introduction to cluster algebras anna...

Introduction to cluster algebras

Anna FeliksonDurham University

a

d

e

b

f

cef=ac+bd

LMS Graduate Student Meeting, London

June 29, 2018

a

Root systems

Frieze patterns

Quiver representations

Dilogarithm identities

Supersymmetric gauge theories

Conformal field theory

Solitons

Mirror symmetry

Poisson geometry

Integrable systems

Tropical geometry

Combinatorics of polytopes

Triangulated surfaces

Teichmuller theory

Hyperbolic geometry

Coxeter groups

Cluster algebras(Fomin, Zelevinsky, 2001)

a

Root systems

Frieze patterns





Solitons

Mirror symmetry

Poisson geometry

Integrable systems

Tropical geometry



Teichmuller theory

Hyperbolic geometry

Coxeter groups


Sergey Fomin

Andrei Zelevinsky

a


Hyperbolic geometry

Coxeter groups

Teichmuller theory


Frieze patterns

Root systems

Grassmannians





Tropical geometry

Poisson geometry

Solitons

Gentle algebras


Mirror symmetry

Integrable systems

0. Prologue: Totally positive matrices

• A minor of a matrix is a determinant of a square submatrix.

• An n× n matrix A is called totally positive

if every minor of A is a positive real number.

Question: How many minors and which ones do one needs to compute

for checking that the matrix is totally positive?

Answer: All is enough, but can do much better:

aaa Example: 2× 2 matrix(a bc d

)– enough to check a, b, c, ad− bc

or d, b, c, ad− bcaaa Example: 3× 3 matrix: out of 19 only need 9.

One can go from one criteria to another by local moves:

asdasdsadadsadeach time substituting one of the functions by a new one.







Answer: All is enough, but we can do better:















aaa Example: 3× 3 matrix: out of 19 only need 9.












d = (ad−bc)+bca

aaa Example: 3× 3 matrix: out of 19 only need 9.












or d, b, c, ad− bc

aaa In general [Fomin, Zelevinsky’ 1999]: need n2.

asdasdsadadsadCan go from one test to another by local moves:


1. Cluster algebras: Quiver mutation

Quiver is an oriented graph without loops and 2-cycles.

Mutation µk of a quiver:

aaa • for every path i→ k → j add an arrow i→ j;

aaa • cancel 2-cycles;

aaa • reverse all arrows incident to k.

Example:

1. Cluster algebras: Quiver mutation

Quiver is an oriented graph without loops and 2-cycles.

Mutation µk of a quiver:

aaa • for every path i→ k → j add an arrow i→ j;

aaa • cancel 2-cycles;

aaa • reverse all arrows incident to k.

Example:

5

3

7

3

5 7 5 7

3235

32

5 7

1. Cluster algebras: Seed mutation

A seed s is a pair (Q, {u1, . . . , un}) where ui = ui(x1, . . . , xn),

A seed s is a pair where asdasdsadasdasd x1, . . . , xn – independent variables.

Initial seed s0 = (Q0, {x1, . . . , xn}.

Mutation: µk(s) = (µk(Q), {u1, . . . , u′k, . . . , un}

u′k =1

uk

∏i→k

ui +∏k→j

uj

Functions ui are called cluster variables.

Definition. Cluster algebra A(Q) is a Q-subalgebra of Q(x1, . . . , xn)

Definition. generated by all cluster variables.




Initial seed s0 = (Q0, {x1, . . . , xn}).

Mutation: µk(s) = (µk(Q), {u1, . . . , u′k, . . . , un})

u′k =1

uk

∏i→k

ui +∏k→j

uj


Definition. Cluster algebra A(Q) is a Q-subalgebra of Q(x1, . . . , xn)

Definition. generated by all cluster variables.




Initial seed s0 = (Q0, {x1, . . . , xn}).

Mutation: µk(s) = (µk(Q), {u1, . . . , u′k, . . . , un})

u′k =1

uk

∏i→k

ui +∏k→j

uj


Definition. Let F be the set of all rational functions in x1, . . . , xn.

The cluster algebra A(Q) is the subring of FDefinition. The cluster algebra aaaa generated by all cluster variables.


How to think about this?

Example:

x1

x3

x4x2

x5

µ1

x′1 = 1x1

(x2x4 + x3x5)

x3

x4x2

x5

Cluster variables: x1, x2, x3, x4, x5, x′1, . . .

1. Cluster algebras: Two remarkable properties

By definition ui = P (x1,...,xn)R(x1,...,xn), where P and R are polynomials.

