Geometry & Topology
- Geom. Topol.
- Volume 20, Number 5 (2016), 2687-2835.
Unified quantum invariants for integral homology spheres associated with simple Lie algebras
For each finite-dimensional, simple, complex Lie algebra and each root of unity (with some mild restriction on the order) one can define the Witten–Reshetikhin–Turaev (WRT) quantum invariant of oriented –manifolds . We construct an invariant of integral homology spheres , with values in , the cyclotomic completion of the polynomial ring , such that the evaluation of at each root of unity gives the WRT quantum invariant of at that root of unity. This result generalizes the case proved by Habiro. It follows that unifies all the quantum invariants of associated with and represents the quantum invariants as a kind of “analytic function” defined on the set of roots of unity. For example, for all roots of unity are determined by a “Taylor expansion” at any root of unity, and also by the values at infinitely many roots of unity of prime power orders. It follows that WRT quantum invariants for all roots of unity are determined by the Ohtsuki series, which can be regarded as the Taylor expansion at , and hence by the Lê–Murakami–Ohtsuki invariant. Another consequence is that the WRT quantum invariants are algebraic integers. The construction of the invariant is done on the level of quantum group, and does not involve any finite-dimensional representation, unlike the definition of the WRT quantum invariant. Thus, our construction gives a unified, “representation-free” definition of the quantum invariants of integral homology spheres.
Geom. Topol., Volume 20, Number 5 (2016), 2687-2835.
Received: 12 March 2015
Accepted: 25 October 2015
First available in Project Euclid: 16 November 2017
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Habiro, Kazuo; Lê, Thang T Q. Unified quantum invariants for integral homology spheres associated with simple Lie algebras. Geom. Topol. 20 (2016), no. 5, 2687--2835. doi:10.2140/gt.2016.20.2687. https://projecteuclid.org/euclid.gt/1510859044