Categorification of skew-symmetrizable cluster algebras

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Categorification of skew-symmetrizable cluster algebras

Laurent Demonet

To cite this version:

Laurent Demonet. Categorification of skew-symmetrizable cluster algebras. 2009. �hal-00414224�

LAURENTDEMONET

Abstrat. We propose anewframeworkfor ategorifying skew-symmetrizableluster alge-

bras. Startingfromanexatstably

2

^{-Calabi-Yauategory}

C

endowedwiththeationofanite group

Γ

^,weonstruta

Γ

-equivariantmutationonthesetofmaximalrigid

Γ

^{-invariantobjets}

of

C

. Using an appropriate luster harater, we anthen attah to these data an expliit skew-symmetrizablelusteralgebra. Asanappliationweprovethelinearindependeneofthe

lustermonomialsinthissetting. Finally,weillustrateouronstrutionwithexamplesassoi-

atedwithpartialagvarietiesandunipotentsubgroupsofKa-Moodygroups,generalizingto

thenonsimply-laedaseseveralresultsofGeiÿ-Leler-Shröer.

Contents

1. Introdution 1

1.1. Cluster algebras 1

1.2. Ationsof groupson ategories 2

1.3. Categoriationof skew-symmetrizable lusteralgebras 4

1.4. Appliations 5

2. Equivariant ategories 5

2.1. Denitionsandrst properties 5

2.2. A

mod k

^r

Γ

^{s-module strutureon}

C Γ

¹⁰

2.3. A

k

^r

Γ

^{s-linearstruture on the}equivariant ategory 11

2.4. Thefuntors r

Γ

^sand

F

¹³

2.5. Approximations 17

2.6. Ation onan exat ategory 18

2.7. Ation ona Frobenius stably

2

^{-Calabi-Yau ategory 20}

2.8. Computationofp

kQ

^q

Γ

^{and p}

Λ

Q^q

Γ

²¹

3. Categoriationof skew-symmetrizable lusteralgebras 22

3.1. Mutationof maximal

Γ

^-stablerigid subategories 23

3.2. Rigidquasi-approximations 28

3.3. Endomorphisms 30

3.4. Exhange matries 36

3.5. Cluster haraters 38

3.6. Linearindependeneof lustermonomials 41

4. Appliations 44

4.1. Reminderaboutroot systems andenveloping algebras 44

4.2.

Sub I

J ^{and partialag varieties 45}

4.3. Categories

C

_M and unipotent groups 56

Aknowledgments 62

Referenes 62

1. Introdution

1.1. Clusteralgebras. In2001,FominandZelevinskyintroduedanewlassofalgebrasalled

luster algebras [FZ1 ℄, [FZ2 ℄ motivated by anonial bases and total positivity [Lus1 ℄, [Lus3℄.

Byonstrution, alusteralgebraisaommutative ringendowed withdistinguished generators

disjoint. On the ontrary, eah luster

A

^{has neighbours obtained by replaing eah of its}

variables

x

_i ^{byanewvariable}

x

¹_i^{. Thenewluster}

µ

_i^p

A

^q

A

^zt

x

_i^uYt

x

¹_i^{u isalledthemutation}

of

A

^{inthediretion}

x

i^{. Moreover, themutations arealways oftheform}

x

_i

x

¹_i

M

_i

M

_i¹

,

where

M

i ^and

M

_i^{1 aremonomials inthevariables of}

A

^otherthan

x

i^{. Theaxioms imply strong}

ompatibilityrelations between themonomialsoftheexhangerelations. Inpartiular,aluster

algebra isfullydeterminedbyaseed, thatis,a singlelusterandits exhange relations withall

its neighbours. In pratie, one usually denes a luster algebra bygiving an initial seed. By

iterating theexhange relations,one an expressevery lustervariableintermsofthevariables

of the initial seed.

Berenstein, Fomin and Zelevinsky have shown that the oordinate rings of many algebrai

varieties attahed to omplex semi-simple Lie groups were endowed with the struture of a

luster algebra [BFZ℄. Other examples have been given by Geiÿ, Leler and Shröer [GLS6 ℄,

[GLS1 ℄.

Sine their emergene, luster algebras have aroused a lot of interest, oming in partiular

fromtheirlinkswithmanyothersubjets: ombinatoris(seeforinstane[CFZ℄,[FST ℄),Poisson

geometry [GSV1℄,[GSV2℄,integrable systems[FZ3℄, Teihmüller spaes [FG℄, and,last butnot

least, representationsof nite-dimensional algebras.

