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

Habiro, Kazuo; Lê, Thang T Q

doi:10.2140/gt.2016.20.2687

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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

DOI: 10.2140/gt.2016.20.2687

Abstract

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 $τ_{M}^{g} (ξ) \in ℂ$ of oriented $3$ –manifolds $M$ . We construct an invariant $J_{M}$ of integral homology spheres $M$ , with values in $\hat{ℤ [q]}$ , the cyclotomic completion of the polynomial ring $ℤ [q]$ , such that the evaluation of $J_{M}$ at each root of unity gives the WRT quantum invariant of $M$ at that root of unity. This result generalizes the case $g = {sl}_{2}$ proved by Habiro. It follows that $J_{M}$ 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, $τ_{M} (ξ)$ 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 $τ_{M} (ξ)$ 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 $τ_{M}^{g} (ξ)$ are algebraic integers. The construction of the invariant $J_{M}$ 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.

Keywords

quantum invariants, integral homology spheres, quantized enveloping algebras, ring of analytic functions on roots of unity

Mathematical Subject Classification 2010

Primary: 57M27

Secondary: 17B37

References

Publication

Received: 12 March 2015

Accepted: 25 October 2015

Published: 7 October 2016

Proposed: Vaughan Jones

Seconded: Robion Kirby, Ciprian Manolescu

Authors

Kazuo Habiro
Research Institute for Mathematical Sciences Kyoto University Kitashirakawa-Oiwakecho, Sakyo-ku Kyoto 606-8502 Japan

Thang T Q Lê
School of Mathematics Georgia Institute of Technology 686 Cherry Street Atlanta, GA 30332-0160 United States

Volume 20, issue 5 (2016)