Five Lectures on Cluster Theory Ray Maresca Abstract

2025-04-30 0 0 538.08KB 38 页 10玖币

侵权投诉

Five Lectures on Cluster Theory

Ray Maresca

Abstract

In this paper, we will present the author’s interpretation and embellishment of ﬁve

lectures on cluster theory given by Kiyoshi Igusa during the Spring semester of 2022 at

Brandeis University. They are meant to be used as an introduction to cluster theory

from a representation-theoretic point of view.

1 Introduction

1.1 Some History

Since its introduction in the early 2000’s by Fomin and Zelevinsky, cluster theory has

been an active area of research. Some of the ﬁrst results in cluster theory, such Fomin and

Zelevinsky’s classiﬁcation of ﬁnite type cluster algebras in [17], bore striking resemblances to

theorems in representation theory such as Gabriel’s theorem in [20] and [21]. In particular,

every seed or cluster in a cluster algebra has some number, say

, cluster variables. This

allows us to represent the variables of a cluster algebra in a graph in which there is an edge

between any two variables that occur in a cluster. In this graph, which we will see in Section

4, we see that if we have

n−

1 cluster variables connected by edges, there are precisely 2

ways to complete the the graph.

On the other hand, many of these phenomena were also being seen in representation theory.

For instance in [40], Skowro´nski showed that every basic tilting module has

indecomposable

summands. Moreover, Happel and Unger showed in [24] that every basic partial tilting

module having

n−

1 indecomposable direct summands can always be completed into a tilting

module and that there are at most two ways that this can be done. Therefore, one may think

that the ‘right’ way to connect cluster theory and representation theory is through tilting

theory, though this fails in several ways, one being that there are more cluster variables then

there are indecomposable rigid modules and more clusters than tilting modules.

To attain a categoriﬁcation of cluster theory in terms of representation theory, we thus

need to extend the module category in some way. This was done by Buan, Marsh, Reineke,

Reiten, and Todorov in [8] where they constructed a larger category called the cluster category

in which the original module category can be embedded. In [8], it was assumed that the

quiver

associated to the initial seed was acyclic, so the path algebra is hereditary, which is

an assumption we will also make throughout this paper. This need not be the case and in [2],

Amoit removes the condition of

being hereditary and constructs the cluster category for

non-hereditary algebras of global dimension 2 and quivers with potential.

There however is still one thing to notice. In the cluster category of a hereditary al-

gebra, we have the following isomorphism from Auslander-Reiten duality: Ext

(

M, N

)

∼

Hom(

N, τM

). One can show that this isomorphism actually also holds in module categories

arXiv:2210.05717v1 [math.RT] 11 Oct 2022

of non-hereditary cluster algebras. Therefore, there is a correspondence between the tilting

objects in the cluster category and modules in the module category of a cluster-tilted algebra.

The issue with this is that these modules may not be partial tilting objects due to having

inﬁnite projective dimension. By dropping the requirement on projective dimension and

loosening rigidity to

-rigidity, Adachi, Iyama, and Reiten introduced

-tilting theory in [1].

1.2 Framework of These Notes

Although

-tilting theory is one of the most active areas of current research, in this paper,

we will focus on classical tilting theory and cluster theory. For a survey on

-tilting theory,

we suggest [43] by Treﬃnger, where many of the references and much of the background

information in these notes were found. In this article, we will illuminate some connections

between cluster theory and representation theory while working through the process of

categorifying cluster theory when the initial seed corresponds to a quiver with no loops or

two cycles. We will do this by working through the author’s interpretation and embellish-

ment of 5 lectures given by Kiyoshi Igusa during the spring semester of 2022 at Brandeis

University which contain several motivating examples and provide some intuition behind

results. One thing to note is that all proofs in the ﬁrst six sections of this paper are meant

to provide the main idea and intuition behind the proof and should be taken as nothing

more than sketches. We will assume that the reader has some background in the founda-

tions of representation theory and suggest [3], [12], [38], and [39] as references for this material.

We will begin these notes with Section 2 in which we give some examples that provide

intuition behind what a cluster algebra is and how clusters and cluster variables are connected

to representation theory. We then provide deﬁnitions of cluster algebras and mutations in

terms of a quiver

. In particular, we will see a connection between cluster variables (charac-

ters) and the Auslander-Reiten (AR) quiver of the corresponding initial quiver. Afterward in

Section 3 we will explicitly provide the correspondence by showing how to associate a cluster

character to a

-module. We will moreover see how the coeﬃcients of this cluster character

are related to the module itself.

In Section 4 we will introduce two questions that we will spend the rest of the notes trying

to answer; namely, which sets of modules are sent to clusters and which algebraic objects

correspond to the initial cluster variables? These two questions motivated the categoriﬁcation

of cluster theory in terms of representation theory. It is here we will introduce the notion of

tilting modules and how they fall short of describing cluster theory in its entirety. We will

need to extend the idea of a tilting module by introducing support tilting modules, shifted

projectives, and silting pairs. We will do this without introducing the bounded derived

category or explicitly creating the cluster category. At this point, we will provide the bijection

between clusters and silting pairs. We will ﬁnish this section by constructing the wall and

chamber structure (stability picture) and show how this structure connects to cluster theory.

After this, in Section 5, we will describe in fact why the stability picture introduced in

the previous section is accurate by showing that rigidity is a Zariski open condition. To do

this, we will introduce the category of 2-term silting complexes. We ﬁnish this section by

introducing the notion of stable barcodes which won’t actually be used for the remainder of

the notes. In the ﬁnal section, Section 6, we introduce the notion of maximal green sequences

using exchange matrices and ice quivers. The deﬁnition of these sequences rely on notions

like sign coherence of

-vectors and

-vectors, which we will also explain in this section.

