A Class of Weak Hopf Algebras

Abstract

We introduce a class of noncommutative and noncocommutative weak Hopf algebras with infinite Ext quivers and study their structure. We decompose them into a direct sum of two algebras. The coalgebra structures of these weak Hopf algebras are described by their Ext quiver. The weak Hopf extension of Hopf algebra 𝐻_𝑛 has a quotient Hopf algebra and a sub-Hopf algebra which are isomorphic to 𝐻_𝑛.

1. Introduction

Weak Hopf algebra was introduced by Li in 1998 as a generalization of Hopf algebras [1]. It had been proved in [1, 2]; for some sorts of finite dimensional weak Hopf algebras 𝐻, the quantum quasidouble 𝐷(𝐻) of 𝐻 is quasibraided equipped with some quasi-R-matrix 𝑅. Hence 𝑅 is a solution of the Quantum Yang-Baxter Equation.

First two examples of noncommutative and noncocommutative weak Hopf algebras were given in [3]. Up to now, many examples of weak Hopf algebras have been found [2, 4–7]. So far, all examples of weak Hopf algebras were based on some Hopf algebras and were constructed by weak extension.

In this paper, we first give a Hopf algebra, denoted by 𝐻_𝑛. By weak extension, we construct a weak Hopf algebra 𝑊(𝑛₁,𝑛₂,𝑛₃) corresponding to 𝐻_𝑛 and study their structure. 𝑊(𝑛₁,𝑛₂,𝑛₃) has a quotient Hopf algebra and a sub-Hopf algebra which are isomorphic to 𝐻_𝑛. And as an algebra, 𝑊(𝑛₁,𝑛₂,𝑛₃) can be decomposed into a direct sum of two algebras, one of which is 𝐻_𝑛. The coalgebra structures of these weak Hopf algebras are described by their Ext quiver [8, 9].

We organize our paper as follows. In Section 2, we introduce the Hopf algebra 𝐻_𝑛. In Section 3, we define a class of weak Hopf algebras 𝑊(𝑛₁,𝑛₂,𝑛₃). In Section 4, we study the structure of 𝑊(𝑛₁,𝑛₂,𝑛₃) and decompose 𝑊(𝑛₁,𝑛₂,𝑛₃) into a direct sum of 𝐻_𝑛 and some algebra of polynomials as an algebra. We give the Ext-quiver of coalgebra of 𝑊(𝑛₁,𝑛₂,𝑛₃) and prove that 𝑊(𝑛₁,𝑛₂,𝑛₃) has a quotient Hopf algebra and a sub-Hopf algebra which are isomorphic to 𝐻_𝑛.

2. A Quiver Hopf Algebra

The Hopf Algebra 𝐹_(𝑞) is defined in [10]. Let 𝑞∈𝑘⧵0. As a 𝑘-algebra 𝐹_(𝑞) is generated by 𝑎,𝑏, and 𝑥 subject to the relations

𝑎𝑏=1,𝑏𝑎=1,𝑥𝑎=𝑞𝑎𝑥,𝑥𝑏=𝑞⁻¹𝑏𝑥.(2.1) The coalgebra structure of 𝐹_(𝑞) is determined by

Δ(𝑎)=𝑎⊗𝜀𝑎(1,)=𝜀Δ((𝑎𝑏))==𝜀𝑏(⊗𝑏)𝑏=,1,Δ(𝜀𝑥()𝑥=)=𝑥⊗0.𝑎+1⊗𝑥.(2.2)

We generalize 𝐹_(𝑞) to 𝐻_𝑛, which is defined as follows. Let 𝑘 be a field, 𝑞∈𝑘⧵0, 𝑖=1,2,…,𝑛. As a 𝑘-algebra 𝐻_𝑛 is generated by 𝐾,𝐾⁻¹, and 𝑋_𝑖,𝑖=1,2,…,𝑛 subject to the relations

