{{A}}_{\mathbb{R}}$ and by $\mathrm{Rep}^{pwf}_{k}(\tilde{{A}}_{\mathbb{R}})$ the subcategory of pointwise finite-dimensional representations.

For each $a<b\in\mathbb{R}$ , consider a representation $\mathbf{T}_{|a,b|}=(\mathbf{T}_{|a,b|}([x]),\mathbf{T}_{|a,b|}([x],[y]))$ of $\tilde{{A}}_{\mathbb{R}}$ , defined by

\textbf{T}_{|a,b|}([x])=\bigoplus_{z\in[x]}{T}_{|a,b|}(z),

\textbf{T}_{|a,b|}([x],[y])=\bigoplus_{z_{1}\in[x],z_{2}\in[y],|z_{1}-z_{2}|<1}{T}_{|a,b|}(z_{1},z_{2}).

Since the representation $T_{|a,b|}$ is indecomposable, $\mathbf{T}_{|a,b|}$ is an indecomposable representation of $\tilde{{A}}_{\mathbb{R}}$ . The representation $\mathbf{T}_{|a,b|}$ are called interval representations.

Hanson and Rock proved the following theorem in [13].

Theorem 3.1 ([13]).

Let $\mathbf{V}$ be a pointwise finite-dimensional nilpotent representation of $\tilde{{A}}_{\mathbb{R}}$ . Then, it can be decomposed into the direct sum of indecomposable interval representations

\mathbf{V}=\bigoplus_{i\in I}\mathbf{T}_{|a_{i},b_{i}|}.

3.2. Isometry Theorem

Let $\mathbf{V}=(\mathbf{V}([x]),\mathbf{V}([x],[y]))$ , $\mathbf{W}=(\mathbf{W}([x]),\mathbf{W}([x],[y]))$ be two representations of $\tilde{{A}}_{\mathbb{R}}$ . For $0\leq\varepsilon<1\in\mathbb{R}$ , a family of linear maps $\alpha([x]):\mathbf{V}([x])\rightarrow\mathbf{W}([x+\varepsilon])$ is called a morphism of degree $\varepsilon$ from $\mathbf{V}$ to $\mathbf{W}$ if the diagram

is commutative for any $x\leq y\in\mathbb{R}$ such that $|x-y|<\frac{1}{2}$ . Denoted by $\textrm{Hom}_{\tilde{{A}}_{\mathbb{R}}}^{\varepsilon}({\mathbb{V}},{\mathbb{W}})$ the set of morphisms of degree $\varepsilon$ from ${\mathbf{V}}$ to ${\mathbf{W}}$ .

An $\varepsilon$ -interleaving between $\mathbf{V}$ and $\mathbf{W}$ is two morphisms $\alpha\in\textrm{Hom}_{\tilde{{A}}_{\mathbb{R}}}^{\varepsilon}({\mathbf{V}},{\mathbf{W}})$ and $\beta\in\textrm{Hom}_{\tilde{{A}}_{\mathbb{R}}}^{\varepsilon}({\mathbf{W}},{\mathbf{V}})$ such that the diagrams

are commutative for any $x\in\mathbb{R}$ .

The interleaving distance of $\tilde{{A}}_{\mathbb{R}}$ representations between $\mathbf{V}$ and $\mathbf{W}$ is defined as

d_{i,\tilde{{A}}_{\mathbb{R}}}(\mathbf{V},\mathbf{W})=\inf\{\varepsilon\geq 0\mid\textrm{there is an $\varepsilon$-interleaving between $\mathbf{V}$ and $\mathbf{W}$}\}.

Consider an equivalence ” $\sim$ ” on $\mathbb{R}^{2}$ , where $(x_{1},x_{2})\sim(y_{1},y_{2})$ if and only if $y_{1}-x_{1}=y_{2}-x_{2}\in\mathbb{Z}$ for any $(x_{1},x_{2}),(y_{1},y_{2})\in\mathbb{R}^{2}$ . Let $\mathbf{V}=\bigoplus_{i\in I}\mathbf{T}_{|a_{i},b_{i}|}$ be a pointwise finite-dimensional nilpotent representation of $\tilde{{A}}_{\mathbb{R}}$ . The multiset $dgm(\mathbf{V})$ in $\mathbb{R}^{2}/{\sim}$ consisting of equivalent classes of points with coordinate $(a_{i},b_{i})$ is called the persistence diagram of $\mathbf{V}$ .

