Factorizable Ribbon Research Articles

A quasi-Hopf algebra for the triplet vertex operator algebra

We give a new factorizable ribbon quasi-Hopf algebra [Formula: see text], whose underlying algebra is that of the restricted quantum group for [Formula: see text] at a [Formula: see text]th root of unity. The representation category of [Formula: see text] is conjecturally ribbon equivalent to that of the triplet vertex operator algebra (VOA) [Formula: see text]. We obtain [Formula: see text] via a simple current extension from the unrolled restricted quantum group at the same root of unity. The representation category of the unrolled quantum group is conjecturally equivalent to that of the singlet VOA [Formula: see text], and our construction is parallel to extending [Formula: see text] to [Formula: see text]. We illustrate the procedure in the simpler example of passing from the Hopf algebra for the group algebra [Formula: see text] to a quasi-Hopf algebra for [Formula: see text], which corresponds to passing from the Heisenberg VOA to a lattice extension.

Hochschild cohomology and the modular group

It has been shown in previous work that the modular group acts projectively on the center of a factorizable ribbon Hopf algebra. The center is the zeroth Hochschild cohomology group. In this article, we extend this projective action of the modular group to an arbitrary Hochschild cohomology group of a factorizable ribbon Hopf algebra, in fact up to homotopy even to a projective action on the entire Hochschild cochain complex.

From non-semisimple Hopf algebras to correlation functions for logarithmic CFT

We use factorizable finite tensor categories, and specifically the representation categories of factorizable ribbon Hopf algebras H, as a laboratory for exploring bulk correlation functions in local logarithmic conformal field theories. For any ribbon Hopf algebra automorphism ω of H, we present a candidate for the space of bulk fields and endow it with a natural structure of a commutative symmetric Frobenius algebra. We derive an expression for the corresponding bulk partition functions as bilinear combinations of irreducible characters; as a crucial ingredient this involves the Cartan matrix of the category. We also show how for any candidate bulk state space of the type we consider, correlation functions of bulk fields for closed oriented world sheets of any genus can be constructed that are invariant under the natural action of the relevant mapping class group.

Higher genus mapping class group invariants from factorizable Hopf algebras

Lyubashenkoʼs construction associates representations of mapping class groups Mapg:n of Riemann surfaces of any genus g with any number n of holes to a factorizable ribbon category. We consider this construction as applied to the category of bimodules over a finite-dimensional factorizable ribbon Hopf algebra H. For any such Hopf algebra we find an invariant of Mapg:n for all values of g and n. More generally, we obtain such invariants for any pair (H,ω), where ω is a ribbon automorphism of H.Our results are motivated by the quest to understand higher genus correlation functions of bulk fields in two-dimensional conformal field theories with chiral algebras that are not necessarily semisimple, so-called logarithmic conformal field theories.

Ambidextrous objects and trace functions for nonsemisimple categories

We provide a necessary and sufficient condition for a simple object in a pivotal k \Bbbk -category to be ambidextrous. In turn, these objects imply the existence of nontrivial trace functions in the category. These functions play an important role in low-dimensional topology as well as in studying the category itself. In particular, we prove they exist for factorizable ribbon Hopf algebras, modular representations of finite groups and their quantum doubles, complex and modular Lie (super)algebras, the ( 1 , p ) (1,p) minimal model in conformal field theory, and quantum groups at a root of unity.

Modular invariant Frobenius algebras from ribbon Hopf algebra automorphisms

For any finite-dimensional factorizable ribbon Hopf algebra H and any ribbon automorphism of H, we establish the existence of the following structure: an H-bimodule Fω and a bimodule morphism Zω from Lyubashenkoʼs Hopf algebra object K for the bimodule category to Fω. This morphism is invariant under the natural action of the mapping class group of the one-punctured torus on the space of bimodule morphisms from K to Fω. We further show that the bimodule Fω can be endowed with a natural structure of a commutative symmetric Frobenius algebra in the monoidal category of H-bimodules, and that it is a special Frobenius algebra iff H is semisimple.The bimodules K and Fω can both be characterized as coends of suitable bifunctors. The morphism Zω is obtained by applying a monodromy operation to the coproduct of Fω; a similar construction for the product of Fω exists as well.Our results are motivated by the quest to understand the bulk state space and the bulk partition function in two-dimensional conformal field theories with chiral algebras that are not necessarily semisimple.

