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Maschke-Type Theorem, Duality Theorem, and theGlobal Dimension for Weak Crossed ProductsBing-Liang Shen aa Department of Mathematics , Shanghai Jiao Tong University , Shanghai , ChinaPublished online: 15 May 2012.

To cite this article: Bing-Liang Shen (2012) Maschke-Type Theorem, Duality Theorem, and the Global Dimension for WeakCrossed Products, Communications in Algebra, 40:5, 1802-1820, DOI: 10.1080/00927872.2011.557812

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Communications in Algebra®, 40: 1802–1820, 2012Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927872.2011.557812

MASCHKE-TYPE THEOREM, DUALITY THEOREM, AND THEGLOBAL DIMENSION FOR WEAK CROSSED PRODUCTS

Bing-liang ShenDepartment of Mathematics, Shanghai Jiao Tong University, Shanghai, China

The notion of a crossed product with a Hopf algebroid was introduced by Böhm andBrzezinski [12]. It is well-known that weak Hopf algebras (see Böhm et al. [10])is an example of Hopf algebroids. Then we get the definition and some property of aweak crossed product A#�H over an algebra A and a weak Hopf algebra H . Nextwe give a Maschke-type theorem for the weak crossed product over a semisimpleweak Hopf algebra H . Furthermore, we obtain an analogue of the Nikshych’s dualitytheorem for weak crossed products. Finally, using this duality theorem, we prove thatthe global dimension of A equals to the global dimension of A#�H if H and H∗ areboth semisimple.

Key Words: Bialgebroid; Duality theorem; Global dimension; Maschke-type theorem; Weak crossedproduct; Weak Hopf algebra.

2000 Mathematics Subject Classification: 16W30.

1. INTRODUCTION

In the theory of the classical Hopf algebras, the Cohen–Fishman Maschke-typetheorem (see [15]) and the Blattner–Cohen–Montgomery duality theorem (see [7, 16])are very celebrated. The study of these aspects can be found in the literature(cf. [1, 8, 19, 26, 27]).

A new notion of Hopf algebroid was introduced by Böhm and Szlachányi[13]. Hopf algebroid was motivated by the study of depth 2 Frobenius extensions.It is encouraging that one can find quantum groupoids in the literature that satisfythe axioms of Hopf algebroids. The weak Hopf algebras with bijective antipode,the examples of Lu-Hopf algebroids in [14] are contained in the category of thisHopf algebroid. A survey of Hopf algebroids and their applications may be foundin [11–13]. The crossed product over a left algebroid was studied and the sufficientand necessary conditions making the crossed product into an algebra are obtainedin [12].

In 1996, Böhm and Szlachányi [9] introduced weak Hopf algebras (or quantumgroupoids) as a generalization of ordinary Hopf algebras and groupoid algebras.A general theory for these objects was subsequently developed in Böhm et al. [10].

Received March 13, 2010; Revised July 2, 2010. Communicated by Q. Wu.Address correspondence to Bing-liang Shen, Department of Mathematics, Shanghai Jiao Tong

University, Shanghai 200240, China; E-mail: bingliangshen@yahoo.com.cn

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Briefly, the axioms of a weak Hopf algebra are the same as the ones for a Hopfalgebra, except that the coproduct of the unit, the product of the counit and theantipode conditions are replaced by weaker properties. The main motivation forstudying weak Hopf algebras comes from quantum field theory, operator algebrasand representation theory. We refer the reader to [20–22] for the further study.

In the categorial level, Alonso Álvarez and González Rodríguez [5] introducedthe notion of a weak crossed product and Alonso Álvarez et al. [3] investigated weakcleft theory for weak Hopf algebras (also see [2] and [4]).

The aim of this article is to continue the study of the weak crossed productA#�H without the categorial point of view and to prove an analogue of the Cohen–Fishman Maschke-type theorem, the Blattner–Cohen–Montgomery duality theoremmentioned as above. The global dimension for the weak crossed product is alsodiscussed.

This article is organized as follows.In Section 2, we recall some definitions and basic results related to weak

Hopf algebras and bialgebroids. From the notion of a crossed product over a leftalgebroid, we get the definition of a weak crossed product A#�H over a weak Hopfalgebra H . The sufficient and necessary conditions making the weak crossed productA#�H into an algebra are obtained.

In Section 3, we give a Maschke-type theorem for the weak crossed productA#�H over a semisimple weak Hopf algebra H with � invertible (see Theorems 3.9and 3.10), which generalizes the well-known Maschke-type theorem in [8, 15].