Let $A$ and $B$ be two multisets in $\mathbb{R}^{2}/{\sim}$ . A partial matching between $A$ and $B$ is a subset $P$ of $A\times B$ such that

(1)

there is at most one $\bar{b}\in B$ such that $(\bar{a},\bar{b})\in P$ for any $\bar{a}\in A$ ;
(2)

there is at most one $\bar{a}\in A$ such that $(\bar{a},\bar{b})\in P$ for any $\bar{b}\in B$ .

The bottleneck cost $c(P)$ of the partial matching $P$ is defined as

c(P)=\max\{\sup_{(\bar{a},\bar{b})\in P}||\bar{a}-\bar{b}||_{\infty},\sup_{\bar{s}=\overline{(s_{x},s_{y})}\in S}{\frac{s_{x}-s_{y}}{2}}\},

where $||\bar{a}-\bar{b}||_{\infty}=\min_{a\in\bar{a},b\in\bar{b}}||a-b||_{\infty}$ and $S$ is the set of unmatched elements in $A\cup B$ . The bottleneck distance is defined as

d_{b,\mathbb{R}^{2}/{\sim}}(A,B)=\inf_{\textrm{partial matching }P}c(P).

The following theorem is the main result in this paper.

Theorem 3.2.

Let $\mathbf{V}$ and $\mathbf{W}$ be two pointwise finite-dimensional nilpotent representations of $\tilde{{A}}_{\mathbb{R}}$ . Then,

d_{b,,\mathbb{R}^{2}/{\sim}}(dgm(\mathbf{V}),dgm(\mathbf{W}))=d_{i,\tilde{{A}}_{\mathbb{R}}}(\mathbf{V},\mathbf{W})

The proof of this theorem will be given in the next section.

4. The proof of the main result

4.1. Continuous quivers of type $A$ with automorphism

In this section, we follow the notation in [12].

Consider a morphism $\sigma:\mathbb{R}\mapsto\mathbb{R}$ sending $x$ to $x+1$ . In this paper, $\textbf{Q}=(A_{\mathbb{R}},\sigma)$ is called a continuous quiver of type $A$ with automorphism $\sigma$ . A representation of Q over $k$ is given by a representation $\mathbb{V}=(\mathbb{V}(x),\mathbb{V}(x,y))$ of $A_{\mathbb{R}}$ such that $\mathbb{V}(x)=\mathbb{V}(\sigma(x))$ for any $x\in\mathbb{R}$ and $\mathbb{V}(x,y)=\mathbb{V}(\sigma(x),\sigma(y))$ for any $x\leq y\in\mathbb{R}$ .

For a representation $\mathbb{V}=(\mathbb{V}(x),\mathbb{V}(x,y))$ of $A_{\mathbb{R}}$ , consider a representation $\sigma^{*}(\mathbb{V})=(\sigma^{*}(\mathbb{V})(x),\sigma^{*}(\mathbb{V})(x,y))$ , where $\sigma^{*}(\mathbb{V})(x)=\mathbb{V}(\sigma(x))$ and $\sigma^{*}(\mathbb{V})(x,y)=\mathbb{V}(\sigma(x,y))$ . Note that $\bigoplus_{k\in\mathbb{Z}}(\sigma^{*})^{k}(\mathbb{V})$ is representation of $\mathbf{Q}$ . Let

\mathbb{T}_{|a,b|}=\bigoplus_{k\in\mathbb{Z}}(\sigma^{*})^{k}(T_{|a,b|}).

Since the representation $T_{|a,b|}$ is indecomposable, so is $\mathbb{T}_{|a,b|}$ . The representation $\mathbb{T}_{|a,b|}$ is called an interval representation of Q.

Denoted by $\mathrm{Rep}_{k}(\textbf{Q})$ the category of representations of Q. Denote $\mathrm{Rep}_{k}^{pwf}(\textbf{Q})$ the subcategory of $\mathrm{Rep}_{k}(\textbf{Q})$ consisting of pointwise finite-dimensional representations.

