Unrestricted Quantum Moduli Algebras. I. The Case of Punctured Spheres

Abstract

Let Σ be a finite type surface, and 𝐺 a complex algebraic simple Lie group with Lie algebra 𝖌. The quantum moduli algebra of (Σ, 𝐺) is a quantization of the ring of functions of 𝑋𝐺(Σ), the variety of 𝐺-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 𝑈q(𝖌)-module-algebras associated to graphs on Σ, where 𝑈q(𝖌) is the quantum group corresponding to 𝐺. We study the structure of the quantum moduli algebra in the case where Σ is a sphere with 𝑛 + 1 open disks removed, 𝑛 ≥ 1, using the graph algebra of the ''daisy'' graph on Σ to make computations easier. We provide new results that hold for arbitrary 𝐺 and generic 𝑞, and develop the theory in the case where 𝑞 = ϵ, a primitive root of unity of odd order, and 𝐺=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 𝒪(𝐺ⁿ). 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 𝒪(𝐺ⁿ). We show that the set of fixed elements of the center under the quantum coadjoint action is a finite extension of ℂ[𝑋𝐺(Σ)] endowed with the Atiyah-Bott-Goldman Poisson structure. Finally, by using Wilson loop operators, we identify the Kauffman bracket skein algebra 𝛫ζ(Σ) at ζ := iϵ¹/² with this quantum moduli algebra specialized at 𝑞 = ϵ. This allows us to recast in the quantum moduli setup some recent results of Bonahon-Wong and Frohman-Kania-Bartoszyńska-Lê on 𝛫ζ(Σ).