We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground fields, and the proof of new results about topological invariants in three dimensions. The central notion is that of a Frobenius algebra in a tensor category A, which reduces to the classical notion if A = F-Vect, where F is a field. An object X ∈ A with two-sided dual X gives rise to a Frobenius algebra in A, and under weak additional conditions we prove a converse: There exists a bicategory E with ObjE = {A, B} such that EndE(A) ⊗ ≃ A and such that there are J, J: B ⇋ A producing the given Frobenius algebra. Many properties (additivity, sphericity, semisimplicity,...) of A carry over to the bicategory E. We define weak monoidal Morita equivalence of tensor categories, denoted A ≈ B, and establish a correspondence between Frobenius algebras in A and tensor categories B ≈ A. While considerably weaker than equivalence of tensor categories, weak monoidal Morita equivalence A ≈ B has remarkable consequences: A and B have equivalent (as braided tensor categories) quantum doubles (‘centers’) and (if A, B are semisimple spherical or ∗-categories) have equal dimensions and give rise the same state sum invariant of closed oriented 3-manifolds as recently defined by Barrett and Westbury. An instructive example is provided by finite dimensional semisimple and cosemisimple Hopf algebras, for which we prove H − mod ≈ ˆH − mod. The present formalism permits a fairly complete analysis of the center of a semisimple spherical category, which is the subject of the companion paper math.CT/0111205. 1

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Any triple (W, L, ρ), where W is a compact closed oriented 3-manifold, L is a link in W and ρ is a flat principal B-bundle over W (B is the Borel subgroup of upper triangular matrices of SL(2, C)), can be encoded by suitable distinguished and decorated triangulations T = (T, H, D). For each T, for each odd integer N ≥ 3, one defines a state sum KN(T), based on the Faddeev-Kashaev quantum dilogarithm at ω = exp(2πi/N), such that KN(W,L, ρ) = KN(T) is a well-defined complex valued invariant. The purely topological, conjectural invariants KN(W,L) proposed earlier by Kashaev correspond to the special case of the trivial flat bundle. Moreover, we extend the definition of these invariants to the case of flat bundles on W \ L with not necessarily trivial holonomy along the meridians of the link’s components, and also to 3-manifolds endowed with a B-flat bundle and with arbitrary non-spherical parametrized boundary components. As a matter of fact the distinguished and decorated triangulations are strongly reminiscent of the way one represents the classical refined scissors congruence class ̂ β(F), belonging to the extended Bloch group, of any given finite volume hyperbolic 3-manifold F by using any hyperbolic ideal triangulation of F. We point out some remarkable specializations of the invariants; among these, the so called Seifert-type invariants, when W = S 3: these seem to be good candidates in order to fully reconstruct the Jones polynomials in the main stream of quantum hyperbolic invariants. Finally, we try to set our results against the heuristic backgroud of the Euclidean analytic continuation of (2+1) quantum gravity with negative cosmological constant, regarded as a gauge theory with the non-compact group SO(3, 1) as gauge group.

Abstract. We define higher Frobenius-Schur indicators for objects in linear pivotal monoidal categories. We prove that they are category invariants, and take values in the cyclotomic integers. We also define a family of natural endomorphisms of the identity endofunctor on a k-linear semisimple rigid monoidal category, which we call the Frobenius-Schur endomorphisms. For a k-linear semisimple pivotal monoidal category — where both notions are defined —, the Frobenius-Schur indicators can be computed as traces of the Frobenius-Schur endomorphisms.