Cluster Algebras
Lara Bossinger
Universidad Nacional Aut´onomade M´exico,Unidad Oaxaca
Pittsburgh September 25 2020, 55 minutes
Cluster Algebras Lara Bossinger 1/ 22 Overview
1 Cluster algebras in Mathematics 2 History 3 Total positivity 4 Quivers, seeds, mutation and cluster algebras 5 Some applications
Cluster Algebras Lara Bossinger 2/ 22 Cluster algebras in Mathematics
Cluster algebras
Cluster Algebras Lara Bossinger 3/ 22 Cluster algebras in Mathematics
Algebra:
Cluster algebras
Cluster Algebras Lara Bossinger 3/ 22 Cluster algebras in Mathematics
Algebra: total positivity, quantum groups, representation theory, homological algebra, categorification, Gr¨obnertheory Cluster algebras
Cluster Algebras Lara Bossinger 3/ 22 Cluster algebras in Mathematics
Algebra: total positivity, quantum groups, representation theory, homological algebra, categorification, Gr¨obnertheory Cluster algebras
Combinatorics/Polyhedral geometry:
Cluster Algebras Lara Bossinger 3/ 22 Cluster algebras in Mathematics
Algebra: total positivity, quantum groups, representation theory, homological algebra, categorification, Gr¨obnertheory Cluster algebras
Combinatorics/Polyhedral geometry: root systems, Stasheff polytopes, associahedra, Newton–Okounkov polytopes, scattering diagrams, dimer models/ plabic graphs
Cluster Algebras Lara Bossinger 3/ 22 Cluster algebras in Mathematics
Algebra: total positivity, quantum groups, representation theory, homological algebra, categorification, Gr¨obnertheory Cluster algebras
Combinatorics/Polyhedral geometry: root systems, Stasheff polytopes, associahedra, Newton–Okounkov polytopes, Geometry: scattering diagrams, dimer models/ plabic graphs
Cluster Algebras Lara Bossinger 3/ 22 Cluster algebras in Mathematics
Algebra: total positivity, quantum groups, representation theory, homological algebra, categorification, Gr¨obnertheory Cluster algebras
Combinatorics/Polyhedral geometry: root systems, Stasheff polytopes, associahedra, Newton–Okounkov polytopes, Geometry: scattering diagrams, dimer models/ tropical and toric geometry, log Calabi–Yau plabic graphs varieties, Donaldson–Thomas theory, Teichm¨uller theory, quantization, integrable systems, symplectic and Poisson geometry, triangulated surfaces
Cluster Algebras Lara Bossinger 3/ 22 Cluster algebras in Mathematics
Algebra: total positivity, quantum groups, representation theory, homological algebra, categorification, Gr¨obnertheory
Topology: Cluster algebras
Combinatorics/Polyhedral geometry: root systems, Stasheff polytopes, associahedra, Newton–Okounkov polytopes, Geometry: scattering diagrams, dimer models/ tropical and toric geometry, log Calabi–Yau plabic graphs varieties, Donaldson–Thomas theory, Teichm¨uller theory, quantization, integrable systems, symplectic and Poisson geometry, triangulated surfaces
Cluster Algebras Lara Bossinger 3/ 22 Cluster algebras in Mathematics
Algebra: total positivity, quantum groups, representation theory, homological algebra, categorification, Gr¨obnertheory
Topology: Cluster algebras knot theory, Jones polynomial, braid groups, Combinatorics/Polyhedral geometry: Lie groups root systems, Stasheff polytopes, associahedra, Newton–Okounkov polytopes, Geometry: scattering diagrams, dimer models/ tropical and toric geometry, log Calabi–Yau plabic graphs varieties, Donaldson–Thomas theory, Teichm¨uller theory, quantization, integrable systems, symplectic and Poisson geometry, triangulated surfaces
Cluster Algebras Lara Bossinger 3/ 22 Cluster algebras in Mathematics
Theoretical Physics: Algebra: total positivity, quantum groups, representation theory, homological algebra, categorification, Gr¨obnertheory
