The integrability of the periodic Full Kostant-Toda on a simple Lie algebra

The full Kostant-Toda hierarchy on the positive flag variety

We study some geometric and combinatorial aspects of the solution to the full Kostant-Toda (f-KT) hierarchy, when the initial data is given by an arbitrary point on the totally non-negative (tnn) flag variety of SL_n(R). The f-KT flows on the tnn flag variety are complete, and their asymptotics are completely determined by the cell decomposition of the tnn flag variety given by Rietsch. We define the f-KT flow on the weight space via the moment map, and show that the closure of each f-KT flow forms an interesting convex polytope generalizing the permutohedron which we call a Bruhat interval po…

The integrability of the 2-Toda lattice on a simple Lie algebra

We define the 2-Toda lattice on every simple Lie algebra g, and we show its Liouville integrability. We show that this lattice is given by a pair of Hamiltonian vector fields, associated with a Poisson bracket which results from an R-matrix of the underlying Lie algebra. We construct a big family of constants of motion which we use to prove the Liouville integrability of the system. We achieve the proof of their integrability by using several results on simple Lie algebras, R-matrices, invariant functions and root systems.

$a_4^{(2)}$ Toda 格的代数完全可积性

Algebraic Complete Integrability of the $a_4^{(2)}$ Toda Lattice

本研究针对扭仿射李代数$a_4^{(2)}$对应的Toda格的代数完全可积性展开，证明其动量映射一般纤维为阿贝尔曲面的仿射部分、其上可积向量场的流为线性，并借助形式Laurent解给出相关几何刻画。

On KP-integrable Hurwitz functions

There is now a renewed interest to the Hurwitz tau-function, counting the isomorphism classes of Belyi pairs, arising in the study of equilateral triangulations and Grothiendicks's dessins d'enfant. It is distinguished by belonging to a particular family of Hurwitz tau-functions, possessing conventional Toda/KP integrability properties. We explain how the variety of recent observations about this function fits into the general theory of matrix model tau-functions. All such quantities possess a number of different descriptions, related in a standard way: these include Toda/KP integrability, sev…