On the full Kostant-Toda hierarchy and its $\ell$-banded reductions for the Lie algebras of type $A, B$ and $G$

Adler-Kostant-Symes systems as Lagrangian gauge theories

It is well known that the integrable Hamiltonian systems defined by the Adler-Kostant-Symes construction correspond via Hamiltonian reduction to systems on cotangent bundles of Lie groups. Generalizing previous results on Toda systems, here a Lagrangian version of the reduction procedure is exhibited for those cases for which the underlying Lie algebra admits an invariant scalar product. This is achieved by constructing a Lagrangian with gauge symmetry in such a way that, by means of the Dirac algorithm, this Lagrangian reproduces the Adler-Kostant-Symes system whose Hamiltonian is the quadrat…

Semi-invariants and Integrals of the Full Symmetric sl(n) Toda Lattice

We consider the full symmetric version of the Lax operator of the Toda lattice which is known as the full symmetric Toda lattice. The phase space of this system is the generic orbit of the coadjoint action of the Borel subgroup B^+(n) of SL(n,R). This system is integrable. We propose a new method of constructing semi-invariants and integrals of the full symmetric Toda lattice. Using only the Toda equations for the Lax eigenvector matrix we prove the existence of the semi-invariants which are Plucker coordinates in the corresponding projective spaces. Then we use these semi-invariants to constr…

Quasigraded Lie Algebras and Modified Toda Field Equations

We construct a family of quasigraded Lie algebras that coincide with the deformations of the loop algebras in "principal" gradation and admit Kostant-Adler-Symes scheme. Using them we obtain new Volterra coupled systems and modified Toda field equations for all series of classical matrix Lie algebras $\mathfrak{g}$.

三角群的 Plancherel 公式的 Poisson 几何

The Poisson Geometry of Plancherel Formulas for Triangular Groups

本文确立了三角群上一般余伴随轨道的典范坐标，推广了Flaschka的极小类型坐标化，发展了与Full Kostant-Toda格子及其Poisson几何的联系，并揭示了Borel Lie群的Plancherel定理与不变量理论的关系。

Explicit Semi-invariants and Integrals of the Full Symmetric sl(n) Toda Lattice

We show how to construct semi-invariants and integrals of the full symmetric sl(n) Toda lattice for all n. Using the Toda equations for the Lax eigenvector matrix we prove the existence of semi-invariants which are homogeneous coordinates in the corresponding projective spaces. Then we use these semi-invariants to construct the integrals. The existence of additional integrals which constitute a full set of independent non-involutive integrals was known but the chopping and Kostant procedures have crucial computational complexities already for low-rank Lax matrices and are practically not appli…

Toda和Laguerre-Freud方程及超几何离散多重正交多项式的tau函数

Toda and Laguerre-Freud equations and tau functions for hypergeometric discrete multiple orthogonal polynomials

本文研究离散多重正交多项式，通过tau函数和带状矩阵描述其性质，证明其提供Toda方程多分量可积扩展及Nijhoff-Capel方程的解，并探讨Laguerre-Freud方程在多种特例下的应用。

Pearson Equations for Discrete Orthogonal Polynomials: I. Generalized Hypergeometric Functions and Toda Equations

The Cholesky factorization of the moment matrix is applied to discrete orthogonal polynomials on the homogeneous lattice. In particular, semiclassical discrete orthogonal polynomials, which are built in terms of a discrete Pearson equation, are studied. The Laguerre-Freud structure semi-infinite matrix that models the shifts by $\pm 1$ in the independent variable of the set of orthogonal polynomials is introduced. In the semiclassical case it is proven that this Laguerre-Freud matrix is banded. From the well known fact that moments of the semiclassical weights are logarithmic derivatives of ge…

具有奇异源的一般 Toda 系统解的总质量

Total masses of solutions to general Toda systems with singular sources

本文研究带原点奇异源的一般单李代数 Toda 系统，借助 DPW、Drinfeld-Sokolov 规范、$W$-不变量及 Kostant 理论确定解的总质量，并将其与 Weyl 群最长元联系起来。

Classical double, R-operators and negative flows of integrable hierarchies

Using classical double G of a Lie algebra g equipped with a classical R-operator we define two sets of mutually commuting functions with respect to the initial Lie-Poisson bracket on g* and its extensions. We consider in details examples of the Lie algebras g with the "Adler--Kostant--Symes" R-operators and the corresponding two sets of mutually commuting functions. Using the constructed commutative hamiltonian flows on different extensions of g we obtain zero-curvature equations with g-valued U-V pairs. Among such the equations are so-called "negative flows" of soliton hierarchies. We illlust…

Iso-spectral deformations of general matrix and their reductions on Lie algebras

We study an iso-spectral deformation of general matrix which is a natural generalization of the Toda lattice equation. We prove the integrability of the deformation, and give an explicit formula for the solution to the initial value problem. The formula is obtained by generalizing the orthogonalization procedure of Szegö. Based on the root spaces for simple Lie algebras, we consider several reductions of the hierarchy. These include not only the integrable systems studied by Bogoyavlensky and Kostant, but also their generalizations which were not known to be integrable before. The behaviors of…

Banach Poisson-Lie 群、Lax 方程与无穷维情形下的 AKS 定理

Banach Poisson-Lie groups, Lax equations and the AKS theorem in infinite dimensions

本文在Banach框架下研究三类括号与Banach Poisson-Lie群的关联，证明适用于无穷维Banach代数上R-矩阵的AKS定理，给出其在Lax方程求解中的应用并以半无穷Toda格为例。