Topology: Cluster algebras knot theory, Jones polynomial, braid groups, Combinatorics/Polyhedral geometry: Lie groups root systems, Stasheff polytopes, associahedra, Newton–Okounkov polytopes, Geometry: scattering diagrams, dimer models/ tropical and toric geometry, log Calabi–Yau plabic graphs varieties, Donaldson–Thomas theory, Teichm¨uller theory, quantization, integrable systems, symplectic and Poisson geometry, triangulated surfaces
Cluster Algebras Lara Bossinger 3/ 22 Cluster algebras in Mathematics
Theoretical Physics: string theory, mirror symmetry, Algebra: scattering amplitudes, Conformal field total positivity, quantum groups, theory, Bethe Ansatz in representation theory, homological Theoremodynamics algebra, categorification, Gr¨obnertheory
Topology: Cluster algebras knot theory, Jones polynomial, braid groups, Combinatorics/Polyhedral geometry: Lie groups root systems, Stasheff polytopes, associahedra, Newton–Okounkov polytopes, Geometry: scattering diagrams, dimer models/ tropical and toric geometry, log Calabi–Yau plabic graphs varieties, Donaldson–Thomas theory, Teichm¨uller theory, quantization, integrable systems, symplectic and Poisson geometry, triangulated surfaces
Cluster Algebras Lara Bossinger 3/ 22 Cluster algebras in Mathematics
Theoretical Physics: Analysis: string theory, mirror symmetry, Algebra: scattering amplitudes, Conformal field total positivity, quantum groups, theory, Bethe Ansatz in representation theory, homological Theoremodynamics algebra, categorification, Gr¨obnertheory
Topology: Cluster algebras knot theory, Jones polynomial, braid groups, Combinatorics/Polyhedral geometry: Lie groups root systems, Stasheff polytopes, associahedra, Newton–Okounkov polytopes, Geometry: scattering diagrams, dimer models/ tropical and toric geometry, log Calabi–Yau plabic graphs varieties, Donaldson–Thomas theory, Teichm¨uller theory, quantization, integrable systems, symplectic and Poisson geometry, triangulated surfaces
Cluster Algebras Lara Bossinger 3/ 22 Cluster algebras in Mathematics
Theoretical Physics: Analysis: string theory, mirror symmetry, ordinary/partial differential Algebra: scattering amplitudes, Conformal field equations, KP solitons total positivity, quantum groups, theory, Bethe Ansatz in representation theory, homological Theoremodynamics algebra, categorification, Gr¨obnertheory
Topology: Cluster algebras knot theory, Jones polynomial, braid groups, Combinatorics/Polyhedral geometry: Lie groups root systems, Stasheff polytopes, associahedra, Newton–Okounkov polytopes, Geometry: scattering diagrams, dimer models/ tropical and toric geometry, log Calabi–Yau plabic graphs varieties, Donaldson–Thomas theory, Teichm¨uller theory, quantization, integrable systems, symplectic and Poisson geometry, triangulated surfaces
Cluster Algebras Lara Bossinger 3/ 22 History
Cluster algebras were first introduced in 2001 by Sergei Fomin and Andrei Zelevinsky. They had observed a mathmatical structure (cluster structure) on various objects related to the study of totally positive matrices, quantum groups and Kashiwara/Lusztig’s canonical basis. Their work gained a lot of attention, first in representation theory, but quickly grew beyond its origins. Today Cluster algebras have their own MSC classifier 13F60, there are 690 papers on Mathscinet with this MSC classification and 1,645 on the arxiv. Since 2003 at least 139 international conferences have been organized on this topic
Cluster Algebras Lara Bossinger 4/ 22 Total Positivity
Definition n×n A matrix M ∈ R is totally positive if all its minors (i.e. determinantes of sub-squarematrices) are positive real numbers.
