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 X G (Σ), the variety of G-characters of π 1 (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 U q (g)-module-algebras associated to graphs on Σ, where U q (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(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 O(G n ). We show that the set of fixed elements of the center under the quantum coadjoint action is a finite extension of C[X G (Σ)] endowed with the Atiyah-Bott-Goldman Poisson structure. Finally, by using Wilson loop operators we identify the Kauffman bracket skein algebra K ζ (Σ) at ζ := iϵ 1/2 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 K ζ (Σ).

Finite presentations for stated skein algebras and lattice gauge field theory

“…This result was generalized in [39] to stated skein algebras as well (see also [11] for generalizations). In [8], Baseilhac and Roche showed that the construction of this so-called Chebyshev-Frobenius morphism is much easier in the context of quantum moduli algebras (that is, using the finite presentations of Theorem 1.1). Even though their study only concerns genus 0 surfaces, their proofs seem to generalize easily to general surfaces, providing simpler proofs for the results in [13; 39].…”

“…4 is a root of unity of odd order, Baseilhac and Roche have proved [8, page 41] that the subalgebra of U q sl 2 -invariant vectors coincides with the center of S ! .m 1 / (denoted by L " 0;1 in [8]). This center is generated by the peripheral curve p encircling the inner puncture p together with the image of the Chebyshev-Frobenius morphism.…”