Unfortunately, beause of the indutive desription of luster algebras, many properties of

thelustervariables whihmight seem elementary areinfatveryhard to prove. For instane:

Conjeture 1.1 (Fomin-Zelevinsky). Cluster monomials (that is produts of luster variables

of a single luster) are linearly independent.

Inseminalartiles,Marsh,Reineke,Zelevinsky[MRZ℄,Buan,Marsh,Reineke,Reiten,Todorov

[BMR ℄andCaldero, Chapoton[CC ℄have shownthattheimportant lassofayli lusteral-

gebrasouldbemodelledwithategoriesonstrutedfromrepresentationsofquivers. Thisgives

inpartiular a global (i.e. non indutive)understanding of these algebras, and givesnew tools

forstudyingthem. Forexample,thisallowedFuandKeller[FK℄toprovethepreviousonjeture

for afamily of lusteralgebras ontaining ayli lusteralgebras.

At the same time, Geiÿ, Leler and Shröer have studied luster algebras assoiated with

Lie groups of type

A

^,

D

^,

E

^{, and have modelled them by ategories of modules over Gelfand-}

Ponomarev preprojetive algebras of the same type. Theyhave shown that luster monomials

formasubset ofthedualsemi-anonialbasis[GLS4 ℄introdued byLusztig[Lus4 ℄,provingthe

above onjetureinthis otherontext.

More reently, Derksen, Weyman and Zelevinsky [DWZ2 ℄, [DWZ1 ℄ have obtained a far-

reahinggeneralization of[MRZ℄,whihalsoontains alltheaboveexamples. Theyhaveshown

that one an ontrol

F

-polynomials and

g

^{-vetors of every luster algebra whose initial seed}

is enoded bya skew-symmetri matrix, using representations of quivers with potentials. This

enabledthemto provethelinearindependeneonjeture,aswellasmanyotheronjetureson

F

-polynomials and

g

^{-vetors formulated in[FZ4℄.}

ButthetheoryofFominandZelevinskyinludesmoregeneralseedsgivenbyskew-symmetri-

zable matries(i.e. produts of a skew-symmetri matrix bya diagonal matrix). For example,

luster algebras assoiated to Lie groups of type

B

^,

C

^,

F

^,

G

^{are only} skew-symmetrizable.

The aim of this artile is to extend theresults of Geiÿ, Leler, Shröer and Fu, Keller to the

skew-symmetrizablease.

1.2. Ations of groups on ategories. It is helpful to view a skew-symmetri matrix

M

€ r

m

r_ij^sP

M

_n^p

Z

^{qasanorientedgraph}

Q

^{(i.e. aquiver)withvertexset}

Q

₀ ^t

1, 2, . . . , n

^uand

m

^r_ij

arrows from

i

^to

j

^if

m

^r_ij ^¡

0

^{(resp. from}

j

^to

i

^if

m

^r_ij  

0

^{). If a group}

Γ

^{ats on}

Q

^{, one an}

assoiatewithitanewmatrix

M

^{indexedbytheorbit set}

Q

0^{

Γ

^,bydening

m

_ij ^asthenumber

of arrows of

Q

^{between a xed vertex}

j

^{of the orbit}

j

^{and any vertex of the orbit}

i

^(ounted

positively if the arrows go from

i

^to

j

^{and negatively if they go from}

j

^to

i

^{). It is easy to see}

that

M

^is skew-symmetrizable. For example,if

M

€

0 0 0 1

0 0 0 1

0 0 0 1

1

1

1 0

Æ

Æ

thenthequiver

Q

^{isof type}

D

₄

1

?

? ?

? ?

? ?

?

2 ^// 4

3.

?? 

 

 

 



There isanarrowfrom

1

^{to the orbitof}

4

^hene

m

₄₁

1

^{and therearethreearrowsfromthe}

orbit of

1

^to

4

^hene

m

₁₄

3

^{. Thus we obtain thematrix}

M

0 3

1 0

of type

G

₂

1

^<

4.

Hene the ation of a group

Γ

^{on a} skew-symmetri matrix

M

€ ^{gives rise to a} skew-symme- trizable matrix

M

^{. If}

M

^{€ is the initial seed of a luster algebra}

A

^r ategoried as before by a ategory

C

,itisnatural to tryto ategorifythelusteralgebra

A

withseed

M

^byaategory

C

¹

onstruted from

C

and thegroup

Γ

^.

