Derived Representations of Quantum Character Varieties

This chapter examines the Poisson structure of the representation variety of the fundamental groupoid of a Riemann surface with punctures and cusps, and the associated decorated character variety.

Quantum Character Varieties

We prove that quantized multiplicative quiver varieties, quantum character varieties, and Kauffman bracket skein algebras each define sheaves of Azumaya algebras over the smooth loci of their corresponding classical moduli spaces. In the case of skein algebras this establishes a strong form of the Unicity Conjecture of Bonahon and Wong. Our proofs exploit a strong compatibility between quantum Hamiltonian reduction and the quantum Frobenius homomorphism as well as a natural nondegeneracy condition satisfied by each of the classical Hamiltonian spaces. We therefore introduce the concepts of Frobenius quantum moment maps, Frobenius Poisson orders and their Hamiltonian reductions, and apply them to the study of Azumaya loci.

We propose a new non-commutative generalization of the representation variety and the character variety of a knot group. Our strategy is to reformulate the construction of the algebra of functions on the space of representations in terms of Hopf algebra objects in a braided category (braided Hopf algebra). The construction works under the assumption that the algebra is braided commutative. The resulting knot invariant is a module with a coadjoint action. Taking the coinvariants yields a new quantum character variety that may be thought of as an alternative to the skein module. We give concrete examples for a few of the simplest knots and links.

We construct an injective algebra homomorphism of the quantum group $$U_q(\mathfrak {sl}_{n+1})$$ into a quantum cluster algebra $$\mathbf {L}_n$$ associated to the moduli space of framed $$PGL_{n+1}$$ -local systems on a marked punctured disk. We obtain a description of the coproduct of $$U_q(\mathfrak {sl}_{n+1})$$ in terms of the corresponding quantum cluster algebra associated to the marked twice punctured disk, and express the action of the R-matrix in terms of a mapping class group element corresponding to the half-Dehn twist rotating one puncture about the other. As a consequence, we realize the algebra automorphism of $$U_q(\mathfrak {sl}_{n+1})^{\otimes 2}$$ given by conjugation by the R-matrix as an explicit sequence of cluster mutations, and derive a refined factorization of the R-matrix into quantum dilogarithms of cluster monomials.

We compute quantum character varieties of arbitrary closed surfaces with boundaries and marked points. These are categorical invariants int _S{mathcal {A}} of a surface S, determined by the choice of a braided tensor category {mathcal {A}}, and computed via factorization homology. We identify the algebraic data governing marked points and boundary components with the notion of a braided module category for {mathcal {A}}, and we describe braided module categories with a generator in terms of certain explicit algebra homomorphisms called quantum moment maps. We then show that the quantum character variety of a decorated surface is obtained from that of the corresponding punctured surface as a quantum Hamiltonian reduction. Characters of braided {mathcal {A}}-modules are objects of the torus category int _{T^2}{mathcal {A}}. We initiate a theory of character sheaves for quantum groups by identifying the torus integral of {mathcal {A}}={text {Rep}}_{q}G with the category {mathcal {D}}_q(G/G)-mod of equivariant quantum {mathcal {D}}-modules. When G=GL_n, we relate the mirabolic version of this category to the representations of the spherical double affine Hecke algebra {mathbb {SH}}_{q,t}.