In fact:

• Laurent phenomenon: R(x1, . . . , xn) is a monomial, R = xd11 . . . xdnn .

aaaaaaaasds It is a miracle as computing u′k = 1uk

(∏i→k

ui +∏k→j

uj) we divide by uk!

• Positivity: P (x1, . . . , xn) has positive coefficients.

aaaaaaaasds It is a miracle as we divide: a3+b3

a+b = a2 − ab+ b2.



In fact:

• Laurent phenomenon [Fomin, Zelevinsky’ 2001]:

R(x1, . . . , xn) is a monomial, R = xd11 . . . xdnn .


(∏i→k

ui +∏k→j


• Positivity conj. by Fomin, Zelevinsky, proved by Schiefler,Lee’2013:

P (x1, . . . , xn) has positive coefficients.


a+b = a2 − ab+ b2.



In fact:

• Laurent phenomenon [Fomin, Zelevinsky’ 2001]:

R(x1, . . . , xn) is a monomial, R = xd11 . . . xdnn .


(∏i→k

ui +∏k→j


• Positivity [Conj.: Fomin, Zelevinsky’ 2001; proved: Lee, Schiffler’ 2013]:

P (x1, . . . , xn) has positive coefficients.


a+b = a2 − ab+ b2.

2. Triangulated polygons

Triangulated polygon −→ Quiver Q

diagonal in triangulation vertex of quiver

two edges of one triangle arrow of quiver

flip of triangulation mutation of quiver



diagonal in triangulation vertex of quiver

two edges of one triangle arrow of quiver


−→ Cluster algebraA(Q)

Label vertices of Q by xi, transform as in x′1 = x2x4+x3x5x1

under mutations.



diagonal in triangulation of quiver

two edges of one triangle of quiver



By sequence of flips can take any triangulation to any other.



diagonal in triangulation of quiver

two edges of one triangle of quiver



By sequence of flips can take any triangulation to any other.

Triangualtions ↔ Seeds of A(Q)

diagonals ↔ cluster variables


Remark: Transformation x′1 = x2x4+x3x5x1

makes sense

for lengths of diagonals in E2:

The lengths of diagonals in an inscribed polygon are not independent variables.This can be fixed by using hyperbolic geometry instead of Euclidean.


Remark: Transformation x′1 = x2x4+x3x5x1

makes sense

for lengths of diagonals in E2:

AB

C

D

Ptolemy Theorem:

Given an inscribed quadrilateral ABCD⊂ E2,

AC ·BD = AB · CD +BC ·AD

The lengths of diagonals in an inscribed polygon are not independent variables.This can be fixed by using hyperbolic geometry instead of Euclidean.

3. Grassmannians Gr2,n

Grassmannian Grk,n(R) = {V | V ∈ Rn, dim V = k}is a space of k-planes in Rn.

k = 1: Gr1,n = { lines in Rn} = RPn−1.

Elements of Gr2,n can be represented by full rank k × n matrices.

k = 2: Gr2,n = { A =

(a11 . . . a1n

a21 . . . a2n

)| rk A = 2}.

This representation is not unique: can take another basis of the 2-plane.

So, Gr2,n = Mat2,n/ ∼, where ∼ stays for changes of basis in R2.

3. Grassmannians Gr2,n

Grassmannian Grk,n(R) = {V | V ∈ Rn, dim V = k}is a space of k-planes in Rn.

k = 1: Gr1,n = { lines in Rn} = RPn−1.

Elements of Grk,n can be represented by full rank k × n matrices.

k = 2: Gr2,n = { A =

(a11 . . . a1n

a21 . . . a2n

)| rk A = 2}.

This representation is not unique: can take another basis of the 2-plane.

So, Gr2,n = Mat2,n/ ∼, where ∼ stays for changes of basis in R2.

3. Grassmannians Gr2,n = {2-planes in Rn }

Determinants ∆ij =∣∣∣∣a1i a1j

a2i a2j

∣∣∣∣ are called Plucker coordinates.

If ∆12 6= 0 then(a11 a12

a21 a22

)−1(a11 . . . a1n

a21 . . . a2n

)=

(1 0 c3 . . . cn0 1 d3 . . . dn

)This represents a point with ∆12 6= 0 uniquely.

Moreover: ci = −∆2i∆12

, di = ∆1i∆12

.

• So, {∆ij} (up to simultaneous scaling) determine a point of Gr2,n.

This works for all points of Gr2,n: ∆ij 6= 0 for some i,j as rkA = 2.