𝐾𝐾⁻¹=1,𝐾⁻¹𝐾=1,𝑋_𝑖𝐾=𝑞𝐾𝑋_𝑖,𝑋_𝑖𝐾⁻¹=𝑞⁻¹𝐾⁻¹𝑋_𝑖.(2.3) The coalgebra structure of 𝐻_𝑛 is determined by

Δ(𝐾)=Δ𝐾⊗𝑋_𝑖𝐾,Δ𝐾⁻¹=𝐾⁻¹⊗𝐾⁻¹,𝜀(𝐾)=𝜀𝐾=₋₁𝑋_𝑖⊗𝐾+1⊗𝑋_𝑖,=1,𝜀𝑋_𝑖=0.(2.4) The antipode 𝑆 is induced by

𝑆(𝐾)=𝐾⁻¹,𝑆𝐾⁻¹=𝐾,𝑆𝑋_𝑖=−𝐾⁻¹𝑋_𝑖.(2.5)

3. A Class of Weak Hopf Algebras

In this section, we construct a class of weak Hopf algebra corresponding to 𝐻_𝑛.

First recall the definition of weak Hopf algebra [1].

Definition 3.1. A 𝑘-bialgebra 𝐻=(𝐻,𝜇,𝜂,Δ,𝜀) is called a weak Hopf algebra if there exists 𝑇∈Hom_𝑘(𝐻,𝐻) such that 𝑖𝑑∗𝑇∗𝑖𝑑=𝑖𝑑 and 𝑇∗𝑖𝑑∗𝑇=𝑇 where 𝑇 is called a weak antipode of 𝐻.

A weak Hopf algebra is called pointed if it is pointed as a coalgebra. If a weak Hopf algebra 𝐻 is pointed, then the set of all group-like elements 𝐺(𝐻) is a regular monoid [6].

Now we construct weak Hopf algebra 𝑊 corresponding to 𝐻_𝑛. The set 𝐺(𝑊) of group-like elements of weak Hopf algebra 𝑊 is a regular monoid which has generators 𝑔, 𝑔, 1, subject to 𝑔𝑔=𝑔𝑔,𝑔²𝑔=𝑔,𝑔²𝑔=𝑔.

To construct all possible weak extension we need the following discussion.

Recall, for any coalgebra 𝐶, that the group-like elements in 𝐶 are the set 𝐺(𝐶)={𝑎∈𝐶∣𝑎≠0andΔ(𝑎)=𝑎⊗𝑎}; necessarily 𝜀(𝑎)=1 for 𝑎∈𝐺(𝐶). Note that a simple subcoalgebra 𝐷 of 𝐶 is one-dimensional ⇔𝐷=𝑘𝑎 for some 𝑎∈𝐺(𝐶). A coalgebra is pointed if all of its simple subcoalgebras are one-dimensional. For 𝑎,𝑏∈𝐺(𝐶), the 𝑎,𝑏-primitive elements in 𝐶 are the set 𝑃_𝑎,𝑏(𝐶)={𝑐∈𝐶∣Δ(𝑐)=𝑐⊗𝑎+𝑏⊗𝑐}; necessarily 𝜀(𝑐)=0 for 𝑐∈𝑃_𝑎,𝑏(𝐶). Note that 𝑘(𝑎−𝑏)={𝑙(𝑎−𝑏)∣𝑙∈𝑘}⊂𝑃_𝑎,𝑏(𝐶); an 𝑎,𝑏-primitive element 𝑐 is nontrivial if 𝑐∉𝑘(𝑎−𝑏)={𝑙(𝑎−𝑏)∣𝑙∈𝑘}. If 𝑎=𝑏=1, the 1,1-primitives are simply called primitive; otherwise they are called skew primitive.

The following result is a generalization of [11].

