A Class of Weak Hopf Algebras

Volume 2010, Article ID 520762,8pages doi:10.1155/2010/520762

Research Article

A Class of Weak Hopf Algebras

Dongming Cheng

1Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027, China

2Department of Mathematics, Henan University of Science and Technology, Luoyang, Henan 471003, China

Correspondence should be addressed to Dongming Cheng,[email protected] Received 24 November 2009; Accepted 27 January 2010

Academic Editor: Palle Jorgensen

Copyrightq2010 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.

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 algebraHnhas a quotient Hopf algebra and a sub-Hopf algebra which are isomorphic toHn.

1. Introduction

Weak Hopf algebra was introduced by Li in 1998 as a generalization of Hopf algebras1. It had been proved in1,2; for some sorts of finite dimensional weak Hopf algebrasH, the quantum quasidoubleDHof H is quasibraided equipped with some quasi-R-matrixR.

HenceRis a solution of the Quantum Yang-Baxter Equation.

First two examples of noncommutative and noncocommutative weak Hopf algebras were given in3. Up to now, many examples of weak Hopf algebras have been found2,4–

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 byHn. By weak extension, we construct a weak Hopf algebraWn1, n₂, n₃corresponding toH_nand study their structure.

Wn1, n₂, n₃has a quotient Hopf algebra and a sub-Hopf algebra which are isomorphic to Hn. And as an algebra,Wn1, n2, n3can be decomposed into a direct sum of two algebras, one of which isH_n. The coalgebra structures of these weak Hopf algebras are described by their Ext quiver8,9.

We organize our paper as follows. In Section 2, we introduce the Hopf algebraHn. InSection 3, we define a class of weak Hopf algebrasWn1, n₂, n₃. InSection 4, we study

the structure ofWn1, n2, n3and decomposeWn1, n2, n3into a direct sum ofHnand some algebra of polynomials as an algebra. We give the Ext-quiver of coalgebra of Wn1, n₂, n₃ and prove thatWn1, n₂, n₃has a quotient Hopf algebra and a sub-Hopf algebra which are isomorphic toHn.

2. A Quiver Hopf Algebra

The Hopf AlgebraF_qis defined in10. Letq∈k\0. As ak-algebraF_qis generated bya, b, andxsubject to the relations

ab1, ba1, xaqax, xbq⁻¹bx. 2.1

The coalgebra structure ofF_qis determined by

Δa a⊗a, Δb b⊗b, Δx x⊗a1⊗x.

ε1 εa εb 1, εx 0. 2.2

We generalizeF_q toHn, which is defined as follows. Letkbe a field,q ∈ k\0,i 1,2, . . . , n. As a k-algebra Hn is generated by K, K⁻¹, and Xi, i 1,2, . . . , nsubject to the relations

KK⁻¹1, K⁻¹K1, X_iKqKX_i, X_iK⁻¹q⁻¹K⁻¹X_i. 2.3

The coalgebra structure ofHnis determined by ΔK K⊗K, Δ

K⁻¹

K⁻¹⊗K⁻¹, ΔXi Xi⊗K1⊗Xi,

εK ε K⁻¹

1, εXi 0.

2.4

The antipodeSis induced by

SK K⁻¹, S K⁻¹

K, SXi −K⁻¹Xi. 2.5

3. A Class of Weak Hopf Algebras

In this section, we construct a class of weak Hopf algebra corresponding toHn. First recall the definition of weak Hopf algebra1.

Definition 3.1. Ak-bialgebraH H, μ, η,Δ, εis called a weak Hopf algebra if there exists T ∈HomkH, Hsuch thatid∗T∗ididandT∗id∗T TwhereTis called a weak antipode ofH.

A weak Hopf algebra is called pointed if it is pointed as a coalgebra. If a weak Hopf algebraHis pointed, then the set of all group-like elementsGHis a regular monoid6.

Now we construct weak Hopf algebraWcorresponding toH_n. The setGWof group- like elements of weak Hopf algebra W is a regular monoid which has generatorsg,g, 1, subject togg gg, g²g g, g²g g.

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

Recall, for any coalgebraC, that the group-like elements inCare the setGC {a∈ C|a /0 andΔa a⊗a}; necessarilyεa 1 fora∈GC. Note that a simple subcoalgebra DofCis one-dimensional⇔ D kafor somea∈GC. A coalgebra is pointed if all of its simple subcoalgebras are one-dimensional. Fora, b ∈GC, thea, b-primitive elements inC are the setPa,bC {c ∈ C | Δc c⊗ab⊗c}; necessarilyεc 0 forc ∈ Pa,bC.

Note thatka−b {la−b| l ∈ k} ⊂ P_a,bC; ana, b-primitive elementcis nontrivial if c /∈ka−b {la−b| l ∈k}. Ifab 1, the 1,1-primitives are simply called primitive;

otherwise they are called skew primitive.

The following result is a generalization of11.

