Fock–Goncharov constructed moduli spaces of G-local systems on decorated surfaces for split semi-simple algebraic groups G, which carry positive structures and generalizes the usual Teichmüller space. They showed that their coordinate rings carry cluster algebra structures when G is of type A, which enables the use of cluster algebra techniques in studying these spaces. Goncharov–Shen extended this result to general G. Using the cluster structure, Goncharov–Shen defined cluster Donaldson–Thomas transformations on general cluster varieties. They conjectured and then proved that the P moduli space has a cluster Donaldson–Thomas transformation. Meanwhile, it is known that the quantum group can be embedded into the corresponding quantum cluster algebras, and it was conjectured by Ip that t...[ Read more ]

Fock–Goncharov constructed moduli spaces of G-local systems on decorated surfaces for split semi-simple algebraic groups G, which carry positive structures and generalizes the usual Teichmüller space. They showed that their coordinate rings carry cluster algebra structures when G is of type A, which enables the use of cluster algebra techniques in studying these spaces. Goncharov–Shen extended this result to general G. Using the cluster structure, Goncharov–Shen defined cluster Donaldson–Thomas transformations on general cluster varieties. They conjectured and then proved that the P moduli space has a cluster Donaldson–Thomas transformation. Meanwhile, it is known that the quantum group can be embedded into the corresponding quantum cluster algebras, and it was conjectured by Ip that the image of the Chevalley generators are polynomials in the cluster variables for any cluster in the mutation class of the quantum cluster algebra. The goal of this thesis is to provide another proof to the conjecture of Goncharov–Shen and prove Ip’s conjecture.