Thisleadsto study

k

^{-additive ategories}

C

on whih agroup

Γ

^atsbyauto-equivalenes. In thissituation, onean forma ategory

C Γ

^{whose objetsarepairs p}

X,

^p

ψ

_g^q_gPΓ^{qonsistingof an}

objet

X

^of

C

isomorphially invariant under

Γ

^{,togetherwithafamily of}isomorphisms

ψ

_g ^from

X

^{to eah of its images by the elements}

g

^of

Γ

^{. One also requires that the}

ψ

g ^{satisfynatural}

ompatibilityonditions. Theategory

C Γ

^{will be alledthe}

Γ

-equivariant ategory. One thenshowsuseful results oftransfer. For example:

if

C

is abelian, then

C Γ

^isabelian;

if

C

is exat andiffor all

g

^P

Γ

^the auto-equivaleneof

C

assoiated to

g

^{is exat,then}

C Γ

^isexat;

if

H

^{is a normal subgroup of}

Γ

^then

Γ

^{

H

^{ats on}

C H

^{and one has an equivalene of}

ategories p

C H

^qp

Γ

^{

H

^q

C Γ

^.

We also prove that

C Γ

^{an be endowed with a natural ation of the ategory}

mod k

^r

Γ

^{s of}

representations of

Γ

^over

k

^.

Theategories

C

^{usedbyGeiÿ,Leler, ShröerandFu,Keller formodellinglusteralgebras}

always have the following essential properties. They are Frobenius ategories (i.e. exat ate-

gorieswith enough injetivesand projetives, and theinjetivesand projetivesarethe same),

and theysatisfy

Ext

¹C^p

X, Y

^q

Ext

¹C^p

Y, X

^q

,

funtorially in

X

^and

Y

^{. To summarize, suh a ategory}

C

issaid to be

2

^{-Calabi-Yau. In this}

framework, the notion of luster-tilting objet introdued by [Iya2℄ is very useful. An objet

X

^is luster-tilting if it is rigid, that is, if

Ext

¹_C^p

X, X

^q

0

^{and if every objet}

Y

^satisfying

Ext

¹_C^p

X, Y

^q

0

^{is in the additive envelope of}

X

^{. If}

C

ategories a skew-symmetri luster algebra

A

r^,the luster-tilting objets modelthe lusters of

A

r^{,and their} indeomposable diret summands orrespondto luster variables.

Our prinipal transfer resultshows that if

C

^is

2

^{-Calabi-Yau, then}

C Γ

^{is also}

2

-Calabi-Yau.

Moreover,thetwonaturaladjointfuntorslinking

C

and

C Γ

^{induereiproalbijetionsbetween}

isomorphismlassesof

Γ

^-stableluster-tiltingobjetsof

C

andisomorphismlassesof

mod k

^r

Γ

^s-

stableluster-tiltingobjets of

C Γ

^.

Inorderto applythesegeneral resultsto theexamplesstudied byGeiÿ,LelerandShröer,

wehaveomputedexpliitlytheategory

C Γ

^{inseveralases[Dem1 ℄. Thisinludesinpartiular}

theasewhen

C

isthe module ategoryof apreprojetive algebra.

1.3. Categoriation of skew-symmetrizable luster algebras. Consider a

2

^-Calabi-Yau

ategory

C

on whih ats a nite group

Γ

^{. For ompleting the} ategoriation, one needs to developatheoryofmutationsof

mod k

^r

Γ

^s-stableluster-tiltingobjetsof

C Γ

^,or,equivalently,of

Γ

^-stable luster-tiltingobjets of

C

. In the skew-symmetri ase(whih an be seenasthease

where

Γ

^{is trivial), itis known that suh a theory is possible assoon as there existsa luster-}

tiltingobjetwhoseassoiatedquiverhasneitherloopsnor

2

^{-yles. Weintrodueinthegeneral}

ase the onepts of

mod k

^r

Γ

^{s-loops and of}

mod k

^r

Γ

^s-

2

^{-yles for a}

mod k

^r

Γ

^{s-stable luster-}

tilting objet. Onethenshows thatif

C Γ

^{admits a}

mod k

^r

Γ

^s-stableluster-tiltingobjethaving neither

mod k

^r

Γ

^{s-loops nor}

mod k

^r

Γ

^s-

2

^{-yles, all}

mod k

^r

Γ

^s-stable luster-tilting objets also havethisproperty. Underthis hypothesis,onean deneamutation operation. Morepreisely,

