Finite quasi-quantum groups of rank two

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.


Introduction
The theory of finite tensor categories [16] 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 [15,32] and references therein. To move on, Etingof and Gelaki proposed in their pioneering work [12] 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 [16]; thirdly, this theory is a natural generalization of the deep and beautiful theory of elementary (or equivalently, finite-dimensional pointed) Hopf algebras, see [1,4,5,18,21].
In [12,13], Etingof and Gelaki obtained a series of classification results about graded elementary quasi-Hopf algebras over cyclic groups of prime order; in [14,17], 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 [6] finite-dimensional elementary quasi-Hopf algebras over cyclic groups whose orders have no small prime divisors. On the other hand, our previous works [22,23,24] 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 [9,10]). 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.
Let H = n≥0 H n be a graded elementary quasi-Hopf algebra. The novel idea of [12,13,17] is that, if H 0 is the group algebra of a cyclic group G = Z Ò , then H can be embedded into a bigger quasi-Hopf algebra H which is twist equivalent to a graded elementary Hopf algebra H ′ with H ′ 0 = kG where G = Z n with n = Ò 2 .
The crux of this fact is essentially due to group cohomology. More precisely, if Φ is a 3-cocycle on G, then its pull-back π * (Φ) along the canonical projection π : G → G vanishes, i.e., a 3-coboundary. In this situation, we say that Φ is resolvable. The first key observation of the present paper is that this fact can be generalized to all abelian groups of form Z Ñ × Z Ò as anticipated in [12]. This relies heavily on our previous work of linear braided Gr-categories [25]. The resolvability of any 3-cocycle on Z Ñ × Z Ò motivates us to pursuit a similar connection between a graded quasi-Hopf algebra H with H 0 = kZ Ñ × Z Ò and an appropriate Hopf algebra H ′ . The second key observation of the present paper is that an explicit connection can be built by overcoming two difficulties, explained in below, which are relatively mild in the case of cyclic groups [12,13,14,17,6].
In accordance with our previous works [22,23,24], we always work on the dual situation. For this, let Å = i≥0 Å i be a coradically graded pointed Majid algebra, then Å 0 is a group Majid algebra (kG, Φ). Similar to the case of pointed Hopf algebras, we may factorize Å as Ê#kG by a quasi-version of the bosonization procedure. Here Ê is the coinvariant subalgebra of Å with respect to the natural coaction of kG. The first difficulty is the generation problem, that is, whether Ê is generated in degree 1. The second difficulty is to determine a suitable resolution π : G → G such that Ê becomes a Nichols algebra in the twisted π : G = Z m × Z n → Z Ñ × Z Ò such that π * (Φ) is a coboundary on G. 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, k is an algebraically closed field with characteristic zero and all vector spaces, linear mappings, (co)algebras and unadorned tensor product ⊗ are over k.

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 [10], and can be given as follows.