Note that there exists an equivalence

\Psi:\mathrm{Rep}_{k}(\tilde{{A}}_{\mathbb{R}})\rightarrow\mathrm{Rep}_{k}(\mathbf{Q})

such that $\Psi({\mathbf{V}})(x)={\mathbf{V}}([x])$ for any $x\in\mathbb{R}$ and $\Psi({\mathbf{V}})(x,y)={\mathbf{V}}([x],[y])$ for any $x<y\in\mathbb{R}$ such that $|x-y|<\frac{1}{2}$ . By definitions, we have

\Psi(\mathbf{T}_{|a,b|})=\mathbb{T}_{|a,b|}.

Let $\mathbb{V}$ be a pointwise finite-dimensional representation of Q. Theorem 3.1 implies that $\mathbb{V}$ can be decomposed into the direct sum of indecomposable interval representations

\mathbb{V}=\bigoplus_{i\in I}\mathbb{T}_{|a_{i},b_{i}|}.

For two representations $\mathbb{V}=(\mathbb{V}(x),\mathbb{V}(x,y))$ , $\mathbb{W}=(\mathbb{W}(x),\mathbb{W}(x,y))$ of $\mathbf{Q}$ and $0\leq\varepsilon\in\mathbb{R}$ , let $\textrm{Hom}_{\mathbf{Q}}^{\varepsilon}({\mathbb{V}},{\mathbb{W}})$ be the set consisting of morphisms $\alpha=(\alpha(x):\mathbb{V}(x)\rightarrow\mathbb{W}(x+\varepsilon))$ of degree $\varepsilon$ such that $\alpha(x)=\alpha(x+1)$ .

An $\varepsilon$ -interleaving of $\mathbf{Q}$ representations between $\mathbb{V}$ and $\mathbb{W}$ is two families of morphisms $\alpha\in\textrm{Hom}_{\mathbf{Q}}^{\varepsilon}({\mathbb{V}},{\mathbb{W}})$ and $\beta\in\textrm{Hom}_{\mathbf{Q}}^{\varepsilon}({\mathbb{W}},{\mathbb{V}})$ such that the following diagrams are commutative for any $x\in\mathbb{R}$ .

The interleaving distance of $\mathbb{V}$ and $\mathbb{W}$ is defined as

d_{i,\mathbf{Q}}(\mathbb{V},\mathbb{W})=\inf\{\varepsilon\geq 0\mid\textrm{there is an $\varepsilon$-interleaving between $\mathbb{V}$ and $\mathbb{W}$}\}.

4.2. The proof of the main result

Let $\sigma:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ be a map sending $(x,y)$ to $(x+1,y+1)$ .

Lemma 4.1.

Let $A$ and $B$ be two $\sigma$ -invariant multisets of points in $\mathbb{R}^{2}$ and denote $\bar{A}=\{\bar{a}|a\in A\}$ and $\bar{B}=\{\bar{b}|b\in B\}$ . Assume that $\bar{A}$ and $\bar{B}$ are finite multisets. Then we have

d_{b,\mathbb{R}^{2}}(A,B)=d_{b,\mathbb{R}^{2}/{\sim}}(\bar{A},\bar{B}),

Proof.

At first, we shall prove

d_{b,\mathbb{R}^{2}}(A,B)\geq d_{b,\mathbb{R}^{2}/{\sim}}(\bar{A},\bar{B}).

(4.1)

Assume that $d_{b,\mathbb{R}^{2}/{\sim}}(\bar{A},\bar{B})=\epsilon$ . Let $P$ be a partial matching between $A$ and $B$ . If $c(P)=\infty$ , then we have $c(P)\geq\epsilon$ . If $c(P)$ is finite, we shall prove $c(P)\geq\epsilon$ , too.

Denoted by $\bar{A}^{\prime}$ the subset of $\bar{A}$ consisting of $\bar{a}\in\bar{A}$ such that all points in $\bar{a}$ are matched points in $A$ . Let $\bar{B}^{\prime\prime}$ be the subset of $\bar{B}$ satisfying that

(1)

for any $\bar{b}\in\bar{B}^{\prime\prime}$ , there exist some $a^{\prime}\in\bar{a}\in\bar{A}^{\prime}$ and some $b^{\prime}\in\bar{b}$ such that $(a^{\prime},b^{\prime})\in P$ ;
(2)

for any $a^{\prime}\in\bar{a}\in\bar{A}^{\prime}$ and $b^{\prime}\in\bar{b}\not\in\bar{B}^{\prime\prime}$ , it holds that $(a^{\prime},b^{\prime})\not\in P$ .

Since $c(P)$ is finite, we have $|\bar{B}^{\prime\prime}|\geq|\bar{A}^{\prime}|$ . Then we can get a bijection $\tau$ between $\bar{A}^{\prime}$ and a subset $\bar{B}^{\prime\prime\prime}$ of $\bar{B}^{\prime\prime}$ satisfying that there are some $a^{\prime}\in\bar{a}$ and some $b^{\prime}\in\tau(\bar{a})$ with $(a^{\prime},b^{\prime})\in P$ for any $\bar{a}\in\bar{A}^{\prime}$ .

Similarly, denoted by $\bar{B}^{\prime}$ the subset of $\bar{B}$ consisting of $\bar{b}\in\bar{B}$ such that all points in $\bar{b}$ are matched points in $B$ . Let $\bar{A}^{\prime\prime}$ be a subset of $\bar{A}$ satisfying that

(1)

for any $\bar{a}\in\bar{A}^{\prime\prime}$ , there exist some $a^{\prime}\in\bar{a}$ and some $b^{\prime}\in\bar{b}\in\bar{B}^{\prime}$ such that $(a^{\prime},b^{\prime})\in P$ ;
(2)

for any $a^{\prime}\in\bar{a}\not\in\bar{A}^{\prime\prime}$ and $b^{\prime}\in\bar{b}\in\bar{B}^{\prime}$ , it holds that $(a^{\prime},b^{\prime})\not\in P$ .

Since $c(P)$ is finite, we have $|\bar{A}^{\prime\prime}|\geq|\bar{B}^{\prime}|$ . Then we can get a bijection $\theta$ between $\bar{B}^{\prime}$ and a subset $\bar{A}^{\prime\prime\prime}$ of $\bar{A}^{\prime\prime}$ satisfying that there are some $a^{\prime}\in\theta(\bar{b})$ and some $b^{\prime}\in\bar{b}$ with $(a^{\prime},b^{\prime})\in P$ for any $\bar{b}\in\bar{B}^{\prime}$ .

Let $\bar{A}^{\prime}=\{A_{1},A_{2},\ldots,A_{s}\}$ . For convenience, assume that $\tau(A_{i})\not\in\bar{B}^{\prime}$ if $i\leq l$ and $\tau(A_{i})\in\bar{B}^{\prime}$ if $i>l$ . Let

\bar{B}^{\prime}=\{\tau(A_{l+1}),\tau(A_{l+2}),\ldots,\tau(A_{s}),B_{1},B_{2},\ldots,B_{t}\}

and

\bar{A}^{\prime}-\theta(\bar{B}^{\prime})=\{A_{m_{1}},\ldots,A_{m_{r}}\}.

For any $1\leq i\leq r$ , we have a sequence

A_{m_{i}},\tau(A_{m_{i}}),\theta\tau(A_{m_{i}}),\tau\theta\tau(A_{m_{i}}),\ldots,X,

where $X=\tau(A_{p})$ with $p\leq l$ or $X=\theta(B_{q})\in\bar{A}-\bar{A}^{\prime}$ . In the first case, let

\bar{P}_{i}=\{(A_{m_{i}},\tau(A_{m_{i}})),(\theta\tau(A_{m_{i}}),\tau\theta\tau(A_{m_{i}})),\ldots,(A_{p},\tau(A_{p}))\}.

In the second case, let

\bar{P}_{i}=\{(A_{m_{i}},\tau(A_{m_{i}})),(\theta\tau(A_{m_{i}}),\tau\theta\tau(A_{m_{i}})),\ldots,(A_{q},B_{q})\}.

Let

\bar{P}_{0}=\{(\theta(X),X)|X\in\bar{B}^{\prime}\textrm{ and $(-,X)\not\in\bar{P}_{i}$ for all $i$}\}.

Let

\bar{P}=\bar{P}_{0}\cup\bar{P}_{1}\cup\cdots\cup\bar{P}_{r}.

Note that $\bar{P}$ is a partial matching between $\bar{A}$ and $\bar{B}$ . Hence, we have $c(\bar{P})\geq\epsilon$ by definition.

Since $||a-b||_{\infty}\geq||\bar{a}-\bar{b}||_{\infty}$ for any $(a,b)\in P$ , we have

\sup_{(a,b)\in{P}}||a-b||_{\infty}\geq\sup_{(\bar{a},\bar{b})\in\bar{P}}||\bar{a}-\bar{b}||_{\infty}.

