Question

How obtain conserved quantities in integrable models in accordance with Liouville's theorem, via Sklyanin Poisson algebra?

Answer 1

In classical integrable models, in the discrete case we have the Sklyanin algebra,

{Ta(u),Tb(v)}=[rab(u,v),Ta(u)Tb(v)].{Ta(u),Tb(v)}=[rab(u,v),Ta(u)Tb(v)].

How to prove that the conserved quantities are generated from ln(τ(u))ln⁡(τ(u)) in the periodic case, which are the integrable models (long-range) or condition which is not this expression which provide us the local conserved quantities in the Arnold-Liouville theorem?

Reference: Hamiltonian Methods in the Theory of Solitons by Ludwig Faddeev, Leon Takhtajan and A.G. Reyman. In special Part 1 - Chapter 3 and Part 2 - Chapter 3 in this book follow more references

Answer 2

The main reason is to prove the formulae(s) given below.

• {Ta(u),Tb(v)}=[rab(u,v),Ta(u)Tb(v)].
• {Ta(u),Tb(v)}=[rab(u,v),Ta(u)Tb(v)].
• (τ(u))ln⁡(τ(u))