(2.6)
Throughout we use the Sweedler sigma notation ∆(a) = a 1 ⊗ a 2 for the coproduct and a 1 ⊗ a 2 ⊗ · · · ⊗ a n+1 for the result of the n-iterated application of ∆ on a.
Example 2.2. Let G be a group and Φ a normalized 3-cocycle on G. It is well known that the group algebra kG is a Hopf algebra with ∆(g) = g ⊗ g, S(g) = g −1 and ε(g) = 1 for any g ∈ G. By extending Φ trilinearly, then Φ : (kG) ⊗3 → k becomes a convolution-invertible map. Define two linear functions α, β : kG → k just by α(g) := ε(g) and β(g) := Now let H be a Majid algebra together with a twisting J. Then one can construct a new Majid algebra H J . By definition, H J = H as a coalgebra and the multiplication " • " on H J is given by for all a, b ∈ H. The associator Φ J and the quasi-antipode (S J , α J , β J ) are given as: for all a, b, c ∈ H.
where Q 0 is the set of vertices, Q 1 is the set of arrows, and s, t : Q 1 −→ Q 0 are two maps assigning respectively the source and the target for each arrow. A path of length l ≥ 1 in the quiver Q is a finitely ordered sequence of l arrows a l · · · a 1 such that s(a i+1 ) = t(a i ) for 1 ≤ i ≤ l − 1. By convention a vertex is called a trivial path of length 0.
The length of paths provides a natural gradation to the path coalgebra. Let Q n denote the set of paths of length n in Q, then kQ = n≥0 kQ n and ∆(kQ n ) ⊆ n=i+j kQ i ⊗ kQ j . Clearly kQ is pointed with the set of group-likes G(kQ) = Q 0 , and has the following coradical filtration Thus kQ 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 Q is said to be a Hopf quiver if the corresponding path coalgebra kQ admits a graded Hopf algebra structure. Hopf quivers can be determined by ramification data of groups. Let G be a group and denote its set of conjugacy classes by C. A ramification datum R of the group G is a formal sum C∈C R C C of conjugacy classes with coefficients in N = {0, 1, 2, · · · }. The corresponding Hopf quiver Q = Q(G, R) is defined as follows: the set of vertices Q 0 is G, and for each x ∈ G and c ∈ C, there are R C arrows going from x to cx. It is clear by definition that Q(G, R) is connected if and only if the the set {c ∈ C|C ∈ C with R C = 0} generates the group G. For a given Hopf quiver Q, the set of graded Hopf structures on kQ is in one-to-one correspondence with the set of kQ 0 -Hopf bimodule structures on kQ 1 .
It is shown in [22] that the path coalgebra kQ admits a graded Majid algebra structure if and only if the quiver Q is a Hopf quiver. Moreover, for a given Hopf quiver Q = Q(G, R), if we fix a Majid algebra structure on kQ 0 = (kG, Φ) with quasi-antipode (S, α, β), then the set of graded Majid algebra structures on kQ with kQ 0 = (kG, Φ, S, α, β) is in one-to-one correspondence with the set of (kG, Φ)-Majid bimodule structures on kQ 1 . Thanks to the Gabriel type theorem given in [22], for an arbitrary pointed Majid algebra H, its graded version gr H 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 Q(H) and call it the Gabriel quiver of H. Therefore, in principle all pointed Majid algebras are able to be constructed on Hopf quivers. Definition 2.5. Let H be a pointed Majid algebra, Q(H) be its Gabriel quiver and R = C∈C R C C be the ramification datum of Q(H). The rank of H is defined to be the natural number C∈C R C |C| where |C| is the cardinality of C. We say that H is connected if Q(H) is connected as a graph.
Example 2.6. Pointed Majid algebras of rank one were studied in [24]. We recall them here in detail as they shed much light on the working philosophy of the present paper.
Consider the following Hopf quiver Q(Z n , g): Now let 0 ≤ s ≤ n − 1 be a natural number, q an n 2 -th primitive root of unity and Õ := q n . Let p l i denote the path in Q(Z n , g) starting from g i with length l. So p 0 i = g i . Let Φ s be the 3-cocycle on Z n defined by Here [x] stands for the integral part of x. For any h ∈ k, define l h = 1 + h + · · · + h l−1 and l! h = 1 h · · · l h . The Gaussian binomial coefficient is defined by l+m ) denote the greatest common divisor of the two natural numbers a, b. Now we are ready to define the rank 1 pointed Majid algebra M (n, s, q). As a coalgebra, The associator, the multiplication, the functions α, β and the antipode are given as follows: for 0 ≤ l, m, t < n 2 (n 2 ,s) and 0 ≤ i, j, k ≤ n − 1, where δ a,b is the Kronecker delta, namely it is equal to 1 if a = b and 0 otherwise, and l ′ means the remainder of l divided by n.
We have the following basic observation. Lemma 2.7. Let H be a connected pointed Majid algebra of rank 2 and Q(H) = Q(G, R) be its Gabriel quiver, then G is an abelian group which can be generated by one or two elements.
Proof. Let 1 denote the unit element of G. By definition, we know that in Q(H) there are exactly two arrows going out from 1. We denote the ending vertices of these two arrows by g and h respectively. As the graph is connected, it follows that G can be generated by g and h. If g and h live in different conjugacy classes, then the definition of Hopf quivers implies that the conjugacy class containing g (resp. h) is just g (resp. h). So both g and h lie in the center of G and thus G is abelian. If g, h live in the same conjugacy class, then again by the definition of Hopf quivers we have This implies that g = h or gh = hg. In either case G is abelian.
2.3. Yetter-Drinfeld modules over (kG, Φ). The definition of Yetter-Drinfeld modules over a general Majid algebra seems cumbersome. However, its formulation becomes much simpler when this Majid algebra is (kG, Φ) with G an abelian group. This special case already suffices for our purpose.
Assume that V is a left kG-comudule with comodule structure map δ L : V → kG ⊗ V . Define g V := {v ∈ V |δ L (v) = g ⊗ v} and thus V = g∈G g V. For the 3-cocycle Φ on G and any g ∈ G, define Direct computation shows that Φ g ∈ Z 2 (G, k * ).
projective G-representation with respect to the 2-cocycle Φ g , namely the G-action ⊲ on g V satisfies The category of all left-left Yetter-Drinfeld modules is denoted by G G YD Φ . Similarly, one can define leftright, right-left and right-right Yetter-Drinfeld modules over (kG, Φ). As the familiar Hopf case, G G YD Φ is a braided tensor category. More precisely, for any M, N ∈ G G YD Φ , the structure maps of M ⊗ N as a left-left Yetter-Drinfeld module are given by for all x, g, h ∈ G and m g ∈ g M , n h ∈ h N . The associativity constraint a and the braiding c of G G YD Φ are given respectively by for all e, f, g ∈ G, u e ∈ e U , v f ∈ f V , w g ∈ g W and U, V, W ∈ G G YD Φ . Remark 2.9. A left-left Yetter-Drinfeld module V over (kG, Φ) is called diagonal if every projective Grepresentation g V is a direct sum of 1-dimensional projective representations. We point out that not like the Hopf case, here the condition of G being abelian can NOT guarantee that every V is diagonal. It turns out that all V ∈ G G YD Φ are diagonal if and only if Φ is an abelian cocycle, see [11] and [30]. We will show in Section 3 that all 3-cocycles on Z m × Z n are abelian.

2.4.
Hopf algebras in braided tensor categories. Let C = (C, ⊗, 1, a, l, r, c) be a braided tensor category, where 1 is the unit object, a (l, or r) is the associativity (left, or right unit) constraint and c is the braiding. An associative algebra in C is an object A of C endowed with a multiplication morphism m : A ⊗ A −→ A and a unit morphism u : If (A, m A , u A ) and (B, m B , u B ) are two algebras in C, then one can define a natural morphism m A⊗B : is again an algebra in C. The resulting algebra is called the braided tensor product of A and B.
Dually, a coassociative coalgebra in C is an object C of C endowed with a comultiplication morphism ∆ : C −→ C ⊗ C and a counit morphism ε : C −→ 1 such that and (D, ∆ D , ε D ) are coalgebras in C, then one can define a suitable morphism ∆ C⊗D : C⊗D −→ (C ⊗ D) ⊗ (C ⊗ D) to get the braided tensor product coalgebra (C ⊗ D, ∆ C⊗D , ε C ⊗ ε D ) in C.
Endowed with braided tensor products, one may naturally define Hopf algebras in braided tensor categories. A sextuplet (H, m, u, ∆, ε, S) is a Hopf algebra in C if (H, m, u) is an algebra in C, (H, ∆, ε) is a coalgebra in C and ∆ : H −→ H ⊗ H and ε : H −→ 1 are algebra maps in C, and S : H −→ H is a morphism, to be called the antipode, subject to As the usual case, we can define ideals of algebras, coideals of coalgebras, Hopf ideals of Hopf algebras, and the corresponding quotient structures in braided tensor categories. We do not include further details. For our purpose, we only record some more facts on the duals of finite-dimensional Hopf algebras in the braided tensor category G G YD Φ . Let H be such a Hopf algebra. Then its left dual * H and right dual H * are again Hopf algebras in G G YD Φ in a natural manner, and in addition, * (H * ) ∼ = ( * H) * ∼ = H. For more details on algebras and Hopf algebras in braided tensor categories, the reader is referred to [29].