(4.2)

Meanwhile, the point $\bar{s}$ is unmatched in $\bar{A}\cup\bar{B}$ implies that $s^{\prime}$ is unmatched in $A\cup B$ for any $s^{\prime}\in\bar{s}$ . Let $S$ and $\bar{S}$ be the sets of unmatched elements in $A\cup B$ and $\bar{A}\cup\bar{B}$ respectively. Hence, we have

\sup_{s=(s_{x},s_{y})\in S}{\frac{s_{x}-s_{y}}{2}}\geq\sup_{\bar{s}=\overline{(s_{x},s_{y})}\in\bar{S}}{\frac{s_{x}-s_{y}}{2}}.

(4.3)

Formulas 4.2 and 4.3 implies that $c(P)\geq c(\bar{P})$ and we get (4.1).

Then, we shall prove

d_{b,\mathbb{R}^{2}}(A,B)\leq d_{b,\mathbb{R}^{2}/{\sim}}(\bar{A},\bar{B}).

(4.4)

Assume that $d_{b,\mathbb{R}^{2}}(A,B)=\epsilon$ . Let $\bar{P}$ be a partial matching between $\bar{A}$ and $\bar{B}$ . For any $(\bar{a},\bar{b})\in\bar{P}$ and any $a^{\prime}\in\bar{a}$ , there exists $b^{\prime}\in\bar{b}$ such that $||a^{\prime}-b^{\prime}||_{\infty}=||\bar{a}-\bar{b}||_{\infty}$ . Let $P$ be the set of all $(a^{\prime},b^{\prime})$ considered above. Since the sets $A$ and $B$ are $\sigma$ -invariant, $\bar{a}$ is a subset of $A$ , $\bar{b}$ is a subset of $B$ and $P$ is a partial matching between ${A}$ and ${B}$ . Hence, we have $c({P})\geq\epsilon$ by definition.

Since $||a^{\prime}-b^{\prime}||_{\infty}=||\bar{a}-\bar{b}||_{\infty}$ for any $(\bar{a},\bar{b})\in\bar{P}$ , we have

\sup_{(a,b)\in{P}}||a^{\prime}-b^{\prime}||_{\infty}=\sup_{(\bar{a},\bar{b})\in\bar{P}}||\bar{a}-\bar{b}||_{\infty}

(4.5)

by constructions. Meanwhile, the point $\bar{s}$ is unmatched in $\bar{A}\cup\bar{B}$ if and only if $s^{\prime}$ is unmatched in $A\cup B$ for any $s^{\prime}\in\bar{s}$ . Let $S$ and $\bar{S}$ be the sets of unmatched elements in $A\cup B$ and $\bar{A}\cup\bar{B}$ respectively. Hence, we have

\sup_{s=(s_{x},s_{y})\in S}{\frac{s_{x}-s_{y}}{2}}=\sup_{\bar{s}=\overline{(s_{x},s_{y})}\in\bar{S}}{\frac{s_{x}-s_{y}}{2}}.

(4.6)

Formulas 4.5 and 4.6 implies that $c(P)=c(\bar{P})$ and we get (4.4).

By Formulas 4.1 and 4.4, we get the desired result.

∎

Proposition 4.2.

For two pointwise finite-dimensional nilpotent representations $\mathbf{V}$ and $\mathbf{W}$ of $\tilde{{A}}_{\mathbb{R}}$ , we have

d_{i,\tilde{{A}}_{\mathbb{R}}}(\mathbf{V},\mathbf{W})\geq d_{b,\mathbb{R}^{2}/{\sim}}(dgm(\mathbf{V}),dgm(\mathbf{W})).

Proof.

Since $\Psi$ is an equivalence between $\mathrm{Rep}^{pwf}_{k}(\tilde{{A}}_{\mathbb{R}})$ and $\mathrm{Rep}^{pwf}_{k}(\mathbf{Q})$ , we have

d_{i,\tilde{{A}}_{\mathbb{R}}}(\mathbf{V},\mathbf{W})=d_{i,\mathbf{Q}}(\Psi(\mathbf{V}),\Psi(\mathbf{W})).

