Integrable Clusters Arkady Berenstein AMS-EMS-SPM Meeting Porto, - - PowerPoint PPT Presentation

Integrable Clusters Arkady Berenstein AMS-EMS-SPM Meeting Porto, 12 June 2015

Integrable Clusters Arkady Berenstein

• A. Berenstein, J, Greenstein, D. Kazhdan, Comptes rendus

Mathematique vol. 353, 5 (2015).

AMS-EMS-SPM Meeting Porto, 12 June 2015

Integrable systems

Informally, a (completely) integrable system in a given a Poisson algebra ❆ is any maximal Poisson-commutative subalgebra ❆✵. In particular, a Hamiltonian ❍ is any element of ❆✵ such that ❆✵ is the Poisson centralizer of ❍. It is well-known that if the bracket on ❆ is symplectic, then ❞✐♠ ❆✵ ❂ ✶

✷ ❞✐♠ ❆ and the map ❙♣❡❝❆ ։ ❙♣❡❝❆✵ is a

Lagrangian foliation.

• Problem. Classify all integrable systems in ❆.

Cluster structures

Upper bounds. Given a field ❋, ❝❤❛r ❋ ❂ ✵, a cluster x ❂ ✭①✶❀ ✿ ✿ ✿ ❀ ①♠✮ is any algebraically independent set in ❋. Each cluster x defines a Laurent polynomial algebra ▲x ❂ Q❬① ✝✶

✶ ❀ ✿ ✿ ✿ ❀ ① ✝✶ ♠ ❪ ❂ ✟❛✷Z♠Q① ❛.

Given ♥ ✔ ♠, a seed is a pair ✭x❀ ⑦ ❇✮, where ⑦ ❇ ❂ ✭❜✶ ✁ ✁ ✁ ❜♥✮ is an integer ♠ ✂ ♥ matrix ( ⑦ ❇ is called an exchange matrix). Define the upper bound algebra ❯✭x❀ ⑦ ❇✮ ✚ ▲x by ❯✭x❀ ⑦ ❇✮ ✿❂

♥

❭

❦❂✶

❯❦✭x❀ ⑦ ❇✮❀ where ❯❦✭x❀ ⑦ ❇✮ is the subalgebra of ▲ generated by x, all ① ✶

✐

, ✐ ✻❂ ❦ and ① ✵

❦ ❂ ① ✶ ❦

✭① ❬❜❦❪✰ ✰ ① ❬❜❦❪✰✮. Here we abbreviate ❬✭❛✶❀ ✿ ✿ ✿ ❀ ❛♠✮❪✰ ❂ ✭♠❛①✭✵❀ ❛✶✮❀ ✿ ✿ ✿ ❀ ♠❛①✭✵❀ ❛♠✮❪.

• Mutations. For each seed ✭x❀ ⑦

❇✮ and ❦ ❂ ✶❀ ✿ ✿ ✿ ❀ ♥ define ✖❦✭x❀ ⑦ ❇✮ ✿❂ ✭x ♥ ❢①❦❣ ❬ ❢① ✵

❦❣❀ ✖❦✭ ⑦

❇✮✮, where ✖❦✭ ⑦ ❇✮✐❥ ❂

✭

❜✐❥ if ❦ ✷ ❢✐❀ ❥ ❣ ❜✐❥ ✰ ❥❜✐❦❥❜❦❥ ✰❜✐❦❥❜❦❥ ❥

✷

• therwise

Theorem 1 (BFZ 2005) Suppose that r❛♥❦ ⑦ ❇ ❂ ♥ and ⑦ ❇❥❬✶❀♥❪✂❬✶❀♥❪ is skew-symmetrizable. Then ✖❦✭ ⑦ ❇✮ satisfies same properties and ❯✭✖❦✭x❀ ⑦ ❇✮✮ ❂ ❯✭x❀ ⑦ ❇✮ for ❦ ❂ ✶❀ ✿ ✿ ✿ ❀ ♥. In particular, for any sequence ✭y❀ ⑦ ❇✵✮ ❂ ✖✐❵ ✁ ✁ ✁ ✖✐✶✭x❀ ⑦ ❇✮, each ②❦ belongs to the Laurent polynomial algebra ▲x.

Poisson clusters. Given an ♠ ✂ ♠ skew symmetric matrix ✄, equip ▲x with the (log-canonical) Poisson bracket via ❢①✐❀ ①❥ ❣ ❂ ✕✐❥ ①✐①❥ . Theorem 2 Suppose that ⑦ ❇ has no zero columns. Then ❯✭x❀ ⑦ ❇✮ is a Poisson subalgebra of ▲x iff ⑦ ❇❚✄ ❂ ✭❉ 0✮ for some ❉ ❂ ❞✐❛❣✭❞✶❀ ✿ ✿ ✿ ❀ ❞♥✮. We say that ✄ as in Theorem 2 is compatible with ⑦ ❇ if ❞✶ ✻❂ ✵❀ ✿ ✿ ✿ ❀ ❞♥ ✻❂ ✵. Lemma If ⑦ ❇ admits a compatible ✄ then ⑦ ❇ is as in Theorem 1. Main example If ❇❀ ❈❀ ❉ ✷ ▼❛t♥✂♥✭Z✮, ❉ ❂ ❞✐❛❣✭❞✶❀ ✿ ✿ ✿ ❀ ❞♥✮, ✭❉❇✮❚ ❂ ❉❇, ❞❡t ❈ ✻❂ ✵, ❞❡t ❉ ✻❂ ✵, then the ✷♥ ✂ ✷♥ matrix ✄ ❂ ✄❉❇❀❈ ❂

✥

❉❈ ✶ ✭❈ ✶✮❚❉ ✭❈ ✶✮❚❉❇❈ ✶

✦

is compatible with the exchange matrix ⑦ ❇ ❂

✥

❇ ❈

✦

.