if

T

^is

mod k

^r

Γ

^s-stable luster-tilting and if

X

^{is the}

mod k

^r

Γ

^{s-orbit of an} indeomposable non projetivediretsummand

X

^of

T

^{,oneonstrutsanother}

mod k

^r

Γ

^s-stableluster-tiltingobjet

T

^{1 obtained byreplaing}

X

^bythe

mod k

^r

Γ

^s-orbit

Y

^ofanother indeomposable objet

Y

^{. One}

denotes

µ

_X^p

T

^q

T

^{1. One an also assoiate to}

T

^a skew-symmetrizable matrix

B

^p

T

^{q whose}

rowsareindexedbythe

mod k

^r

Γ

^s-orbits

X

^ofindeomposable summandsof

T

^andtheolumns

by the

mod k

^r

Γ

^s-orbits

X

^of indeomposable non projetive fatorsof

T

^{. The oeients}

b

_XY

are the numbers of arrows in the Gabriel quiver of

End

C^p

T

^{q between a xed} indeomposable objet

Y

^of

Y

^{and any} indeomposable objet

X

^of

X

^{(the arrows from}

X

^to

Y

^beingounted

positively andthe arrows from

Y

^to

X

^beingountednegatively). We then show(see theorem 3.42):

Theorem A. The mutation of

Γ

^-stable luster-tilting objets of

C

agrees with the mutation dened ombinatorially by Fomin andZelevinsky for skew-symmetrizable matries. That is,

B

^p

µ

_X^p

T

^qq

µ

_X^p

B

^p

T

^qq

,

where, in the right-hand side, by abuse of notation,

µ

_X ^{is the matrix mutation of Fomin and}

Zelevinsky.

Viatheabove-mentioned bijetion between

mod k

^r

Γ

^s-stableluster-tilting objetsof

C Γ

^and

Γ

^-stable luster-tiltingobjets of

C

, one an assoiate to eah

Γ

^-stable luster-tilting objet

T

of

C

a matrix whih willbealso denoted by

B

^p

T

^q.

Finally,inordertoattahto

C

^and

Γ

^{alusteralgebra,weintrodueanotionof}

Γ

-equivariant lusterharater. Intheskew-symmetriase,aordingtotheworkofCaldero-Chapoton [CC℄,

of Caldero-Keller [CK2 ℄, [CK1℄, of Palu [Pal℄, of Dehy-Keller [DK℄ and of Fu-Keller [FK ℄ one

anassigntoeveryobjet

X

^of

C

aLaurent polynomialinthelustervariables ofaninitialseed of the luster algebra

A

ategoried by

C

. If this initial seed is

Γ

^{-stable, one an identify the}

lustervariableswhihbelongtothesame

Γ

^{-orbit. This}speializationoftheLaurentpolynomial assoiatedto

X

^{onlydependson the}

Γ

^-orbit

X

^of

X

^{,andisdenoted by}

P

_X^{. Onededues from}

thepreviousonstrutionthatif

T

^isa

Γ

^-stableluster-tiltingobjetin

C

^,andif

A

^istheluster

algebra whoseinitial seedhasskew-symmetrizable matrix

B

^p

T

^{q,thentheluster variables of}

A

areof the form

P

_X ^where

X

^{isthe orbit of an} indeomposable summand ofa

Γ

^{-stableluster-}

tilting objetof

C

^{. In this situation,we say thatthe pair p}

C , Γ

^{q is a} ategoriation of

A

^{. One}

an thengeneralize the resultof Fu andKeller (see orollary3.61):

Theorem B. Let

B

^{be an}

m

n

^{matrix with}skew-symmetrizable prinipal part. If

B

^{has full}

rank and if the luster algebra

A

p

B

^{q has a} ategoriation p

C , Γ

^{q, then the luster monomials}

are linearly independent.

