m at h . G T ] 8 N ov 1 99 8 GROUP CATEGORIES AND THEIR FIELD THEORIES
نویسنده
چکیده
A group-category is an additively semisimple category with a monoidal product structure in which the simple objects are invertible. For example in the category of representations of a group, 1-dimensional representations are the invertible simple objects. This paper gives a detailed exploration of “topological quantum field theories” for group-categories, in hopes of finding clues to a better understanding of the general situation. Group-categories are classified in several ways extending results of Frölich and Kerler. Topological field theories based on homology and cohomology are constructed, and these are shown to include theories obtained from group-categories by Reshetikhin-Turaev constructions. Braided-commutative categories most naturally give theories on 4-manifold thickenings of 2-complexes; the usual 3-manifold theories are obtained from these by normalizing them (using results of Kirby) to depend mostly on the boundary of the thickening. This is worked out for group-categories, and in particular we determine when the normalization is possible
منابع مشابه
ar X iv : 0 71 0 . 57 61 v 2 [ m at h . G T ] 8 N ov 2 00 7 ON EXOTIC MODULAR TENSOR CATEGORIES
It has been conjectured that every (2+1)-TQFT is a Chern-SimonsWitten (CSW) theory labelled by a pair (G, λ), where G is a compact Lie group, and λ ∈ H(BG;Z) a cohomology class. We study two TQFTs constructed from Jones’ subfactor theory which are believed to be counterexamples to this conjecture: one is the quantum double of the even sectors of the E6 subfactor, and the other is the quantum do...
متن کاملOn Reducibility of Mapping Class Group Representations: the Su(n) Case
We review and extend the results of [1] that give a condition for reducibility of quantum representations of mapping class groups constructed from Reshetikhin-Turaev type topological quantum field theories based on modular categories. This criterion is derived using methods developed to describe rational conformal field theories, making use of Frobenius algebras and their representations in mod...
متن کاملar X iv : 0 90 6 . 34 15 v 1 [ m at h . Q A ] 1 8 Ju n 20 09 QUIVERS , QUASI - QUANTUM GROUPS AND FINITE TENSOR CATEGORIES
We study finite quasi-quantum groups in their quiver setting developed recently by the first author. We obtain a classification of finite-dimensional pointed Majid algebras of finite corepresentation type, or equivalently a classification of elementary quasi-Hopf algebras of finite representation type, over the field of complex numbers. By the Tannaka-Krein duality principle, this provides a cl...
متن کاملar X iv : 0 90 8 . 10 44 v 1 [ m at h . Q A ] 7 A ug 2 00 9 Modular invariants for group - theoretical modular data
We classify indecomposable commutative separable (special Frobenius) algebras and their local modules in (untwisted) group-theoretical modular categories. This gives a description of modular invariants for grouptheoretical modular data. As a bi-product we provide an answer to the question when (and in how many ways) two group-theoretical modular categories are equivalent as ribbon categories.
متن کاملBraided Near-group Categories
A near-group category is an additively semisimple category with a product such that all but one of the simple objects is invertible. We classify braided structures on near-group categories, and give explicit numerical formulas for their associativity and commutativity morphisms. 1. Results We consider near-group categories; that is, semisimple monoidal categories, whose set of simples consists ...
متن کامل