International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 520762 | 8 pages | https://doi.org/10.1155/2010/520762

A Class of Weak Hopf Algebras

Academic Editor: Palle Jorgensen
Received24 Nov 2009
Accepted27 Jan 2010
Published04 Mar 2010

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 𝐻𝑛 has a quotient Hopf algebra and a sub-Hopf algebra which are isomorphic to 𝐻𝑛.

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 𝐻𝑛. By weak extension, we construct a weak Hopf algebra 𝑊(𝑛1,𝑛2,𝑛3) corresponding to 𝐻𝑛 and study their structure. 𝑊(𝑛1,𝑛2,𝑛3) has a quotient Hopf algebra and a sub-Hopf algebra which are isomorphic to 𝐻𝑛. And as an algebra, 𝑊(𝑛1,𝑛2,𝑛3) can be decomposed into a direct sum of two algebras, one of which is 𝐻𝑛. 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 𝐻𝑛. 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 𝐻𝑛 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 𝐻𝑛.

2. A Quiver Hopf Algebra

The Hopf Algebra 𝐹(ğ‘ž) is defined in [10]. Let ğ‘žâˆˆğ‘˜â§µ0. As a 𝑘-algebra 𝐹(ğ‘ž) is generated by ğ‘Ž,𝑏, and 𝑥 subject to the relations

ğ‘Žğ‘=1,ğ‘ğ‘Ž=1,ğ‘¥ğ‘Ž=ğ‘žğ‘Žğ‘¥,𝑥𝑏=ğ‘žâˆ’1𝑏𝑥.(2.1) The coalgebra structure of 𝐹(ğ‘ž) is determined by

𝜀Δ(ğ‘Ž)=ğ‘ŽâŠ—ğ‘Ž,Δ(𝑏)=𝑏⊗𝑏,Δ(𝑥)=ğ‘¥âŠ—ğ‘Ž+1⊗𝑥.(1)=𝜀(ğ‘Ž)=𝜀(𝑏)=1,𝜀(𝑥)=0.(2.2)

We generalize 𝐹(ğ‘ž) to 𝐻𝑛, which is defined as follows. Let 𝑘 be a field, ğ‘žâˆˆğ‘˜â§µ0, 𝑖=1,2,…,𝑛. As a 𝑘-algebra 𝐻𝑛 is generated by 𝐾,𝐾−1, and 𝑋𝑖,𝑖=1,2,…,𝑛 subject to the relations

𝐾𝐾−1=1,𝐾−1𝐾=1,𝑋𝑖𝐾=ğ‘žğ¾ğ‘‹ğ‘–,𝑋𝑖𝐾−1=ğ‘žâˆ’1𝐾−1𝑋𝑖.(2.3) The coalgebra structure of 𝐻𝑛 is determined by

𝐾Δ(𝐾)=𝐾⊗𝐾,Δ−1=𝐾−1⊗𝐾−1,Δ𝑋𝑖=𝑋𝑖⊗𝐾+1⊗𝑋𝑖,𝐾𝜀(𝐾)=𝜀−1𝑋=1,𝜀𝑖=0.(2.4) The antipode 𝑆 is induced by

𝑆(𝐾)=𝐾−1𝐾,𝑆−1𝑋=𝐾,𝑆𝑖=−𝐾−1𝑋𝑖.(2.5)

