quasi-Hopf Algebras Research Articles

Double wreath quasi-Hopf algebras

For a quasi-bialgebra H, we show that the category C of H-bimodules is duoidal and that the so called u-coaugmented bimonoids in C are exactly the quasi-bialgebras with a coalgebra projection. When H is a quasi-Hopf algebra with bijective antipode, we prove that the u-coaugmented bimonoids in C can also be described as what we will call double wreath quasi-Hopf algebras, objects determined by H and pre-bialgebras R within the category of Yetter-Drinfeld modules over H. A particular class of double wreath quasi-Hopf algebras is obtained by deforming with a 2-cocycle the multiplication of a Radford biproduct quasi-Hopf algebra. Other classes of this type are given by the symplectic fermion quasi-Hopf algebras.

Extensions of the category of comodules of the Taft algebra

We construct a family of non-equivalent pairwise extensions of the category of comodules of the Taft algebra, which are equivalent to representation categories of non-triangular quasi-Hopf algebras.

Slack Hopf monads

Hopf monads generalise Hopf algebras. They clarify several aspects of the theory of Hopf algebras and capture several related structures such as weak Hopf algebras and Hopf algebroids. However, important parts of Hopf algebra theory are not reached by Hopf monads, most noticeably Drinfeld's quasi-Hopf algebras. In this paper we introduce a generalisation of Hopf monads, that we call slack Hopf monads. This generalisation retains a clean theory and is flexible enough to encompass quasi-Hopf algebras as examples. A slack Hopf monad is a colax magma monad T on a magma category C such that the forgetful functor UT:CT→C ‘slackly’ preserves internal Homs. We give a number of different descriptions of slack Hopf monads, and study special cases such as slack Hopf monads on cartesian categories and k-linear exact slack Hopf monads on Vectk, that is comagma algebras such that a modified fusion operator is invertible. In particular, we characterise quasi-Hopf algebras in terms of slackness.

Exact factorizations and extensions of finite tensor categories

We extend [13] to the nonsemisimple case. We define and study exact factorizations B=A•C of a finite tensor category B into a product of two tensor subcategories A,C, and relate exact factorizations of finite tensor categories to exact sequences of finite tensor categories with respect to exact module categories [8]. We apply our results to study exact factorizations of quasi-Hopf algebras, and extensions of a finite group scheme theoretical tensor category [14] by another one. We also provide several examples to illustrate our results.

Modular quasi-Hopf algebras and groups with one involution

In a previous paper the authors constructed a class of quasi-Hopf algebras Dω(G,A) associated to a finite group G, generalizing the twisted quantum double construction. We gave necessary and sufficient conditions, cohomological in nature, that the corresponding module category Rep(Dω(G,A)) is a modular tensor category. In the present paper we verify the cohomological conditions for the class of groups G which contain a unique involution, and in this way we obtain an explicit construction of a new class of modular quasi-Hopf algebras. We develop the basic theory for general finite groups G, and also a parallel theory concerned with the question of when Rep(Dω(G,A)) is super-modular rather than modular. We give some explicit examples involving binary polyhedral groups and some sporadic simple groups.

The symplectic fermion ribbon quasi-Hopf algebra and the [formula omitted]-action on its centre

We introduce a family of factorisable ribbon quasi-Hopf algebras Q(N) for N a positive integer: as an algebra, Q(N) is the semidirect product of CZ2 with the direct sum of a Graßmann and a Clifford algebra in 2N generators. We show that RepQ(N) is ribbon equivalent to the symplectic fermion category SF(N) that was computed in [54] from conformal blocks of the corresponding logarithmic conformal field theory. The latter category in turn is conjecturally ribbon equivalent to representations of Vev, the even part of the symplectic fermion vertex operator super algebra.Using the formalism developed in [20] we compute the projective SL(2,Z)-action on the centre of Q(N) as obtained from Lyubashenko's general theory of mapping class group actions for factorisable finite ribbon categories. This allows us to test a conjectural non-semisimple version of the modular Verlinde formula: we verify that the SL(2,Z)-action computed from Q(N) agrees projectively with that on pseudo trace functions of Vev.

