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
–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
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