Lemma 3.2. Let 𝑊 be the weak Hopf algebra defined above. One has 𝑔𝑃_𝑎,𝑏(𝑊)⊆𝑃_{𝑔𝑎,𝑔𝑏}(𝑊),𝑔𝑃_𝑎,𝑏(𝑊)⊆𝑃_{𝑔𝑎,𝑔𝑏}(𝑊).(3.1)

Proof. Let 𝑢∈𝑃_𝑎,𝑏(𝑊), then Δ(𝑢)=𝑢⊗𝑎+𝑏⊗𝑢. Hence, Δ(𝑔𝑢)===Δ𝑔(𝑔𝑢(⊗⊗𝑔)𝑔𝑔Δ)𝑎((+𝑢𝑢)⊗𝑔𝑏𝑎⊗+𝑔𝑏𝑢⊗∈𝑢𝑃)_{𝑔𝑎,𝑔𝑏}(𝑊).(3.2) The second inclusion is proved similarly.

Corollary. For 𝑊, one has ddiimm𝑃𝑃_{𝑔𝑖+1,𝑔𝑖𝑔𝑖,𝑔𝑖+1}(𝑊)=dim𝑃_{𝑔𝑖,𝑔𝑖−1}(𝑊),𝑖≥2,dim𝑃_𝑔,𝑔2(𝑊)=dim𝑃_𝑔^𝑖−1_,𝑔^𝑖(𝑊),𝑖≥2,(𝑊)=dim𝑃_{𝑔𝑔,𝑔}(𝑊)=dim𝑃_{𝑔,𝑔𝑔}(𝑊)=dim𝑃_𝑔2,𝑔(𝑊).(3.3)

Proof. We only prove the first equation. In fact, the map 𝜑∶𝑃_{𝑔𝑖,𝑔𝑖−1}(𝑊)→𝑃_{𝑔𝑖+1,𝑔𝑖}(𝑊), 𝑢↦𝑔𝑢 is a linear map with inverse 𝜓∶𝑃_{𝑔𝑖+1,𝑔𝑖}(𝑊)↦𝑃_{𝑔𝑖,𝑔𝑖−1}(𝑊), 𝑣↦𝑔𝑣. Hence, 𝑃_{𝑔𝑖,𝑔𝑖−1}(𝑊) and 𝑃_{𝑔𝑖+1,𝑔𝑖}(𝑊) are isomorphic as vector spaces.

Since all the dimensions in Corollary 3.3 are same, we have the following corollary.

Corollary 3.4. One has dim𝑃_1,𝑔(𝑊)≤dim𝑃_{𝑔,𝑔𝑔}(𝑊),dim𝑃_𝑔,1(𝑊)≤dim𝑃_{𝑔,𝑔𝑔}(𝑊).(3.4)

Proof. The map 𝜑∶𝑃_𝑔,1(𝑊)→𝑃_{𝑔,𝑔𝑔}(𝑊), 𝑢↦𝑔𝑔𝑢 is a linear map. If 𝜑(𝑢)=𝑔𝑔𝑢=𝑙(𝑔−𝑔𝑔), for some 𝑙∈𝑘, then 𝑢∈𝑘𝐺(𝑊), the vector space spanned by all group-like elements, because 𝑊 is graded. Hence, 𝑢=𝑙(𝑔−1). Therefore, the linear map 𝜑 is an injection. Consequently, dim𝑃_1,𝑔(𝑊)≤dim𝑃_{𝑔,𝑔𝑔}(𝑊).(3.5) The proof of the second inequality is similar.

By the above discussion we know that weak Hopf algebra 𝑊 is determined by 𝑃_1,𝑔(𝑊), 𝑃_𝑔,1(𝑊), and 𝑃_{𝑔,𝑔𝑔}(𝑊). Take 𝑥₁,…,𝑥_𝑛1 to be linearly independent nontrivial elements in 𝑃_1,𝑔(𝑊), and 𝑦₁,…,𝑦_𝑛2 linearly independent nontrivial elements in 𝑃_𝑔,1(𝑊). Let

𝑃_{𝑔,𝑔𝑔}(𝑊)=𝑔𝑔𝑃_1,𝑔(𝑊)+𝑔𝑃_𝑔,1(𝑊)⊕𝑉,(3.6) and 𝑧₁,…,𝑧_𝑛3 a basis of 𝑉. Then 𝑊 is determined by 𝑥₁,…,𝑥_𝑛1, 𝑦₁,…,𝑦_𝑛2, 𝑧₁,…,𝑧_𝑛3.

