Representation-reduced stated skein modules and algebras

Abstract

For any marked three manifold (M,N) and any quantum parameter q12 (a nonzero complex number), we use Sq1/2(M,N) to denote the stated skein module of (M,N). When q12 is a root of unity of odd order, the commutative algebra S1(M,N) acts on Sq1/2(M,N). For any maximal ideal ρ of S1(M,N), define Sq1/2(M,N)ρ=Sq1/2(M,N)⊗S1(M,N)(S1(M,N)/ρ).We prove the splitting map for Sq1/2(M,N) respects the S1(M,N)-module structure, so it reduces to the splitting map for Sq1/2(M,N)ρ. We prove the splitting map for Sq1/2(M,N)ρ is injective if there exists at least one component of N such that this component and the boundary of the splitting disk belong to the same component of ∂M. We also prove the representation-reduced stated skein module of the marked handlebody is an irreducible Azumaya representation of the stated skein algebra of its boundary.Let M be an oriented connected closed three manifold. For any positive integer k, we use Mk to denote the marked three manifold obtained from M by removing k open three dimensional balls and adding one marking to each newly created sphere boundary component. We prove dimCSq1/2(Mk)ρ=1 for any maximal ideal ρ of S1(Mk).