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 ♠ Q ① ❛ . 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 � ✶ , ✐ ❦ ❂ ① � ✶ ✭ ① ❬ ❜ ❦ ❪ ✰ ✰ ① ❬ � ❜ ❦ ❪ ✰ ✮ . Here we abbreviate ✐ ✻ ❂ ❦ and ① ✵ ❦ ❬✭ ❛ ✶ ❀ ✿ ✿ ✿ ❀ ❛ ♠ ✮❪ ✰ ❂ ✭♠❛①✭✵ ❀ ❛ ✶ ✮ ❀ ✿ ✿ ✿ ❀ ♠❛①✭✵ ❀ ❛ ♠ ✮❪ .

Mutations. For each seed ✭ x ❀ ⑦ ❇ ✮ and ❦ ❂ ✶ ❀ ✿ ✿ ✿ ❀ ♥ define ✖ ❦ ✭ x ❀ ⑦ ❇ ✮ ✿❂ ✭ x ♥ ❢ ① ❦ ❣ ❬ ❢ ① ✵ ❦ ❣ ❀ ✖ ❦ ✭ ⑦ ❇ ✮✮ , where ✭ � ❜ ✐❥ if ❦ ✷ ❢ ✐ ❀ ❥ ❣ ✖ ❦ ✭ ⑦ ❇ ✮ ✐❥ ❂ ❜ ✐❥ ✰ ❥ ❜ ✐❦ ❥ ❜ ❦❥ ✰ ❜ ✐❦ ❥ ❜ ❦❥ ❥ otherwise ✷ Theorem 1 (BFZ 2005) Suppose that r❛♥❦ ⑦ ❇ ❂ ♥ and ❇ ❥ ❬✶ ❀ ♥ ❪ ✂ ❬✶ ❀ ♥ ❪ is skew-symmetrizable. Then ✖ ❦ ✭ ⑦ 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 ✥ ✦ � ❉❈ � ✶ 0 ✄ ❂ ✄ ❉❇ ❀ ❈ ❂ 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 ❆ ✵ . ✶ ✷ in Given a skew field ❋ and a central transcendental element q ❋ , a quantum cluster X ❂ ✭ ❳ ✶ ❀ ✿ ✿ ✿ ❀ ❳ ♠ ✮ is any algebraically independent subset such that ❳ ✐ ❳ ❥ ❂ q ✕ ✐❥ ❳ ❥ ❳ ✐ for all ✶ ✔ ✐ ❀ ❥ ✔ ♠ and some skew-symmetric matrix ✄ X ❂ ✭ ✕ ✐❥ ✮ . ✶ ✷ ✮ ❳ ❛ , where we This defines a quantum torus ▲ X ❂ ❛ ✷ Z ♠ Q ✭ q ▲ ✶ P abbreviate ❳ ❛ ❂ q ✐ ❁ ❥ ✕ ❥✐ ❛ ✐ ❛ ❥ ❳ ❛ ✶ 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 .