Lemma 3.2. LetWbe the weak Hopf algebra defined above. One has

gP_a,bW⊆P_ga,gbW, gP_a,bW⊆P_ga,gbW. 3.1

Proof. Letu∈P_a,bW, thenΔu u⊗ab⊗u. Hence, Δ

gu Δ

g Δu

g⊗g

u⊗ab⊗u

gu⊗gagb⊗gu∈Pga,gbW.

3.2

The second inclusion is proved similarly.

Corollary 3.3. ForW, one has

dimP_gⁱ¹_,gⁱW dimP_gⁱ_,gⁱ⁻¹W, i≥2, dimP_gⁱ_,gⁱ¹W dimP_gⁱ⁻¹_,gⁱW, i≥2,

dimP_g,g²W dimPgg,gW dimPg,ggW dimPg²,gW.

3.3

Proof. We only prove the first equation. In fact, the mapϕ:P_gⁱ_,gⁱ⁻¹W → P_gⁱ¹_,gⁱW,u→gu is a linear map with inverseψ : P_gi1,gⁱW → P_gⁱ_,gi−1W,v → gv. Hence, P_gⁱ_,gi−1Wand P_gi1,gⁱWare isomorphic as vector spaces.

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

Corollary 3.4. One has

dimP1,gW≤dimPg,ggW, dimPg,1W≤dimPg,ggW. 3.4

Proof. The mapϕ:Pg,1W → Pg,ggW,u→gguis a linear map. Ifϕu ggulg−gg, for somel∈k, thenu∈kGW, the vector space spanned by all group-like elements, because Wis graded. Hence,ulg−1. Therefore, the linear mapϕis an injection. Consequently,

dimP_1,gW≤dimP_g,ggW. 3.5

The proof of the second inequality is similar.

By the above discussion we know that weak Hopf algebraWis determined byP1,gW, P_g,1W, and P_g,ggW. Take x₁, . . . , x_n1 to be linearly independent nontrivial elements in P_1,gW, andy₁, . . . , y_n2linearly independent nontrivial elements inP_g,1W. Let

P_g,ggW

ggP_1,gW gP_g,1W

⊕V, 3.6

andz1, . . . , zn3a basis ofV. ThenWis determined byx1, . . . , xn1,y1, . . . , yn2,z1, . . . , zn3. To summarize, we define weak Hopf algebraWn1, n2, n3 corresponding to Hn as follows.

Definition 3.5. Letkbe a field. For any positive integersn1, n2, n3, and nonzero elementq∈k, we defineWn1, n2, n3to be associative algebra over field k generated by 1, g, g, xi, yj, zk, i1,2, . . . , n₁, j 1,2, . . . , n₂, k1,2, . . . , n₃,subject to

gggg, gg²g, g²g g, 3.7

gxiqxig, gxiq⁻¹xig, i1,2, . . . , n1, 3.8 gy_jqy_jg, gy_jq⁻¹y_jg, j1,2, . . . , n₂, 3.9 gz_kg qz_k, k1,2, . . . , n₃. 3.10 Wn1, n2, n3can be endowed with coalgebra structure by

Δ g

g⊗g, 3.11 Δxi xi⊗g1⊗xi, 3.12 Δ

y_j

y_j⊗1g⊗y_j, 3.13 Δzk zk⊗ggg⊗zk, 3.14 ε1 ε

g ε

g

1, εxi 0, ε y_j

0, εzk 0, 3.15

while the weak antipodeTis induced by T1 1, T

g

g, T g

g, 3.16 Txi −xig, T

yj

−gyj, Tzk −zkg, 3.17

Theorem 3.6. For any positive integersn1, n2, n3,Wn1, n2, n3is a weak Hopf algebra.

Proof. First we must check that the coproductΔis an algebra map. It suffices to prove thatΔ preserves the relations3.7–3.10. It is easy to see thatΔpreserves the relations3.7. And

Δ gx_i

g⊗g

x_i⊗g1⊗x_i gxi⊗g²g⊗gxi

qx_ig

⊗g²g⊗ qx_ig q

xi⊗g1⊗xi

g⊗g

Δ qx_ig

, Δ

gy_j

g⊗g

y_j⊗1g⊗y_j gyj⊗ggg⊗gyj

qy_jg

⊗ggg⊗ qy_jg q

y_j⊗1g⊗y_j g⊗g Δ

qyjg , Δ

gz_kg

g⊗g

z_k⊗ggg⊗z_k g⊗g gzkg⊗ggggggg⊗gzkg

qzk

⊗ggg⊗

qzk

Δ

qz_k .

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 ofT ∗id∗T is the same as that ofT, and the action ofid∗T ∗id is the same as that ofid.

Since

Δ⊗idΔxi Δ⊗id

xi⊗g1⊗xi

x_i⊗g1⊗x_i

⊗g1⊗1⊗x_i xi⊗g⊗g1⊗xi⊗g1⊗1⊗xi,

3.19

we get

id∗T∗idxi xigg

−xig

gxixiidxi, T∗id∗Txi

−xig

ggx_ig

−xig −xi

ggg

−xig Txi.