To summarize, we define weak Hopf algebra 𝑊(𝑛₁,𝑛₂,𝑛₃) corresponding to 𝐻_𝑛 as follows.

Definition 3.5. Let 𝑘 be a field. For any positive integers 𝑛₁,𝑛₂,𝑛₃, and nonzero element 𝑞∈𝑘, we define 𝑊(𝑛₁,𝑛₂,𝑛₃) to be associative algebra over field 𝑘 generated by 1,𝑔,𝑔,𝑥_𝑖,𝑦_𝑗,𝑧_𝑘, 𝑖=1,2,…,𝑛₁,𝑗=1,2,…,𝑛₂,𝑘=1,2,…,𝑛₃, subject to 𝑔𝑥_𝑖𝑔𝑔=𝑔𝑔,𝑔𝑔²=𝑔,𝑔²𝑔=𝑔,(3.7)𝑔𝑦_𝑗=𝑞𝑥_𝑖𝑔,𝑔𝑥_𝑖=𝑞⁻¹𝑥_𝑖𝑔,𝑖=1,2,…,𝑛₁,(3.8)=𝑞𝑦_𝑗𝑔𝑔𝑧,_𝑘𝑔𝑦_𝑗=𝑞⁻¹𝑦_𝑗𝑔,𝑗=1,2,…,𝑛₂,(3.9)𝑔=𝑞𝑧_𝑘,𝑘=1,2,…,𝑛₃.(3.10)

𝑊(𝑛₁,𝑛₂,𝑛₃) can be endowed with coalgebra structure by

Δ𝑥Δ(𝑔)=𝑔⊗𝑔,(3.11)_𝑖Δ𝑦=𝑥_𝑖⊗𝑔+1⊗𝑥_𝑖,(3.12)_𝑗Δ𝑧=𝑦_𝑗⊗1+𝑔⊗𝑦_𝑗,(3.13)_𝑘𝜀(1)=𝜀(𝑔)=𝜀=𝑧_𝑘⊗𝑔+𝑔𝑔⊗𝑧_𝑘,(3.14)𝑔=1,𝜀𝑥_𝑖=0,𝜀𝑦_𝑗=0,𝜀𝑧_𝑘=0,(3.15) while the weak antipode 𝑇 is induced by

𝑇𝑥𝑇(1)=1,𝑇(𝑔)=𝑔,𝑇𝑔=𝑔,(3.16)_𝑖=−𝑥_𝑖𝑔,𝑇𝑦_𝑗=−𝑔𝑦_𝑗,𝑇𝑧_𝑘=−𝑧_𝑘𝑔,(3.17)

Theorem 3.6. For any positive integers 𝑛₁,𝑛₂,𝑛₃, 𝑊(𝑛₁,𝑛₂,𝑛₃) is a weak Hopf algebra.

Proof. First we must check that the coproduct Δ is an algebra map. It suffices to prove that Δ preserves the relations (3.7)–(3.10). It is easy to see that Δ preserves the relations (3.7). And Δ𝑔𝑥_𝑖==𝑔(𝑔𝑥_𝑖⊗𝑔)𝑥_𝑖⊗𝑔+1⊗𝑥_𝑖⊗𝑔²+𝑔⊗𝑔𝑥_𝑖==𝑞𝑞𝑥𝑥_𝑖𝑖𝑔⊗𝑔²+𝑔⊗𝑞𝑥_𝑖𝑔=Δ𝑞𝑥⊗_𝑖𝑔+1⊗𝑥_𝑖(𝑔⊗𝑔)Δ𝑔𝑦_𝑗𝑔,==(𝑔𝑔𝑦_𝑗⊗𝑔)𝑦_𝑗⊗1+𝑔⊗𝑦_𝑗⊗𝑔+𝑔𝑔⊗𝑔𝑦_𝑗==𝑞𝑞𝑦𝑦_𝑗𝑗𝑔⊗𝑔+𝑔𝑔⊗𝑞𝑦_𝑗𝑔=Δ𝑞𝑦⊗_𝑗1+𝑔⊗𝑦_𝑗(𝑔⊗𝑔)Δ𝑔𝑧_𝑘𝑔,𝑔==𝑔(𝑔𝑧_𝑘⊗𝑔)𝑧_𝑘⊗𝑔+𝑔𝑔⊗𝑧_𝑘𝑔⊗𝑔=𝑞𝑧𝑔_𝑘⊗𝑔𝑔𝑔+𝑔𝑔𝑔𝑔⊗𝑔𝑧_𝑘𝑔=Δ𝑞𝑧_𝑘⊗𝑔+𝑔𝑔⊗𝑞𝑧_𝑘.(3.18) Next we prove that 𝑇 is the weak antipode. It suffices to prove that for each generator g,𝑔,𝑥_𝑖,𝑦_𝑗,𝑧_𝑘, the action of 𝑇∗𝑖𝑑∗𝑇 is the same as that of 𝑇, and the action of 𝑖𝑑∗𝑇∗𝑖𝑑 is the same as that of 𝑖𝑑.
Since
(Δ⊗𝑖𝑑)Δ𝑥_𝑖==(Δ𝑥_𝑖⊗𝑖𝑑)𝑥_𝑖⊗𝑔+1⊗𝑥_𝑖⊗𝑔+1⊗𝑥_𝑖⊗𝑔+1⊗1⊗𝑥_𝑖=𝑥_𝑖⊗𝑔⊗𝑔+1⊗𝑥_𝑖⊗𝑔+1⊗1⊗𝑥_𝑖,(3.19) we get ((𝑖𝑇𝑑∗∗𝑖𝑇𝑑∗∗𝑖𝑇𝑑))𝑥𝑥_𝑖𝑖=𝑥_𝑖𝑔𝑔+−𝑥_𝑖𝑔𝑔+𝑥_𝑖=𝑥_𝑖=𝑖𝑑𝑥_𝑖,==−−𝑥𝑥_𝑖𝑖𝑔𝑔𝑔+𝑥_𝑖𝑔+−𝑥_𝑖𝑔𝑔𝑔𝑔=−𝑥_𝑖𝑔=𝑇𝑥_𝑖.(3.20) Since (Δ⊗𝑖𝑑)Δ𝑦_𝑗==(Δ𝑦_𝑗⊗𝑖𝑑)𝑦_𝑗⊗1+𝑔⊗𝑦_𝑗⊗1+𝑔⊗𝑦_𝑗⊗1+𝑔⊗𝑔⊗𝑦_𝑗=𝑦_𝑗⊗1⊗1+𝑔⊗𝑦_𝑗⊗1+𝑔⊗𝑔⊗𝑦_𝑗,(3.21) it follows that ((𝑖𝑇𝑑∗∗𝑖𝑇𝑑∗∗𝑖𝑇𝑑))𝑦𝑦_𝑗𝑗=𝑦_𝑗+𝑔−𝑔𝑦_𝑗+𝑔𝑔𝑦_𝑗=𝑦_𝑗=𝑖𝑑𝑦_𝑗,=−𝑔𝑦_𝑗+𝑔𝑦_𝑗+𝑔𝑔−𝑔𝑦_𝑗=−𝑔𝑦_𝑗=𝑇𝑦_𝑗.(3.22) Since (Δ⊗𝑖𝑑)Δ𝑧_𝑘==(Δ𝑧_𝑘⊗𝑖𝑑)𝑧_𝑘⊗𝑔+𝑔𝑔⊗𝑧_𝑘⊗𝑔+𝑔𝑔⊗𝑧_𝑘⊗𝑔+𝑔𝑔⊗𝑔𝑔⊗𝑧_𝑘𝑧_𝑘=𝑧_𝑘⊗𝑔⊗𝑔+𝑔𝑔⊗𝑧_𝑘⊗𝑔+𝑔𝑔⊗𝑔𝑔⊗𝑧_𝑘,𝑔𝑔𝑔=𝑞⁻¹𝑔𝑧_𝑘𝑔𝑔𝑔=𝑞⁻¹𝑔𝑧_𝑘𝑔=𝑧_𝑘,𝑔𝑧_𝑘=𝑞⁻¹𝑔𝑔𝑔𝑧_𝑘𝑔=𝑞⁻¹𝑔𝑧_𝑘𝑔=𝑧_𝑘,(3.23) we get (𝑖𝑑∗𝑇∗𝑖𝑑)𝑧_𝑘=𝑧_𝑘𝑔𝑔+𝑔𝑔−𝑧_𝑘𝑔𝑔+𝑔𝑔𝑔𝑔𝑧_𝑘=𝑧_𝑘−𝑞𝑔𝑧_𝑘𝑔+𝑔𝑔𝑧_𝑘(𝑇∗𝑖𝑑∗𝑇)𝑧_𝑘=𝑧_𝑘−𝑧_𝑘+𝑧_𝑘=𝑧_𝑘=𝑖𝑑𝑧_𝑘,==−−𝑧𝑧_𝑘𝑘𝑔𝑔𝑔+𝑔𝑔𝑧_𝑘𝑔+𝑔𝑔𝑔𝑔−𝑧_𝑘𝑔𝑔+𝑧_𝑘𝑔+−𝑧_𝑘𝑔=−𝑧_𝑘𝑔=𝑇𝑧_𝑘.(3.24)

