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IntroductionQuantum cluster algebras

Triangular bases

Monoidal categori�cations and triangular bases of

quantum cluster algebras

Fan Qin

Conference on Representation Theory, Shanghai, 2015

Fan Qin Monoidal categori�cations of cluster algebras


Triangular bases

Outline

1 IntroductionCluster algebrasMonoidal categori�cation conjectures

2 Quantum cluster algebrasExample and propertiesLaurent expansionsComapare with characters

3 Triangular basesConstructionMain theoremExamples of cluster algebras



Triangular bases

Cluster algebrasMonoidal categori�cation conjectures

Cluster algebrasCluster algebras are combinatorial objects.

Cluster algebra: Z-subalgebra of a Laurent polynomial ring

Cluster variables = generators de�ned recursively by mutations

Seeds (local charts) = collections of generators + matrices

Cluster monomials = monomials of cluster variables in thesame seeds

Invented by [Fomin-Zelevinsky, 2000] as an combinatorial approachto the dual canonical basis of quantum groups in the sense ofLusztig and Kashiwara.

slow progress for many years



Triangular bases


Cluster algebrasCluster algebras' appearance.

Fruitful in many other areas:

Combinatorics

Representation theory of �nite dimensional algebras,2-Calabi-Yau categories

Higher Teichmüller theory [Fock-Goncharov]

Poisson geometry [Gekhtman-Shapiro-Vainstein]



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Cluster algebrasCluster algebras' appearance.

Discrete dynamical systems:

[Francesco-Kedem] [FZ][Inoue-Iyama-Kuniba-Nakanishi-Suzuki] [Keller]proof of the periodicity conjecture of Y-system

commutative/non-commutative algebraic geometry:

Bridgeland's stability conditions of 3-Calabi-Yau categories,Donaldson-Thomas invariants[Kontsevich-Soibelman],Tropical geometry, Gromov-Witten theory[Gross-Hacking-Keel-Kontsevich] [G-Pandharipande][G-P-Siebert]

A new Mathematics Subject Classi�cation: 13F60



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Monoidal categori�cation ConjecturesRead cluster algebras from monoidal categories

[Hernandez-Leclerc, 09] proposed the monoidal categori�cationapproach to a cluster algebra A :

A C Monoidal cateogry+ ⊕· ⊗

A ' K0(C ) Grohendieck ringcluster monomials ⊂ simple objects

good basis = {simples}

Find the monoidal category such that A ' K0(C )?

The cluster monoidals are simples?



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Monoidal categori�cation ConjecturesWorld of cluster algebras

Type I Type II

All cluster algebras

After quantization, the quantum cluster algebras are related to,

in Type I, the quantum groups of symmetric Cartan type:Aq ' K0(KLR−alg f .d . mod) (∼ Uq(n)∗,gr )

[Geiss-Leclerc-Schröer];[Khovanov-Lauda] [Rouquier]

in Type II, the quantum a�ne algebras of type ADE :Aq ' Kt(Uq(g) f .d . mod) t-deformed Grothendieck ring

[Hernandez-Leclerc];[Varagnolo-Vasserot] [Nakajima][H.]

Generalize type II: partial results[HL][Gleitz][Zhang-Duan-Li-Luo]



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Monoidal categori�cation ConjecturesConjectures and results

Monoidal categori�catoin Conjecture

The cluster monomials are simples?

Theorem ([Lampe] [Hernandez-Leclerc] [Nakajima] [Kimura-Q.])

Partial results for type I and type II.

Theorem (Q., 15)

For type II and some of type I: The monoidal categori�cation

conjecture is true. The Fock-Goncharov conjecture is also true.

Theorem (Kang-Kashiwara-Kim-Oh, 15)

For type I: The monoidal categori�cation conjecture is true.



Triangular bases

Example and propertiesLaurent expansionsComapare with characters

A rank 2 example

Example (Quantum cluster variables)

Take matrices B =

(0 −11 0

), Λ =

(0 −11 0

).

Initial cluster variable: X1,X2.Quantum torus T : Luarent polynomial ring (Z[q±

12 ][X±

1,X±

2],+, ·)

q-twisted product X g ∗X h = q12gΛhTX g+h,

bar involution qsX g = q−sX g .The quantum cluster variables {Xk} by mutations:

Xk ∗Xk+2 = q−12Xk+1 +1, ∀k ∈ Z.

