Geometric construction of cluster algebras and cluster ...algebra. Expectation: the positive part of the quantized enveloping algebra has a (quantum) cluster algebra structure, with

Geometric construction of

cluster algebras and cluster categories

Karin BaurETH Zurich

[email protected]

12 October, 2007

Karin Baur Zurich, October 2007

Overview

◮ Motivation

◮ Triangulated surfaces ←→ cluster algebras

◮ Translation quivers ←→ cluster categories

◮ Connections, future directions.

Plan of talk 1


Motivation

Cluster algebras arose from the study of two related problems.

Problem 1 Canonical basis

Understand the canonical basis (Lusztig), or crystal basis (Kashiwara)of quantized enveloping algebras associated to a semisimple complex Liealgebra.Expectation: the positive part of the quantized enveloping algebra has a(quantum) cluster algebra structure, with the cluster monomials formingpart of the dual canonical basis.

This picture motivated the definition of cluster variables.

Motivation 2


Problem 2 Total positivity

An invertible matrix/R is totally positive if all its minors are positive. Thisnotion can be extended to all reductive groups (Lusztig).To check total positivity for an upper uni-triangular matrix, only a certaincollection of minors needs to be checked. The minimal sets of such all havethe same cardinality. When one of them is removed, it can be replaced by aunique alternative minor. The two minors are connected through a certainrelation.

This exchange (mutation) motivated the definition of cluster mutation.

Motivation 3


Connections, Applications

◮ Y -systems in thermodynamic Bethe Ansatz, (Frenkel-Szenes 1995)Fomin-Zelevinsky 2003, Szenes 2007;

◮ Poisson geometry, Teichmuller spaces, Gekhtman-Shapiro-Vainshtein2003/2005, Fomin-Goncharov 2006;

◮ Stasheff polytopes, associahedra, Chapoton-Fomin-Zelevinsky 2002;

◮ ad-nilpotent ideals of Borel subalgebras in Lie algebras, Panushev 2004;

◮ Preprojective algebra models, Geiss-Leclerc-Schroer 2005;

◮ Quiver representations, tilting theory, e.g. BMRRT 2005.

Cluster algebras - examples of connections to other fields 4


Surfaces and triangulations

Let S be a connected oriented Riemann surface with boundary. Fix a finiteset M of marked points on S. Marked points in the interior of S are calledpunctures.

We consider triangulations of S whose vertices are at the marked pointsin M and whose edges are pairwise non-intersecting curves, so-called arcsconnecting some of the marked points.

Assume: M non-empty, each boundary component has marked points.Exclude the following (S,M):

◮ a monogon with ≤ 1 puncture

◮ an unpunctured digon or triangle.

Triangulated surfaces - after S. Fomin, M. Shapiro, D. Thurston, Cluster algebras and triangulated surfaces I 5


(a) Once-punctured triangle

(b) Annulus

Figure 1: Examples of triangulations

Triangulated surfaces 6


Surfaces

The pair (S,M): defined by the genus of S, the number of boundarycomponents, the number of marked points on each boundary componentand the number of punctures.

A curve in S (up to isotopy rel. M) is called an arc γ in (S,M) if

◮ the endpoints of γ are marked points in M;

◮ γ does not intersect itself (but its endpoints might coincide);

◮ relative interior of γ: disjoint from the boundary of S;

◮ γ does not cut out an unpunctured monogon/digon.

Triangulated surfaces 7


Triangulations

The set of all arcs in (S,M) is usually infinite. It is finite if and only if(S,M) is a disk with at most one puncture.

Two arcs are called compatible if they do not intersect in the interior of S.

An ideal triangulation is a maximal collection T of pairwise compatiblearcs.

The arcs of T cut S into ideal triangles. These triangles may be self-folded,e.g. along the horizontal arc in the picture:

Ideal triangulations 8


Example

The once-punctured triangle has 10 ideal triangulations, the four offigure 1, with the rotations of the last three (by 120◦ and 240◦).

Ideal triangulations 9


The rank of (S,M)

The number of arcs in an ideal triangulation is an invariant of (S,M), wecall it the rank of (S,M). The rank of (S,M) is

6g + 3b + 3p + c− 6

where g is the genus of S, b the number of boundary components, p thenumber of punctures, c the number of marked points on the boundary.

E.g.: The rank of the punctured triangle is

6 · 0 + 3 · 1 + 3 · 1 + 3− 6 = 3

Rank of a punctured surface 10


◮ Rank = 1unpunctured square (type A1);

◮ Rank = 2unpunctured pentagon (type A2);once-punctured digon (type A1 ×A1);annulus with one marked point on each boundary component;

◮ Rank = 3unpunctured hexagon (type A3);once-punctured triangle (type A3 = D3);annulus with one marked point on one bd component, two on the other;once-punctured torus.

Small rank cases 11


If T is an ideal triangulation of (S,M) and p an arc of T , we can replacep by an arc q through a so-called flip or Whitehead move:

Whitehead moves 12


If T is an ideal triangulation of (S,M) and p an arc of T , we can replacep by an arc q through a so-called flip or Whitehead move:

b

qa

p

d

c

For any two ideal triangulations T and T ′ exists a sequence of flips leadingfrom T to T ′ (Hatcher 1991).

Whitehead moves 12


To an ideal triangulation T we associate a square matrix B(T ) with rowsand columns labeled by the arcs (say 1, 2, . . . , n):

B(T ) =∑

∆

B∆

where the B∆ are defined by

b∆ij =

1 if ∆ has sides i and j where j is a clockwise neighbour of i;−1 if ∆ has sides i and j where i is a clockwise neighbour of j;

0 otherwise.

The matrix B(T ) is skew-symmetric with entries 0,±1,±2

The matrix of a triangulation 13


Caveat

If △ is a self-folded triangle folded along an arc i (or j), then, in the

right hand side of the definition of b△ij above, such an arc i (or j) has to bereplaced by the enclosing loop l(i) (or l(j)).

l(i)i



Example of a matrix B(T )

Example: B(T ) for the triangulated punctured triangle

12 3

D

D

D

1

2

3

is:

BD1︸︷︷︸= 0

+BD2+BD3 =

0 1 −1−1 0 11 −1 0

+

0 0 00 0 −10 1 0

=

0 1 −1−1 0 01 0 0



Cluster algebras (Fomin-Zelevinsky 2001)

A cluster algebra is a subring of F = Q(u1, . . . , um), associated to a seed(x,B).

◮ A seed: a cluster x= (x1, . . . , xm) is a transcendence basis of F over Q,B = (byz)y,z an m×m matrix (sign-symmetric, integer coefficients).

◮ Mutation at z ∈ x, µz: (x,B) 7→ (x’,B’): x’= x − z ∪ z′ where z′ isdefined via the exchange relation (B’ defined via “matrix mutation”):

zz′ =∏

x∈xbxz>0

xbxz +∏

x∈xbxz<0

x−bxz

Note: µ is involutive, i.e. µz′(µz((x,B))) =(x,B).

Crash course in cluster algebras 16


Two seeds (x,B) and (x’,B’) are mutation-equivalent if one can beobtained from the other through a sequence of mutations.The cluster variables are defined to be the union of all clusters of amutation-equivalence class (of a given seed).Finally, the corresponding cluster algebra is generated by all the clustervariables.

Example: (Type A2). We start with the pair x= (x1, x2), B=

(0 1−1 0

)

.

First mutate at x1: from x1x′1 = 1 + x2 we get x′

1 = 1+x2x1

.

Mutation at x2 then gives x′2 = x1+1

x2.

And then x′′1 = x1+x2+1

x1x2; x′′

2 = x1, x′′′1 = x2.

So there are five cluster variables.

Cluster algebras 17


Properties of cluster algebras

Main results

◮ Laurent phenomenon: Cluster algebra ⊂ Z[x±1 , . . . , x±

m](i.e. every element of the cluster algebra is an integer Laurent polynomialin the variables of x);

◮ Classification of finite type cluster algebras by roots systems;

◮ Realizations of algebras of regular functions on double Bruhat cells interms of cluster algebras.

ExamplesCoordinate rings of SL2, SL3;Plucker coordinates on Gr2,n+3.

Cluster algebras 18


Cluster algebras and quivers

A Quiver Γ = (Γ0, Γ1) is an oriented graph with vertices Γ0 and arrowsΓ1: E.g.

1α−→ 2

β−→ 3

with Γ0 = {1, 2, 3} and Γ1 = {α, β};

If the matrix B of a seed (x, B) is skew-symmetric, then the seeddetermines a quiver:

◮ Label the rows (columns) by {1, 2, . . . , m}, set Γ0 = {1, 2, . . . , m}.

◮ Draw bxy arrows from x to y if bxy > 0 (for x, y ∈ Γ0).

Cluster algebras and quivers 19


Example The matrix B=

(0 1−1 0

)

from earlier gives the quiver:

1 −→ 2

This process is revertible: a quiver gives a skew-symmetric matrix.

Cluster algebras and quivers 20


Back to triangulated surfaces

Consider a disc (S,M) with at most one puncture, T a triangulation.

Question: T ; seed?

B(T ) ; cluster algebra 21


Back to triangulated surfaces

Consider a disc (S,M) with at most one puncture, T a triangulation.

Question: T ; seed?

Define a quiver Γ(T ) from T :

Γ0(T ): the arcs {1, 2, . . . , n} of T .Γ1(T ): there is an arrow i → j whenever the arcs i and j of T bound acommon triangle and j is the clock-wise neighbour of i.

Define a cluster xT = (x1, . . . , xn) by sending i 7→ xi. And let B(Γ) be thematrix associated to Γ(T ).

Thus, we have T ; a seed (xT , B(Γ)), hence obtain a cluster algebra.

B(T ) ; cluster algebra 21


Example of Γ(T )

12

1

2

B(Γ) =

(0 1−1 0

)

, seed: ((x1, x2), B(Γ))

Example of a quiver 22


Cluster categories

Certain quotients of derived categories of algebras.

Cluster categories 23


Cluster categories

Certain quotients of derived categories of algebras.

They were introduced in 2005 by BMRRT (Buan-Marsh-Reineke-Reiten-Todorov).

A graphical description (of type An) was given by CCS (Caldero-Chapoton-Schiffler) in 2005, by Schiffler (of type Dn) in 2006.

Aim: model cluster algebras using the representation theory of quivers.

Cluster categories: provide insight into cluster algebras. Have led to thedefinition of cluster-tilting theory.



Construction

Q quiver, underlying graph: ADE, think of type A.

Db(Q): bounded derived category of fin.dim. kQ-modules (k = k).

Shape of the quiver of Db(Q) is Q× Z

There are two graph automorphisms:τ (“Auslander-Reiten translate”)[1] (the “shift”)



Cluster category, C : orbit category of Db(Q) under a canonicalautomorphism. Independent of orientation of Q.

C := Db(Q)/τ−1 ◦ [1]



Cluster category, C : orbit category of Db(Q) under a canonicalautomorphism. Independent of orientation of Q.

C := Db(Q)/τ−1 ◦ [1]

The m-cluster category (Keller, 2005)

Cm := Db(Q)/τ−1 ◦ [m]



Properties of Cm

◮ triangulated; Calabi-Yau category of dimension m + 1 (Keller);

◮ Krull-Schmidt (BMRRT).

Studied by Keller, Reiten, Thomas, Wralsen, Zhu, B-Marsh, Assem-Brustle-Schiffler-Todorov, Amiot, Wralsen, etc.

Goal Describe Cm using diagonals of a polygon (type An) and arcs in apunctured polygon (type Dn).

m-cluster categories 26


Example Hexagon

Define a quiver given as follows:

Γ0: diagonals (ij)Γ1: arrows (ij) → (i, j + 1), (ij) → (i + 1, j), provided the image is adiagonal (i, j ∈ Z6).

Translation τ : (ij)→ (i− 1, j − 1) (60◦ anti-clockwise about center)

6

1 2

3

15

13

5 4

It is an example of a (stable) translation quiver: a finite quiver (no loops)with a map τ on the vertices (called the translation).

Translation quivers 27


A translation quiver

The quiver from the hexagon is

15 26 13

14 25 36 14

13 24 35 46 15

Translation quivers 28


Cluster category via polygons

The quiver of the cluster category C = C(An−1) is equal to the quiverobtained from an (n + 2)-gon.

(Caldero-Chapoton-Schiffler 2005).

Approach Caldero-Chapoton-Schiffler 29


Generalized diagonals

For n, m ∈ N let Π be an nm + 2-gon, label the vertices 1, 2, . . . , nm + 2.

An m-diagonal is a diagonal (ij) dividing Π into an mj + 2-gon and anm(n− j) + 2-gon (1 ≤ j ≤ n−1

2 ).

Example n = 3, m = 2:

Π is an octagon, mj + 2 = 4 and m(n− j) + 2 = 6.

m-diagonals 30


Example Octagon

(n = 3,m = 2)

Subdivide the octagon into quadrilaterals and hexagons.

21

3

4

56

7

8

Maximal sets of non-crossing 2-diagonals: {(16), (36)}, {(16), (25)},{(16), (14)} and rotated version of these.

m-diagonals 31


Quiver Γ(n,m), type A

We define a quiver Γ(m,n) = (Γ, τm) as follows:

Γ0: m-diagonalsΓ1: (ij) → (ij′) if (ij), Bjj′ and (ij′) span an m + 2-gon (Bjj′ is theboundary j to j′).

ij

ij’

i

j

j’

m+2−gon

τm: rotation anti-clockwise (about center), angle m 2πnm+2.

An A-type quiver 32


Properties

◮ If m = 1, we recover the usual diagonals.

◮ Γ(n,m) is a stable translation quiver.

Towards modeling the m-cluster category 33


Properties

◮ If m = 1, we recover the usual diagonals.

◮ Γ(n,m) is a stable translation quiver.

E.g. for n = 3, m = 2:

16 38 25 47 16

14 36 58 27 14

Towards modeling the m-cluster category 33


Description of the m-cluster category

TheoremThe quiver of the m-cluster category Cm(An−1) is Γ(m,n).

(CCS for m = 1, B-Marsh for m ≥ 1)

(Proof uses Happels description of (the AR-quiver of) Db(An−1) andcombinatorial analysis of Γ(n,m). )

In above example: C2(A2) = Db(A2)/τ−1 ◦ [2].

The m-cluster category given via polygons 34


Type Dn - Quiver Γ⊙(n,m)

Γ0: m-arcs of a punctured nm−m + 1-gonΓ1: m-moves

E.g (ij)→ (ik) if (ij), Bjk (boundary j to k) and (ik) span a (degenerate)m + 2-gon.

��

move (i)

2

6

7

1

3

4

5

��

move (ii)

τm: rotation anti-clockwise (about center).

D-type quiver 35


The m-th power of a translation quiver

(Γ, τ) a translation quiver as before. Its m-th power, Γm:

Γm0 = Γ0 (same vertices)

Γm1 : paths of length m (unique direction)

Let τm be τ ◦ τ ◦ · · · ◦ τ (m times).

Then we define the m-th power of (Γ, τ) to be the pair (Γm, τm).

Observe: (Γm, τm) is again a stable translation quiver.It is not connected in general.

m-th power of a quiver 36


Example

Type A5, gives C(3, 2) from earlier.

17 28 13

16 27 38 14

15 26 37 48 15

14 25 36 47 58 16

13 24 35 46 57 68 17

There are two other components: Db(A3)/[1] twice.

37


Example, continued

16 38 25 47 16

14 36 58 27 14

17 13

15 37 15

13 35 57 17

28 24

26 48 26

24 46 68 24

38


The m-cluster category via the cluster category

We obtain the following (B-Marsh):

Theorem:Γ(n,m) is a connected component of (Γ(nm, 1))m = (Γ(cluster category))m

◮ We have a corresponding result for type D

◮ Other components?

m-th power of a quiver 39


Conclusions

Part I: Triangulations give rise to cluster algebras;

Triangulation T 7−→ Γ(T ) 7−→ seed ((x1, . . . , xn), B(Γ)){1, 2, . . . , n}

Conclusions 40


Conclusions

Part I: Triangulations give rise to cluster algebras;

Triangulation T 7−→ Γ(T ) 7−→ seed ((x1, . . . , xn), B(Γ)){1, 2, . . . , n}

Part II: Polygons describe cluster categories;

diagonals ←→ indec objects of Cm-diagonals ←→ indec. objects of Cm.

DiagonalsΓm

7−→ indec. objects of Cm.

Conclusions 40


Connections and future directions

◮ Work in progress (w. Marsh): link between cluster algebra combinatoricsand perfect matchings (for vertices and edges of a triangulation). Useswork of Conway-Coxeter and Broline-Crowe-Isaacs on frieze patterns.

◮ Y -systems can be defined in general for a pair (G,H) of Dynkin types.Zamolodchikovs periodicity conjecture - proved for G =A1 and H =An

by Frenkel-Szenes 1995, Gliozzi-Tateo 1996, and for H any Dynkin typeby F-Z 2003 (via cluster algebras).G =Ak, H =An (Szenes, Volkov, both 2006).Open: (1) m-cluster picture ! Y -system theory?(2) Work on matchings leads to (periodic) systems with G =A1, H =A∞.Link to Y -systems? (3) Periodicity for arbitrary Dynkin types?

Outlook 41


◮ Approach with surfaces (S,M) works for types A, D (Fomin-Shapiro-Thurston 2005) and B, C with modifications (Ch-F-Z 2002)Open: type E.

◮ Jorgensen (2007) obtains m-cluster categories as quotient categories(type A, D) of cluster categories via deletion of rows (τ -orbits). Theyinherit a triangulated structure.Question: how to explain the triangulated structure of m-clustercategories via the m-th power of a quiver?

Outlook 42

Geometric construction of cluster algebras and cluster ...algebra. Expectation: the positive part of the quantized enveloping algebra has a (quantum) cluster algebra structure, with

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