Unrestricted Quantum Moduli Algebras. I. The Case of Punctured Spheres - Baseilhac, Stéphane; Roche, Philippe (Symmetry, Integrability and Geometry: Methods and Applications, 2022)
Let Σ be a finite type surface, and a complex algebraic simple Lie group with Lie algebra . The quantum moduli algebra of (Σ, ) is a quantization of the ring of functions of (Σ), the variety of -characters of π₁(Σ), introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche in the mid '90s. It can be realized as the invariant subalgebra of so-called graph algebras, which are q()-module-algebras associated to graphs on Σ, where q() is the quantum group corresponding to . We study the structure of the quantum moduli algebra in the case where Σ is a sphere with + 1 open disks removed, ≥ 1, using the graph algebra of the ''daisy'' graph on Σ to make computations easier. We provide new results that hold for arbitrary and generic , and develop the theory in the case where = ϵ, a primitive root of unity of odd order, and =SL(2, ℂ). In such a situation, we introduce a Frobenius morphism that provides a natural identification of the center of the daisy graph algebra with a finite extension of the coordinate ring (ⁿ). We extend the quantum coadjoint action of De-Concini-Kac-Procesi to the daisy graph algebra, and show that the associated Poisson structure on the center corresponds by the Frobenius morphism to the Fock-Rosly Poisson structure on (ⁿ). We show that the set of fixed elements of the center under the quantum coadjoint action is a finite extension of ℂ[(Σ)] endowed with the Atiyah-Bott-Goldman Poisson structure. Finally, by using Wilson loop operators, we identify the Kauffman bracket skein algebra ζ(Σ) at ζ := iϵ¹/² with this quantum moduli algebra specialized at = ϵ. This allows us to recast in the quantum moduli setup some recent results of Bonahon-Wong and Frohman-Kania-Bartoszyńska-Lê on ζ(Σ).