Example (Quantum cluster algebra of rank 2)

Seeds (local charts): ({Xk , Xk+1},(−1)k+1B,(−1)k+1Λ).

Quantum cluster algebra Aq = Z[q±12 ][Xk ]k∈Z.



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A rank 2 example

Example (Previous example)

X3 = X (−1,1) +X (−1,0)(= X−11·X2 +X−1

1)

X4 = X (0,−1) +X (−1,−1) +X (−1,0).



Triangular bases


General cluster algebras

De�nition (Berenstein-Zelevinsky, 05)

In general, for any given skew-symmetrizable m×n matrix B ,m ≥ n, and a compatible skew-symmetric m×m matrix Λ, we cande�ne the quantum cluser algebra

Aq = Aq((X1, . . . ,Xm),B,Λ).

Theorem (Laurent phenomenon[Fomin-Zelevinsky][Berenstein-Zelevinsky])

Any cluster variable is a Laurent polynomial in Z[X±1, . . . ,X±m ].

Any quantum cluster variable is a Laurent polynomial in

Z[q±12 ][X±

1, . . . ,X±m ].



Triangular bases


Cluster expansions

Theorem ([Derksen-Weyman-Zelevinsky][Plamondon][Nagao][Gross-Hacking-Keel-Kontsevich],+[Tran])

De�ne Yk = X (Bek )T , then any quantum cluster variable is always a

Laurent polynomial of the form:

X g (1+ ∑06=v∈Nn

cvYv ), cv ∈ Z[q±

12 ].


Recall B =

(0 −11 0

), we have Y1 = X2, Y2 = X−1

1,

X3 = X (−1,1)(1+Y1), X4 = X (0,−1)(1+Y2 +Y1Y2)



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Degrees and partial order

De�nition

A Laurent polynomial Z of the form X g (1+ ∑06=v∈Nn cvYv ) is said

to be pointed at the degree g .In this case, we denote degZ = g .

De�nition

We say degrees g ≥ g ′ if degX gY v = degX g ′ for some v ∈ Nn.

Then the above Z has the unique maximal degree g .


Recall Y1 = X2, Y2 = X−11

,X3 = X (−1,1)(1+Y1), X4 = X (0,−1)(1+Y2 +Y1Y2)degX4 > degX3= degX4Y2Y

2

1.



Triangular bases


Characters of simples of quatum a�ne algebras

Uq(g): Quantum a�ne algebraUε (Lg): specialized quantum loop algebra, ε ∈ C∗generic .w : vector (mulitiplicities of the roots of Drinfeld polynomials)S(w): �nite dim. simple l-highest weight module

De�nition

We have the characterχS(w) = Y w (1+ ∑06=v∈Nn avA

v ), av ∈ N,Y w , Av : some formal Laurent monomialsav : dimension of l-weight space

Dominiance order: Y w dominates Y wAv

t-deformation: we can re�ne the coe�cients to av ∈ N[t±].[Varagnolo-Vasserot] [Nakajima] [Hernandez]



Triangular bases


Compare Laurent polynomials

[Hernandez-Leclerc] suggests we compare the Laurent expansion ofa (quantum) cluster variable

X g (1+ ∑06=v∈Nn cvYv ), cv ∈ Z[q±

12 ].

with the (t-deformed) character of a simpleχS(w) = Y w (1+ ∑06=v∈Nn avA

v ), av ∈ N[t±].

Fact

We can identify the dominance order Y w > Y wAv with our partial

order degX g > degX gY v .

Monoidal categori�catoin Conjecture

Can we identify the two Laurent polynomials?



Triangular bases

ConstructionMain theoremExamples of cluster algebras

Triangular basis for an initial seed

Choose and work in an initial seed ({X1, . . . ,Xn, . . . ,Xm},B,Λ).Assume that there exists quantum cluser varaibles Ik , 1≤ k ≤ n,such that prn deg Ik =−ek(prn = projection to the �rst n-coordinates).

De�nition

The triangular basis L is the basis of Aq such that

1 Xi , Ik ∈L

2 elements of L are bar-invariant

3 (Parametrization) elements of L have unique maximal degreeswith coe�cient 1, such that deg : L ' Zm.