Nichols algebras in
In simple terms, Nichols algebras are the analogue of the usual symmetric algebras in more general braided tensor categories. They can be defined by various equivalent ways, see [2,4].
Here we adopt the defining method in terms of the universal property.
This induces a natural N-graded structure on T Φ (V ). Define a comultiplication on T Φ (V ) by ∆(X) = X ⊗ 1 + 1 ⊗ X, ∀X ∈ V, a counit by ε(X) = 0, and an antipode by S(X) = −X. It is routine to verify that these provide an N-graded Hopf algebra structure on T Φ (V ) in the braided tensor category G G YD Φ . Definition 2.10. The Nichols algebra B(V ) of V is defined to be the quotient Hopf algebra where I is the unique maximal graded Hopf ideal generated by homogeneous elements of degree greater than or equal to 2.
Then clearly the group Majid algebras (kG, Φ) and (kG, Φ∂(J)) are twist equivalent with twisting offered by extending J bilinearly. It is well known that their associated Yetter-Drinfeld categories G G YD Φ and G G YD Φ∂(J) are tensor equivalent [8]. More precisely, the tensor functor (F J , ϕ 0 , ϕ 2 ) : where U J as a G-comodule is the same as U, but with a new G-action obtained by twisting that on U via J as follows (2.19) g ⊲ J X = J(g, x) J(x, g) g ⊲ X, ∀g ∈ G, X ∈ x U .
Naturally, this tensor equivalence maps algebras in G G YD Φ to algebras in G G YD Φ∂(J) . In particular, the Nichols algebra B(V ) is mapped to B(V ) J which is again a Nichols algebra in G G YD Φ∂(J) . Note that the multiplication of the latter, denoted by "•", is given by In addition, we have the following obvious but useful observation: 2.6. Bosonization for pointed Majid algebras. The theory of bosonization in a broad context can be found in [29] in terms of braided diagrams. For our purpose, it is enough to focus on the situation of graded pointed Majid algebras. For the sake of completeness and later applications, we record in the following some explicit concepts, notations and results without proof.
In the rest of the paper, we always assume that is a connected coradically graded pointed Majid algebra with unit 1. So Å 0 = (kG, Φ) for some group G together with a 3-cocycle Φ on G. Let π : Å → Å 0 be the canonical projection. Then Å is a kG-bicomodule naturally through Thus there is a G-bigrading on Å, that is, We only deal with homogeneous elements with respect to this G-bigrading unless stated otherwise. For example, whenever we write ∆(X) = X 1 ⊗ X 2 , all X, X 1 , X 2 are assumed homogeneous. For the convenience of the exposition, we make a convention: given Define the subalgebra of Å consisting of coinvariants as Clearly 1 ∈ Ê and Ê inherits from Å a left G-coaction, i.e., Ê = g∈G g Ê. There is also a (kG, Φ)-action on Ê given by Moreover, there are several natural operations on Ê inherited from Å as follows: Then it is routine to verify that (Ê, M, u, As before, we only need to deal with G-homogeneous elements. As a convention, homogeneous elements in H are denoted by capital letters, say X, Y, Z, . . . , and the associated degrees are denoted by their lower cases, say x, y, z, . . . .
For our purpose, we also assume that H is N-graded with H 0 = k. If X ∈ H n , then we say that X has length n. Moreover, we assume that both gradings are compatible in the sense that For example, the Hopf algebra Ê in G G YD Φ considered above satisfies these assumptions as Ê = i∈N Ê i is coradically graded. For any X ∈ H, we write its comultiplication as Proposition 2.12. Keep the assumptions on H as above. Define on H ⊗ kG a product by and a coproduct by Then H ⊗ kG becomes a graded Majid algebra with a quasi-antipode (S, α, β) given by here g, h ∈ G and X, Y are homogeneous elements of length ≥ 1.
In the following, by H#kG we denote the resulting Majid algebra defined on H ⊗ kG.
Proposition 2.13. Let Å and Ê be as before, and Ê#kG be the Majid algebra as defined in the previous proposition. Then the map is an isomorphism of Majid algebras.

2.7.
Generators of abelian groups. For applications in Section 5, we also need to recall some elementary results given in [27] about generators of abelian groups. Given two generators g, h of m × n = g 1 , g 2 |g m 1 = g n 2 = 1, g 1 g 2 = g 2 g 1 with m|n, we know that there are integers a, b, c, d such that g = g a 1 g b 2 , h = g c 1 g d 2 and g, h generate m × n . The question is that can we simplify the expression of g, h? That is, up to an automorphism of m × n , deduce the integers a, b, c, d as simple as possible. To this end, we call two generators The following two lemmas, which are [27,Corollary 4.3] and [27, Proposition 4.1] respectively, answer the above question.
Lemma 2.14. Let g, h be two generators of m × n with m|n. Assume the order of h is n, then there are standard generators g 1 , g 2 of m × n such that Lemma 2.15. Assume that g and h generate the abelian group m × n with m|n, then there are integers m 1 , m 2 , n 1 , n 2 , a, b such that (i) m = m 1 n 1 , n = m 2 n 2 , m 1 |m 2 , n 1 |n 2 , (m 2 , n 2 ) = 1;

3-cocycles on Z Ñ × Z Ò and their resolutions
The aim of this section is to show that every 3-cocycle Φ on Z Ñ × Z Ò is abelian in the sense of [11] and can be "resolved" in a bigger abelian group G, namely there exists a group epimorphism π : G → Z Ñ × Z Ò such that the pull-back π * (Φ) is a coboundary on G.
3.1. Abelian cocycles. The original definition of abelian cocycles was given in [11]. For our purpose, we prefer the following equivalent definition via twisted quantum doubles appeared in [30]. So first we need to recall the definition of twisted quantum doubles [8]. The twisted quantum double D ω (G) of G with respect to the 3-cocycle ω on G is the semisimple quasi-Hopf algebra with underlying vector space (kG) * ⊗ kG in which multiplication, comultiplication ∆, associator φ, counit ε, antipode S, α and β are given by where {e(g)|g ∈ G} is the dual basis of {g|g ∈ G}, δ g,1 is the Kronecker delta, g x = x −1 gx, and for any x, y, g ∈ G.
It is well known that M is a left D ω (G)-module if and only if M is a left-left Yetter-Drinfeld module over (kG, ω) as defined in Subsection 2.3.
With this, it is not hard to find that

