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Title:  Poisson structures and degenerations of integrable systems related to y (gl2) 
Author(s):  Rawig, Siraprapa 
Director of Research:  Bergvelt, Maarten 
Doctoral Committee Chair(s):  Nevins, Thomas 
Doctoral Committee Member(s):  Yong, Alexander; Loja Fernandes, Rui 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Toda lattice
Integrable systems taufunctions 
Abstract:  The Toda lattice is an important dynamical system studied in the theory of integrable systems. It is known that the Toda lattice is related to other integrable systems: the DST system and the XXX model. We will generalize these three systems by attaching a Hamiltonian system to a nonincreasing sequence ▁k=(k_0,k_1,...,k_N) such that k_i −k_(i+1)≤2. The Toda system corresponds to the constant sequence k_i = k, the DST system to k_i=k_(i+1)+1, and the XXX system to k_i=k_(i+1)+2. We will express the variables in all these systems in terms of τfunctions, and use this to give the relation between the 2 × 2 and N × N Lax matrix descriptions of the systems. We show that all these systems are completely integrable, giving explicit actionangle variables. In the past, these three systems were studied using independent sets of variables. Since we can express any system corresponding to such k using the same set of variables (τfunctions), we prove that all these systems are in fact isomorphic to the Toda system, and hence to each other. This seems not to have been known before. Sklyanin showed that the Toda lattice, the DST system, and the XXX model are related by degenerations. We will use deformed τfunctions to deform the systems attached to sequences k as above. The deformed systems will correspond to two sequences ▁k,▁s. Then we define explicit degenerations of these deformed systems, generalizing Sklyanin’s results to our deformed systems. 
Issue Date:  20190913 
Type:  Text 
URI:  http://hdl.handle.net/2142/106418 
Rights Information:  Copyright 2019 Siraprapa Rawig 
Date Available in IDEALS:  20200302 
Date Deposited:  201912 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois