THESIS
2023
1 online resource (viii, 77 pages) : illustrations
Abstract
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...[
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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.
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