(4.7)

Note $\textrm{Hom}_{\mathbf{Q}}^{\varepsilon}(\Psi(\mathbf{V}),\Psi(\mathbf{W}))$ is a subset of $\textrm{Hom}_{{A}_{\mathbb{R}}}^{\varepsilon}(\Psi(\mathbf{V}),\Psi(\mathbf{W}))$ . Hence,

d_{i,\mathbf{Q}}(\Psi(\mathbf{V}),\Psi(\mathbf{W}))\geq d_{i,{A}_{\mathbb{R}}}(\Psi(\mathbf{V}),\Psi(\mathbf{W})).

By using Theorem 2.1, we have

d_{i,{A}_{\mathbb{R}}}(\Psi(\mathbf{V}),\Psi(\mathbf{W}))=d_{b,\mathbb{R}^{2}}(dgm(\Psi(\mathbf{V})),dgm(\Psi(\mathbf{W}))).

Since $dgm(\Psi(\mathbf{V}))$ and $dgm(\Psi(\mathbf{W}))$ are $\sigma$ -invariant multisets such that $\overline{dgm(\Psi(\mathbf{V}))}$ and $\overline{dgm(\Psi(\mathbf{W}))}$ are finite, Lemma 4.1 implies that

d_{b,\mathbb{R}^{2}}(dgm(\Psi(\mathbf{V})),\Psi(dgm(\mathbf{W})))=d_{b,\mathbb{R}^{2}/{\sim}}(dgm(\mathbf{V}),dgm(\mathbf{W})).

(4.8)

Hence, we get the desired result.

∎

Lemma 4.3 ([17]).

Let $|p,q|$ , $|r,s|$ be two intervals. Then,

d_{i,{A}_{\mathbb{R}}}(T_{|p,q|},T_{|r,s|})\leq||a-b||_{\infty},

where $a=(p,q)$ , $b=(r,s)$ are the corresponding points in $\mathbb{R}^{2}$ .

Lemma 4.4.

Let $|p,q|$ , $|r,s|$ be two intervals. Then,

d_{i,\mathbf{Q}}(\mathbb{T}_{|p,q|},\mathbb{T}_{|r,s|})\leq||\bar{a}-\bar{b}||_{\infty}.

where $a=(p,q)$ , $b=(r,s)$ are the corresponding points in $\mathbb{R}^{2}$ and $\bar{a}$ , $\bar{b}$ are the equivalent classes in $\mathbb{R}^{2}/{\sim}$ .

Proof.

By definition of $||\bar{a}-\bar{b}||_{\infty}$ , we can choose $a^{\prime}=(p^{\prime},q^{\prime})\in\bar{a}$ and $b^{\prime}=(r^{\prime},s^{\prime})\in\bar{b}$ such that

||a^{\prime}-b^{\prime}||_{\infty}=||\bar{a}-\bar{b}||_{\infty}.

(4.9)

By Lemma 4.3, there is a $\varepsilon$ -interleaving $(\alpha,\beta)$ of $T_{|p^{\prime},q^{\prime}|}$ and $T_{|r^{\prime},s^{\prime}|}$ such that

\varepsilon\leq||a^{\prime}-b^{\prime}||_{\infty}.

(4.10)

Since

\mathbb{T}_{|p,q|}=\bigoplus_{k\in\mathbb{Z}}(\sigma^{*})^{k}(T_{|p^{\prime},q^{\prime}|})

and

\mathbb{T}_{|r,s|}=\bigoplus_{k\in\mathbb{Z}}(\sigma^{*})^{k}(T_{|r^{\prime},s^{\prime}|}),

the pair $(\bigoplus_{k\in\mathbb{Z}}(\sigma^{*})^{k}(\alpha),\bigoplus_{k\in\mathbb{Z}}(\sigma^{*})^{k}(\beta))$ of maps is a $\varepsilon$ -interleaving of $\mathbb{T}_{|p,q|}$ and $\mathbb{T}_{|r,s|}$ . Hence,

d_{i,\mathbf{Q}}(\mathbb{T}_{|p,q|},\mathbb{T}_{|r,s|})\leq\varepsilon.

(4.11)

Then, Formulas (4.9), (4.10) and (4.11) imply the desired result.

∎

Similarly to the proof of the converse stability part of Theorem 2.1 in [17], we have the following lemma.

Lemma 4.5.