3. A Class of Weak Hopf Algebras

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

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∈Hom𝑘(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 𝐻𝑛. 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)={ğ‘Žâˆˆğ¶âˆ£ğ‘Žâ‰ 0andΔ(ğ‘Ž)=ğ‘ŽâŠ—ğ‘Ž}; necessarily 𝜀(ğ‘Ž)=1 for ğ‘Žâˆˆğº(𝐶). Note that a simple subcoalgebra 𝐷 of 𝐶 is one-dimensional ⇔𝐷=ğ‘˜ğ‘Ž for some ğ‘Žâˆˆğº(𝐶). A coalgebra is pointed if all of its simple subcoalgebras are one-dimensional. For ğ‘Ž,𝑏∈𝐺(𝐶), the ğ‘Ž,𝑏-primitive elements in 𝐶 are the set ğ‘ƒğ‘Ž,𝑏(𝐶)={𝑐∈𝐶∣Δ(𝑐)=ğ‘âŠ—ğ‘Ž+𝑏⊗𝑐}; necessarily 𝜀(𝑐)=0 for ğ‘âˆˆğ‘ƒğ‘Ž,𝑏(𝐶). Note that 𝑘(ğ‘Žâˆ’ğ‘)={𝑙(ğ‘Žâˆ’ğ‘)∣𝑙∈𝑘}âŠ‚ğ‘ƒğ‘Ž,𝑏(𝐶); an ğ‘Ž,𝑏-primitive element 𝑐 is nontrivial if 𝑐∉𝑘(ğ‘Žâˆ’ğ‘)={𝑙(ğ‘Žâˆ’ğ‘)∣𝑙∈𝑘}. If ğ‘Ž=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 ğ‘”ğ‘ƒğ‘Ž,𝑏(𝑊)âŠ†ğ‘ƒğ‘”ğ‘Ž,𝑔𝑏(𝑊),ğ‘”ğ‘ƒğ‘Ž,𝑏(𝑊)âŠ†ğ‘ƒğ‘”ğ‘Ž,𝑔𝑏(𝑊).(3.1)

Proof. Let ğ‘¢âˆˆğ‘ƒğ‘Ž,𝑏(𝑊), then Δ(𝑢)=ğ‘¢âŠ—ğ‘Ž+𝑏⊗𝑢. Hence, =Δ(𝑔𝑢)=Δ(𝑔)Δ(𝑢)(𝑔⊗𝑔)(ğ‘¢âŠ—ğ‘Ž+𝑏⊗𝑢)=ğ‘”ğ‘¢âŠ—ğ‘”ğ‘Ž+ğ‘”ğ‘âŠ—ğ‘”ğ‘¢âˆˆğ‘ƒğ‘”ğ‘Ž,𝑔𝑏(𝑊).(3.2) The second inclusion is proved similarly.

Corollary. For 𝑊, one has dim𝑃𝑔𝑖+1,𝑔𝑖(𝑊)=dim𝑃𝑔𝑖,𝑔𝑖−1(𝑊),𝑖≥2,dim𝑃𝑔𝑖,𝑔𝑖+1(𝑊)=dim𝑃𝑔𝑖−1,𝑔𝑖(𝑊),𝑖≥2,dim𝑃𝑔,𝑔2(𝑊)=dim𝑃𝑔𝑔,𝑔(𝑊)=dim𝑃𝑔,𝑔𝑔(𝑊)=dim𝑃𝑔2,𝑔(𝑊).(3.3)

Proof. We only prove the first equation. In fact, the map 𝜑∶𝑃𝑔𝑖,𝑔𝑖−1(𝑊)→𝑃𝑔𝑖+1,𝑔𝑖(𝑊), 𝑢↦𝑔𝑢 is a linear map with inverse 𝜓∶𝑃𝑔𝑖+1,𝑔𝑖(𝑊)↦𝑃𝑔𝑖,𝑔𝑖−1(𝑊), 𝑣↦𝑔𝑣. Hence, 𝑃𝑔𝑖,𝑔𝑖−1(𝑊) and 𝑃𝑔𝑖+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 dim𝑃1,𝑔(𝑊)≤dim𝑃𝑔,𝑔𝑔(𝑊),dim𝑃𝑔,1(𝑊)≤dim𝑃𝑔,𝑔𝑔(𝑊).(3.4)

Proof. The map 𝜑∶𝑃𝑔,1(𝑊)→𝑃𝑔,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, dim𝑃1,𝑔(𝑊)≤dim𝑃𝑔,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 𝑃1,𝑔(𝑊), 𝑃𝑔,1(𝑊), and 𝑃𝑔,𝑔𝑔(𝑊). Take 𝑥1,…,𝑥𝑛1 to be linearly independent nontrivial elements in 𝑃1,𝑔(𝑊), and 𝑦1,…,𝑦𝑛2 linearly independent nontrivial elements in 𝑃𝑔,1(𝑊). Let

𝑃𝑔,𝑔𝑔𝑔(𝑊)=𝑔𝑃1,𝑔(𝑊)+𝑔𝑃𝑔,1(𝑊)⊕𝑉,(3.6) and 𝑧1,…,𝑧𝑛3 a basis of 𝑉. Then 𝑊 is determined by 𝑥1,…,𝑥𝑛1, 𝑦1,…,𝑦𝑛2, 𝑧1,…,𝑧𝑛3.