4. The Structure of 𝑊(𝑛₁,𝑛₂,𝑛₃)

In this section we study the algebra and coalgebra structure of 𝑊(𝑛₁,𝑛₂,𝑛₃).

It is easy to prove that the elements 𝑔𝑔 and 1−𝑔𝑔 are a pair of orthogonal central idempotents. Set 𝑊₁=𝑊(𝑛₁,𝑛₂,𝑛₃)𝑔𝑔, 𝑊₂=𝑊(𝑛₁,𝑛₂,𝑛₃)(1−𝑔𝑔). We have the following.

Theorem 4.1. 𝑊(𝑛₁,𝑛₂,𝑛₃) can be written as a direct sum of two-sided ideals 𝑊(𝑛₁,𝑛₂,𝑛₃)=𝑊₁⨁𝑊₂. And one has the following. (1)As an algebra, 𝑊₁ is isomorphic to 𝐻_𝑛, where 𝑛=𝑛₁+𝑛₂+𝑛₃.(2)As an algebra, 𝑊₂ is isomorphic to the free associative algebra 𝑘⟨𝑌₁,…,𝑌_𝑡⟩ of 𝑡 generators, where 𝑡=𝑛₁+𝑛₂.

Proof. (1)  Since 𝑔𝑔 and 1−𝑔𝑔 are a pair of orthogonal central idempotents,
𝑊𝑛₁,𝑛₂,𝑛₃=𝑊𝑛₁,𝑛₂,𝑛₃𝑔𝑔⊕𝑊𝑛₁,𝑛₂,𝑛₃1−𝑔𝑔=𝑊₁⊕𝑊₂.(4.1) The isomorphism 𝑊₁→𝐻_𝑛 is induced by 𝑥_𝑖𝑔𝑔↦𝑋_𝑖, 𝑦_𝑗𝑔𝑔↦𝑋_𝑛1+𝑗, 𝑧_𝑘𝑔𝑔↦𝑋_{𝑛1+𝑛2+𝑘}, 𝑔𝑔↦1, 𝑔²𝑔↦𝐾.
(2)  Note that 𝑧_𝑘(1−𝑔𝑔)=0 and 𝑥_𝑖(1−𝑔𝑔)𝑦_𝑗(1−𝑔𝑔)=𝑦_𝑗(1−𝑔𝑔)𝑥_𝑖(1−𝑔𝑔). Since 𝑥_𝑖(1−𝑔𝑔),𝑦_𝑗(1−𝑔𝑔) are generators of 𝑊₂, the isomorphism 𝑊₂→𝑘⟨𝑌₁,…,𝑌_𝑡⟩ is defined by (1−𝑔²)↦1, 𝑥_𝑖(1−𝑔²)↦𝑌_𝑖, 𝑦_𝑗(1−𝑔²)↦𝑌_𝑛1+𝑗.