Throughout the notes, we will attempt to provide as much referencing as possible to both

history and proofs of the results.

Before we begin, we would like to remark that not all connections between cluster and

representation theory are made in these notes. For instance, there is a beautiful connection

between functorially ﬁnite torsion classes in mod-

and clusters, namely that they are in

bijection. One way to see this is through the fact that the dual graph of the stability picture

is precisely the Hasse quiver of functorially ﬁnite torsion classes in mod-

. For more on

the lattice of torsion classes, we suggest Thomas’s exposition [41]. For more details on the

bijection, Treﬃnger’s survey [43] is a great place to start.

2 Cluster Theory

2.1 Examples

Before presenting the formal deﬁnition of a cluster algebra, we provide some intuition,

motivation, and examples. Intuitively, ‘clusters’ are sets of

objects which can be mutated.

Each of these

objects are a cluster variable or cluster character. Throughout these lectures,

we will provide two methods of thinking about clusters, the former is the original deﬁnition

and the later is a categoriﬁcation of it.

(a) Clusters are transcendence bases for Q(x1, x2, . . . , xn) given by a quiver Q.

(b) Clusters are objects in a category of the form T=T1⊕T2⊕ · · · ⊕ Tn.

One relationship between the above two methods is through something called the ‘cluster

character’ denoted by

(

)

∈Q

(

x1, x2, . . . , xn

). We begin our studies of cluster algebras

using method (a) through an example.

Until otherwise stated, let

be the quiver 2

→

←

3. Note that this quiver consists of

only

descending

arrows, that is, there is an arrow from

if and only if

j < i

. Below

are three depictions of the Auslander-Reiten (AR) quiver for this quiver. Note that the maps

between the projectives are ascending with respect to their vertices; that is, there is an arrow

from

in the AR quiver if and only if

i < j

. This is one reason to always take the

arrows to be descending when the quiver does not have oriented cycles. Moreover, note that

the quiver formed by the three projectives in the AR quiver is the opposite quiver of

. On

the top left we have the standard projective/injective at vertex

notation. On the top right

we have another standard notation indicating the tops and socles of the modules on the left.

Finally, below these two is a depiction of the AR quiver using dimension vectors. In each

quiver the dotted lines indicate the AR translate τ:







2 3



(1,1,0)

(0,0,1)

(1,0,0)

(1,1,1)

(1,0,1)

(0,1,0)

The form that uses the dimension vectors is computationally useful and sheds some light

onto the relationship between AR theory and cluster theory. Recall that given a short exact

sequence of modules 0

→A→B→C→

0, the dimension vectors satisfy the equality

dim

+ dim

= dim

. Since each mesh in the above AR quiver forms an almost split

sequence, which is short exact, we can construct the AR quiver using this relationship and in

some sense extend it as follows.

(1,1,0)

(0,0,1)

(−1,0,−1)

(1,0,0)

(1,1,1)

(−1,0,0)

(1,0,1)

(0,1,0)

(−1,−1,0)

These newly added vectors are not dimension vectors in the usual sense since they

contain negative entries. From a representation theoretic point of view, we will see in

Section 4 that these negative ‘dimension vectors’ correspond to ‘shifted projectives’ in

the categoriﬁcation method of studying cluster algebras. Now to the dimension vector

(

i1, i2, . . . , in

), we associate the symbol

xi1

1xi2

2. . . xin

. Then from the aforementioned additive

relationship given by the dimension vectors, we have that given a short exact sequence of

modules 0

→A→B→C→

0, the symbols satisfy the equality

xdimAxdimC

xdimB

. In

this notation, by

xdimA

we mean

i=1 xdimAi

where

dimAi

is the

th entry in the dimension

vector of

. Then the above AR quiver can be rewritten in terms of the symbols as follows.

x1x2



x1x3

x1x2x3



x1x3

;;

x1x2

2.2 Cluster Variables

These symbols are the cluster variables or cluster characters for the cluster algebra whose

initial quiver is

. Though we have not yet deﬁned the cluster variables, they satisfy a nice

property. As was shown by Caldero and Chapoton in [9], given an almost split sequence of

modules 0

→A→B→C→

0, the corresponding cluster characters satisfy the relationship

χ(A)χ(C) = χ(B)+1

where

(

) =

(

⊕iBi

) =

Piχ

(

). Note that if the sequence is neither split nor almost

split, we would need to add more than one on the right hand side and if the sequence is

split, we would not need to add anything, providing some intuition on why the sequences are

almost split but not split. We will see in Section 3 where this plus 1 is coming from. Using

this formula for cluster characters, we can reconstruct the AR quiver for our type

quiver

, along with those ‘shifted projectives’, for the corresponding path algebra by beginning

with the opposite quiver as follows.



>><<

To ﬁll the left most oval, we need a cluster variable

such that

x1y

x2x3

+ 1, so

y=x2x3+1

x1. We get:

!!!!



x2x3+1

====

Continuing in this way will provide us the entire AR quiver and more for

in terms of the

cluster variables, though it will get quite messy quickly. To reduce unnecessary computations,

we examine a simpler example. Let Qnow be the quiver 1 ←2. Then the AR quiver is

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

FiveLecturesonClusterTheoryRayMarescaAbstractInthispaper,wewillpresenttheauthor'sinterpretationandembellishmentofvelecturesonclustertheorygivenbyKiyoshiIgusaduringtheSpringsemesterof2022atBrandeisUniversity.Theyaremeanttobeusedasanintroductiontoclustertheoryfromarepresentation-theoreticpointofview....

展开>> 收起<<

Five Lectures on Cluster Theory Ray Maresca Abstract.pdf

共38页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Five Lectures on Cluster Theory Ray Maresca Abstract

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: