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Affine cluster algebras G. Dupont University of Lyon november 6th, 2008 G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 1 / 52
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Affine cluster algebras

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Page 1: Affine cluster algebras

Affine cluster algebras

G. Dupont

University of Lyon

november 6th, 2008

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 1 / 52

Page 2: Affine cluster algebras

Context

Introduction : S. Fomin et A. Zelevinsky (Cluster Algebras I :Foundations, J. Amer. Math. Soc. 2001)

Motivation : Framework for a combinatorial study of

total positivity in algebraic groups,canonical bases in quantum groups.

Connections :Combinatorics,Lie Theory,Poisson Geometry,Representation theory...

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 2 / 52

Page 3: Affine cluster algebras

Problematics

Problem : Find and compute bases in cluster algebras.

Canonical Bases:Sherman-Zelevinsky (A2, A1,1),

Cerulli Irelli (A2,1).

Bases :Caldero-Keller (finite type),Geiss-Leclerc-Schroer (general, abstract).

Strategy: Give an unified and explicit method to compute bases incluster algebras using representation theory.

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 3 / 52

Page 4: Affine cluster algebras

Contents

1 Cluster algebras and cluster categories

2 Affine cluster algebras

3 Generalized Chebyshev polynomials

4 Generic variables

5 Further directions

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 4 / 52

Page 5: Affine cluster algebras

Seeds and clusters

A seed is a pair (Q, x) such that:

Q = (Q0,Q1) is a quiver without loops and 2-cycles;

x = (xi : i ∈ Q0) is a Q0-tuple of indeterminates over Z, calledcluster of the seed (Q, x).

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 5 / 52

Page 6: Affine cluster algebras

Mutation of seeds

For every k ∈ Q0, µk(Q, x) = (Q ′, x′) is the new seed given by:

Q Q ′

ir //

s ��::: j

kt

AA���i

r+st // j

t�����

ks

]]:::

ir // j

t�����

ks

]]:::i

r−st //

s ��::: j

kt

AA���

andx′ = x \ {xk} t

{x ′k}

wherexkx ′k =

∏i−→ k∈Q1

xi +∏

k−→ i∈Q1

xi .

We denote by (Q, x) ∼mut (R, y) the generated equivalence relation.

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 6 / 52

Page 7: Affine cluster algebras

Acyclic cluster algebras

Let (Q,u) be a seed with Q ayclic.

Definition

The cluster algebra A(Q) with initial seed (Q,u) is

A(Q) = Z[x |x ∈ c s.t. (R, c) ∼mut (Q,u)] ⊂ Q(u)

The c occuring are called the clusters of A(Q),The x ∈ c are called the cluster variables of A(Q).

Cl(Q) = {cluster variables} .

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 7 / 52

Page 8: Affine cluster algebras

The Laurent phenomenon

Theorem, Fomin-Zelevinsky, 2001

A(Q) ⊂ Z[u±1].

If x ∈ Z[u±1], the denominator vector den(x) ∈ ZQ0 of x is given by

x =P(u)

uden(x)

in its irreducible form.

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 8 / 52

Page 9: Affine cluster algebras

Cluster monomials

Definition

A cluster monomial is a monomial in cluster variables belonging to a samecluster.We set

M(Q) = {cluster monomials in A(Q)} .

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 9 / 52

Page 10: Affine cluster algebras

Cluster algebras of finite type

Definition

A cluster algebra A(Q) is said to be of finite type if |Cl(Q)| <∞.

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 10 / 52

Page 11: Affine cluster algebras

Simply-laced Dynkin diagrams

Anc cc c c cc c Dn

c cc c c c��@@

cc

E6c cc cc cc cc E7

c cc cc cc cc ccE8c cc cc cc cc cc cc

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 11 / 52

Page 12: Affine cluster algebras

Finite type classification

Theorem, F.Z., 2002

A(Q) is of finite type if and only if Q is a Dynkin quiver.In this case, den induces a 1-1 correspondence

den : Cl(Q)−→Φ>0(Q) t (−Π(Q)).

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 12 / 52

Page 13: Affine cluster algebras

A Z-basis in finite type

Theorem, Caldero-Keller, 2005

If A(Q) is of finite type, then M(Q) is a Z-basis in A(Q).

Fact, Sherman-Zelevinsky

In general, M(Q) does not span A(Q).

Conjecture, Zelevinsky

In general, M(Q) is linearly independent over Z.

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 13 / 52

Page 14: Affine cluster algebras

The cluster category

Let k = C, Q be an acyclic quiver

kQ-mod ' rep(Q).

Definition, BMRRT, 2004

The cluster category of Q is the orbit category of the auto-functorF = τ−1[1] in the bounded derived category Db(kQ) of kQ-mod.

CQ = Db(kQ)/F .

Theorem, K, BMRRT, 2004

CQ is a triangulated category;

Ext1CQ (X ,Y ) ' DExt1

CQ (Y ,X ) (2-Calabi-Yau);

ind(CQ) = ind(kQ-mod) t {Pi [1] : i ∈ Q0} .

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 14 / 52

Page 15: Affine cluster algebras

The quiver grassmannian

Let M be a kQ-module and e ∈ ZQ0 . We write

Gre(M) = {N ⊂ M : dim N = e}

the quiver grassmannian.

We denote by χ the Euler-Poincare characteristic.

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 15 / 52

Page 16: Affine cluster algebras

The Caldero-Chapoton map

Definition, Caldero-Chapoton

The Caldero-Chapoton map is the map X? : Ob(CQ)−→Z[u±1]:

If M,N are in Ob(CQ), then XM⊕N = XMXN ;

If M ' Pi [1], then XPi [1] = ui ;

If M is an indecomposable module, then

XM =∑

e

χ(Gre(M))∏i∈Q0

u−〈e,dimSi 〉−〈dimSi ,dimM−e〉i . (1)

Equality (1) holds for any kQ-module M.

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 16 / 52

Page 17: Affine cluster algebras

From cluster categories to cluster algebras

Theorem, Caldero-Keller

X? induces a 1-1 correspondence

{indecomposable rigid objects in CQ}∼−→ Cl(Q).

Moreover, the map{{maximal rigid objects in CQ}

∼−→ {clusters in A(Q)}T =

⊕i∈Q0

Ti 7→ {XTi: i ∈ Q0}

is a 1-1 correspondence.

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 17 / 52

Page 18: Affine cluster algebras

Cluster monomials and rigid objects

Corollary

X? induces a 1-1 correspondence

{ rigid objects in CQ}∼−→M(Q).

Corollary

den induces a 1-1 correspondence

Cl(Q)∼−→ Φre,Sc(Q) t (−Π(Q))

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 18 / 52

Page 19: Affine cluster algebras

The one-dimensional multiplication formula

Theorem, CK

Let M,N be indecomposable objects in CQ such thatdim Ext1

CQ (M,N) = 1. Then

XMXN = XB + XB′

where B and B ′ are the unique objects such that there exists non-splittriangles

M−→B−→N−→M[1],

N−→B ′−→M−→N[1]

in CQ .

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 19 / 52

Page 20: Affine cluster algebras

Contents

1 Cluster algebras and cluster categories

2 Affine cluster algebras

3 Generalized Chebyshev polynomials

4 Generic variables

5 Further directions

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 20 / 52

Page 21: Affine cluster algebras

Motivation

Finite-tame-wild classification theorem

Affine quivers are minimal among representation-infinite quivers

Representation theory of affine quivers is well-known

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 21 / 52

Page 22: Affine cluster algebras

Simply laced affine diagrams

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c c c cc c c c @

@

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c Dn c��c@@c c c c��

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cc

E6c cc cc cc ccc

E7c cc cc cc cc cc cc

E8c cc cc cc cc cc cc cc

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 22 / 52

Page 23: Affine cluster algebras

Affine cluster algebras

Definition

A quiver Q is called affine if it is acyclic and if its underlying diagram is anaffine diagram.

Definition

A cluster algebra A(Q) is called affine if Q is an affine quiver.

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 23 / 52

Page 24: Affine cluster algebras

Affine root systems

Φ>0(Q) = Φre>0(Q) t N∗δ

ΦSc(Q) = Φre,Sc(Q) t {δ}

Kac’s theorem

Let d ∈ NQ0 . Then

∃M indecomposable in rep(Q,d) iff d ∈ Φ>0(Q);

∃!M indecomposable in rep(Q,d) iffd ∈ Φre>0(Q);

There exists a 1-parameter family of pairwise non-isomorphicindecomposable representations in rep(Q, nδ) for every n ≥ 1.

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 24 / 52

Page 25: Affine cluster algebras

The Auslander-Reiten quiver of kQ-mod

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G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 25 / 52

Page 26: Affine cluster algebras

Tubes in Γ(kQ)

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G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 26 / 52

Page 27: Affine cluster algebras

Tubes in Γ(kQ)

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G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 26 / 52

Page 28: Affine cluster algebras

Tubes in Γ(kQ)

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G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 26 / 52

Page 29: Affine cluster algebras

Contents

1 Cluster algebras and cluster categories

2 Affine cluster algebras

3 Generalized Chebyshev polynomials

4 Generic variables

5 Further directions

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 27 / 52

Page 30: Affine cluster algebras

Motivation

Problem: Understand X? on regular components.

Strategy: Use the combinatorial description of regular componentsin order to have a combinatorial description of the behaviour of X?.

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 28 / 52

Page 31: Affine cluster algebras

Generalized Chebyshev polynomials

Let xi , i ≥ 1 be indeterminates over Z.

Definition

The n-th generalized Chebyshev polynomial Pn is given by

Pn(x1, . . . , xn) = det

xn 1 (0)

1. . .

. . .. . .

. . . 1(0) 1 x1

∈ Z[x1, . . . , xn]

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 29 / 52

Page 32: Affine cluster algebras

A tube

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G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 30 / 52

Page 33: Affine cluster algebras

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G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 30 / 52

Page 34: Affine cluster algebras

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G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 30 / 52

Page 35: Affine cluster algebras

Example in type A3,1

Let Q be an affine quiver of type A3,1.

2 // 3

��========

Q : 1

@@��������// 4

Γ(kQ) contains an unique exceptional tube T0 and rg(T0) = 3.We denote by E0,E1,E2 the quasi-simple modules in T0.

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 31 / 52

Page 36: Affine cluster algebras

Example: Quasi-simples in the exceptional tube of A3,1

0 // k0

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E0 : 0

@@��������// 0

k0 // 0

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E1 : 0

@@��������// 0

0 // 0

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E2 : k

0

@@�������� 1 // k

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 32 / 52

Page 37: Affine cluster algebras

The exceptional tube of A3,1

•E0

•E1

•E2

•E0

• E(2)0

• M0

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G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 33 / 52

Page 38: Affine cluster algebras

Variables in the exceptional tube of A3,1

x0 = XE0 =u2 + u4

u3, x1 = XE1 =

u1 + u3

u2,

x2 = XE2 =1 + u1u3 + u2u4

u1u4.

XE

(2)0

=u1u2 + u1u4 + u3u4

u2u3

XM0 =u2

1u3u4 + u21u2u3 + u1u2

3u4 + u1u4 + u1u2 + u3u4 + u2u3u24

u1u2u3u4

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 34 / 52

Page 39: Affine cluster algebras

Variables in the exceptional tube of A3,1

x0 = XE0 =u2 + u4

u3, x1 = XE1 =

u1 + u3

u2,

x2 = XE2 =1 + u1u3 + u2u4

u1u4.

