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Topological invariants from nonrestricted quantum groups

Nathan Geer and Bertrand Patureau-Mirand

Algebraic & Geometric Topology 13 (2013) 3305–3363

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].

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