Quantum integrability from set theoretic Yang-Baxter & reflection equations

提出的研究计划旨在汇集数学物理，特别是量子可积性领域的思想，以及纯代数，特别是辫群，支撑和环理论领域的思想。该提案考虑了一类特殊的一维相互作用n体量子系统，称为可积量子自旋链。可积量子系统的特征是存在一组相互交换的代数对象，通常与相关的自由度一样多。这组交换对象保证了量子系统的精确可解性。这意味着系统的一些基本物理性质，比如能量特征值，原则上可以精确地计算出来，并且可以用Bethe ansatz方程组的解来表示。用于构建可积量子自旋链和解析其光谱的主要方法是量子逆散射法（QISM），这是一种优雅的代数技术，可以自然地得到Bethe ansatz方程，从而得到自旋链的能谱。QISM还直接导致了被称为量子群或量子代数的准三角形霍普夫代数的发明。Yang-Baxter方程是量子可积性理论中的一个关键对象，因为该方程的不同解产生不同类型的量子自旋链和不同的代数约束集合，即量子代数。代数约束保证了可交换代数对象的存在性，保证了关联系统的量子可积性。在这个项目中，我们专注于YBE的一类特殊解，即集合论解，它也提供了Artin辫群的某些商的表示。为了描述YBE的所有有限的、对合的、集合论的解，我们发展了一种特殊的代数结构来推广幂零环，称为括号。已经很好地证明了每一个括号都提供了YBE的一个集合论解，并且YBE的每一个非退化的、对合的集合论解都可以从一个括号中得到。提出的研究计划的中心目标是研究与由YBE的集合论解构建的量子可积系统相关的代数和物理方面。从代数的角度来看，从大括号中产生的量子群的表示理论的研究是关键目标之一。我们还研究了某些二次代数，如反射代数，并得到了可能的可积边界条件的分类。这些发现将导致具有周期性和开放边界条件的新型物理自旋链系统的识别。另一个关键问题是检查我们是否可以将YBE的括号型解表示为Drinfeld扭转。Hopf代数的“扭曲”是产生另一个Hopf代数的代数作用。对于某些特殊的集合论解，我们已经得到了这种扭曲的显式表达式。我们的基本目标之一是将这些发现推广到包括更大类别的集合论解，并研究这种扭曲在新兴量子群对称性中的作用。从物理角度来看，最终目标是识别由集合论解构造的开放和周期可积量子自旋链的本征值和本征态。我们将通过实现广义Bethe ansatz技术来系统地研究这个问题，这将导致一组新颖的Bethe ansatz方程和相关量子自旋链的谱。有了频谱和相关的Bethe ansatz方程，我们将能够计算物理上相关的量，如能量、散射振幅和算子期望值。

The proposed research program aims at bringing together ideas from mathematical physics and in particular the domain of quantum integrability, and pure algebra specifically the areas of braid groups, braces and ring theory. The proposal regards a special class of one dimensional interacting N-body quantum systems known as integrable quantum spin chains. Integrable quantum systems are characterized by the existence of a set of mutually commuting algebraic objects, usually as many as the associated degrees of freedom. This set of commuting objects ensures the exact solvability of the quantum system. This means that some of the fundamental physical properties of the system, such as the energy eigenvalues can be in principle computed exactly and can be expressed in terms of solutions of a system of equations known as Bethe ansatz equations.The main methodology used for the construction of integrable quantum spin chains and the resolution of their spectra is the Quantum Inverse Scattering Method (QISM), an elegant algebraic technique that naturally yields the Bethe ansatz equations and consequently the energy spectrum of the spin chains. The QISM has also led directly to the invention of quasitriangular Hopf algebras known as quantum groups or quantum algebras. The Yang-Baxter equation is a key object in the theory of quantum integrability, given that distinct solutions of the equation generate different types of quantum spin chains and distinct sets of algebraic constraints, i.e. quantum algebras. The algebraic constraints guarantee the existence of mutually commuting algebraic objects, ensuring the quantum integrabiltiy of the associated system. In this project we focus on a particular class of solutions of the YBE known as set theoretic solutions, which also provide representations of certain quotients of Artin's braid group. A special algebraic structure that generalizes nilpotent rings, called a brace was developed in order to describe all finite, involutive, set-theoretic solutions of the YBE. It is well established that every brace provides a set theoretic solution of the YBE, and every non-degenerate, involutive set theoretic solution of the YBE can be obtained from a brace.The central aim of the proposed research program is to investigate both algebraic and physical aspects associated to quantum integrable systems constructed from set theoretic solutions of the YBE. From the algebraic point of view the study of the representation theory of the quantum groups emerging from braces is one of the key objectives. We also aim at investigating certain quadratic algebras, such as the refection algebra, and obtain a classification of possible integrable boundary conditions. These findings will lead to the identification of new classes of physical spin chain systems with periodic and open boundary conditions. Another key issue is to examine whether we can express brace type solutions of the YBE as Drinfeld twists. The 'twisting' of a Hopf algebra is an algebraic action that produces yet another Hopf algebra. Explicit expressions of such twists have been derived for some special classes of set theoretic solutions. One of our fundamental objectives is to generalize these findings to include larger classes of set theoretic solutions and also investigate the role of such twists on the emerging quantum group symmetries. From a physical viewpoint the ultimate goal is the identification of the eigenvalues and eigenstates of open and periodic integrable quantum spin chains constructed from set theoretic solutions. We will systematically pursue this problem by implementing generalized Bethe ansatz techniques that will lead to sets of novel Bethe ansatz equations and the spectrum of the associated quantum spin chains. Having at our disposal the spectrum and the associated Bethe ansatz equations we will be able to compute physically relevant quantities, such as energy, scattering amplitudes and operator expectation values.