XE

(2)0

=u1u2 + u1u4 + u3u4

u2u3

XM0 =u2

1u3u4 + u21u2u3 + u1u2

3u4 + u1u4 + u1u2 + u3u4 + u2u3u24

u1u2u3u4

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 34 / 52

Page 40: Affine cluster algebras

Variables in the exceptional tube of A3,1

x0 = XE0 =u2 + u4

u3, x1 = XE1 =

u1 + u3

u2,

x2 = XE2 =1 + u1u3 + u2u4

u1u4.

XE

(2)0

=u1u2 + u1u4 + u3u4

u2u3

XM0 =u2

1u3u4 + u21u2u3 + u1u2

3u4 + u1u4 + u1u2 + u3u4 + u2u3u24

u1u2u3u4

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 34 / 52

Page 41: Affine cluster algebras

Contents

1 Cluster algebras and cluster categories

2 Affine cluster algebras

3 Generalized Chebyshev polynomials

4 Generic variables

5 Further directions

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 35 / 52

Page 42: Affine cluster algebras

Motivations

“Generalizing” cluster monomials,

Analogue of the dual semicanonical basis.

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 36 / 52

Page 43: Affine cluster algebras

Existence of generic variables

Lemma, D. 2008

Let Q be an acyclic quiver and d ∈ NQ0 . Then, there exists an open densesubset Ud ⊂ rep(Q,d) such that X? is constant over Ud.We denote by Xd the value of X? on this open subset.

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 37 / 52

Page 44: Affine cluster algebras

Definition of generic variables

Definition

Let d ∈ ZQ0 . We setXd = X[d]+

∏di<0

u−dii

the generic variable of dimension d.

B′(Q) ={

Xd : d ∈ ZQ0

}

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 38 / 52

Page 45: Affine cluster algebras

Generic variables and cluster monomials

Proposition, D. 2008

Let Q be an acyclic quiver. Then

M(Q) ⊂ B′(Q).

Moreover, if Q is Dynkin, then

M(Q) = B′(Q).

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 39 / 52

Page 46: Affine cluster algebras

Canonical decomposition and generic variables

Proposition, D. 2008

Let Q be an acyclic quiver, d ∈ NQ0 and d = d1 ⊕ · · · ⊕ dn its canonicaldecomposition. Then,

Xd =n∏

i=1

Xdi.

It thus suffices to compute Xd for d ∈ ΦSc.

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 40 / 52

Page 47: Affine cluster algebras

Explicitness of generic variables

Proposition, D. 2008

Let Q be an affine quiver and d ∈ ΦSc(Q).

If d ∈ ΦSc,re(Q), then Xd ∈M(Q);

Otherwise, d = δ and Xδ = XMλfor any λ ∈ P1

0.

Corollary, D. 2008

Let Q be an affine quiver. Then,

B′(Q) =M(Q) t {X nδ XE : n ≥ 1,E ∈ ER}

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 41 / 52

Page 48: Affine cluster algebras

The difference property

Definition

Let Q be an affine quiver. We say that Q satisfies the difference propertyif for every indecomposable kQ-modules M,Mλ in rep(Q, δ) belongingrespectively to an exceptional and an homogeneous tube, we have:

XMλ= XM − Xq.radM/q.socM .

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 42 / 52

Page 49: Affine cluster algebras

The difference property

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G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 43 / 52

Page 50: Affine cluster algebras

The difference property for type A

Theorem, D. 2008

Let Q be an affine quiver of type A. Then Q satisfies the differenceproperty.

Conjecture

Every affine quiver satisfies the difference property.

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 44 / 52

Page 51: Affine cluster algebras

Generic variables and cluster algebras

Lemma, D. 2008

Let Q be an affine quiver satisfying the difference property. Then,

Z[XM : M ∈ Ob(CQ)] = A(Q).