For two representations $\mathbb{V}$ and $\mathbb{W}$ of $\mathbf{Q}$ , we have

d_{i,\mathbf{Q}}(\mathbb{V},\mathbb{W})\leq d_{b,\mathbb{R}^{2}}(dgm(\mathbb{V}),dgm(\mathbb{W})).

Proof.

Let $\mathbb{V}$ and $\mathbb{W}$ be two representations of Q and $d_{b,\mathbb{R}^{2}}(dgm(\mathbb{V}),dgm(\mathbb{W}))=\varepsilon$ . By definitions, the representations $\mathbb{V}$ and $\mathbb{W}$ can be wrote as direct sums of indecomposable interval representations as follows

\mathbb{V}=\bigoplus_{i\in I_{1}}\mathbb{V}_{i}\oplus\bigoplus_{j\in J}\mathbb{V}_{j},

\mathbb{W}=\bigoplus_{i\in I_{2}}\mathbb{W}_{i}\oplus\bigoplus_{j\in J}\mathbb{W}_{j},

and there is a partial matching between the direct summands of $\mathbb{V}$ and those of $\mathbb{W}$ such that

(1)

$\mathbb{V}_{j}$ is matched with $\mathbb{W}_{j}$ for any $j\in J$ ,
(2)

$\mathbb{V}_{i}$ is unmatched for any $i\in I_{1}$ ,
(3)

$\mathbb{W}_{i}$ is unmatched for any $i\in I_{2}$ .

By Lemma 4.4, we have $d_{i,\mathbf{Q}}(\mathbb{V}_{j},\mathbb{W}_{j})\leq\varepsilon$ .

If $\mathbb{V}=\mathbb{V}_{1}\bigoplus\mathbb{V}_{2}$ and $\mathbb{W}=\mathbb{W}_{1}\bigoplus\mathbb{W}_{2}$ , then

d_{i,\mathbf{Q}}(\mathbb{V}_{1}\bigoplus\mathbb{V}_{2},\mathbb{W}_{1}\bigoplus\mathbb{W}_{2})\leq max\{d_{i,\mathbf{Q}}(\mathbb{V}_{1},\mathbb{W}_{1}),d_{i,\mathbf{Q}}(\mathbb{V}_{2},\mathbb{W}_{2})\}.

Hence, we have $d_{i,\mathbf{Q}}(\mathbb{V},\mathbb{W})\leq\varepsilon$ .

∎

Proposition 4.6.

For two representations $\mathbf{V}$ and $\mathbf{W}$ of $\tilde{{A}}_{\mathbb{R}}$ . we have

d_{i,\tilde{{A}}_{\mathbb{R}}}(\mathbf{V},\mathbf{W})\leq d_{b,\mathbb{R}^{2}/{\sim}}(dgm(\mathbf{V}),dgm(\mathbf{W}))

Proof.

By Lemma 4.5, we have

d_{i,\mathbf{Q}}(\Psi(\mathbf{V}),\Psi(\mathbf{W}))\leq d_{b,\mathbb{R}^{2}}(dgm(\Psi(\mathbf{V})),dgm(\Psi(\mathbf{W}))).

Then, the desired result is implied by Formulas (4.7) and (4.8).

∎

Propositions 4.2 and 4.6 imply Theorem 3.2.

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Isometry Theorem for Continuous Quiver of Type A~\tilde{A}

Abstract.

Key words and phrases:

2010 Mathematics Subject Classification:

1. Introduction

2. Continuous quivers of type A{A}

2.1. Continuous quivers of type A{A}

2.2. Isometry Theorem

Theorem 2.1 ([17]).

3. Continuous quivers of type A~\tilde{A}

3.1. Continuous quivers of type A~\tilde{A}

Theorem 3.1 ([13]).

3.2. Isometry Theorem

Theorem 3.2.

4. The proof of the main result

4.1. Continuous quivers of type AA with automorphism

4.2. The proof of the main result

Lemma 4.1.

Proof.

Proposition 4.2.

Proof.

Lemma 4.3 ([17]).

Lemma 4.4.

Proof.

Lemma 4.5.

Proof.

Proposition 4.6.

Proof.

References

Isometry Theorem for Continuous Quiver of Type $\tilde{A}$

2. Continuous quivers of type ${A}$

2.1. Continuous quivers of type ${A}$

3. Continuous quivers of type $\tilde{A}$

3.1. Continuous quivers of type $\tilde{A}$

4.1. Continuous quivers of type $A$ with automorphism