Example a b Take M = . Then M is totally positive if and only if a, b, c, d and c d ∆ = ad − cd ∈ R>0. 1 Observe that d = a (∆ + cd). Hence, it suffices to verify that a, b, c, ∆ ∈ R>0. The set {a, b, c, ∆} is a positivity test. Question: How can we efficiently test for total positivity?
Cluster Algebras Lara Bossinger 5/ 22 n Totally positive Grassmannian Gr2(C )>0
n n Gr2(C ) := {V ⊂ C | dim V = 2} and its elements can be represented by v1 v2 ... vn 2×n matrices: for V = hv, wi define MV = ∈ C w1 w2 ... wn (unique up to rescaling of the rows). Definition n For i, j ∈ {1, 2,..., n} with i < j define for V ∈ Gr2(C ) with MV as above vi vj pij (V ) := det . wi wj n The totally positive Grassmannian Gr2(C )>0 consist of those points n V ∈ Gr2(C ) for which pij (V ) ∈ R>0 for all i, j. n Question: There are 2 Pl¨ucker coordinates. How many do we have to n test to know that a given point lies in Gr2(C )>0?
Cluster Algebras Lara Bossinger 6/ 22 Ptolemy and Pl¨ucker relations Pl¨ucker coordinates are not independent.
Cluster Algebras Lara Bossinger 7/ 22 Ptolemy and Pl¨ucker relations Pl¨ucker coordinates are not independent. One verifies that for i < j < k < l ∈ {1,..., n} : pik pjl = pij pkl + pil pjk .
Cluster Algebras Lara Bossinger 7/ 22 Ptolemy and Pl¨ucker relations Pl¨ucker coordinates are not independent. One verifies that for i < j < k < l ∈ {1,..., n} : pik pjl = pij pkl + pil pjk . Can vizualize the Pl¨ucker relation:
Cluster Algebras Lara Bossinger 7/ 22 Ptolemy and Pl¨ucker relations Pl¨ucker coordinates are not independent. One verifies that for i < j < k < l ∈ {1,..., n} : pik pjl = pij pkl + pil pjk . Can vizualize the Pl¨ucker relation:
pjk pjk j k pik pjl = pij pkl + pil pjk j k
pij pik pkl pij pjl pkl
i l i l pil pil
Cluster Algebras Lara Bossinger 7/ 22 Ptolemy and Pl¨ucker relations Pl¨ucker coordinates are not independent. One verifies that for i < j < k < l ∈ {1,..., n} : pik pjl = pij pkl + pil pjk . Can vizualize the Pl¨ucker relation:
pjk pjk j k pik pjl = pij pkl + pil pjk j k
pij pik pkl pij pjl pkl
i l i l pil pil
→ compare to the Ptolemy relation:
Cluster Algebras Lara Bossinger 7/ 22 Ptolemy and Pl¨ucker relations Pl¨ucker coordinates are not independent. One verifies that for i < j < k < l ∈ {1,..., n} : pik pjl = pij pkl + pil pjk . Can vizualize the Pl¨ucker relation:
pjk pjk j k pik pjl = pij pkl + pil pjk j k
pij pik pkl pij pjl pkl
i l i l pil pil
→ compare to the Ptolemy relation: AC · BD = AB · CD + BC · AD
Cluster Algebras Lara Bossinger 7/ 22 n TP tests for Gr2(C ) efficient TP 1−1 triangulations Consequence: n ←→ . tests for Gr2(C ) of an n-gon
Cluster Algebras Lara Bossinger 8/ 22 n TP tests for Gr2(C ) efficient TP 1−1 triangulations Consequence: n ←→ . tests for Gr2(C ) of an n-gon n → it suffices to check a set of 2(n − 2) + 1 = dim Gr2(C ) + 1 independent Pl¨ucker coordinates for positivity.
Cluster Algebras Lara Bossinger 8/ 22 n TP tests for Gr2(C ) efficient TP 1−1 triangulations Consequence: n ←→ . tests for Gr2(C ) of an n-gon n → it suffices to check a set of 2(n − 2) + 1 = dim Gr2(C ) + 1 independent Pl¨ucker coordinates for positivity.
1 p12 p15
2 p13 p14 5 p23 p45 3 4 p34
Cluster Algebras Lara Bossinger 8/ 22 n TP tests for Gr2(C ) efficient TP 1−1 triangulations Consequence: n ←→ . tests for Gr2(C ) of an n-gon n → it suffices to check a set of 2(n − 2) + 1 = dim Gr2(C ) + 1 independent Pl¨ucker coordinates for positivity.
1 p12 p15
2 p13 p14 5 p23 p45 3 4 p34 1
2 5
3 4
Cluster Algebras Lara Bossinger 8/ 22 n TP tests for Gr2(C ) efficient TP 1−1 triangulations Consequence: n ←→ . tests for Gr2(C ) of an n-gon n → it suffices to check a set of 2(n − 2) + 1 = dim Gr2(C ) + 1 independent Pl¨ucker coordinates for positivity.
1 p12 p15
2 p13 p14 5 p23 p45 3 4 p34 1
2 5
3 4
1
2 5
3 4
Cluster Algebras Lara Bossinger 8/ 22 n TP tests for Gr2(C ) efficient TP 1−1 triangulations Consequence: n ←→ . tests for Gr2(C ) of an n-gon n → it suffices to check a set of 2(n − 2) + 1 = dim Gr2(C ) + 1 independent Pl¨ucker coordinates for positivity.
1 p12 p15
2 p13 p14 5 p23 p45 3 4 p34 1
2 5
3 4
1 1
2 5 2 5
3 4 3 4
Cluster Algebras Lara Bossinger 8/ 22 n TP tests for Gr2(C ) efficient TP 1−1 triangulations Consequence: n ←→ . tests for Gr2(C ) of an n-gon n → it suffices to check a set of 2(n − 2) + 1 = dim Gr2(C ) + 1 independent Pl¨ucker coordinates for positivity.
1 p12 p15
2 p13 p14 5 p23 p45 3 4 p34 1 1
2 5 2 5
3 4 3 4
1 1
2 5 2 5
3 4 3 4
Cluster Algebras Lara Bossinger 8/ 22 Quivers and mutation
A quiver Q is a directed graph, consisting of a finite set of vertices and arrows between them. Technical assumption: Q does not have any loops or 2-cycles. We split the vertex set into mutable vertices {1,..., n} and frozen vertices {n + 1,..., m}, e.g. 1 ⇒ 2 → 3. Definition (Quiver mutation) Given a quiver Q and a mutable vertex k, the mutation in direction k µk (Q) is a quiver obtained from Q in three steps: 1 for every path i → k → j add an arrow i → j; 2 invert every arrow incident to k; 3 remove a maximal set of 2-cycles.
→ Quiver mutation is an involution: µk (µk (Q)) = Q.
Cluster Algebras Lara Bossinger 9/ 22 Example: quiver mutation in Keller’s mutation app
1 Q :42 o 5 1 O o @ O
1 / 3
2 Q2 : 2 @ @
1 o 3
Cluster Algebras Lara Bossinger 10/ 22 Iteration of mutations: the n-regular tree
• • 5 4 • • 1 • • 3 5 4 • • 1 • 2 2 • • 3 • Q 5 4 • 1 • 3 • 2 • 4 • 5 5 • 1 • 2 3 • • 4 • • 1 • 3 2 • •
Cluster Algebras Lara Bossinger 11/ 22 Seeds and mutation
A seed s is a pair (x, Q), where x = (x1,..., xn, xn+1,..., xn+m) is a collection of variables called a cluster and Q a quiver with n mutable and m frozen vertices. Definition (Seed mutation) Given a seed s = (x, Q) and a mutable vertex k of Q, the mutation in direction k µk (s) is the pair (µk (x), µk (Q)), where 0 µk (x) = x \{xk } ∪ {xk } and Q Q 0 i→k∈Q xi + k→j∈Q xj xk := . xk
→ Seed mutation is an involution: µk (µk (s)) = s.