3.20

Since

Δ⊗idΔ y_j

Δ⊗id

y_j⊗1g⊗y_j

yj⊗1g⊗yj

⊗1g⊗g⊗yj

y_j⊗1⊗1g⊗y_j⊗1g⊗g⊗y_j,

3.21

it follows that

id∗T∗id yj

yjg

−gyj

ggyjyjid yj

, T∗id∗T

y_j

−gyj

gy_jgg

−gyj

−gyjT

y_j

. 3.22

Since

Δ⊗idΔzk Δ⊗id

z_k⊗ggg⊗z_k

zk⊗ggg⊗zk

⊗ggg⊗gg⊗zk

z_k⊗g⊗ggg⊗z_k⊗ggg⊗gg⊗z_k, zkggq⁻¹gzkggg q⁻¹gzkgzk,

ggz_kq⁻¹gggz_kgq⁻¹gz_kgz_k,

3.23

we get

id∗T∗idzk zkgggg

−zkg

gggggzk

z_k−qgz_kgggz_k

z_k−z_kz_kz_kidzk, T∗id∗Tzk

−zkg

ggggzkggggg

−zkg

−zkg

z_kg

−zkg

−zkgTzk.

3.24

4. The Structure of W n

, n

, n

In this section we study the algebra and coalgebra structure ofWn1, n₂, n₃.

It is easy to prove that the elementsgg and 1−gg are a pair of orthogonal central idempotents. SetW1Wn1, n2, n3gg,W2Wn1, n2, n31−gg. We have the following.

Theorem 4.1. Wn1, n₂, n₃can be written as a direct sum of two-sided ideals Wn1, n₂, n₃ W₁

W₂. And one has the following.

1As an algebra,W1is isomorphic toHn, wherenn1n2n3.

2As an algebra,W2is isomorphic to the free associative algebrakY1, . . . , Ytoftgenerators, wheretn₁n₂.

Proof. 1Sinceggand 1−ggare a pair of orthogonal central idempotents, Wn1, n2, n3 Wn1, n2, n3gg⊕Wn1, n2, n3

1−gg

W1⊕W2. 4.1 The isomorphism W₁ → H_n is induced by x_igg → X_i,y_jgg → X_n1j,z_kgg → X_n1n2k, gg→1,g²g→K.

2Note thatzk1−gg 0 andxi1−ggyj1−gg yj1−ggxi1−gg. Since x_i1−gg, yj1−ggare generators ofW₂, the isomorphismW₂ → kY1, . . . , Y_tis defined by1−g²→1,x_i1−g²→Y_i,y_j1−g²→Y_n1j.

A weak Hopf idealJof a weak Hopf algebraHis a bi-ideal such thatTJ⊂J, where Tis the weak antipode ofH. It is easy to see thatH/Jhas a natural structure of a weak Hopf algebra.

Theorem 4.2. The idealJinWn1, n₂, n₃generated by 1−ggis a weak Hopf ideal. And the quotient weak Hopf algebraWn1, n2, n3/Jis a Hopf algebra, which is isomorphic toHn, wherenn1n2 n₃.

Proof. Since

Δ 1−gg

1⊗1−gg⊗gg

1⊗1−gg⊗1gg⊗1−gg⊗gg

1−gg

⊗1gg⊗ 1−gg

, T

1−gg

T1−T g

T g

1−gg,

4.2

Jis a weak Hopf ideal inWn1, n2, n3.

The isomorphism Wn1, n₂, n₃/J → H_n is defined by g J → K,g J → K⁻¹, x_iJ →X_i,gy_jJ →X_n1j,z_kJ→X_n1n2k.

Now we give the Ext quiver ofWn1, n₂, n₃. For the definition and calculation of Ext quiver, we refer to5,8,9,12.

The Ext quiver ofWn1, n₂, n₃is shown inFigure 1. The multiplicity of arrowg· → ·1 isn₁. The multiplicity of arrow 1· → ·gisn₂. The multiplicity of other arrows is alln.

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

Proof. The isomorphismH → Hnis induced bygg →1,g → K,g →K⁻¹,xi →Xi,gyj → Xn1j,zk→Xn1n2k.

Remark 4.4. The isomorphisms described inTheorem 4.1are not isomorphisms of bialgebras.

Remark 4.5. The weak Hopf algebras discussed in4,5also have quotient Hopf algebras and sub-Hopf algebras which are isomorphic to the related Hopf algebras.

· · ·

g³ g² g

· · · g³ g² g 1

gg

Figure 1: Ext quiver ofWn1, n2, n3.

· · · g³ g² g

· · · g³ g² g

gg

Figure 2: A subquiver of Ext quiver ofWn1, n2, n3.

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 China10571153.

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