# Agata Smoktunowicz has been awarded an EPSRC research grant entitled Quantum Integrability from Set Theoretic Yang-Baxter & Reflection Equations

Agata Smoktunowicz has been awarded an EPSRC research grant entitled Quantum Integrability from Set Theoretic Yang-Baxter & Reflection Equations. The grant, in collaboration with Anastasia Doikou and Robert Weston, Heriot-Watt University, 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 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 Yang Baxter Equations. 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.