Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 025, 78 pages      arXiv:1912.02440      https://doi.org/10.3842/SIGMA.2022.025

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

Stéphane Baseilhac and Philippe Roche
IMAG, Univ Montpellier, CNRS, Montpellier, France

Received April 09, 2021, in final form March 07, 2022; Published online March 29, 2022

Abstract
Let $\Sigma$ be a finite type surface, and $G$ a complex algebraic simple Lie group with Lie algebra $\mathfrak{g}$. The quantum moduli algebra of $(\Sigma,G)$ is a quantization of the ring of functions of $X_G(\Sigma)$, the variety of $G$-characters of $\pi_1(\Sigma)$, 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 $U_q(\mathfrak{g})$-module-algebras associated to graphs on $\Sigma$, where $U_q(\mathfrak{g})$ is the quantum group corresponding to $G$. We study the structure of the quantum moduli algebra in the case where $\Sigma$ is a sphere with $n+1$ open disks removed, $n\geq 1$, using the graph algebra of the ''daisy'' graph on $\Sigma$ to make computations easier. We provide new results that hold for arbitrary $G$ and generic $q$, and develop the theory in the case where $q=\epsilon$, a primitive root of unity of odd order, and $G={\rm SL}(2,{\mathbb C})$. 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 $\mathcal{O}(G^n)$. 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 $\mathcal{O}(G^n)$. We show that the set of fixed elements of the center under the quantum coadjoint action is a finite extension of ${\mathbb C}[X_G(\Sigma)]$ endowed with the Atiyah-Bott-Goldman Poisson structure. Finally, by using Wilson loop operators we identify the Kauffman bracket skein algebra $K_{\zeta}(\Sigma)$ at $\zeta:={\rm i}\epsilon^{1/2}$ with this quantum moduli algebra specialized at $q=\epsilon$. This allows us to recast in the quantum moduli setup some recent results of Bonahon-Wong and Frohman-Kania-Bartoszyńska-Lê on $K_{\zeta}(\Sigma)$.

Key words: quantum groups; invariant theory; character varieties; skein algebras.

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