Ribbon Hopf Algebra Research Articles

Kerler–Lyubashenko Functors on 4-Dimensional 2-Handlebodies

Abstract We construct a braided monoidal functor $J_4$ from Bobtcheva and Piergallini’s category $4\textrm {HB}$ of connected 4-dimensional 2-handlebodies (up to 2-deformations) to an arbitrary unimodular ribbon category $\mathscr {C}$, which is not required to be semisimple. The main example of target category is provided by ${H}\textrm{-mod}$, the category of left modules over a unimodular ribbon Hopf algebra $H$. The source category $4\textrm {HB}$ is freely generated, as a braided monoidal category, by a Bobtcheva--Piergallini Hopf (BPH) algebra object, and this is sent by the Kerler–Lyubashenko functor $J_4$ to the end $\int _{X \in \mathscr {C}} X \otimes X^*$ in $\mathscr {C}$, which is given by the adjoint representation in the case of ${H}\textrm{-mod}$. When $\mathscr {C}$ is factorizable, we show that the construction only depends on the boundary and signature of handlebodies and thus projects to a functor $J_3^{\sigma }$ defined on Kerler’s category $3\textrm {Cob}^{\sigma }$ of connected framed 3-dimensional cobordisms. When $H^*$ is not semisimple and $H$ is not factorizable, our functor $J_4$ has the potential of detecting diffeomorphisms that are not 2-deformations.

Modified graded Hennings invariants from unrolled quantum groups and modified integral

The second author constructed a topological ribbon Hopf algebra from the unrolled quantum group associated with the super Lie algebra sl ( 2 | 1 ) . We generalize this fact to the context of unrolled quantum groups and construct the associated topological ribbon Hopf algebras. Then we use such an algebra, the discrete Fourier transforms , a symmetrized graded integral and a modified trace to define a modified graded Hennings invariant of 3-manifolds endowed with a cohomology class and which contains a ribbon graph. Finally, we use the notion of a modified integral to extend this invariant to manifolds without ribbon graphs inside and show that it recovers the invariant of [6] .

Hopf algebra gauge theory on a ribbon graph

We generalize gauge theory on a graph so that the gauge group becomes a finite-dimensional ribbon Hopf algebra, the graph becomes a ribbon graph, and gauge-theoretic concepts such as connections, gauge transformations and observables are replaced by linearized analogs. Starting from physical considerations, we derive an axiomatic definition of Hopf algebra gauge theory, including locality conditions under which the theory for a general ribbon graph can be assembled from local data in the neighborhood of each vertex. For a vertex neighborhood with [Formula: see text] incoming edge ends, the algebra of non-commutative ‘functions’ of connections is dual to a two-sided twist deformation of the [Formula: see text]-fold tensor power of the gauge Hopf algebra. We show these algebras assemble to give an algebra of functions and gauge-invariant subalgebra of ‘observables’ that coincide with those obtained in the combinatorial quantization of Chern–Simons theory, thus providing an axiomatic derivation of the latter. We then discuss holonomy in a Hopf algebra gauge theory and show that for semisimple Hopf algebras this gives, for each path in the embedded graph, a map from connections into the gauge Hopf algebra, depending functorially on the path. Curvatures — holonomies around the faces canonically associated to the ribbon graph — then correspond to central elements of the algebra of observables, and define a set of commuting projectors onto the subalgebra of observables on flat connections. The algebras of observables for all connections or for flat connections are topological invariants, depending only on the topology, respectively, of the punctured or closed surface canonically obtained by gluing annuli or discs along edges of the ribbon graph.

A topological origin of quantum symmetric pairs

It is well known that braided monoidal categories are the categorical algebras of the little two-dimensional disks operad. We introduce involutive little disks operads, which are {mathbb {Z}}_2-orbifold versions of the little disks operads. We classify their categorical algebras and describe these explicitly in terms of a finite list of functors, natural isomorphisms and coherence equations. In dimension two, the categorical algebras are braided monoidal categories with an anti-involution together with a pointed module category carrying a universal solution to the (twisted) reflection equation. Examples of these are obtained from the categories of representations of a ribbon Hopf algebra with an involution and a quasi-triangular coideal subalgebra, such as a quantum group and a quantum symmetric pair coideal subalgebra.

Ribbon Hopf Superalgebras and Drinfel’d Double

Finite dimensional ribbon Hopf (super) algebras play an important role in constructing invariants of 3-manifolds. In the present paper, the authors give a necessary and sufficient condition for the Drinfel’d double of a finite dimensional Hopf superalgebra to have a ribbon element. The criterion can be seen as a generalization of Kauffman and Radford’s result in the non-super situation to the ℤ2-graded situation, however, the derivation of the result in the ℤ2-graded case will be much more complicated.

The universal quantum invariant and colored ideal triangulations

The Drinfeld double of a finite dimensional Hopf algebra is a quasi-triangular Hopf algebra with the canonical element as the universal $R$-matrix, and one can obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum invariant of framed links is constructed using a ribbon Hopf algebra. In that construction, a copy of the universal $R$-matrix is attached to each crossing, and invariance under the Reidemeister III move is shown by the quantum Yang-Baxter equation of the universal $R$-matrix. On the other hand, the Heisenberg double of a finite dimensional Hopf algebra has the canonical element (the $S$-tensor) satisfying the pentagon relation. In this paper we reconstruct the universal quantum invariant using the Heisenberg double, and extend it to an invariant of equivalence classes of colored ideal triangulations of $3$-manifolds up to colored moves. In this construction, a copy of the $S$-tensor is attached to each tetrahedron, and invariance under the colored Pachner $(2,3)$ moves is shown by the pentagon relation of the $S$-tensor.

