Algebraic & Geometric Topology

Topological invariants from nonrestricted quantum groups

Nathan Geer and Bertrand Patureau-Mirand

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We introduce the notion of a relative spherical category. We prove that such a category gives rise to the generalized Kashaev and Turaev–Viro-type 3–manifold invariants defined in [J. Reine Angew. Math. 673 (2012) 69–123] and [Adv. Math. 228 (2011) 1163–1202], respectively. In this case we show that these invariants are equal and extend to what we call a relative homotopy quantum field theory which is a branch of the topological quantum field theory founded by E Witten and M Atiyah. Our main examples of relative spherical categories are the categories of finite-dimensional weight modules over nonrestricted quantum groups considered by C De Concini, V Kac, C Procesi, N Reshetikhin and M Rosso. These categories are not semisimple and have an infinite number of nonisomorphic irreducible modules all having vanishing quantum dimensions. We also show that these categories have associated ribbon categories which gives rise to renormalized link invariants. In the case of sl2 these link invariants are the Alexander-type multivariable invariants defined by Y Akutsu, T Deguchi and T Ohtsuki [J. Knot Theory Ramifications 1 (1992) 161–184].

Article information

Algebr. Geom. Topol., Volume 13, Number 6 (2013), 3305-3363.

Received: 3 July 2012
Revised: 17 May 2013
Accepted: 20 May 2013
First available in Project Euclid: 19 December 2017

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Zentralblatt MATH identifier

Primary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23] 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds

unrestricted quantum groups homotopy quantum field theory psi hat systems


Geer, Nathan; Patureau-Mirand, Bertrand. Topological invariants from nonrestricted quantum groups. Algebr. Geom. Topol. 13 (2013), no. 6, 3305--3363. doi:10.2140/agt.2013.13.3305.

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