A weak Hopf ideal 𝐽 of a weak Hopf algebra 𝐻 is a bi-ideal such that 𝑇(𝐽)⊂𝐽, where 𝑇 is the weak antipode of 𝐻. It is easy to see that 𝐻/𝐽 has a natural structure of a weak Hopf algebra.

Theorem 4.2. The ideal 𝐽 in 𝑊(𝑛₁,𝑛₂,𝑛₃) generated by 1−𝑔𝑔 is a weak Hopf ideal. And the quotient weak Hopf algebra 𝑊(𝑛₁,𝑛₂,𝑛₃)/𝐽 is a Hopf algebra, which is isomorphic to 𝐻_𝑛, where 𝑛=𝑛₁+𝑛₂+𝑛₃.

Proof. Since Δ1−𝑔𝑔==11⊗⊗11−−𝑔𝑔𝑔⊗𝑔𝑔=1−𝑔𝑔⊗1+𝑔𝑔⊗1−𝑔𝑔⊗𝑔𝑔𝑇1−𝑔𝑔⊗1+𝑔𝑔⊗1−𝑔𝑔,𝑔=𝑇(1)−𝑇𝑔𝑇(𝑔)=1−𝑔𝑔,(4.2)𝐽 is a weak Hopf ideal in 𝑊(𝑛₁,𝑛₂,𝑛₃).
The isomorphism 𝑊(𝑛₁,𝑛₂,𝑛₃)/𝐽→𝐻_𝑛 is defined by 𝑔+𝐽↦𝐾, 𝑔+𝐽↦𝐾⁻¹, 𝑥_𝑖+𝐽↦𝑋_𝑖, 𝑔𝑦_𝑗+𝐽↦𝑋_𝑛1+𝑗, 𝑧_𝑘+𝐽↦𝑋_{𝑛1+𝑛2+𝑘}.

Now we give the Ext quiver of 𝑊(𝑛₁,𝑛₂,𝑛₃). For the definition and calculation of Ext quiver, we refer to [5, 8, 9, 12].

Figure 1 

Ext quiver of 𝑊(𝑛1,𝑛2,𝑛3).

The Ext quiver of 𝑊(𝑛₁,𝑛₂,𝑛₃) is shown in Figure 1. The multiplicity of arrow 𝑔⋅→⋅1 is 𝑛₁. The multiplicity of arrow 1⋅→⋅𝑔 is 𝑛₂. The multiplicity of other arrows is all 𝑛.

Theorem 4.3. The sub-coalgebra 𝐻 related to the subquiver in Figure 2 is isomorphic to 𝐻_𝑛 as coalgebra.

Figure 2 

A subquiver of Ext quiver of 𝑊(𝑛1,𝑛2,𝑛3).

Proof. The isomorphism 𝐻→𝐻_𝑛 is induced by 𝑔𝑔↦1, 𝑔↦𝐾, 𝑔↦𝐾⁻¹, 𝑥_𝑖↦𝑋_𝑖, 𝑔𝑦_𝑗↦𝑋_𝑛1+𝑗, 𝑧_𝑘↦𝑋_{𝑛1+𝑛2+𝑘}.

Remark 4.4. The isomorphisms described in Theorem 4.1 are not isomorphisms of bialgebras.

Remark 4.5. The weak Hopf algebras discussed in [4, 5] also have quotient Hopf algebras and sub-Hopf algebras which are isomorphic to the related Hopf algebras.

Acknowledgments

This research is supported by Doctor scientific research start fund of Henan University of Science and Technology, supported by SRF of Henan University of Science and Technology (2006zy007), and partly supported by NNSF of China (10571153).