3.3.
Resolutions. One of our key observations is that any 3-cocycle Φ on G can be "resolved" in a suitable bigger abelian group G. More precisely, we may take G = Z m × Z n = g 1 × g 2 with m = Ñ 2 , n = Ò 2 and the canonical epimorphism π : G → , g 1 → 1 , g 2 → 2 . By pulling back the 3-cocycles on G along π one gets 3-cocycles on G. Therefore, for any a, b, d, the map becomes a 3-cocycle on G. The observation is that π * (Φ a,b,d ) is indeed a coboundary. In fact, consider the following map where y ′ 1 is the remainder of y 1 divided by Ñ (resp. y ′′ 2 is the remainder of y 2 divided by Ò). Here we require that ζ Ñ m = ζ Ò ÑÒ = ζ Ñ and ζ Ò n = ζ Ñ ÑÒ = ζ Ò . Of course, this requirement can be easily satisfied. For example, for t ∈ N. Thus, we have Proof. Indeed, by direct computation

Generation in degree one
Throughout this section, Å is a finite-dimensional connected coradically graded pointed Majid algebra of rank 2. The aim of this section is to prove that Å is generated by Å 0 and Å 1 . Recall that, as before, and we can assume that Thanks to Proposition 2.13, we have Å = Ê#kG.
Note that Ê = i∈N Ê i is also coradically graded. The main result of this section can be stated as follows: The proof of this proposition is divided into three steps and each of them appears as a subsection.
4.1. The l -grading of B(V ). In this subsection, G stands for an arbitrary finite abelian group and Φ a 3-cocycle on G. Let V be a diagonal Yetter-Drinfeld module in G G YD Φ with dimension l. Let V = l i=1 kX i be a decomposition of V into the direct sum of 1-dimensional Yetter-Drinfeld modules. Let Z l be the free abelian group of rank l and e i (1 ≤ i ≤ l) be the canonical generators of Z l .
Proof. Obviously, there is a Z l -grading on the tensor algebra T Φ (V ) ∈ G G YD Φ by assigning deg X i = e i . Let I = ⊕ i≥i0 I i be the maximal graded Hopf ideal generated by homogeneous elements of degree greater than or equal to 2. To prove that B(V ) is Z l -graded, it amounts to prove that I is Z l -graded. This will be done by induction on the N-degree.
First let X ∈ I be a homogenous element with minimal degree i 0 . Since ∆(X) ∈ T (V ) ⊗ I + I ⊗ T (V ), X must be a primitive element, i.e., ∆(X) = X ⊗ 1 + 1 ⊗ X. Suppose X = X 1 + X 2 + · · · + X n , where X i is Z l -homogenous, and X i and So, each X i is a primitive element and hence must be contained in I by the maximality of I. Therefore, I i0 is Z l -graded.
Then suppose that I k := ⊕ i0≤i≤k I i is Z l -graded. We shall prove that I k+1 = ⊕ i0≤i≤k+1 I i is also Z l -graded. Let X ∈ I k+1 and X = X 1 +X 2 +· · ·+X n , with each X i being Z l -homogenous and X i and X j having different If there was an X i / ∈ I k+1 , then I + X i is a Hopf ideal properly containing I, which contradicts to the maximality of I. It follows that X i ∈ I k+1 for all 1 ≤ i ≤ n and hence I k+1 is also Z l -graded by the assumption on X. We complete the proof of the lemma. Now return to our Majid algebra Å. Since it is assumed of rank 2, dim Ê 1 = 2. By Corollary 3.4, Ê 1 is a diagonal Yetter-Drinfeld module over (kG, Φ). Therefore, we may write as the direct sum of two 1-dimensional Yetter-Drinfeld modules. As in Section 3, consider a bigger abelian group G = Z m × Z n = g 1 × g 2 with m = Ñ 2 , n = Ò 2 and the canonical epimorphism: Observe that π has a section ι : k → kG, is not a group morphism. Let δ L and ⊲ be the comodule and module structure maps of Ê ∈ YD Φ . Define for all g ∈ G and Z ∈ Ê 1 . Through this way, Ê 1 ∈ G G YD π * (Φ) and this can be verified by direct computation: for all e, f ∈ G and δ L (Z) = z ⊗ Z for Z ∈ Ê 1 . We denote this new Yetter-Drinfeld module by Ê 1 in order to distinguish from the original one Ê 1 ∈ YD Φ . From these, we have two essentially identical Nichols algebras B(Ê 1 ) ∈ YD Φ and B( Ê 1 ) ∈ G G YD π * (Φ) which however live in different environment. Proof. Let F : It is easy to show that F also preserves the comultiplication of T Φ (Ê 1 ) and T π * (Φ) ( Ê 1 ). Note that F induces a one to one correspondence between the set of Z 2 -graded Hopf ideals of T Φ (Ê 1 ) and that of T π * (Φ) ( Ê 1 ).

4.2.
Twisted version of ordinary Nichols algebras. Again, first suppose G is an arbitrary finite abelian group. Let V be a diagonal Yetter-Drinfeld module in G G YD and let V = N i=1 kX i be a decomposition of V into 1-dimensional Yetter-Drinfeld modules. We use δ L and ⊲ to denote the comodule and module structure maps of V . Then there are g i ∈ G and q ij ∈ k * such that δ L (X i ) = g i ⊗ X i and g i ⊲ X j = q ij X j . Let J be a 2-cochain on G and let Φ denote its differential ∂(J). Recall that B(V ) J is defined in Subsection 2.5 and we have B(V ) J ∼ = B(V J ) as Nichols algebras in G G YD ∂(J) . Now assume that B(V ) ∈ G G YD is finite-dimensional, then B(V ) = T (V )/I where I is the Hopf ideal of T (V ) generated by the polynomials listed in [7, Theorem 3.1]. In the following, let S denote the set of these polynomials. Preserve the notations of Subsection 2.5. Define a map Ψ : It is easy to see that Ψ is an isomorphism of linear spaces. Proof. We claim that for any homogeneous E, F ∈ T Φ (V J ). We prove this by induction on the length of F.
If the length of F is 1, i.e., F ∈ {X 1 , X 2 , . . . , X N }, then (4.2) is just a special case of (4.1). Next assume (4.2) holds for all F of length < n. If F is of length n, then we can write F = F ′ X i for some 1 ≤ i ≤ N. And we have Proof. It suffices to prove that (V ⊕ kZ) J ∼ = V J ⊕ kΨ −1 (Z) as objects in G G YD Φ . But this is clear.
The following lemma is a summary of the important results in [7,Section 4], which is also crucial for the present paper.  [7,Theorem 4.13] we know that the subalgebra of B(U ) generated by Z is infinite-dimensional. Therefore B(U ) is also infinite-dimensional.
In the rest of this subsection, we return to the case of Å, i.e., = Z Ñ × Z Ò and Φ = Φ a,b,d , J = J a,b,d .
The following result is a generalization of [7,Theorem 4.13] to pointed Majid algebras.
Proposition 4.8. Let R = ⊕ i≥0 R i be a finite-dimensional graded (not necessarily coradically graded) Hopf algebra in YD Φ such that R 0 = k1 and dim k R 1 = 2. If R is generated by R 1 , then R = B(R 1 ).
Proof. Let I be an ideal of T Φ (R 1 ) such that R = T Φ (R 1 )/I. Clearly, we have a surjective Hopf map YD is a usual Nichols algebra. Now assume that θ is not an isomorphism, then there should be some polynomials in Ψ −1 (S), which are not contained in I by Lemma 4.5. Suppose that Ψ −1 (Z) is one of those with minimal length. From the proof of Lemma 4.2, we know that Ψ −1 (Z) must be a primitive element in R. Let U = R 1 ⊕ kΨ −1 (Z), then by the preceding assumption there is an embedding of linear spaces B(U ) → R.
We already know that B(R 1 ) J −1 is a finite-dimensional Nichols algebra in G G YD. By Lemma 2.11, there Hence B(U ) is infinite-dimensional, which contradicts to the assumption that R is finite-dimensional. Thus θ is an isomorphism and R is the Nichols algebra B(R 1 ). Lemma 4.9. Let R = ⊕ i≥0 R i be a graded Hopf algebra in YD Φ with R 0 = k1 and P (R) = R 1 . Then

Dual
For f 1 , f 2 , · · · , f s ∈ R * 1 and X ∈ R s , we have where π − → s = (· · · (π ⊗ π) · · · π s ) and π : R → R 1 is the canonical projection. Hence as linear maps, m Therefore, to prove the claim it is enough to show that P (R) = R 1 if and only if Γ s is injective for all s ≥ 2. On the one hand, assume that the Γ s are injective. If X ∈ R s is a primitive element for some s ≥ 2, then Γ s (X) = 0 by definition, and hence X = 0. So it follows that P (R) = R 1 .
On the other hand, assume that P (R) = R 1 . We will use induction to prove that each Γ s is injective for all s ≥ 2. If s = 2, for an X ∈ R 2 we have ∆(X) = X ⊗ 1 which implies X 1 ⊗ X 2 = 0 since Γ l are injective for l < s + 1 by induction. Since l is arbitrary, we have ∆(X) ∈ R s+1 ⊗ R 0 ⊕ R 0 ⊗ R s+1 . Hence X must be primitive, which implies X = 0.

Corollary 4.10.
Let H be a finite-dimensional pointed Majid algebras of rank 2, then H is generated by group-like and skew-primitive elements.

Proof. It is clear that H is generated by group-like and skew-primitive elements if and only if gr(H) is. So we
may assume that H is coradically graded. Let H 0 = k , then the coinvariant subalgebra Ê is a graded Hopf algebra in YD Φ . By Proposition 4.1, Ê is generated by primitives. This clearly leads to the claim.

Classification procedure and the main result
In this section, a theoretical procedure to classify graded connected pointed Majid algebras of rank 2 is provided. The procedure is applied to get our main classification result, that is, Theorem 5.13.

General setup.
In this section, we always assume that Å is a finite-dimensional connected coradically graded pointed Majid algebra of rank 2. Keep the notations of Section 4. Recall that, Å 0 = (kG, Φ) where G = Z Ñ × Z Ò = 1 × 2 with Ñ|Ò and Φ = Φ a,b,d for some 0 ≤ a, b ≤ Ñ − 1, 0 ≤ c ≤ Ò − 1. By the bosonization procedure, we have Å = Ê#kG and dim Ê 1 = 2 since Å is assumed of rank 2. By Corollary 3.4, Ê 1 is a diagonal Yetter-Drinfeld module over (kG, Φ) . Therefore, we may write as a direct sum of two 1-dimensional Yetter-Drinfeld modules. As in Subsection 3.3, we consider the bigger abelian group G = Z m × Z n = g 1 × g 2 with m = Ñ 2 , n = Ò 2 and the canonical epimorphism with a typical section ι : k → kG, 5.2. From Majid algebras to Hopf algebras. It was shown in Subsection 2.6 that the coinvariant subalgebra Ê of Å is a Hopf algebra in YD Φ . Moreover, Ê = B(Ê 1 ) ∈ YD Φ by Proposition 4.1. Thanks to Lemma 4.3, Ê is also a Hopf algebra in G G YD π * (Φ) by extending the following Yetter-Drinfeld module structure on Ê 1 to Ê: for all g ∈ G and X ∈ Ê 1 .
The following observation is contained implicitly in the proof of Proposition 4.1. As it is the crux of our classification procedure, we include an explicit proof here. Proof. Note that the associator of M is provided by π * (Φ a,b,d ) for some 0 ≤ a, b < Ñ, 0 ≤ d < Ò. Then by Proposition 3.5, there exists a 2-cochain J on G such that π * (Φ a,b,d ) = ∂J. Clearly, M J −1 is an ordinary Hopf algebra, see Subsection 2.1. Finally, the connectedness of M implies that of M J −1 .
The following example, though of rank 1, provides an explanation of the previous proposition.

Now Ê becomes a Hopf algebra in Zn
Zn YD π * (Φ) and we get the Majid algebra Ê#kZ n by the bosonization. Note that the associator of Ê#kZ n is π * (Φ), which is exactly the differential of the following 2-cochain J : kZ n ⊗ kZ n → k, (g i , g j ) → q i(j−j ′ ) .
Therefore, the associator of (Ê#kZ n ) J −1 is trivial by Subsection 2.1, and thus (Ê#kZ n ) J −1 is an ordinary Hopf algebra. Remark 5.3. Let Ê be a Hopf algebra in G G YD Φ . Note that the bosonization Ê#kG is an ordinary Hopf algebra if and only if each g Ê is a linear representation of G. As a matter of fact, in the above example the Z n -action on Ê in (Ê#kZ n ) J −1 is given by From this, we always have That is, we turn the projective representation into the usual linear representation. That is, (Ê#kZ n ) J −1 is a usual Hopf algebra, which in fact is the familiar Taft algebra T n .
So far, we have achieved the following one-way road map: This offers us a possible chance to take advantage of the successful theory of finite-dimensional pointed Hopf algebras to Majid algebras. Of course, our next task is to get a "return ticket" from ordinary pointed Hopf algebras to genuine pointed Majid algebras.

5.3.
From Hopf algebras to Majid algebras. Although Ê is a "twisted" version of a Nichols algebra, it is completely not clear yet when a "twisted" version of a Nichols algebra is indeed the coinvariant subalgebra of a genuine Majid algebra. The aim of this subsection is to find an answer to this question.
According to Figure I, take B = B(V ) a finite-dimensional diagonal Nichols algebra of rank 2 as classified in [19]. Fix a decomposition V = kX 1 ⊕ kX 2 as a direct sum of 1-dimensional Yetter-Drinfeld modules. Find two square integers m = Ñ 2 , n = Ò 2 with m | n and the abelian group G = Z m × Z n = g 1 × g 2 such that B is a Nichols algebra in G G YD. Take J = J a,b,d as given in (3.2) and consider where (B J , ρ L , ◮) is regarded as a Nichols algebra in G G YD ∂(J) . As before, let Let Φ = Φ a,b,d as given in (3.1) which is a 3-cocycle on . Keep the notations of Subsection 5.2. What we need to do is to reverse the first step of the procedure in Subsection 5.2, that is, find If this is the case, we call (B J , ρ L , ◮) the induced Nichols algebra of (B J , δ L , ⊲).
It is not hard to find that Claim: B J is a Nichols algebra in YD Φ such that its induced Nichols algebra is B J in G G YD ∂(J) if and only if I is a Majid ideal.
Proof of the claim: "⇐" Assume that I is a Majid ideal, so H J /I = B J ⊗ k is a Majid algebra. This implies B J is a Nichols algebra in YD Φ . Simple computation shows that the induced Nichols algebra of this B J is exactly (B J , ρ L , ◮).
"⇒" By assumption, B J #kG is a Majid algebra and we have a canonical epimorphism It is easy to see that I ⊆ Ker(F ). By Clearly, if both g Ñ 1 and g Ò 2 lie in the center then I is an ideal. Conversely, assume that I is an ideal. Then • denotes the multiplication of H J . Since we always have Therefore, So (α − 1)X 1 ∈ I, this forces that α = 1, that is, g Ñ 1 commutes with X 1 . Similarly, g Ñ 1 commutes with X 2 .
Thus g Ñ 1 lies in the center. By the same method one can show that g Ò 2 also belongs to the center.
Example 5.5. Let T n = T Ò 2 be the Taft algebra of dimension Ò 4 . By definition, T n is generated by two elements x, g subject to Its comultiplication is given as Consider the right ideal I generated by g Ò − 1. That is, It is not hard to see that I is a coideal but is NOT an ideal. Define a twisting J in the following way Consider the twisted Majid algebra T J n and we denote the new product by •. In this algebra, we have n . This implies the right ideal I generated by g Ò − 1 is a Majid ideal in T J n , hence the quotient T J n /I is a Majid algebra. It is not hard to verify that T J n /I ∼ = M (n, 1, q), where the latter was considered in Example 5.2. Using the same method, the readers can realize all M (n, s, q) constructed in Example 2.6 as the quotients of twisted Taft algebras. Now we give a criteria to determine when both g Ñ 1 and g Ò 2 lie in the center of B J #kG. At first, we set several parameters. Assume that Since we always assume that h 1 , h 2 generate the group G, there are s i , t i ∈ N such that With these preparations, we have Proof. This is a consequence of direct computations. Indeed,   Keep the notations of Subsection 5.3. Call B(V ) a connected rank 2 Nichols algebra in G G YD if the ordinary pointed Hopf algebra B(V )#kG is connected. As before, let V = kX 1 ⊕ kX 2 be the direct sum of 1-dimensional Yetter-Drinfeld modules. Therefore, there exist h i ∈ G and q ij ∈ k * such that where σ L and ⋄ are structure maps of V ∈ G G YD. Assume that h 1 = g α1 1 g α2 2 , h 2 = g β1 1 g β2 2 for some α i , β i ∈ N. As h 1 and h 2 generate G (due to the connectedness of B(V )#kG), there exist 0 ≤ s 1 , t 1 < m, 0 ≤ s 2 , t 2 < n such that g 1 = h s1 1 h s2 2 , g 2 = h t1 1 h t2 2 . Therefore, we have Note that which clearly are equivalent to following congruence equations Using (5.7) again, by multiplying the first (resp. second) congruence equation by s 1 (resp. s 2 ) and take the sum of them, we have here as before, x ′ is the remainder of x divided by Ñ (resp. x ′′ is the remainder of x divided by Ò).
In summary, we have The last problem is to determine when equations (5.6) do have a solution. This is settled as follows.
Proposition 5.10. Keep the above notations. Equations (5.6) have a solution if and only if (5.14) Proof. "⇐" It is enough to show that (5.13) is exactly a solution of (5.11). Consider the last two equations in (5.11), we have Therefore, the last two equations of (5.11) are satisfied. Next we consider the first two equations of (5.11).
To show this, we need the following claim.