In Section 4, we prove an analogue of the Nikshych’s duality theorem forweak crossed products: Let H be a finite dimensional weak Hopf algebra and A#�Hbe a weak crossed product with � invertible. Then there is a canonical isomorphismbetween the algebras �A#�H�#H∗ and End�A#�H�A.

In Section 5, using the above duality theorem, we consider the globaldimension of weak crossed products. If H and H∗ are semisimple, thengl.dim�A#�H� = gl.dim�A�, which generalizes the results of crossed products and(weak) smash products.

2. BASIC DEFINITIONS AND RESULTS

Throughout this article, k denotes a fixed field, the tensor product ⊗ = ⊗k, andHom is always assumed to be over k. For an algebra A and a coalgebra C, we havethe convolution algebra Conv�C�A� = Hom�C�A� as spaces, with the multiplicationgiven by

�f ∗ g��c� = mA�f ⊗ g��C�c� = f�c1�g�c2��

for all f� g ∈ Hom�C�A�� c ∈ C.

2.1. Weak Bialgebras

Recall from Böhm et al. [10] that a weak k-bialgebra H is both a k-algebra(m��) and a k-coalgebra (�� �) such that ��hk� = ��h���k�, for all h� k ∈ H , and

�2�1� = 11 ⊗ 121′1 ⊗ 1′2 = 11 ⊗ 1′112 ⊗ 1′2� (2.1)

��hkl� = ��hk1���k2l� = ��hk2���k1l�� (2.2)

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for all h� k� l ∈ H , where 1′ stands for another copy of 1. We use the Sweedler’snotation (see [18, 24]) for the comultiplication. Namely,

��h� = h1 ⊗ h2�

We summarize the elementary properties of weak bialgebras. The maps �t� �s:H −→ H defined by

�t�h� = ��11h�12 �s�h� = 11��h12��

�t and �s are called the target map and source map, and their images Ht and Hs arecalled the target and source space. They can be described as follows:

Ht = h ∈ H � �t�h� = h� = h ∈ H ���h� = 11h⊗ 12 = h11 ⊗ 12��

Hs = h ∈ H � �s�h� = h� = h ∈ H ���h� = 11 ⊗ h12 = 11 ⊗ 12h��

For all g� h� k ∈ H , we also have

�t�h��s�k� = �s�k��t�h��

and its dual property

�s�h1�⊗ �t�h2� = �s�h2�⊗ �t�h1��

Finally, �t�1� = �s�1� = 1 and �t�h��t�g� = �t��t�h�g� �s�h��s�g� = �s�h�s�g��� Thisimplies that Ht and Hs are subalgebras of H .

2.2. Weak Hopf Algebras

A weak Hopf algebra H is a weak bialgebra together with a k-linear map S �H −→ H (called the antipode) satisfying

S ∗ idH = �s� idH ∗ S = �t� S ∗ idH ∗ S = S�

where ∗ is the convolution product. It follows immediately that

S = �s ∗ S = S ∗ �t�

If the antipode exists, then it is unique. The antipode S is both an anti-algebraand an anti-coalgebra morphism. If H is a finite-dimensional weak Hopf algebraover k, then S is automatically bijective and the dual H∗ = Hom�H� k� has a naturalstructure of a weak Hopf algebra with the structure operations dual to those of H .We always assume H is finite dimensional. Now we recall some properties about S.

Let H be a weak Hopf algebra. Then we have the following conclusionsby [10]:

(1) �t � S = �t � �s = S � �s� �s � S = �s � �t = S � �t;

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(2) x1 ⊗ x2S�x3� = x1 ⊗ �t�x2� = 11x⊗ 12;(3) S�x1�x2 ⊗ x3 = �s�x1�⊗ x2 = 11 ⊗ x12;(4) x1 ⊗ S�x2�x3 = x1 ⊗ �s�x2� = x11 ⊗ S�12�;(5) x1S�x2�⊗ x3 = �t�x1�⊗ x2 = S�11�⊗ 12x;(6) x1y ⊗ x2 = x1 ⊗ x2S�y�� for all y ∈ Hs;(7) x1 ⊗ zx2 = S�z�x1 ⊗ x2� for all z ∈ Ht�

Let H be a weak Hopf algebra with a bijective antipode SH . Then Hcop is alsoa weak Hopf algebra with antipode S−1 (here, S−1 is the composite-inverse of theantipode SH ). At this time

S−1�h2�h1 = S−1�s�h� = ��h11�12 � �t�h�� h2S−1�h1� = S−1�t�h� = 11��12h� � �s�h��

2.3. Bialgebroids

We recall from [13] the notion of a left bialgebroid.