Renormalized Hennings Invariants and 2 + 1-TQFTs

We construct non-semisimple $2+1$-TQFTs yielding mapping class group representations in Lyubashenko's spaces. In order to do this, we first generalize Beliakova, Blanchet and Geer's logarithmic Hennings invariants based on quantum $\mathfrak{sl}_2$ to the setting of finite-dimensional non-degenerate unimodular ribbon Hopf algebras. The tools used for this construction are a Hennings-augmented Reshetikhin-Turaev functor and modified traces. When the Hopf algebra is factorizable, we further show that the universal construction of Blanchet, Habegger, Masbaum and Vogel produces a $2+1$-TQFT on a not completely rigid monoidal subcategory of cobordisms.

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.

$|Z_{\mathrm{Kup}}|=|Z_{\mathrm{Henn}}|^2$ for lens spaces

M. Hennings and G. Kuperberg defined quantum invariants Z_{\rm Henn} and Z_{\rm Kup} of closed oriented 3-manifolds based on certain Hopf algebras, respectively. We prove that |Z_{\rm Kup}|=|Z_{\rm Henn}|^2 for lens spaces when both invariants are based on factorizable finite dimensional ribbon Hopf algebras.

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.

Ribbon Hopf algebras from group character rings

We study the diagram alphabet of knot moves associated with the character rings of certain matrix groups. The primary object is the Hopf algebra Char-GL of characters of the finite dimensional polynomial representations of the complex group in the inductive limit, realized as the ring of symmetric functions on countably many variables . Isomorphic as spaces are the character rings Char-O and Char-Sp of the classical matrix subgroups of , the orthogonal and symplectic groups. We also analyse the formal character rings Char-H of algebraic subgroups of , comprised of matrix transformations leaving invariant a fixed but arbitrary tensor of Young symmetry type , which have been introduced earlier (Fauser et al.) (these include the orthogonal and symplectic groups as special cases). The set of tangle diagrams encoding manipulations of the group and subgroup characters has many elements deriving from products, coproducts, units and counits as well as different types of branching operators. From these elements we assemble for each a crossing tangle which satisfies the braid relation and which is nontrivial, in spite of the commutative and co-commutative setting. We identify structural elements and verify the axioms to establish that each Char-H ring is a ribbon Hopf algebra. The corresponding knot invariant operators are rather weak, giving merely a measure of the writhe.

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.

Convex PBW-type Lyndon bases and restricted two-parameter quantum groups of type [formula omitted

We construct convex PBW-type Lyndon bases for two-parameter quantum groups U r , s ( s o 2 n + 1 ) with detailed commutation relations. It turns out that under a certain condition, the restricted two-parameter quantum group u r , s ( s o 2 n + 1 ) ( r , s are roots of unity) is a Drinfel’d double. All Hopf isomorphisms of u r , s ( s o 2 n + 1 ) , as well as u r , s ( s l n ) , are determined. Finally, necessary and sufficient conditions for u r , s ( s o 2 n + 1 ) to be a ribbon Hopf algebra are singled out by describing the left and right integrals.

Convex PBW-type Lyndon basis and restricted two-parameter quantum groups of typeG2

From the 2-parameter quantum group Ur,s(G2) defined by Hu and Shi in 2007, we construct finite-dimensional pointed Hopf algebras ur,s(G2) (that is, restricted 2-parameter quantum groups); these turn out to be Drinfel�d doubles. Crucial is a detailed combinatorial construction of the convex PBW-type Lyndon basis for type G2 in the 2-parameter quantum version. We exhibit the possible commutation relations among quantum root vectors. Then we show that the restricted quantum groups are ribbon Hopf algebras under certain conditions, by determining their left and right integrals. We also determine all the Hopf algebra isomorphisms of ur,s(G2) by describing its sets of left (right) skew-primitive elements.

Quantum D-modules, Elliptic Braid Groups, and Double Affine Hecke Algebras

We build representations of the elliptic braid group from the data of a quantum D-module M over a ribbon Hopf algebra U. The construction is modelled on, and generalizes, similar constructions by Lyubashenko and Majid, and also certain geometric constructions of Calaque, Enriquez, and Etingof concerning trigonometric Cherednik algebras. In this context, the former construction is the special case where M is the basic representation, while the latter construction can be recovered as a quasi-classical limit of U=U_t(sl_N), as t limits 1. In the latter case, we produce representations of the double affine Hecke algebra of type A_{n-1}, for each n.

Generic Modules Over a Class of Drinfeld's Quantum Doubles

Let k be a field and A n (ω) be the Taft's n 2-dimensional Hopf algebras. When n is odd, the Drinfeld quantum double D(A n (ω)) of A n (ω) is a Ribbon Hopf algebra. In the previous articles, we constructed an n 4-dimensional Hopf algebra H n (p, q) which is isomorphic to D(A n (ω)) if p ≠ 0 and q = ω−1, and studied the finite dimensional representations of H n (1, q). We showed that the basic algebra of any nonsimple block of H n (1, q) is independent of n. In this article, we examine the infinite representations of H 2(1, − 1), or equivalently of H n (1, q)≃D(A n (ω)) for any n ≥ 2. We investigate the indecomposable and algebraically compact modules over H 2(1, − 1), describe the structures of these modules and classify them under the elementary equivalence.

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 .