高阶$q$ -Dolan-Grady关系与量子可积系统
The higher order $q$-Dolan-Grady relations and quantum integrable systems
摘要：
本文首先回顾了最近引入的代数结构（三对角代数、$q$-Onsager代数、广义$q-$-Onsager代数）、相关表示理论（三对角对、Leonard对、正交多项式）、这些代数的一些性质以及格点上相关量子可积模型（单位根处的$XXZ$开自旋链）的分析之间的联系。然后，介绍了论文的主要成果：（i）对于与$\widehat{sl_2}$和ADE型单缀仿射李代数相联系的$q-$ Onsager代数类，在各种情况下给出并证明了Lusztig关系的高阶类似;（ii）对于单位根上的开$XXZ$自旋链，引入了新的元素（即$q-$ Onsager生成元的可分多项式），并研究了它们的一些性质.这两个元素与$q-$ Onsager代数的两个基本元素一起生成一个新的代数，它可以被理解为$q-$ Onsager代数的Lusztig量子群的模拟。提出了一些观点。
英文摘要：
In this thesis, the connection between recently introduced algebraic structures (tridiagonal algebra, $q$-Onsager algebra, generalized $q-$Onsager algebras), related representation theory (tridiagonal pair, Leonard pair, orthogonal polynomials), some properties of these algebras and the analysis of related quantum integrable models on the lattice (the $XXZ$ open spin chain at roots of unity) is first reviewed. Then, the main results of the thesis are described: (i) for the class of $q-$Onsager algebras associated with $\widehat{sl_2}$ and ADE type simply-laced affine Lie algebras, higher order analogs of Lusztig's relations are conjectured and proven in various cases, (ii) for the open $XXZ$ spin chain at roots of unity, new elements (that are divided polynomials of $q-$Onsager generators) are introduced and some of their properties are studied. These two elements together with the two basic elements of the $q-$Onsager algebra generate a new algebra, which can be understood as an analog of Lusztig's quantum group for the $q-$Onsager algebra. Some perspectives are presented.
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