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

Definition 3.5. Let 𝑘 be a field. For any positive integers 𝑛1,𝑛2,𝑛3, and nonzero element ğ‘žâˆˆğ‘˜, we define 𝑊(𝑛1,𝑛2,𝑛3) to be associative algebra over field 𝑘 generated by 1,𝑔,𝑔,𝑥𝑖,𝑦𝑗,𝑧𝑘, 𝑖=1,2,…,𝑛1,𝑗=1,2,…,𝑛2,𝑘=1,2,…,𝑛3, subject to 𝑔𝑔=𝑔𝑔,𝑔𝑔2=𝑔,𝑔2𝑔=𝑔,(3.7)𝑔𝑥𝑖=ğ‘žğ‘¥ğ‘–ğ‘”,𝑔𝑥𝑖=ğ‘žâˆ’1𝑥𝑖𝑔,𝑖=1,2,…,𝑛1,(3.8)𝑔𝑦𝑗=ğ‘žğ‘¦ğ‘—ğ‘”,𝑔𝑦𝑗=ğ‘žâˆ’1𝑦𝑗𝑔,𝑗=1,2,…,𝑛2,(3.9)𝑔𝑧𝑘𝑔=ğ‘žğ‘§ğ‘˜,𝑘=1,2,…,𝑛3.(3.10)

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

Δ𝑥Δ(𝑔)=𝑔⊗𝑔,(3.11)𝑖=𝑥𝑖⊗𝑔+1⊗𝑥𝑖Δ𝑦,(3.12)𝑗=𝑦𝑗⊗1+𝑔⊗𝑦𝑗Δ𝑧,(3.13)𝑘=𝑧𝑘⊗𝑔+𝑔𝑔⊗𝑧𝑘,(3.14)𝜀(1)=𝜀(𝑔)=𝜀𝑔𝑥=1,𝜀𝑖𝑦=0,𝜀𝑗𝑧=0,𝜀𝑘=0,(3.15) while the weak antipode 𝑇 is induced by

𝑇(1)=1,𝑇(𝑔)=𝑔,𝑇𝑔𝑇𝑥=𝑔,(3.16)𝑖=−𝑥𝑖𝑦𝑔,𝑇𝑗=−𝑔𝑦𝑗𝑧,𝑇𝑘=−𝑧𝑘𝑔,(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𝑔𝑥𝑖=𝑥(𝑔⊗𝑔)𝑖⊗𝑔+1⊗𝑥𝑖=𝑔𝑥𝑖⊗𝑔2+𝑔⊗𝑔𝑥𝑖=î€·ğ‘žğ‘¥ğ‘–ğ‘”î€¸âŠ—ğ‘”2+ğ‘”âŠ—ğ‘žğ‘¥ğ‘–ğ‘”î€¸î€·ğ‘¥=ğ‘žğ‘–âŠ—ğ‘”+1⊗𝑥𝑖(𝑔⊗𝑔)=Î”ğ‘žğ‘¥ğ‘–ğ‘”î€¸,Δ𝑔𝑦𝑗𝑦=(𝑔⊗𝑔)𝑗⊗1+𝑔⊗𝑦𝑗=𝑔𝑦𝑗⊗𝑔+𝑔𝑔⊗𝑔𝑦𝑗=î€·ğ‘žğ‘¦ğ‘—ğ‘”î€¸âŠ—ğ‘”+ğ‘”î€·ğ‘”âŠ—ğ‘žğ‘¦ğ‘—ğ‘”î€¸î€·ğ‘¦=ğ‘žğ‘—âŠ—1+𝑔⊗𝑦𝑗(𝑔⊗𝑔)=Î”ğ‘žğ‘¦ğ‘—ğ‘”î€¸,Δ𝑔𝑧𝑘𝑔𝑧=(𝑔⊗𝑔)𝑘⊗𝑔+𝑔𝑔⊗𝑧𝑘𝑔⊗𝑔=𝑔𝑧𝑘𝑔⊗𝑔𝑔𝑔+𝑔𝑔𝑔𝑔⊗𝑔𝑧𝑘𝑔=î€·ğ‘žğ‘§ğ‘˜î€¸âŠ—ğ‘”+ğ‘”î€·ğ‘”âŠ—ğ‘žğ‘§ğ‘˜î€¸î€·=Î”ğ‘žğ‘§ğ‘˜î€¸.(3.18) Next we prove that 𝑇 is the weak antipode. It suffices to prove that for each generator g,𝑔,𝑥𝑖,𝑦𝑗,𝑧𝑘, 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
𝑥(Δ⊗𝑖𝑑)Δ𝑖=𝑥(Δ⊗𝑖𝑑)𝑖⊗𝑔+1⊗𝑥𝑖=𝑥𝑖⊗𝑔+1⊗𝑥𝑖⊗𝑔+1⊗1⊗𝑥𝑖=𝑥𝑖⊗𝑔⊗𝑔+1⊗𝑥𝑖⊗𝑔+1⊗1⊗𝑥𝑖,(3.19) we get 𝑥(𝑖𝑑∗𝑇∗𝑖𝑑)𝑖=𝑥𝑖𝑔𝑔+−𝑥𝑖𝑔𝑔+𝑥𝑖=𝑥𝑖𝑥=𝑖𝑑𝑖,(𝑥𝑇∗𝑖𝑑∗𝑇)𝑖=−𝑥𝑖𝑔𝑔𝑔+𝑥𝑖𝑔+−𝑥𝑖𝑔=−𝑥𝑖𝑔𝑔𝑔=−𝑥𝑖𝑥𝑔=𝑇𝑖.(3.20) Since 𝑦(Δ⊗𝑖𝑑)Δ𝑗=𝑦(Δ⊗𝑖𝑑)𝑗⊗1+𝑔⊗𝑦𝑗=𝑦𝑗⊗1+𝑔⊗𝑦𝑗⊗1+𝑔⊗𝑔⊗𝑦𝑗=𝑦𝑗⊗1⊗1+𝑔⊗𝑦𝑗⊗1+𝑔⊗𝑔⊗𝑦𝑗,(3.21) it follows that 𝑦(𝑖𝑑∗𝑇∗𝑖𝑑)𝑗=𝑦𝑗+𝑔−𝑔𝑦𝑗+𝑔𝑔𝑦𝑗=𝑦𝑗𝑦=𝑖𝑑𝑗,𝑦(𝑇∗𝑖𝑑∗𝑇)𝑗=−𝑔𝑦𝑗+𝑔𝑦𝑗+𝑔𝑔−𝑔𝑦𝑗=−𝑔𝑦𝑗𝑦=𝑇𝑗.(3.22) Since 𝑧(Δ⊗𝑖𝑑)Δ𝑘=𝑧(Δ⊗𝑖𝑑)𝑘⊗𝑔+𝑔𝑔⊗𝑧𝑘=𝑧𝑘⊗𝑔+𝑔𝑔⊗𝑧𝑘⊗𝑔+𝑔𝑔⊗𝑔𝑔⊗𝑧𝑘=𝑧𝑘⊗𝑔⊗𝑔+𝑔𝑔⊗𝑧𝑘⊗𝑔+𝑔𝑔⊗𝑔𝑔⊗𝑧𝑘,𝑧𝑘𝑔𝑔=ğ‘žâˆ’1𝑔𝑧𝑘𝑔𝑔𝑔=ğ‘žâˆ’1𝑔𝑧𝑘𝑔=𝑧𝑘,𝑔𝑔𝑧𝑘=ğ‘žâˆ’1𝑔𝑔𝑔𝑧𝑘𝑔=ğ‘žâˆ’1𝑔𝑧𝑘𝑔=𝑧𝑘,(3.23) we get 𝑧(𝑖𝑑∗𝑇∗𝑖𝑑)𝑘=𝑧𝑘𝑔𝑔+𝑔𝑔−𝑧𝑘𝑔𝑔+𝑔𝑔𝑔𝑔𝑧𝑘=ğ‘§ğ‘˜âˆ’ğ‘žğ‘”ğ‘§ğ‘˜ğ‘”+𝑔𝑔𝑧𝑘=𝑧𝑘−𝑧𝑘+𝑧𝑘=𝑧𝑘𝑧=𝑖𝑑𝑘,𝑧(𝑇∗𝑖𝑑∗𝑇)𝑘=−𝑧𝑘𝑔𝑔𝑔+𝑔𝑔𝑧𝑘𝑔+𝑔𝑔𝑔𝑔−𝑧𝑘𝑔=−𝑧𝑘𝑔+𝑧𝑘𝑔+−𝑧𝑘𝑔=−𝑧𝑘𝑧𝑔=𝑇𝑘.(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 𝐻𝑛, where 𝑛=𝑛1+𝑛2+𝑛3.(2)As an algebra, 𝑊2 is isomorphic to the free associative algebra 𝑘⟨𝑌1,…,𝑌𝑡⟩ 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𝑛=𝑊1,𝑛2,𝑛3𝑔𝑛𝑔⊕𝑊1,𝑛2,𝑛31−𝑔𝑔=𝑊1⊕𝑊2.(4.1) The isomorphism 𝑊1→𝐻𝑛 is induced by 𝑥𝑖𝑔𝑔↦𝑋𝑖, 𝑦𝑗𝑔𝑔↦𝑋𝑛1+𝑗, 𝑧𝑘𝑔𝑔↦𝑋𝑛1+𝑛2+𝑘, 𝑔𝑔↦1, 𝑔2𝑔↦𝐾.
(2)  Note that 𝑧𝑘(1−𝑔𝑔)=0 and 𝑥𝑖(1−𝑔𝑔)𝑦𝑗(1−𝑔𝑔)=𝑦𝑗(1−𝑔𝑔)𝑥𝑖(1−𝑔𝑔). Since 𝑥𝑖(1−𝑔𝑔),𝑦𝑗(1−𝑔𝑔) are generators of 𝑊2, the isomorphism 𝑊2→𝑘⟨𝑌1,…,𝑌𝑡⟩ is defined by (1−𝑔2)↦1, 𝑥𝑖(1−𝑔2)↦𝑌𝑖, 𝑦𝑗(1−𝑔2)↦𝑌𝑛1+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 𝐻𝑛, where 𝑛=𝑛1+𝑛2+𝑛3.

