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• W3105301818 endingPage "2835" @default.
• W3105301818 startingPage "2687" @default.
• W3105301818 abstract "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 $tau_M^{mathfrak g}(xi)in mathbb C$ of oriented 3-manifolds $M$. In the present paper we construct an invariant $J_M$ of integral homology spheres $M$ with values in the cyclotomic completion $widehat {mathbb Z [q]}$ of the polynomial ring $mathbb Z [q]$, 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 ${mathfrak g}=sl_2$ proved by the first author. 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, $tau_M(xi)$ 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 $tau_M(xi)$ 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 Le-Murakami-Ohtsuki invariant. Another consequence is that the WRT quantum invariants $tau_M^{ mathfrak g}(xi)$ 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." @default.
• W3105301818 created "2020-11-23" @default.
• W3105301818 creator A5018027659 @default.
• W3105301818 creator A5024859920 @default.
• W3105301818 date "2016-10-07" @default.
• W3105301818 modified "2023-09-24" @default.
• W3105301818 title "Unified quantum invariants for integral homology spheres associated with simple Lie algebras" @default.
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• W3105301818 doi "https://doi.org/10.2140/gt.2016.20.2687" @default.
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