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

Let $\Sigma$ be a finite type surface, and $G$ complex algebraic simple Lie group with algebra $\mathfrak{g}$. The quantum moduli of $(\Sigma,G)$ is quantization the ring functions $X_G(\Sigma)$, variety $G$-characters $\pi_1(\Sigma)$, introduced by Alekseev-Grosse-Schomerus Buffenoir-Roche in mid '90s. It can realized as invariant subalgebra so-called graph algebras, which are $U_q(\mathfrak{g})$-module-algebras associated to graphs on $\Sigma$, where $U_q(\mathfrak{g})$ corresponding $G$. We study structure case sphere $n+1$ open disks removed, $n\geq 1$, using "daisy" make computations easier. provide new results that hold for arbitrary generic $q$, develop theory $q=\epsilon$, primitive root unity odd order, $G={\rm SL}(2,{\mathbb C})$. In such situation we introduce Frobenius morphism provides natural identification center daisy extension coordinate $\mathcal{O}(G^n)$. extend coadjoint action De-Concini-Kac-Procesi algebra, show Poisson corresponds Fock-Rosly set fixed elements under ${\mathbb C}[X_G(\Sigma)]$ endowed Atiyah-Bott-Goldman structure. Finally, Wilson loop operators identify Kauffman bracket skein $K_{\zeta}(\Sigma)$ at $\zeta:={\rm i}\epsilon^{1/2}$ this specialized $q=\epsilon$.