Proof. Since Δ1−𝑔𝑔=1⊗1−𝑔𝑔⊗𝑔𝑔=1⊗1−𝑔𝑔⊗1+𝑔𝑔⊗1−𝑔𝑔⊗𝑔𝑔=1−𝑔𝑔⊗1+𝑔𝑔⊗1−𝑔𝑔,𝑇1−𝑔𝑔=𝑇(1)−𝑇𝑔𝑇(𝑔)=1−𝑔𝑔,(4.2)𝐽 is a weak Hopf ideal in 𝑊(𝑛1,𝑛2,𝑛3).
The isomorphism 𝑊(𝑛1,𝑛2,𝑛3)/𝐽→𝐻𝑛 is defined by 𝑔+𝐽↦𝐾, 𝑔+𝐽↦𝐾−1, 𝑥𝑖+𝐽↦𝑋𝑖, 𝑔𝑦𝑗+𝐽↦𝑋𝑛1+𝑗, 𝑧𝑘+𝐽↦𝑋𝑛1+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].

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 𝐻𝑛 as coalgebra.

Proof. The isomorphism 𝐻→𝐻𝑛 is induced by 𝑔𝑔↦1, 𝑔↦𝐾, 𝑔↦𝐾−1, 𝑥𝑖↦𝑋𝑖, 𝑔𝑦𝑗↦𝑋𝑛1+𝑗, 𝑧𝑘↦𝑋𝑛1+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).

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Copyright © 2010 Dongming Cheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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