Cluster Algebras Lara Bossinger 12/ 22 1 =∼ 1+x 2 ← x x2 ← x1 x1 2
2 1
1+x2 → 1+x1+x2 1+x1 → x x1 x1x2 x2 1
1 2 1+x1 ← 1+x1+x2 x2 x1x2
1 + x + x 1 + x x x + 1 + x + x 1 + x 1 + 1 2 ÷ 2 = 1 2 1 2 = 1 x1x2 x1 x2(1 + x2) x2
Example: seed mutation
x1 → x2
Cluster Algebras Lara Bossinger 13/ 22 =∼ 1+x 2 ← x x2 ← x1 x1 2
2 1
1+x2 → 1+x1+x2 1+x1 → x x1 x1x2 x2 1
1 2 1+x1 ← 1+x1+x2 x2 x1x2
1 + x + x 1 + x x x + 1 + x + x 1 + x 1 + 1 2 ÷ 2 = 1 2 1 2 = 1 x1x2 x1 x2(1 + x2) x2
Example: seed mutation
x1 → x2
1
Cluster Algebras Lara Bossinger 13/ 22 =∼
x2 ← x1
2 1
1+x2 → 1+x1+x2 1+x1 → x x1 x1x2 x2 1
1 2 1+x1 ← 1+x1+x2 x2 x1x2
1 + x + x 1 + x x x + 1 + x + x 1 + x 1 + 1 2 ÷ 2 = 1 2 1 2 = 1 x1x2 x1 x2(1 + x2) x2
Example: seed mutation
x1 → x2
1
1+x2 ← x x1 2
Cluster Algebras Lara Bossinger 13/ 22 =∼
x2 ← x1
1
1+x2 → 1+x1+x2 1+x1 → x x1 x1x2 x2 1
1 2 1+x1 ← 1+x1+x2 x2 x1x2
1 + x + x 1 + x x x + 1 + x + x 1 + x 1 + 1 2 ÷ 2 = 1 2 1 2 = 1 x1x2 x1 x2(1 + x2) x2
Example: seed mutation
x1 → x2
1
1+x2 ← x x1 2
2
Cluster Algebras Lara Bossinger 13/ 22 =∼
x2 ← x1
1
1+x1 → x x2 1
1 2 1+x1 ← 1+x1+x2 x2 x1x2
1 + x + x 1 + x x x + 1 + x + x 1 + x 1 + 1 2 ÷ 2 = 1 2 1 2 = 1 x1x2 x1 x2(1 + x2) x2
Example: seed mutation
x1 → x2
1
1+x2 ← x x1 2
2
1+x2 → 1+x1+x2 x1 x1x2
Cluster Algebras Lara Bossinger 13/ 22 =∼
x2 ← x1
1
1+x1 → x x2 1
2 1+x1 ← 1+x1+x2 x2 x1x2
1 + x + x 1 + x x x + 1 + x + x 1 + x 1 + 1 2 ÷ 2 = 1 2 1 2 = 1 x1x2 x1 x2(1 + x2) x2
Example: seed mutation
x1 → x2
1
1+x2 ← x x1 2
2
1+x2 → 1+x1+x2 x1 x1x2
1
Cluster Algebras Lara Bossinger 13/ 22 =∼
x2 ← x1
1
1+x1 → x x2 1
2 1+x1 ← 1+x1+x2 x2 x1x2
x x + 1 + x + x 1 + x = 1 2 1 2 = 1 x2(1 + x2) x2
Example: seed mutation
x1 → x2
1
1+x2 ← x x1 2
2
1+x2 → 1+x1+x2 x1 x1x2
1
1 + x + x 1 + x 1 + 1 2 ÷ 2 x1x2 x1
Cluster Algebras Lara Bossinger 13/ 22 =∼
x2 ← x1
1
1+x1 → x x2 1
2 1+x1 ← 1+x1+x2 x2 x1x2
1 + x = 1 x2
Example: seed mutation
x1 → x2
1
1+x2 ← x x1 2
2
1+x2 → 1+x1+x2 x1 x1x2
1
cluster magic! 1 + x + x 1 + x x x + 1 + x + x 1 + 1 2 ÷ 2 = 1 2 1 2 x1x2 x1 x2(1 + x2)
Cluster Algebras Lara Bossinger 13/ 22 =∼
x2 ← x1
1
1+x1 → x x2 1
2 1+x1 ← 1+x1+x2 x2 x1x2
Example: seed mutation
x1 → x2
1
1+x2 ← x x1 2
2
1+x2 → 1+x1+x2 x1 x1x2
1
cluster magic! 1 + x + x 1 + x x x + 1 + x + x 1 + x 1 + 1 2 ÷ 2 = 1 2 1 2 = 1 x1x2 x1 x2(1 + x2) x2
Cluster Algebras Lara Bossinger 13/ 22 =∼
x2 ← x1
1
1+x1 → x x2 1
2
Example: seed mutation
x1 → x2
1
1+x2 ← x x1 2
2
1+x2 → 1+x1+x2 x1 x1x2
1 1+x1 ← 1+x1+x2 x2 x1x2 cluster magic! 1 + x + x 1 + x x x + 1 + x + x 1 + x 1 + 1 2 ÷ 2 = 1 2 1 2 = 1 x1x2 x1 x2(1 + x2) x2
Cluster Algebras Lara Bossinger 13/ 22 =∼
x2 ← x1
1
1+x1 → x x2 1
1 + x + x 1 + x x x + 1 + x + x 1 + x 1 + 1 2 ÷ 2 = 1 2 1 2 = 1 x1x2 x1 x2(1 + x2) x2
Example: seed mutation
x1 → x2
1
1+x2 ← x x1 2
2
1+x2 → 1+x1+x2 x1 x1x2
1 2 1+x1 ← 1+x1+x2 x2 x1x2
Cluster Algebras Lara Bossinger 13/ 22 =∼
x2 ← x1
1
1 + x + x 1 + x x x + 1 + x + x 1 + x 1 + 1 2 ÷ 2 = 1 2 1 2 = 1 x1x2 x1 x2(1 + x2) x2
Example: seed mutation
x1 → x2
1
1+x2 ← x x1 2
2
1+x2 → 1+x1+x2 1+x1 → x x1 x1x2 x2 1
1 2 1+x1 ← 1+x1+x2 x2 x1x2
Cluster Algebras Lara Bossinger 13/ 22 =∼
x2 ← x1
1 + x + x 1 + x x x + 1 + x + x 1 + x 1 + 1 2 ÷ 2 = 1 2 1 2 = 1 x1x2 x1 x2(1 + x2) x2
Example: seed mutation
x1 → x2
1
1+x2 ← x x1 2
2 1
1+x2 → 1+x1+x2 1+x1 → x x1 x1x2 x2 1
1 2 1+x1 ← 1+x1+x2 x2 x1x2
Cluster Algebras Lara Bossinger 13/ 22 =∼
1 + x + x 1 + x x x + 1 + x + x 1 + x 1 + 1 2 ÷ 2 = 1 2 1 2 = 1 x1x2 x1 x2(1 + x2) x2
Example: seed mutation
x1 → x2
1
1+x 2 ← x x2 ← x1 x1 2
2 1
1+x2 → 1+x1+x2 1+x1 → x x1 x1x2 x2 1
1 2 1+x1 ← 1+x1+x2 x2 x1x2
Cluster Algebras Lara Bossinger 13/ 22 1 + x + x 1 + x x x + 1 + x + x 1 + x 1 + 1 2 ÷ 2 = 1 2 1 2 = 1 x1x2 x1 x2(1 + x2) x2
Example: seed mutation
x1 → x2 1 =∼ 1+x 2 ← x x2 ← x1 x1 2
2 1
1+x2 → 1+x1+x2 1+x1 → x x1 x1x2 x2 1
1 2 1+x1 ← 1+x1+x2 x2 x1x2
Cluster Algebras Lara Bossinger 13/ 22 Cluster algebra Let F = C(x1,..., xn+m) be the field of rational functions in the variables x1,..., xn+m. 0 0 0 If s = (x, Q) is a seed with x = (x1,..., xn+m) and let s = (x , Q ) be a seed obtained from s by a sequence of mutations, then the cluster 0 0 0 0 0 x = (x1,..., xn, xn+1,..., xn+m) satisfies
0 0 0 0 C(x1,..., xn, xn+1,..., xn+m) = F.