• {∆ij} are not independent: ∆ij

∆12= cidj − cjdi = −∆2,i∆1j

∆212

+∆2j∆ij

∆212

⇔ ∆2j∆1i = ∆12∆ij + ∆2i∆1j



a2i a2j


If ∆12 6= 0 then(a11 a12

a21 a22

)−1(a11 . . . a1n

a21 . . . a2n

)=

(1 0 c3 . . . cn0 1 d3 . . . dn



, di = ∆1i∆12

.




∆12= cidj − cjdi = −∆2,i∆1j

∆212

+∆2j∆ij

∆212

⇔ ∆2j∆1i = ∆12∆ij + ∆2i∆1j j1

2 i



a2i a2j


If ∆12 6= 0 then(a11 a12

a21 a22

)−1(a11 . . . a1n

a21 . . . a2n

)=

(1 0 c3 . . . cn0 1 d3 . . . dn



, di = ∆1i∆12

.




∆12= cidj − cjdi = −∆2,i∆1j

∆212

+∆2j∆ij

∆212

⇔ ∆2j∆1i = ∆12∆ij + ∆2i∆1j j1

2 i

Similarly, ∆ik∆jl = ∆ij∆kl + ∆il∆jk for i < j < k < l.l

j k

i



a2i a2j


They are subject to Plucker relations:

∆ik∆jl = ∆ij∆kl + ∆il∆jk l

j k

i

How many of them are enough to know? Dimension requires 2n− 3.

Take any triangulation T of an n-gon.

Then {∆ij | ij = side or diagonal in T} is sufficient to find all ∆lk:

asda apply Plucker (=Ptolemy) relations to resolve crossings!

Get a cluster algebra structure on Gr2,n withasdasdaaa cluster variables ↔ diagonals of n-gonasdasdaaa aaasdsdsiiiiiseeds ↔ triangulations of n-gon



a2i a2j




j k

i







a2i a2j




j k

i





Get a cluster algebra structure on Gr2,n withasdasdaaa cluster variables ↔ diagonals of n-gonasdasdaaa aaasdsdsiiiiiseeds ↔ triangulations of n-gon



a2i a2j




j k

i





A totally positive Grassmannian Grtp2,n is a subset of Gr2,n

where ∆ij > 0 for all i, j.By positivity of cluster variables,

only need to check initial variables.

4. Finite type

A cluster algebra is of finite type if it contains finitely many cluster variables.

Theorem [Fomin, Zelevinsky’ 2002] A(Q) is of finite type iff

Q is mutation-equivalent to an orientation of a Dynkin diagram

Note: Dynkin diagrams describe:

finite reflection groups, semisimple Lie algebras, surface singularities...

Cluster algebra related toan n-gon and Grassmannian G2,n

4. Finite type




An Dn

E6 E7 E8




4. Finite type




An Dn

E6 E7 E8




is of type An:

4. Finite type and finite mutation type



Q is mutation-equivalent to an orientation of a Dynkin diagram.

A cluster algebra A(Q) is of finite mutation type if there are finitely many

quivers in the mutation class of Q.

Theorem [A.F, M.Shapiro, P.Tumarkin’ 2008]

Let Q be a connected quiver of finite mutation type with |Q| > 2. Then

- either Q is obtained from a triangulated surface;

- or Q is mut.-equivalent to one of the following 11 quivers:

4. Finite type and finite mutation type



Q is mutation-equivalent to an orientation of a Dynkin diagram.

A cluster algebra A(Q) is of finite mutation type if there are finitely many

quivers in the mutation class of Q.

Theorem [A.F, M.Shapiro, P.Tumarkin’ 2008]

A connected quiver Q s.t. |Q| > 2 is of finite mutation type iff

- either Q is obtained from a triangulated surface;

- or Q is mut.-equivalent to one of the following 11 quivers:

5. Bonus: proof of Ptolemy theorem

ef=ac+bd

(∆ ABC)f .

(∆ BCD)a. BDA)(∆b.

ac bd

ef

C

De

d

a

f

c

b

A

B

af bf

β γ

γ

α+δ γ+δ

α

αβ

αδ

δ

β α

δγ

β

β+γ

(Proof borrowed from cut-the-knot portal)

5. Bonus: proof of Ptolemy theorem

ef=ac+bd

(∆ ABC)f .

BDA)(∆b.(∆ BCD)a.

ac bd

ef

C

De

d

a

f

c

b

A

B

af bf

β γ

γ

α+δ γ+δ

α

αβ

αδ

δ

β α

δγ

β

β+γ

Thanks!

(Proof borrowed from cut-the-knot portal)

introduction to cluster algebras anna...

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