Quantum integrability from set theoretic Yang-Baxter & reflection equations

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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.