Definition The cluster algebra defined by the initial quiver Q is the F-subalgebra
S 0 AQ := h (x0,Q0)∼(x,Q) x i ⊂ F.
Theorem (Fomin–Zelevinsky 2001)
The cluster algebra AQ only depends on the mutation class of Q.
Cluster Algebras Lara Bossinger 14/ 22 Example: cluster algebra
x1 → x2 1 =∼ 1+x 2 ← x x2 ← x1 x1 2
2 1
1+x2 → 1+x1+x2 1+x1 → x x1 x1x2 x2 1
1 2 1+x1 ← 1+x1+x2 x2 x1x2 D E A = x , x , 1+x2 , 1+x1+x2 , 1+x1 ⊂ (x , x ). Q 1 2 x1 x1x2 x2 C 1 2
Cluster Algebras Lara Bossinger 15/ 22 Structure Theorems Theorem (Fomin–Zelevinsky 2001) All cluster variables are Laurent polynomials in the cluster variables of the initial seed with integer coefficients. More precisely, they are contained in
± ± Z[x1 ,..., xn , xn+1,..., xn+m].
Positivity Conjecture (Fomin–Zelevinsky 2001) ± ± All cluster variables are contained in N[x1 ,..., xn , xn+1,..., xn+m].
Theorem (Gross–Hacking–Keel–Kontsevich 2014) The positivity conjecture is true.
Gross–Hacking–Keel–Kontsevich view cluster algebras as rings of functions on certain log Calabi–Yau varieties (cluster varieties) and use tools from birational geometry and mirror symmetry.
Cluster Algebras Lara Bossinger 16/ 22 Finite type classification Theorem (Fomin–Zelevinsky 2003)
A cluster algebra AQ is of finite type (i.e. the set of its cluster variables is finite) if and only if (the mutable part of) Q is mutation equivalent to an orientation of a type ADE Dynkin diagram: An : • • ··· •
•
Dn : • • ··· • •
E6 : •
• • • • •
E7 : •
• • • • • •
E8 : •
• • • • • • •
Cluster Algebras Lara Bossinger 17/ 22 n Grassmannian Gr2(C ) and Ptolemy
n n Recall the 2 Pl¨ucker coordinates pij for Gr2(C ) and the correspondence: efficient TP 1−1 triangulations n ←→ . tests for Gr2(C ) of an n-gon
We can pass from a triangulation T to a quiver QT as follows:
1 mutable vertices of QT ↔ diagonals;
2 frozen vertices of QT ↔ boundary edges; 3 add arrows inside every triangle:
4 forget arrows between frozen vertices.
