## Algebraic & Geometric Topology

### Topological invariants from nonrestricted quantum groups

#### Abstract

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

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

Dates
Revised: 17 May 2013
Accepted: 20 May 2013
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715735

Digital Object Identifier
doi:10.2140/agt.2013.13.3305

Mathematical Reviews number (MathSciNet)
MR3248736

Zentralblatt MATH identifier
1273.17018

#### Citation

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. https://projecteuclid.org/euclid.agt/1513715735

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