3-Dimensional TQFTs from non-semisimple modular categories

We use modified traces to renormalize Lyubashenko’s closed 3-manifold invariants coming from twist non-degenerate finite unimodular ribbon categories. Our construction produces new topological invariants which we upgrade to 2 + 1-TQFTs under the additional assumption of factorizability. The resulting functors provide monoidal extensions of Lyubashenko’s mapping class group representations, as discussed in De Renzi et al. (Commun Contemp Math, 2021. https://doi.org/10.1142/S0219199721500917). This general framework encompasses important examples of non-semisimple modular categories coming from the representation theory of quasi-Hopf algebras, which were left out of previous non-semisimple TQFT constructions.

The Kontsevich integral for bottom tangles in handlebodies

Using an extension of the Kontsevich integral to tangles in handlebodies similar to a construction given by Andersen, Mattes and Reshetikhin, we construct a functor Z\colon \mathcal{B}\to \hat{A} , where \mathcal{B} is the category of bottom tangles in handlebodies and \hat{A} is the degree-completion of the category \mathbf{A} of Jacobi diagrams in handlebodies. As a symmetric monoidal linear category, \mathbf{A} is the linear PROP governing “Casimir Hopf algebras”, which are cocommutative Hopf algebras equipped with a primitive invariant symmetric 2 -tensor. The functor Z induces a canonical isomorphism \operatorname{gr}\mathcal{B}\cong\mathbf{A} , where \operatorname{gr}\mathcal{B} is the associated graded of the Vassiliev–Goussarov filtration on \mathcal{B} . To each Drinfeld associator \varphi we associate a ribbon quasi-Hopf algebra H_\varphi in \hat{A} , and we prove that the braided Hopf algebra resulting from H_\varphi by “transmutation” is precisely the image by Z of a canonical Hopf algebra in the braided category \mathcal{B} . Finally, we explain how Z refines the LMO functor, which is a TQFT-like functor extending the Le–Murakami–Ohtsuki invariant.

Dynamical spin chains in 4D mathcal{N} = 2 SCFTs

This is the first in a series of papers devoted to the study of spin chains capturing the spectral problem of 4d mathcal{N} = 2 SCFTs in the planar limit. At one loop and in the quantum plane limit, we discover a quasi-Hopf symmetry algebra, defined by the R-matrix read off from the superpotential. This implies that when orbifolding the mathcal{N} = 4 symmetry algebra down to the mathcal{N} = 2 one and then marginaly deforming, the broken generators are not lost, but get upgraded to quantum generators. Importantly, we demonstrate that these chains are dynamical, in the sense that their Hamiltonian depends on a parameter which is dynamically determined along the chain. At one loop we map the holomorphic SU(3) scalar sector to a dynamical 15-vertex model, which corresponds to an RSOS model, whose adjacency graph can be read off from the gauge theory quiver/brane tiling. One scalar SU(2) sub-sector is described by an alternating nearest-neighbour Hamiltonian, while another choice of SU(2) sub-sector leads to a dynamical dilute Temperley-Lieb model. These sectors have a common vacuum state, around which the magnon dispersion relations are naturally uniformised by elliptic functions. Concretely, for the ℤ2 quiver theory we study these dynamical chains by solving the one- and two-magnon problems with the coordinate Bethe ansatz approach. We confirm our analytic results by numerical comparison with the explicit diagonalisation of the Hamiltonian for short closed chains.

Log-Modular Quantum Groups at Even Roots of Unity and the Quantum Frobenius I

We construct log-modular quantum groups at even order roots of unity, both as finite-dimensional ribbon quasi-Hopf algebras and as finite ribbon tensor categories, via a de-equivariantization procedure. The existence of such quantum groups had been predicted by certain conformal field theory considerations, but constructions had not appeared until recently. We show that our quantum groups can be identified with those of Creutzig-Gainutdinov-Runkel in type $$A_1$$ , and Gainutdinov-Lentner-Ohrmann in arbitrary Dynkin type. We discuss conjectural relations with vertex operator algebras at (1, p)-central charge. For example, we explain how one can (conjecturally) employ known linear equivalences between the triplet vertex algebra and quantum $$\mathfrak {sl}_2$$ , in conjunction with a natural $${{\,\mathrm{PSL}\,}}_2$$ -action on quantum $$\mathfrak {sl}_2$$ provided by our de-equivariantization construction, in order to deduce linear equivalences between “extended” quantum groups, the singlet vertex operator algebra, and the (1, p)-Virasoro logarithmic minimal model. We assume some restrictions on the order of our root of unity outside of type $$A_1$$ , which we intend to eliminate in a subsequent paper.

Yetter–Drinfeld category for the quasi-Turaev group coalgebra and cocycle deformation

Let [Formula: see text] be a group and [Formula: see text] a quasi-Turaev group coalgebra over [Formula: see text]. In this paper, we firstly construct the category [Formula: see text] of left-right Yetter–Drinfeld modules over [Formula: see text], generalizing both of the Yetter–Drinfeld modules over Hopf group coalgebra and quasi-Hopf algebra, and prove that this category is isomorphic to the center of the representation category of [Formula: see text]. Next, we prove that the full subcategory [Formula: see text] consisting of all finite dimensional Yetter–Drinfeld modules over [Formula: see text] is autonomous. Finally, when [Formula: see text] is reduced to a Turaev group coalgebra, we introduce the dual 2-cocycle [Formula: see text] of [Formula: see text], and dual cocycle deformation [Formula: see text], and show that [Formula: see text] is isomorphic to [Formula: see text] as braided monoidal category.

Computing modular data for pointed fusion categories

A formula for the modular data of $\mathcal{Z}(Vec^{\omega}G)$ was given by Coste, Gannon and Ruelle in arXiv:hep-th/0001158, but without an explicit proof for arbitrary 3-cocycles. This paper presents a derivation using the representation category of the quasi Hopf algebra $D^{\omega}G$. Further, we have written code to compute this modular data for many pairs of small finite groups and 3-cocycles. This code is optimised using Galois symmetries of the S and T matrices. We have posted a database of modular data for the Drinfeld center of every Morita equivalence class of pointed fusion categories of dimension less than 64.

Adjunctions Between Hom and Tensor as Endofunctors of (bi-)Module Category of Comodule Algebras Over a Quasi-Hopf Algebra

Adjunctions Between Hom and Tensor as Endofunctors of (bi-)Module Category of Comodule Algebras Over a Quasi-Hopf Algebra

Some ribbon elements for the quasi-Hopf algebra Dω(H)

We construct an explicit isomorphism between the quasitriangular quasi-Hopf algebra [Formula: see text] defined in [D. Bulacu and F. Panaite, A generalization of the quasi-Hopf algebra [Formula: see text], Commun. Algebra 26 (1998) 4125–4141] and a certain quantum double quasi-Hopf algebra. We give also new characterizations for a quasitriangular quasi-Hopf algebra to be ribbon and use them to construct some ribbon elements for [Formula: see text].

Categorical aspects of cointegrals on quasi-Hopf algebras

We discuss relations between some category-theoretical notions for a finite tensor category and cointegrals on a quasi-Hopf algebra. Specifically, for a finite-dimensional quasi-Hopf algebra H, we give an explicit description of categorical cointegrals of the category MH of left H-modules in terms of cointegrals on H. Provided that H is unimodular, we also express the Frobenius structure of the ‘adjoint algebra’ in the Yetter-Drinfeld category YHHD by using an integral in H and a cointegral on H. Finally, we give a description of the twisted module trace for projective H-modules in terms of cointegrals on H.

