Representations of the Kauffman bracket skein algebra II: punctured surfaces

Abstract

In earlier work, we constructed invariants of irreducible representations of the Kauffman skein algebra of a surface. We introduce here an inverse construction, which to a set of possible invariants associates an irreducible representation that realizes these invariants. The current article is restricted to surfaces with at least one puncture, a condition that will be lifted in subsequent work of the authors that relies on this one. A step in the proof is of independent interest, and describes the algebraic structure of the Thurston intersection form on the space of integer weight systems for a train track.

22 pages. Version 2: The article was much reorganized, for compatibility with the subsequent article [BonWon6] in the same series; the results are unchanged. Version 3: This new version takes into account the possible impact of sign reversal symmetries, overlooked in the earlier versions, on the uniqueness properties needed for [BonWon6]; the manuscript is now ready for journal submission

References in corpus (4)

• Representations of the Kauffman bracket skein algebra I: invariants and miraculous cancellations
• On Kauffman Bracket Skein Modules at Root of Unity
• Frobenius Algebras Derived from the Kauffman Bracket Skein Algebra
• The Witten-Reshetikhin-Turaev representation of the Kauffman skein algebra

Cited by in corpus (8)

• Classical shadows of stated skein representations at roots of unity
• Unrestricted Quantum Moduli Algebras. I. The Case of Punctured Spheres
• Shear-shape cocycles for measured laminations and ergodic theory of the earthquake flow
• ${\mathrm{SL}}_2$ quantum trace in quantum Teichmüller theory via writhe
• Cauchy-compact flat spacetimes with extreme BTZ
• An embedding of skein algebras of surfaces into quantum tori from Dehn-Thurston coordinates
• Kauffman bracket intertwiners and the volume conjecture
• Mapping Class Group Representations Derived from Stated Skein Algebras