Corollary, D. 2008

Let Q be an affine quiver satisfying the difference property. Then,

B′(Q) ⊂ A(Q).

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 45 / 52

Page 52: Affine cluster algebras

The semicanonical basis

Theorem, D. 2008

Let Q be an affine quiver such that every quiver reflection-equivalent to Qsatisfies the difference property. Then, B′(Q) is a Z-basis in A(Q).

Corollary, D. 2008

Let Q be an affine quiver of type A. Then, B′(Q) is a Z-basis in A(Q).

Conjecture

Let Q be an affine quiver. Then, B′(Q) is a Z-basis in A(Q).

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 46 / 52

Page 53: Affine cluster algebras

Representations in rep(Q, δ) for A3,1

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k1 // k

λ

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Mλ : k

1@@������� 1 // k

If λ 6= 0, Mλ is a quasi-simple in an homogeneous tube.M0 is in T0 and

q.socM0 ' E0, q.radM0 ' E(2)0 .

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 47 / 52

Page 54: Affine cluster algebras

The exceptional tube of A3,1

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•E1

•E2

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• M0m

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G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 48 / 52

Page 55: Affine cluster algebras

Example of difference property for type A3,1

e 0 [0001] [0010] [0110] [0011] [0111] [1111]

Gre(M0) 0 S4 E0 E(2)0 E0 ⊕ S4 E

(2)0 ⊕ S4 M0

χ(Gre(M0)) 1 1 1 1 1 1 1

Gre(Mλ) 0 S4 ∅ ∅ P3 P2 Mλ

χ(Gre(Mλ)) 1 1 0 0 1 1 1

XM0 =u2

1u3u4 + u21u2u3 + u1u2

3u4 + u1u4 + u1u2 + u3u4 + u2u3u24

u1u2u3u4

XMλ=

u21u2u3 + u1u2 + u1u4 + u3u4 + u2u3u2

4

u1u2u3u4

XM0 = XMλ+

u2 + u4

u3= XMλ

+ XE0

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 49 / 52

Page 56: Affine cluster algebras

Example of difference property for type A3,1

e 0 [0001] [0010] [0110] [0011] [0111] [1111]

Gre(M0) 0 S4 E0 E(2)0 E0 ⊕ S4 E

(2)0 ⊕ S4 M0

χ(Gre(M0)) 1 1 1 1 1 1 1

Gre(Mλ) 0 S4 ∅ ∅ P3 P2 Mλ

χ(Gre(Mλ)) 1 1 0 0 1 1 1

XM0 =u2

1u3u4 + u21u2u3 + u1u2

3u4 + u1u4 + u1u2 + u3u4 + u2u3u24

u1u2u3u4

XMλ=

u21u2u3 + u1u2 + u1u4 + u3u4 + u2u3u2

4

u1u2u3u4

XM0 = XMλ+

u2 + u4

u3= XMλ

+ XE0

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 49 / 52

Page 57: Affine cluster algebras

The semicanonical basis of A(A3,1)

x0 = XE0 , x1 = XE1 , x2 = XE2 ,

y0 = XE

(2)0

, y1 = XE

(2)1

, y2 = XE

(2)2

,

z = XMλ

Alors,

B′(Q) =M(Q) t {znx ri y s

i : n > 0, r , s ≥ 0, i = 0, 1, 2}

est une Z-base de A(Q).

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 50 / 52

Page 58: Affine cluster algebras

Contents

1 Cluster algebras and cluster categories

2 Affine cluster algebras

3 Generalized Chebyshev polynomials

4 Generic variables

5 Further directions

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 51 / 52

Page 59: Affine cluster algebras

Further directions

Canonical bases for affine quivers,

Cluster algebras with coefficients,

Semicanonical bases for wild quivers,

Connections with the dual semicanonical basis.

G. Dupont (University of Lyon) Affine cluster algebras november 6th, 2008 52 / 52