Claim: We always have Proof of this claim: Note that the reason for equation (5.7) fulfilled is that if g 1 = g s 1 g t 2 , g 2 = g x 1 g y 2 then s ≡ 1 (mod m), t ≡ 0 (mod n), x ≡ 0 (mod m), y ≡ 1 (mod n). So, to show (5.15) it is enough to show that will imply that s ≡ 1 (mod m), t ≡ 0 (mod m), x ≡ 0 (mod m), y ≡ 1 (mod m). We only consider the case h 1 = h s 1 h t 2 since the other one is similar. Clearly, it is equivalent to showing that if h s 1 h t 2 = 1 then m|s and m|t. We use Lemma 2.15 to prove this. Here we use notations of Lemma 2.15 freely. In order to not cause confusion, we point out that our h 1 (resp. h 2 ) is just the element g (resp. h) of Lemma 2.15. Therefore by Lemma 2.15, g s h t = 1 implies that m 1 |t, m 2 |(s + tb), n 1 |s, n 2 |(sa + t).
We return to the proof of this proposition. Now, we have where in the third "≡" we use equation (5.15). The second equation of (5.11) can be verified similarly: ≡ 0 (mod Ò).
"⇒" By the proof of the sufficiency, we know that Therefore,   The main aim of this section is to provide an explicit presentation of Å(V, G), or equivalently, of B(V ) J a,b,d .
In principle, the idea is simple: first we recall the classification results about rank 2 Nichols algebras given by Heckenberger [19,20], then we carry out the twisting process.
6.1. Generalized Dynkin diagrams and full binary trees. Let E = {e i |1 ≤ i ≤ d} be a canonical basis of Z d , and χ be a bicharacter of Z d . The numbers q ij = χ(e i , e j ) are called the structure constants of χ with respect to E.
Definition 6.1. The generalized Dynkin diagram of the pair (χ, E) is a nondirected graph D χ,E with the following properties: (1) There is a bijective map φ from I = {1, 2, . . . , d} to the set of vertices of D χ,E .
(2) For all 1 ≤ i ≤ d, the vertex φ(i) is labelled by q ii .
(3) For all 1 ≤ i, j ≤ d, the number n ij of edges between φ(i) and φ(j) is either 0 or 1. If i = j or q ij q ji = 1 then n ij = 0, otherwise n ij = 1 and the edge is labelled by q ij q ji .
Obviously a generalized Dynkin diagram is completely determined by the matrix (q ij ) d×d , which also arises naturally in the context of Yetter-Drinfeld modules and Nichols algebras. Recall that for a diagonal for some g i ∈ G and q ij ∈ k * . Hence we obtain a matrix (q ij ) and the corresponding generalized Dynkin diagram D (qij ) . We also call D (qij ) the generalized Dynkin diagram of V or B(V ).
Suppose V ∈ G G YD and let J be a 2-cochain on G. Recall that B(V ) J is a Nichols algebra in G G YD ∂(J) by Lemma 2.11. It is natural to observe that Proof. By easy computation using (2.19).
To describe the structure of rank 2 Nichols algebras of diagonal type, we need the notion of full binary trees (see [19] and Appendix A). A binary tree T is a tree such that each node has at most two children. T is called full if each node of T has exactly zero or two children. Let r(T ) denote the root of the binary tree T . We use N 0 (T ) and N 2 (T ) to present the set of nodes of T which have zero and two children respectively, and In a similar manner one defines the right godfather a R of a. Moreover, we can define a function l : N (T ) → N inductively by l(a) = l(a R ) + 1 if a R L = a, 1 otherwise.
6.2. Classification of rank 2 Nichols algebras. Let V be a diagonal Yetter-Drinfeld module of dimension 2 over an abelian group G. Let {X 1 , X 2 } be a basis of V such that (6.1) holds for certain g i ∈ G and q ij ∈ k * for 1 ≤ i, j ≤ 2. Choose a basis E = {e 1 , e 2 } of Z 2 . Then there exist a unique group homomorphism ϕ : Z 2 → G and a unique bicharacter χ : Clearly, the generalized Dynkin diagram of V is the same as D χ,E .
Let T (V ) be the tensor algebra of V , then T (V ) admits a Z 2 -grading by extending For brevity, here and below we write |X| = deg(X) for the Z 2 -homogenous X ∈ T (V ).
From now on, for convenience we will write χ(a, b) instead of χ(|τ (a)|, |τ (a)|) for all a, b ∈ N (T ). For a ∈ N (T ), define In the following, by R n we denote the set of primitive n-th roots of unity and set R = n≥2 R n . With these notations, we can state Heckenberger's classification result as follows.  Table 6.1, where D is a generalized Dynkin diagram with all parameters taken in R and T is a full binary tree. Moreover, the Nichols algebra B(V ) associated with (D, T ) is the quotient of the tensor algebra T (V ) by the ideal generated by the following set In this subsection, the notations of Section 5 are used freely. Especially, we take = Z Ñ × Z Ò = 1 × 2 , G = Z m × Z n = g 1 × g 2 with m = Ñ 2 , n = Ò 2 . Moreover, let V = kX 1 ⊕ kX 2 be the direct sum of 1-dimensional Yetter-Drinfeld modules of (kG, π * (Φ)). Therefore, there exist h i ∈ G and q ij ∈ k * such that Consider the tensor algebra T π * (Φ) (V ) in G G YD π * (Φ) . The G-comodule structure of V induces a G-grading on T π * (Φ) (V ). For any homogenous element X ∈ T π * (Φ) (V ), let δ(X) denote its G-degree, that is δ L (X) = δ(X) ⊗ X. Note that δ(X) is also denoted simply by x in Sections 4 and 5. We introduce this new notation mainly for the awkward situation when there is no sense of lowercase, for example, a long expression.
Now the structure of B(V ) J can be described explicitly in the following proposition.
Proposition 6.5. Suppose that B(V ) is a finite-dimensional rank 2 Nichols algebra in G G YD associated with the pair (D, T ).
Proof. Recall that B(V ) J is identical to B(V ) as a linear space and its multiplication is obtained by a twisting from that of the latter, i.e., E • F = J(e, f )EF for any E, F ∈ B(V ).
Define a map Ψ : Evidently Ψ is an isomorphism of linear spaces. By the proof of Lemma 4.5, we have By the definition of τ * , one can easily show that Ψ(τ * (a)) = τ (a) for any a ∈ N (T ). With this fact and (6.7), it is immediately that Ψ maps relations (6.5) to relations (6.4), hence Ψ induces an isomorphism Ψ : for any E, F ∈ T Φ (V )/I. This completes the proof of the proposition.
Remark 6.6. It is clear that to give the structure of B(V ) J we need to know the group G and the bicharacter at first. Therefore it is more convenient for us to translate Table 6.1 to the following Table 6.2.