Cluster Algebras Lara Bossinger 18/ 22 5 Example: Gr2(C )
1
T 2 5
3 4
Cluster Algebras Lara Bossinger 19/ 22 5 Example: Gr2(C )
1 1
T 2 5 2 5 QT
3 4 3 4
Cluster Algebras Lara Bossinger 19/ 22 5 Example: Gr2(C )
1 1
T 2 5 2 5 QT
3 4 3 4
flip13 1
flip13(T ) 2 5
3 4
Cluster Algebras Lara Bossinger 19/ 22 5 Example: Gr2(C )
1 1
T 2 5 2 5 QT
3 4 3 4
flip13 1 1
2 5 2 5 Q flip13(T ) flip13(T )
3 4 3 4
Cluster Algebras Lara Bossinger 19/ 22 5 Example: Gr2(C )
1 1
T 2 5 2 5 QT
3 4 3 4
flip13 µ13 1 1
2 5 2 5 Q = µ (Q ) flip13(T ) flip13(T ) 13 T
3 4 3 4
Cluster Algebras Lara Bossinger 19/ 22 n Example: cluster algebra for Gr2(C ) We have a bijection triangulations seeds of the cluster , flip ←→1−1 , mutation . of an n-gon algebra AQT In particular: {seeds} ←→1−1 {efficient TP tests}. The corresponding cluster algebra is
AQT = C[pij : 1 ≤ i < j ≤ n]/Pl¨ucker relations.
Remark n Similar results hold for Grk (C ), GLn, SLn, (partial) flag varieties, Schubert varieties, double Bruhat cells, ...
Remark The cluster structure can be used to get (for example) Landau–Ginzburg models, toric degenerations and Newton–Okounkov bodies.
Cluster Algebras Lara Bossinger 20/ 22 More cluster structures
1 Marked bordered surfaces:
I fix a triangulation → coordinates for the Teichm¨ullerspace, I triangulation ↔ quiver, I flip of triangulation ↔ quiver mutation ↔ Ptolemy relation. 2 2-bridge knots and links:
I 2-bridge link ↔ continued fraction ↔ snake graph ↔ cluster variable of some cluster algebra, I Jones polynomial = specialization of the Laurent polynomial of the cluster variable.
3 2 2 2 Markov equation: x1 + x2 + x3 = 3x1x2x3, I solutions are called Markov triples,
I take Q = 2 and (x1, x2, x3) = (1, 1, 1), @ @
1 oo 3 0 0 0 0 0 I (x , Q) ∼ (x, Q) then x = (x1, x2, x3) is a Markov triple. 4 ...
Cluster Algebras Lara Bossinger 21/ 22 References
FZ02 Sergey Fomin and Andrei Zelevinsky. Cluster algebras. I. Foundations. J. Amer. Math. Soc., 15(2):497–529, 2002. FZ03 Sergey Fomin and Andrei Zelevinsky. Cluster algebras. II. Finite type classification. Invent. Math. 154 (2003), no. 1, 63–121. BFZ05 Arkady Berenstein, Sergey Fomin and Andrei Zelevinsky. Cluster algebras. III. Upper bounds and double Bruhat cells. Duke Math. J. 126 (2005), no. 1, 1–52. FZ07 Sergey Fomin and Andrei Zelevinsky. Cluster algebras. IV. Coefficients. Compos. Math. 143, no. 1, 112–164 (2007) FG07 Vladimir V. Fock and Alexander Goncharov. Dual Teichm¨ullerand lamination spaces. Handbook of Teichm¨ullertheory. Vol. I, 647–684, IRMA Lect. Math. Theor. Phys., 11, Eur. Math. Soc., Z¨urich,(2007) GHKK18 Mark Gross, Paul Hacking, Sean Keel, and Maxim Kontsevich. Canonical bases for cluster algebras. J. Amer. Math. Soc., 31(2):497–608 (2018) Pen87 Robert C. Penner. The decorated Teichm¨uller space of punctured surfaces. Comm. Math. Phys. Volume 113, Number 2 (1987), 299-339. Sco06 Joshua S. Scott. Grassmannians and cluster algebras. Proc. London Math. Soc. (3) 92 (2006), no. 2, 345–380. LS19 Kyungyong Lee and Ralf Schiffler. Cluster algebras and Jones polynomials. Selecta Math. (N.S.) 25 4 Paper No. 58, 41 (2019)
Cluster Algebras Lara Bossinger 22/ 22