Definition 2.1 (Böhm and Szlachányi [13]). A left bialgebroid �L consists of thedata �A� L� sL� tL� L� �L�. The A and L are associative unital rings, the total and baserings, respectively. The sL � L → A and tL � Lop → A are ring homomorphisms suchthat the images of L in A commute making A an L-L-bimodule via

l · a · l′ �= sL�l�tL�l′�a�

Introducing the Sweedler’s notation L�a� ≡ a1 ⊗L a2 ∈ A⊗L A, the identities

a1tL�l�⊗L a2 = a1 ⊗L a2sL�l��

L�1A� = 1A ⊗L 1A�

L�ab� = L�a� L�b��

�L�1A� = 1L�

�L�asL � �L�b�� = �L�ab� = �L�atL � �L�b��

are required for all l ∈ L and a� b ∈ A, where �L � A → L is a homomorphism.

For the definitions of right bialgebroids and Hopf algebroids, refer to [13].

Example 2.2. Let H be a weak Hopf algebra. Then:

(1) �H�Ht� id� S−1� �� �t� is a left bialgebroid;

(2) �H�Hs� id� S� �� �s� is a right bialgebroid.

The other properties of bialgebroids can be seen in [11–13].

2.4. Crossed Product A#�H

We review the notion of crossed products with Hopf algebroids.

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Definition 2.3 (Böhm and Brzezinski [12]). Let � = �H� L� s� t� � �� be a leftbialgebroid and A an L-ring with unit map � � L → A. � measures A if there existsa k-linear map, called a measuring, H ⊗ A → A� h⊗ b → h · b such that, for allh ∈ H� l ∈ L� b� b′ ∈ A:

(a) h · 1A = ����h��;(b) �t�l�h� · b = �h · b���l� and �s�l�h� · b = ��l��h · b�;(c) h · �bb′� = �h1 · b��h2 · b′�.

Definition 2.4 (Böhm and Brzezinski [12]). Let � = �H� L� s� t� � �� be a leftbialgebroid and � � L → A an L-ring, measured by �. An A-valued 2-cocycle � on� is a k-linear map H ⊗Lop H → A (where the right and left Lop-module structureson H are given by right and left multiplication by t�l�, respectively) satisfying thefollowing equation:

(a) ��s�l�h� k� = ��l���h� k� and ��t�l�h� k� = ��h� k���l�(b) �h1 · ��l����h2� k� = ��h� s�l�k�(c) ��1� h� = ����h�� = ��h� 1�(d) �h1 · ��k1�m1����h2� k2m2� = ��h1� k1���h2k2�m��

for all h� k�m ∈ H� l ∈ L.A �-measured L-ring A is called a �-twisted left �-module if a 2-cocycle �

satisfies:

(e) 1H · b = b;(f) �h1 · �k1 · b����h2� k2� = ��h1� k1��h2k2 · b��for all h� k ∈ H� b ∈ A.

Lemma 2.5 (Böhm and Brzezinski [12]). Let � = �H� L� s� t� � �� be a leftbialgebroid and � � L → A an L-ring, measured by �. Let � � H ⊗Lop H → A be a map,satisfying properties (a) and (b) in Definition 2.4. Consider the k-module A⊗L H ,where the left L-module structure on H is given by multiplication by s�l� on the left.A⊗L H is an associative algebra with unit 1A ⊗L 1H and product

�A⊗L H�⊗ �A⊗L H� → A⊗L H�

�b ⊗L h�⊗ �b′ ⊗L h′� → b�h1 · b′���h2� h

′1�⊗L h3h

′2�

if and only if � is a 2-cocycle and A is a �-twisted �-module. The resulting associativealgebra is called a crossed product of A with � and is denoted by A#�H .

Let H be a weak Hopf algebra. From Example 2.2, we get the definition ofthe weak crossed product over a weak Hopf algebra H .