Structure constants related to symmetric Hopf algebras

Let k be an algebraically closed field of characteristic 0. In this paper we continue our study of structure constants for semisimple Hopf algebras H whose character algebra is commutative, and for non-semisimple factorizable ribbon Hopf algebras. This is done from the point of view of symmetric algebras, such as group algebras. In particular we consider general fusion rules which are structure constants associated to products of irreducible characters and structure constants associated to generalizations of class sums and conjugacy classes. Our methods are reminiscent on one hand of the methods employed when analyzing tilting modules for certain quantum groups and on the other hand of “splitting modules” for certain Drinfeld doubles. The family of irreducible characters is divided in two according to the vanishing of their quantum dimension. The fusion rules on the character algebra are computed with respect to this division.

Characters and a Verlinde-type formula for symmetric Hopf algebras

We study certain aspects of finite-dimensional non-semisimple symmetric Hopf algebras H and their duals H ∗ . We focus on the set I ( H ) of characters of projective H-modules which is an ideal of the algebra of cocommutative elements of H ∗ . This ideal corresponds via a symmetrizing form to the projective center (Higman ideal) of H which turns out to be Λ ▪ H , where Λ is an integral of H and ▪ is the left adjoint action of H on itself. We describe Λ ▪ H via primitive and central primitive idempotents of H. We also show that it is stable under the quantum Fourier transform. Our best results are obtained when H is a factorizable ribbon Hopf algebra over an algebraically closed field of characteristic 0. In this case Λ ▪ H is also the image of I ( H ) under a “translated” Drinfel'd map. We use this fact to prove the existence of a Steinberg-like character. The above ingredients are used to prove a Verlinde-type formula for Λ ▪ H .

Factorizable ribbon quantum groups in logarithmic conformal field theories

We review the properties of quantum groups occurring as the Kazhdan-Lusztig dual to logarithmic conformal field theory models. These quantum groups at even roots of unity are not quasitriangular but are factorizable and have a ribbon structure; the modular group representation on their center coincides with the representation on generalized characters of the chiral algebra in logarithmic conformal field models.

Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity

An example of a finite dimensional factorizable ribbon Hopf ℂ-algebra is given by a quotientH=u q (g) of the quantized universal enveloping algebraU q (g) at a root of unityq of odd degree. The mapping class groupM g,1 of a surface of genusg with one hole projectively acts by automorphisms in theH-moduleH *⊗g, ifH * is endowed with the coadjointH-module structure. There exists a projective representation of the mapping class groupM g,n of a surface of genusg withn holes labeled by finite dimensionalH-modulesX 1, ...,X n in the vector space Hom H (X 1 ⊗ ... ⊗X n ,H *⊗g). An invariant of closed oriented 3-manifolds is constructed. Modifications of these constructions for a class of ribbon Hopf algebras satisfying weaker conditions than factorizability (including most ofu q (g) at roots of unityq of even degree) are described.

Braided Groups and Quantum Fourier Transform

We show that acting on every finite-dimensional factorizable ribbon Hopf algebra H there are invertible operators S, T obeying the modular identities (ST)3 = λS2, where λ is a constant. The class includes the finite-dimensional quantum groups uq(g) associated to complex simple Lie algebras. We give the example of uq(sl(2)) at a root of unity in detail, as well as an example relating to anyons. The operator S plays the role of "quantum Fourier Transform" and acts naturally on H viewed by transmutation as a braided group H (a braided-cocommutative Hopf algebra in a braided category). It obeys S2 = s−1, where s is the antipode of H. The results follow as an application of previous category-theoretical constructions.