UC Berkeley

In this thesis we address several questions involving quantum groups, quantum cluster algebras, and integrable systems, and provide some novel examples of the very useful interplay between these subjects. In the Chapter 2, we introduce the classical reflection equation (CRE), and give a construction of integrable Hamiltonian systems on $G/K$, where $G$ is a quasitriangular Poisson Lie group and $K$ is a Lie subgroup arising as the fixed point set of a group automorphism $\sigma$ of $G$ satisfying the CRE. As an application, we provide a detailed treatment of the algebraic integrability of the XXZ spin chain with reflecting boundary conditions.

In Chapter 3, we study doubles of Hopf algebras and dual pairs of quantum moment maps. For any Hopf algebra $A$, we construct a natural generalization of the (quantized) Grothendieck-Springer resolution; the standard resolution corresponds to taking $A$ a quantum Borel subalgebra. In this latter case, we apply the general construction to yield an algebra embedding of the Drinfeld-Jimbo quantum group $U_q(\mathfrak{g})$ into a quantum torus algebra which is a central extension of the quantum coordinate ring of the reduced big double Bruhat cell in the corresponding simply-connected group $G$.

Chapter 4 gives an alternative geometric description of this quantum torus embedding. Namely, we construct an embedding of $U_q(\mathfrak{sl}_n)$ into a quantum cluster chart on a quantum character variety associated to a marked punctured disk. We obtain a description of the coproduct of $U_q(\mathfrak{sl}_{n})$ in terms of a quantum character variety associated to the marked twice punctured disk, and express the action of the $R$-matrix in terms of a mapping class group element corresponding to the half-Dehn twist rotating one puncture about the other. As a consequence, we realize the algebra automorphism of $U_q(\mathfrak{sl}_{n})^{\otimes 2}$ given by conjugation by the $R$-matrix as an explicit sequence of cluster mutations, and derive a refined factorization of the $R$-matrix into quantum dilogarithms of cluster monomials.

We conclude by mentioning some applications of our cluster realization of quantum groups to the decomposition of tensor products of positive representations, and the construction of a modular functor from quantum higher Teichmuller theory.