2023/03/14 by Thang T. Q. Lê, Lê, Thang T. Q., Changyuan Yu +1
Mathematics · #57M25 #57N10 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT)
paper · pdf · doi:10.48550/arxiv.2303.08082
openalex publication_date 2023/03/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We establish the existence of several quantum trace maps. The simplest one is an algebra map between two quantizations of the algebra of regular functions on the SLn-character variety of a surface \mathfrakS equipped with an ideal triangulation λ. The first is the (stated) SLn-skein algebra \mathscrS(\mathfrakS). The second X(\mathfrakS,λ) is the Fock and Goncharov's quantization of their X-moduli space. The quantum trace is an algebra homomorphism trX:\mathscrS(\mathfrakS)→X(\mathfrakS,λ) where the reduced skein algebra \mathscrS(\mathfrakS) is a quotient of \mathscrS(\mathfrakS). When the quantum parameter is 1, the quantum trace trX coincides with the classical Fock-Goncharov homomorphism. This is a generalization of the Bonahon-Wong quantum trace map for the case n=2. We then define the extended Fock-Goncharov algebra X(\mathfrakS,λ) and show that trX can be lifted to trX:\mathscrS(\mathfrakS)\toX(\mathfrakS,λ). We show that both trX and trX are natural with respect to the change of triangulations. When each connected component of \mathfrakS has non-empty boundary and no interior ideal point, we define a quantization of the Fock-Goncharov A-moduli space A(\mathfrakS,λ) and its extension A(\mathfrakS,λ). We then show that there exist quantum traces trA:\mathscrS(\mathfrakS)→A(\mathfrakS,λ) and trA:\mathscrS(\mathfrakS)\hookrightarrowA(\mathfrakS,λ), where the second map is injective, while the first is injective at least when \mathfrakS is a polygon. They are equivalent to the X-versions but have better algebraic properties.