Topological invariants from nonrestricted quantum groups

Geer, Nathan¹ ; Patureau-Mirand, Bertrand²

¹Mathematics and Statistics, Utah State University, 3900 Old Main Hill, Logan, UT 84322-3900, USA
²LMAM, Université de Bretagne-Sud, BP 573, 56017 Vannes, France

Algebraic and Geometric Topology, Tome 13 (2013) no. 6, pp. 3305-3363

Résumé

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].

Détail

DOI : 10.2140/agt.2013.13.3305

Classification : 17B37, 57M25, 57M27
Keywords: unrestricted quantum groups, homotopy quantum field theory, psi hat systems

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