Boost generator in AdS3 integrable superstrings for general braiding

In this paper we find a host of boost operators for a very general choice of coproducts in AdS3-inspired scattering theories, focusing on the massless sector, with and without an added trigonometric deformation. We find that the boost coproducts are exact symmetries of the R-matrices we construct, besides fulfilling the relations of modified Poincaŕe-type superalgebras. In the process, we discover an ambiguity in determining the boost coproduct which allows us to derive differential constraints on our R-matrices. In one particular case of the trigonometric deformation, we find a non-coassociative structure which satisfies the axioms of a quasi-Hopf algebra.

Frobenius cowreaths and Morita contexts

Abstract We prove a uniqueness type theorem for (weak, total) integrals on a Frobenius cowreath in a monoidal category. When the cowreath is, moreover, pre-Galois, we construct a Morita context relating the subalgebra of coinvariants and a certain wreath algebra. Then we see that the strictness of the Morita context is related to the Galois property of the cowreath and the existence of a weak total integral on it. We apply our results to quasi-Hopf algebras.

Monadic cointegrals and applications to quasi-Hopf algebras

For $\mathcal{C}$ a finite tensor category we consider four versions of the central monad, $A_1, \dots, A_4$ on $\mathcal{C}$. Two of them are Hopf monads, and for $\mathcal{C}$ pivotal, so are the remaining two. In that case all $A_i$ are isomorphic as Hopf monads. We define a monadic cointegral for $A_i$ to be an $A_i$-module morphism $\mathbf{1} \to A_i(D)$, where $D$ is the distinguished invertible object of $\mathcal{C}$. We relate monadic cointegrals to the categorical cointegral introduced by Shimizu (2019), and, in case $\mathcal{C}$ is braided, to an integral for the braided Hopf algebra $\mathcal{L} = \int^X X^\vee \otimes X$ in $\mathcal{C}$ studied by Lyubashenko (1995). Our main motivation stems from the application to finite dimensional quasi-Hopf algebras $H$. For the category of finite-dimensional $H$-modules, we relate the four monadic cointegrals (two of which require $H$ to be pivotal) to four existing notions of cointegrals for quasi-Hopf algebras: the usual left/right cointegrals of Hausser and Nill (1994), as well as so-called $\gamma$-symmetrised cointegrals in the pivotal case, for $\gamma$ the modulus of $H$. For (not necessarily semisimple) modular tensor categories $\mathcal{C}$, Lyubashenko gave actions of surface mapping class groups on certain Hom-spaces of $\mathcal{C}$, in particular of $SL(2,\mathbb{Z})$ on $\mathcal{C}(\mathcal{L},\mathbf{1})$. In the case of a factorisable ribbon quasi-Hopf algebra, we give a simple expression for the action of $S$ and $T$ which uses the monadic cointegral.

Modified traces for quasi-Hopf algebras

Let H be a finite-dimensional unimodular pivotal quasi-Hopf algebra over a field k, and let H-mod be the pivotal tensor category of finite-dimensional H-modules. We give a bijection between left (resp. right) modified traces on the tensor ideal H-pmod of projective modules and left (resp. right) cointegrals for H. The non-zero left/right modified traces are non-degenerate, and we show that non-degenerate left/right modified traces can only exist for unimodular H. This generalises results of Beliakova, Blanchet, and Gainutdinov [1] from Hopf algebras to quasi-Hopf algebras. As an example we compute cointegrals and modified traces for the family of symplectic fermion quasi-Hopf algebras.

Frobenius-Perron Dimensions of Integral $\mathbb {Z}_{+}$-rings and Applications

We introduce the notion of the Frobenius-Perron dimension of an integral \(\mathbb {Z}_{+}\)-ring and give some applications of this notion to classification of finite dimensional quasi-Hopf algebras with a unique nontrivial simple module, and of quasi-Hopf and Hopf algebras of prime dimension p.