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

Habiro, Kazuo¹ ; Lê, Thang T Q²

¹ Research Institute for Mathematical Sciences, Kyoto University, Kitashirakawa-Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan
² School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332-0160, United States

Geometry & topology, Tome 20 (2016) no. 5, pp. 2687-2835.

Résumé

For each finite-dimensional, simple, complex Lie algebra $\mathfrak{g}$ and each root of unity $\xi$ (with some mild restriction on the order) one can define the Witten–Reshetikhin–Turaev (WRT) quantum invariant ${\tau }_M^{\mathfrak{g}}\left(\xi \right)\in ℂ$ of oriented $3$–manifolds $M$. We construct an invariant $J_M$ of integral homology spheres $M$, with values in $\widehat{\mathbb{Z}\left[q\right]}$, the cyclotomic completion of the polynomial ring $\mathbb{Z}\left[q\right]$, 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 $\mathfrak{g}={sl}_2$ proved by Habiro. It follows that $J_M$ unifies all the quantum invariants of $M$ associated with $\mathfrak{g}$ and represents the quantum invariants as a kind of “analytic function” defined on the set of roots of unity. For example, ${\tau }_M\left(\xi \right)$ 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 ${\tau }_M\left(\xi \right)$ 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 ${\tau }_M^{\mathfrak{g}}\left(\xi \right)$ 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.

Détail

DOI : 10.2140/gt.2016.20.2687

Classification : 57M27, 17B37
Keywords: quantum invariants, integral homology spheres, quantized enveloping algebras, ring of analytic functions on roots of unity

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