Volume 13, issue 6 (2013)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 6, 3213–3852
Issue 5, 2565–3212
Issue 4, 1917–2564
Issue 3, 1283–1916
Issue 2, 645–1281
Issue 1, 1–643

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
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

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