# Finite quasi-quantum groups of rank two

## Abstract

This is a contribution to the structure theory of finite pointed quasi-quantum groups. We classify all finite-dimensional connected graded pointed Majid algebras of rank two which are not twist equivalent to ordinary pointed Hopf algebras.

## 1. Introduction

The theory of finite tensor categories Reference 19 has aroused much interest in recent years. Among which, a proper classification theory is highly welcome and certainly is very challenging. As the general classification problem seems still far out of reach, it is necessary to narrow the scope and focus first on some interesting classes. In this respect, fusion and multi-fusion categories, that is, semisimple finite tensor and multi-tensor categories, are first investigated in depth, see Reference 18Reference 36 and references therein. To move on, Etingof and Gelaki proposed in their pioneering work Reference 15 to classify finite pointed tensor categories which are nonsemisimple. By pointed it is meant that the simple objects are invertible. There are multifold reasons for this restriction: firstly, this kind of reduction is standard and powerful in representation theory; secondly, this class of tensor categories are essentially concrete, i.e., they admit quasi-fiber functors and they can be realized as the module categories of finite-dimensional elementary quasi-Hopf algebras by the Tannakian formalism Reference 19; thirdly, this theory is a natural generalization of the deep and beautiful theory of elementary (or equivalently, finite-dimensional pointed) Hopf algebras, see Reference 1Reference 4Reference 5Reference 21Reference 24.

In Reference 15Reference 16, Etingof and Gelaki obtained a series of classification results about graded elementary quasi-Hopf algebras over cyclic groups of prime order; in Reference 17Reference 20, they studied graded elementary quasi-Hopf algebras over general cyclic groups and their liftings. One main achievement of this series of works is a complete classification of elementary quasi-Hopf algebras of rank 1. More importantly, a novel method of constructing genuine quasi-Hopf algebras from known pointed Hopf algebras is invented. Along the same vein, Angiono classified in Reference 6 finite-dimensional elementary quasi-Hopf algebras over cyclic groups whose orders have no small prime divisors. On the other hand, our previous works Reference 25Reference 26Reference 27 introduce many useful ideas and tools from the representation theory of finite-dimensional algebras into the theory of pointed tensor categories and quasi-quantum groups (including quasi-Hopf algebras and their duals in accordance with the philosophy of Drinfeld’s theory of quantum groups Reference 11Reference 12). In particular, a quiver framework is set up and a general method of constructing quasi-quantum groups and pointed tensor categories via projective representations of finite groups and quiver representation theory is provided.

However, except for some sporadic examples Reference 16, so far all finite quasi-quantum groups obtained in the literature are either over cyclic groups or of rank 1 (in fact, mostly both). An obvious reason, to the authors, is that a uniform expression of 3-cocycles over non-cyclic groups was not available. This prevents us from a desired control of the associators of quasi-quantum groups. With the explicit unified formulas of 3-cocycles on general finite abelian groups recently offered in Reference 28Reference 29, now it seems possible to pursuit the classification of finite quasi-quantum groups and finite pointed tensor categories in a much greater scope. As a crucial step to move forward, we should first tackle the classification of finite quasi-quantum groups of rank 2 as clearly suggested by the successful development strategy of the classification theory of finite-dimensional pointed Hopf algebras, see Reference 22Reference 23. This is the main aim of the present paper.

Let be a graded elementary quasi-Hopf algebra and the rank of is defined to be (since is free over it is an integer). The novel idea of ,Reference 15Reference 16Reference 20 is that, if is the group algebra of a cyclic group then can be embedded into a bigger quasi-Hopf algebra which is twist equivalent to a graded elementary Hopf algebra with where with The crux of this fact is essentially due to group cohomology. More precisely, if is a 3-cocycle on then its pull-back along the canonical projection vanishes, i.e., a 3-coboundary. In this situation, we say that is *resolvable*. In order to tackle the classification problem of rank 2 finite quasi-quantum groups, naturally we need to extend the aforementioned results on a cyclic group to the direct product of two cyclic groups. The first key observation of the present paper is that any 3-cocycle on the direct product of two cyclic groups is resolvable, as anticipated in Reference 15. This relies heavily on our previous work of linear braided Gr-categories Reference 28.

The resolvability of the 3-cocycles on motivates us to pursuit a similar connection between a graded quasi-Hopf algebra with and an appropriate Hopf algebra The second key observation of the present paper is that an explicit connection can be built by overcoming two difficulties, explained below, which are relatively mild in the case of cyclic groups Reference 6Reference 15Reference 16Reference 17Reference 20. In accordance with our previous works Reference 25Reference 26Reference 27, we always work on the dual situation. For this, let be a connected coradically graded pointed Majid algebra of rank 2 (see Definition 2.6 for the definition of rank) then is a group Majid algebra with by Lemma 2.9 (note that cyclic group is a special case). Similar to the case of pointed Hopf algebras, we may factorize as by a quasi-version of the bosonization procedure. Here is the coinvariant subalgebra of with respect to the natural coaction of The first difficulty is the generation problem, that is, whether is generated in degree 1. In fact, this problem is a generalization of the well-known Andruskiewitsch-Schneider conjecture for finite-dimensional pointed Hopf algebras. The second difficulty is to determine a suitable resolution such that becomes a Nichols algebra in the twisted Yetter-Drinfeld category Fortunately, we overcome these two difficulties well:

As is tensor equivalent to by Theorem A, Theorem B will facilitate applications to Majid algebras the theory of finite-dimensional pointed Hopf algebras. In other words, we get the following diagram:

In order to take full advantage of the theory of finite-dimensional pointed Hopf algebras, in particular Heckenberger’s well-known classification result of finite-dimensional rank Nichols algebras Reference 22Reference 23 for the present purpose, we still need to answer the following question: For which finite-dimensional pointed Hopf algebra of rank can one reverse the above diagram to get a pointed Majid algebra? Our third key observation is that this question can be reduced to solving some elementary congruence equations, see ,Equation 5.11. We show that such congruence equations have at most one solution and we give a simple criterion to determine when they do. Therefore, we complete the above one-way diagram into to a circuit in below.

This diagram finally leads to the desired classification of finite quasi-quantum groups of rank 2 (see Definition 5.12 for the definition of a Nichols algebra of Majid type):

Based on this, we can present the Majid algebra by explicit generators and relations with a help of Heckenberger’s classification of finite-dimensional rank Nichols algebras, see Proposition 6.5. Some more explicit classification results of typical type over the direct product of any two cyclic groups and a complete list over an arbitrary cyclic group can thus be obtained, see Theorems 7.2 and 8.3.

The paper is organized as follows. Some necessary concepts, notations and facts are collected in Section 2. In particular, Nichols algebras in twisted Yetter-Drinfeld categories are introduced. Section 3 is devoted to the resolvability of an arbitrary -cocycle on The result of this section is one of several key ingredients of our classification procedure. The generation problem for finite-dimensional pointed Majid algebras of rank 2 is established in Section 4. Our classification procedure and the main result are given in Section 5. Sections 6, 7, and 8 are designed to give explicit classification results based on the previous sections. There is also an appendix of the list of full binary trees used in Sections 6–8.

Throughout of this paper, is an algebraically closed field with characteristic zero and all vector spaces, linear mappings, (co)algebras and unadorned tensor product are over

## 2. Preliminaries

This section is devoted to some preliminary concepts, notations and facts.

### 2.1. Majid algebras

The concept of Majid algebras is dual to that of quasi-Hopf algebras Reference 12, and can be given as follows.

A Majid algebra is said to be *pointed*, if the underlying coalgebra is so. Given a pointed Majid algebra let be its coradical filtration, and

the corresponding coradically graded coalgebra. Then naturally inherits from a graded Majid algebra structure. The corresponding graded associator satisfies for all homogeneous unless they all lie in Similar condition holds for and In particular, is a Majid subalgebra and it turns out to be the Majid algebra for the set of group-like elements of We call a pointed Majid algebra *coradically graded* if as Majid algebras. One can also see Reference 25 for more details on pointed Majid algebras.

Now let be a Majid algebra together with a twisting Then one can construct a new Majid algebra . By definition, . as a coalgebra and the multiplication on is given by

for all The associator . and the quasi-antipode are given as:

for all .

### 2.2. Quiver setting for pointed Majid algebras and ranks

A quiver is a quadruple where is the set of vertices, is the set of arrows, and are two maps assigning respectively the source and the target for each arrow. A path of length in the quiver is a finitely ordered sequence of arrows such that for By convention a vertex is called a trivial path of length

For a quiver the associated path coalgebra is the spanned by the set of paths with counit and comultiplication maps defined by -space for each and for each nontrivial path

The length of paths provides a natural gradation to the path coalgebra. Let denote the set of paths of length in then and Clearly is pointed with the set of group-likes and has the following coradical filtration

Thus is coradically graded. The path coalgebras of quivers can be presented as cotensor coalgebras, so they are cofree in the category of pointed coalgebras and enjoy a universal mapping property.

A quiver is said to be a Hopf quiver if the corresponding path coalgebra admits a graded Hopf algebra structure. Hopf quivers can be determined by ramification data of groups. Let be a group and denote its set of conjugacy classes by A ramification datum of the group is a formal sum of conjugacy classes with coefficients in The corresponding Hopf quiver is defined as follows: the set of vertices is and for each and there are arrows going from to It is clear by definition that is connected if and only if the set generates the group For a given Hopf quiver the set of graded Hopf structures on is in one-to-one correspondence with the set of bimodule structures on -Hopf

It is shown in Reference 25, Theorem 3.1 that the path coalgebra admits a graded Majid algebra structure if and only if the quiver is a Hopf quiver. Moreover, for a given Hopf quiver if we fix a Majid algebra structure on with quasi-antipode then the set of graded Majid algebra structures on with is in one-to-one correspondence with the set of bimodule structures on -Majid Thanks to the Gabriel type theorem given in Reference 25, Theorem 3.4, for an arbitrary pointed Majid algebra its graded version can be realized uniquely as a large Majid subalgebra of some graded Majid algebra structure on a Hopf quiver. By “large” it is meant that the Majid subalgebra contains the set of vertices and arrows of the Hopf quiver. We denote this unique quiver by and call it the *Gabriel quiver* of Therefore, in principle all pointed Majid algebras are able to be constructed on Hopf quivers. .