A cluster realization of $U_q(\mathfrak{s}{\mathfrak{l}}_{\mathfrak{n}})$ from quantum character varieties

Abstract

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 $PG{L}_{n+1}$-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.

42 pages; v3: main result was strengthened, see Proposition 4.10, introduction and section 1 were rewritten, section 5 was added, some proofs were simplified, some more referenced were added

References in corpus (3)

• The quantum dilogarithm and representations quantum cluster varieties
• Cluster structures on quantum coordinate rings
• Continuous tensor categories from quantum groups I: algebraic aspects

Cited by in corpus (2)

• Continuous tensor categories from quantum groups I: algebraic aspects
• Cluster Realization of $U_q(\mathfrak{g})$ and Factorization of the Universal $R$-Matrix