#### Volume 13, issue 6 (2013)

 Download this article For screen For printing
 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
Topological invariants from nonrestricted quantum groups

### Nathan Geer and Bertrand Patureau-Mirand

Algebraic & Geometric Topology 13 (2013) 3305–3363
##### 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 ${\mathfrak{s}\mathfrak{l}}_{2}$ 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].

##### Keywords
unrestricted quantum groups, homotopy quantum field theory, psi hat systems
##### Mathematical Subject Classification 2010
Primary: 17B37, 57M25, 57M27
##### Publication
Received: 3 July 2012
Revised: 17 May 2013
Accepted: 20 May 2013
Published: 10 October 2013
##### Authors
 Nathan Geer Mathematics and Statistics Utah State University 3900 Old Main Hill Logan, UT 84322-3900 USA Bertrand Patureau-Mirand LMAM Université de Bretagne-Sud BP 573 56017 Vannes France