Unicity for representations of the Kauffman bracket skein algebra. (English) Zbl 1491.57014
Summary: This paper resolves the unicity conjecture of F. Bonahon and H. Wong [Quantum Topol. 10, No. 2, 325–398 (2019; Zbl 1447.57017)] for the Kauffman bracket skein algebras of all oriented finite type surfaces at all roots of unity. The proof is a consequence of a general unicity theorem that says that the irreducible representations of a prime affine \(k\)-algebra over an algebraically closed field \(k\), that is finitely generated as a module over its center, are generically classified by their central characters. The center of the Kauffman bracket skein algebra of any orientable surface at any root of unity is characterized, and it is proved that the skein algebra is finitely generated as a module over its center. It is shown that for any orientable surface the center of the skein algebra at any root of unity is the coordinate ring of an affine algebraic variety.
MSC:
| 57K31 | Invariants of 3-manifolds (including skein modules, character varieties) |
| 57K16 | Finite-type and quantum invariants, topological quantum field theories (TQFT) |
| 16S99 | Associative rings and algebras arising under various constructions |
Citations:
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