As a result, we obtain a proof of onjeture 1.1 for a large family of skew-symmetrizable

luster algebras.

gebras. Let

G

^{be a} semi-simple onneted and simply-onneted Lie group of simply-laed Dynkindiagram

∆

^,andlet

Λ

^{betheassoiated}preprojetivealgebra. Geiÿ,LelerandShröer have shown that the subategories

Sub I

J ^of

mod Λ

^{indue luster strutures on the multi-}

homogeneous oordinaterings of partialag varietiesassoiated to

G

^{[GLS6 ℄. Ourwork allows}

to extend this result to the ase where

G

^{orrespondsto a non} simply-laed Dynkin diagram.

In partiular, one obtains a proof of theonjeture 1.1 for these luster algebras, and one an

omplete the lassiation of partial ag varieties whose luster struture is of nite type (i.e.

admit anite numberof lusters). In partiular,this provestheonjeture of[GLS6 , Ÿ14℄.

Let

G

^{be a Ka-Moody group of symmetri Cartan matrix, and let}

Λ

^{be the assoiated}

preprojetive algebra. Geiÿ, Leler and Shröer have introdued ertain subategories

C

_M of

mod Λ

^{and shownthatthey indue lusterstrutureson theoordinate ring ofsome unipotent}

subgroups and unipotent ells of

G

^{[GLS1 ℄ (see also [BIRS℄ whih gives a dierent denition}

of similarsubategories). Our work allows to extend theseresults to the Ka-Moodygroups

G

with symmetrizable Cartan matries. In partiular, one obtains for all these examples a proof

of theonjeture1.1. Asa partiular aseof this onstrution,weget(see theorem4.47):

Theorem C. For every ayli luster algebra without oeient

A

^{,there is a ategory}

C

^and

a nite group

Γ

^{ating on}

C

whih ategorify

A

up to speialization of oeients to

1

^{. This}

holds inpartiular for luster algebras of nite type.

Note that the works of Geiÿ, Leler and Shröer use as a ruial fat the existene of the

dualsemianonial basisonstruted byLusztigfor the oordinateringof amaximalunipotent

subgroup of

G

^{. But, when}

G

^{is not of} simply-laed type, there is no available onstrution of semianonial bases. Our result an be interpreted as giving a part of the dual semianonial

basisinthenon simply-laedase, namely thesetof lustermonomials.

2. Equivariant ategories

For referenes about monoidal ategories and module ategories over a monoidal ategory,

seefor example [BK ℄,[CP ℄, [Kas ℄ and[Ost℄.

2.1. Denitionsand rstproperties. Let

k

^beaeld,

C

a

k

^-ategory,

Hom

^{-niteand Krull-}

Shmidt (whih means that the endomorphism rings of indeomposable objets are loal, or

equivalently, that every idempotent splits). Let

Γ

^{be a nite group whose ardinality is not}

divisiblebytheharateristiof

k

^{. Let}

Γ

mod k

^p

Γ

^{qbethemonoidalategoryof}

k

^p

Γ

^q-modules,

where

k

^p

Γ

^{q is the Hopf algebra of}

k

^{-valuedfuntions on the group}

Γ

^{. Remarkthat the simple}

objets in

Γ

^arethe one-dimensional

k

^p

Γ

^{q-modules given byevaluation mapsat eahelement}

g

of

Γ

^{. If}

g

^P

Γ

^,theorrespondingsimple objetin

Γ

^{will bedenotedby}

g

^{. Withthisnotation,it}

iseasyto hekthatthemonoidalstruture issimply

g

^b

h

gh

^,wherefor

g, h

^P

Γ

^,wedenote

by

gh

^thesimple

k

^p

Γ

^q-moduleorrespondingto

gh

^P

Γ

^.

Denition 2.1. Anation of

Γ

^on

C

^{isastruture of}

Γ

^{-module ategoryon}

C

^.

Remark 2.2. If one onsiders, asin [RR, p. 254℄, a group morphism

ρ

^from

Γ

^{to the group of}

autofuntors of

C

^{,oneobtains a strit}

Γ

^{-module struturebysetting}

g

^b

ρ

^p

g

^q.

We now introdue a ategory of

Γ

^{-invariant objets of}

C

^{. The naive idea of onsidering the}

full subategory of

C

of invariant objets does not work beause almost none of the desired properties arepreserved.

Denition 2.3. Let

C

^{be endowed with an ation of}

Γ

^{. The}

Γ

-equivariant ategory of

C

^is

the ategory whose objets are pairs p

X, ψ

^{q, where}

X

^P

C

, and

ψ

^p

ψ

_g^q_g ^P

Γ

^{is a family of}