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"Invariants of 3-manifolds via link polynomials and quantum groups", Nicolai Reshetikhin, Vladimir Turaev, 1991

Reviewed August 31, 2023

Citation: Reshetikhin, Nicolai, and Vladimir G. Turaev. "Invariants of 3-manifolds via link polynomials and quantum groups." Inventiones mathematicae 103.1 (1991): 547-597.

Web: https://link.springer.com/article/10.1007/BF01239527

Tags: Foundational, Mathematical, TQFT, Modular-tensor-categories, Quantum-groups

This paper gave the first construction of non-trivial TQFTs, now known as the Reshetikhin-Turaev TQFT construction. To quote the authors again: "our invariants may be viewed as a mathematical realization of the Witten's program". In particular, they gave a procedure to turn any modular Hopf algebra into a TQFT. This procedure was then re-worked by Turaev in

> Turaev, Vladimir G. Quantum invariants of knots and 3-manifolds. de Gruyter, 2010.

where it was showed that these constructions should take general modular tensor categories as inputs, not modular Hopf algebras. It is this re-worked construction which people most commonly refer to as the Reshetikhin-Turaev construction. Modular Hopf algebras were studied a bit more in the years following this paper, such as in

> Ramgoolam, Sanjaye. "New Modular Hopf Algebras related to rational $ k $$\widehat {sl (2)} $." arXiv preprint hep-th/9301121 (1993).

but the language of modular tensor categories proved to be superior and now modular Hopf algebras are all but forgotten.