Integrable seeds

We say that ✭X❀ ⑦ ❇✮ is an integrable seed if ⑦ ❇ ❂

✥

❇ ❈

✦

as in Main example. Lemma For each integrable seed ✭x❀ ⑦ ❇✮ the algebra ❆✵ ❂ Q❬①✶❀ ✿ ✿ ✿ ❀ ①♥❪ is an integrable system in the Poisson algebra ❆ ❂ ❯✭x❀ ⑦ ❇✮ ✚ ▲x whose Poisson bracket given by ✄❉❇❀❈.

• Problem. Find all integrable seeds ✭x✵❀ ⑦

❇✵✮ mutation equivalent to a given integrable seed ✭x❀ ⑦ ❇✮. Main Theorem Let ✭x❀ ⑦ ❇✮ is be a principal (i.e., ❈ ❂ ■♥, the identity matrix) integrable seed. Then all seeds ✭x✵❀ ⑦ ❇✵✮ mutation equivalent to ✭x❀ ⑦ ❇✮ are integrable. Proof uses the sign coherence conjecture (now theorem), which asserts that in each exchange matrix

✥

❇✵ ❈

✦

❂ ✖✐❵ ✁ ✁ ✁ ✖✐✶

✥

❇ ■♥

✦

each column of ❈ is either in ✭Z✕✵✮♥ or in ✭Z✕✵✮♥.

Quantum story

A quantum integrable system in a given algebra ❆ is any maximal commutative subalgebra ❆✵. Given a skew field ❋ and a central transcendental element q

✶ ✷ in

❋, a quantum cluster X ❂ ✭❳✶❀ ✿ ✿ ✿ ❀ ❳♠✮ is any algebraically independent subset such that ❳✐❳❥ ❂ q✕✐❥ ❳❥ ❳✐ for all ✶ ✔ ✐❀ ❥ ✔ ♠ and some skew-symmetric matrix ✄X ❂ ✭✕✐❥ ✮. This defines a quantum torus ▲X ❂

▲

❛✷Z♠ Q✭q

✶ ✷ ✮❳ ❛, where we

abbreviate ❳ ❛ ❂ q

✶ ✷

P

✐❁❥ ✕❥✐❛✐❛❥ ❳ ❛✶

✶ ✁ ✁ ✁ ❳ ❛♠ ♠

for ❛ ✷ Z♠ (so that ❳ ❛❳ ❜ ❂ q

✶ ✷✄✭❛❀❜✮❳ ❛✰❜).

A pair ✭X❀ ⑦ ❇✮ is a quantum seed if ✄X is compatible with ⑦ ❇. The quantum upper cluster algebra ❯✭X❀ ❇✮ is the intersection

♥

❚

❦❂✶

❯❦✭X❀ ⑦ ❇✮, where ❯❦✭X❀ ⑦ ❇✮ is the subalgebra of the quantum torus ▲X generated by X, all ❳ ✶

✐

, ✐ ✻❂ ❦ and ❳ ✵

❦ ❂ ❳ ❬❜❦❪✰❡❦ ✰ ❳ ❬❜❦❪✰❡❦ . Here we abbreviate ❳ ❡❦ ❂ ❳❦.

For each quantum seed ✭X❀ ⑦ ❇✮ and ❦ ❂ ✶❀ ✿ ✿ ✿ ❀ ♥ define ✖❦✭X❀ ⑦ ❇✮ ✿❂ ✭X ♥ ❢❳❦❣ ❬ ❢❳ ✵

❦❣❀ ✖❦✭ ⑦

❇✮✮. Theorem 3 (BZ 2005) For any quantum seed ✭X❀ ❇✮ and ❦ ❂ ✶❀ ✿ ✿ ✿ ❀ ♥ one has: (a) ✖❦✭X❀ ❇✮ is also a quantum seed. (b) ❯✭✖❦✭x❀ ⑦ ❇✮✮ ❂ ❯✭x❀ ⑦ ❇✮.

We say that ✭X❀ ⑦ ❇✮ is an quantum integrable seed if ⑦ ❇ ❂

✥

❇ ❈

✦

as in Main example. Lemma For each quantum integrable seed ✭X❀ ⑦ ❇✮ the algebra ❆✵ ❂ Q✭q✮❬❳✶❀ ✿ ✿ ✿ ❀ ❳♥❪ is an quantum integrable system in ❆ ❂ ❯✭X❀ ⑦ ❇✮. q-Main Theorem Let ✭X❀ ⑦ ❇✮ be an integrable principal quantum

• seed. Then all quantum seeds mutation equivalent to ✭X❀ ⑦

❇✮ are also integrable.