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Unified quantum invariants for integral homology spheres associated with simple Lie algebras

Kazuo Habiro and Thang T Q Lê

Geometry & Topology 20 (2016) 2687–2835

For each finite-dimensional, simple, complex Lie algebra g and each root of unity ξ (with some mild restriction on the order) one can define the Witten–Reshetikhin–Turaev (WRT) quantum invariant τMg(ξ) of oriented 3–manifolds M. We construct an invariant JM of integral homology spheres M, with values in [q]̂, the cyclotomic completion of the polynomial ring [q], such that the evaluation of JM at each root of unity gives the WRT quantum invariant of M at that root of unity. This result generalizes the case g = sl2 proved by Habiro. It follows that JM unifies all the quantum invariants of M associated with g and represents the quantum invariants as a kind of “analytic function” defined on the set of roots of unity. For example, τM(ξ) 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 τM(ξ) for all roots of unity are determined by the Ohtsuki series, which can be regarded as the Taylor expansion at q = 1, and hence by the Lê–Murakami–Ohtsuki invariant. Another consequence is that the WRT quantum invariants τMg(ξ) are algebraic integers. The construction of the invariant JM 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.

quantum invariants, integral homology spheres, quantized enveloping algebras, ring of analytic functions on roots of unity
Mathematical Subject Classification 2010
Primary: 57M27
Secondary: 17B37
Received: 12 March 2015
Accepted: 25 October 2015
Published: 7 October 2016
Proposed: Vaughan Jones
Seconded: Robion Kirby, Ciprian Manolescu
Kazuo Habiro
Research Institute for Mathematical Sciences
Kyoto University
Kitashirakawa-Oiwakecho, Sakyo-ku
Kyoto 606-8502
Thang T Q Lê
School of Mathematics
Georgia Institute of Technology
686 Cherry Street
Atlanta, GA 30332-0160
United States