4 (Triangularity) ∀Xi ,b1 ∈L , there exists some s ∈ Z2such that

qsXi ∗b1 = b2 + ∑b ab ·b,where b2,b ∈L , coe�cients ab ∈ q−

12Z[q−

12 ],

degXi +degb1 = degb2 > degb.Fan Qin Monoidal categori�cations of cluster algebras


Triangular bases


Triangularity

Triangular decomposition of Xi ∗b1 into L is inspired by

The transition from PBW basis to canonical basis is unitriangular

qsE1 ∗E2 ∗ . . .∗El = b2 + ∑ab ·bwhere param(b2) = ∑i param(Ei ), param(b) < param(b2),

ab ∈ q−12Z[q−

12 ].

This property is very similar to

Theorem (Leclerc's conjecture, proved in [KKKO 14])

Let X ,b1 ∈B∗ be dual canonical basis elements, such that

X ∗X ∈ B∗, then there exists some s ∈ Z2such that

X ∗b1 = q−sb2 + ∑b ab ·b+q−tb3,qsab,q

s−t ∈ q−12Z[q−

12 ], and qt−s ,qtab ∈ q

12Z[q

12 ].



Triangular bases


Common triangular basis and Fock-Goncharov conjecture

A triangular basis, if it exists, is unique.

The notion of the triangular basis depends on the choseninitial seed (local chart).In particular, the degree of a basis element will di�er when wechange the initial seed.

De�nition

A basis L is called the common triangular basis, if it is thetriangular basis for any seed and,moreover, its parametrazation in di�erent seeds veri�es theFock-Goncharov conjecture:

L ' Zm

|| l mutation of tropical Z-pointsL ' Zm

.



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Main Theorem

Type I Type II

All cluster algebras

Theorem ([Q. 15])

For some of type I (adaptable Coxeter element case) and all type II,

the basis of simples produces the common triangular basis of the

quantum cluster algebra, which also veri�es the Fock-Goncharov

conjecture.

Corollary

Monoidal categori�cation conjecture is true in these cases..



Triangular bases


Idea of the proof

How to �nd the common tringular basis?

Verify that the basis of simples produces the triangular basis ofa good initial seed.

Try to proceed by induction on seeds:Key obervation: for a basis with positive structure constants,

If it contains all cluster variables (monoidal categori�cationconjecture), then it has good parametrizations(Fock-Goncharov conjecture)If it has good parametrizations, then it contains all clustervariables.

In practice, we do induction on seeds to show that the basisproduced by simples satisify both conjectures.



Triangular bases


Cluster algebra associated with a quiver

For any m ≥ n ∈ N and quiver Q (�nite oriented graph) withvertices {1, . . . ,n, . . . ,m}, we can de�ne an m×n matrix B = (bij):bij = |arrows i → j |− |arrows j → i |.Therefore, we can associate cluster algebra with any quiver.We can further impose a quantization if rkB = n.


Choose m = n = 2. The following quiver gives us the 2×2 matrix

B =

(0 −11 0

).

2 1

By choosing a special quantization, we get the previous quantumcluster algebra.



Triangular bases


Cluster algebra associated with a quiver

Example

Choose m = 4, n = 2. The following quiver gives us the 4×2

matrix B =

0 −11 0−1 10 −1

.

4

3

2

1



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Cluster algebra: Type II

Example (Type II: Uq(sl3)-mod)

The following quiver arising from a level 3 subcategory ofUq(sl3)-mod.

1

2

3

4

5

6

7

8



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Cluster algebra: Type I adaptable word

Example (Type I: adaptable word)

The following quiver is associated with the adaptable word

i = (2,1,2,1,2,1,2,1) and the Cartan matrix C =

(2 −2−2 2

).

1

2

3

4

5

6

7

82

22

22

22



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Cluster algebra: Type I and Type II

Example (Type I and type II)

The following quiver is associated with the adaptable wordi = (1,2,1,3,2,1,4,3,2,1) and the Cartan matrix

C =

2 −1 0 0−1 2 −1 00 −1 2 −10 0 −1 2

.

It also arises from a subcategory of Uε (sl5)-mod.

1

2

3

4

5

6

7

8

9

10



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Cluster algebra: Type I non-adaptable word

Example (Type I: non-adaptable word)

The following quiver is associated with the non-adaptable wordi = (2,3,2,1,2,1,3,1,2,1) and the Cartan matrix

C =

2 −3 −2−3 2 −2−2 −2 2

.

It is not included in our Theorem.

1

2

3

4

5

6

7

8

9

10

3

22

333

222

23

2


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