Examples: the standard case
Keep the notations of Sections 5 and 6. We call a rank 2 graded pointed Majid algebra Å = Å(V, G) over By Figure I, we have the corresponding Nichols algebra B(V ) ∈ G G YD in which G = Z m × Z n and (7.2) σ L (X 1 ) = g 1 ⊗ X 1 , σ L (X 2 ) = g 2 ⊗ X 2 .
In this case, we also call the Nichols algebra B(V ) ∈ G G YD standard. As an explicit example to explain our theoretical results obtained so far, the aim of the present section is to classify standard rank 2 graded pointed Majid algebras. Moreover, we will show that if Ñ = Ò then Å is always standard. Proof. It is a direct consequence of Proposition 5.10 and (7.3).
Therefore, Table 7.1 gives all the possible standard rank 2 Nichols algebras of Majid type in G G YD. On the other hand, take a Nichols algebra B(V ) in Table 7.1 and from which we get a, b, d uniquely due to Proposition 5.8. As a matter of fact, in this case we have  Structure constants of Dynkin diagrams Binary tree 1. q 12 q 21 = 1 T 1 2. q 12 q 21 = q −1 11 , q 11 = q 22 T 1 3. . 10.
7.2. Ñ = Ò implies standard. It turns out that the condition Ñ = Ò will always put us in the standard situation. More precisely, we have Lemma 7.3. Let Å be a rank 2 Majid algebra over Z Ñ × Z Ò . If Ñ = Ò, then Å is standard.
Proof. It suffices to show that any two generators of Z Ñ ×Z Ñ must be standard. In fact, let h 1 , h 2 ∈ Z Ñ ×Z Ñ and assume that h 1 , h 2 generate Z Ñ ×Z Ñ . Therefore, h Ñ 1 = h Ñ 2 = 1 and thus h 1 , h 2 can only generate elements of the form h s 1 h t 2 with 0 ≤ s < m, 0 ≤ t < m. On the other hand, such elements should exhaust the whole Z Ñ × Z Ñ by assumption. This forces h 1 ∩ h 2 = {1} and so h 1 , h 2 make a set of standard generators. Table 7.3. Finite-dimensional rank 2 Nichlos algebras over G G YD for G = Z 9 × Z 9 .
(1) Take a Nichols algebra B(V ) in Table 7.4. Then B(V ) J a,b,d is a Nichols in YD Φ a,b,d and thus B(V ) J a,b,d #k is a connected graded rank 2 pointed Majid algebra over = Z p × Z p . (2) Any finite-dimensional connected graded rank 2 pointed Majid algebra over Z p × Z p is isomorphic to B(V ) J a,b,d #k for some B(V ) given in Table 7.4. Table 7.4. Finite-dimensional rank 2 Nichlos algebras over G G YD for G = Z p 2 × Z p 2 .

8.
Examples: finite rank 2 quasi-quantum groups over Z Ò This section is devoted to a complete list of finite-dimensional connected rank 2 pointed Majid algebras over an arbitrary cyclic group Z Ò . For the rest of the paper, = Z Ò and G = Z n with n = Ò 2 . As before, we start with the following lemma.
Lemma 8.1. Suppose h 1 , h 2 ∈ Z n and h 1 , h 2 generate Z n . Then there exists a generator g of Z n such that h 1 = g s , h 2 = g t and (s, t) = 1.
Proof. Let g ′ be a generator of Z n , then h 1 = g ′s , h 2 = g ′t for some 0 ≤ s, t < n. Since h 1 , h 2 generate Z n , so we have h 1 , h 2 = g ′(s,t) = Z n . This implies that ((s, t), n) = 1, hence g = g ′(s,t) is another generator.
Again keep the notations of Sections 5 and 6. Let B(V ) be a finite-dimensional rank 2 Nichols algebras in G G YD with G = Z n = g . By Lemma 8.1, we can assume that h 1 = g s , h 2 = g t with (s, t) = 1.
(1) Take a Nichols algebra B(V ) in Table 8.1. Then B(V ) J d is a Nichols algebra in YD Φ d and thus B(V ) J d #k is a connected graded rank 2 pointed Majid algebra over = Z Ò .
(2) Any finite-dimensional connected graded rank 2 pointed Majid algebra over Z Ò is isomorphic to B(V ) J d #k for some B(V ) given in Table 8.1.
Proof. The proof can be carried out in the same way as that of Proposition 7.4. We just work on two typical examples, namely case 2 and the first item of case 8, to explain our equations about structure constants in Table 8.1.
Remarks 8.4. We conclude the paper with several remarks.
(1) In Theorem 8.3, if n is a prime number and n > 2, then one can check Table 8.1 case by case and conclude that there are no genuine rank 2 graded pointed Majid algebras. This fact offers another explanation to a seemingly mysterious (at least to the authors) result of Etingof and Gelaki [13, Theorem 3.1], which states that elementary graded genuine quasi-Hopf algebras over a cyclic group of prime( = 2) order are of rank ≤ 1. (2) If n has no small prime divisors, namely 2, 3, 5, 7, then essentially the finite-dimensional grtaded pointed Majid algebras constructed in Theorem 8.3 already appeared in the recent work of Angiono [6], only in the dual version. (3) The idea of the present paper should be useful in a much broader context. In particular, the results of the rank 2 case may be extended to diagonal Nichols algebras in G G YD Φ of higher ranks in a similar working philosophy of the theory of pointed Hopf algebras. This shall be dealt with in our forthcoming work.