Definition 2.6. Let H be a weak Hopf algebra and A an algebra. H measures Aif there exists a k-linear map, called a measuring, H ⊗ A → A� h⊗ b → h · b suchthat, for all h ∈ H� l ∈ Ht� b� b

′ ∈ A,

h · 1A = �t�h� · 1A� (2.3)

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�S−1�l�h� · b = �h · b��l · 1A� and �lh� · b = �l · 1A��h · b�� (2.4)

h · �bb′� = �h1 · b��h2 · b′�� (2.5)

Definition 2.7 (cf. Rodríguez Raposo [23]). Let H be a weak Hopf algebrameasuring A. An A-valued 2-cocycle � on H is a k-linear map H ⊗Ht

op H → A(where the right and left Ht

op-module structures on H are given by right and leftmultiplication by S−1�l�, respectively) satisfying

��hS−1�l�� k� = ��h� S−1�l�k�� (2.6)

��lh� k� = �l · 1A���h� k� and ��S−1�l�h� k� = ��h� k��l · 1A�� (2.7)

�h1 · �l · 1A����h2� k� = ��h� lk�� (2.8)

��1� h� = �t�h� · 1A = ��h� 1�� (2.9)

�h1 · ��k1�m1����h2� k2m2� = ��h1� k1���h2k2�m�� (2.10)

for all h� k�m ∈ H� l ∈ Ht.

An H-measured algebra A is called a �-twisted left H-module if a 2-cocycle �satisfies

1H · b = b� (2.11)

�h1 · �k1 · b����h2� k2� = ��h1� k1��h2k2 · b�� (2.12)

for all h� k ∈ H� b ∈ A.

Corollary 2.8. Let H be a weak Hopf algebra measuring A. Let � � H ⊗Htop H → A

be a map, satisfying properties (2.7)–(2.8) in Definition 2.7. Consider the k-space A⊗Ht

H , where the left Ht-module structure on H is given by multiplication on the left andthe right Ht-module structure on A is given by a · z = a�z · 1A� for all a ∈ A� z ∈ Ht.A⊗Ht

H is an associative algebra with unit 1A ⊗Ht1H and product

�A⊗HtH�⊗ �A⊗Ht

H� → A⊗HtH�

�b ⊗Hth�⊗ �b′ ⊗Ht

h′� → b�h1 · b′���h2� h′1�⊗Ht

h3h′2�

if and only if � is an A-valued 2-cocycle � on H and A is a �-twisted left H-module.The resulting associative algebra is called a weak crossed product of A with H and isdenoted by A#�H .

3. THE MASCHKE-TYPE THEOREM FOR WEAK CROSSED PRODUCTS

In this section, we first state some properties of a weak crossed product A#�Hover a weak Hopf algebra H . Next we will give a Maschke-type theorem for theweak crossed product A#�H over a semisimple weak Hopf algebra H .

Referring to Böhm and Brzezinski [12], we get the notion of an inverse of a2-cocycle � and its identities.

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Definition 3.1. Let H be a weak Hopf algebra measuring A. An A-valued 2-cocycle � on H is invertible if there exists a k-linear map � � H ⊗Ht

H → A (where theright and left Ht-module structures on H are given by right and left multiplication,respectively), satisfying

��lh� k� = �l · 1A���h� k� and ��S−1�l�h� k� = ��h� k��l · 1A�� (3.1)

��h1� k��h2 · �l · 1A�� = ��h� S−1�l�k�� (3.2)

��h1� k1���h2� k2� = h · �k · 1A� and ��h1� k1���h2� k2� = hk · 1A� (3.3)

for all h� k ∈ H and l ∈ Ht. The map � is called an inverse of �.

Proposition 3.2. Let H be a weak Hopf algebra measuring A, and � an invertibleA-valued 2-cocycle on H . Then an inverse � of � is unique, and the following equationshold:

��1H� h� = �t�h� · 1A = ��h� 1H�� (3.4)

��h� k� = ��t�h1k1� · 1A���h2� k2�� (3.5)

��h� k� = ��h1� k1��h2 · �k2 · 1A��� (3.6)

h · ��k�m� = ��h1� k1���h2k2�m1���h3� k3m2�� (3.7)

h · ��k�m� = ��h1� k1m1���h2k2�m2���h3� k3�� (3.8)

for all h� k�m ∈ H .

From the properties of � and the multiplication of weak crossed product, wecan easily get the following equations.

Proposition 3.3. Let A#�H be a weak crossed product over a weak Hopf algebra H .Then for all a� b ∈ A, h� k ∈ H , and y ∈ Hs,

�a#�y��b#�k� = ab#�yk�

�a#�h��b#�1H� = a�h1 · b�#�h2�

�a#�h��1A#�k� = a��h1� k1�#�h2k2

hold.

Definition 3.4. Let H be a weak Hopf algebra and A an algebra. We define thefollowing set:

WC�H�A� = u ∈ Hom�H�A� � u ∗ v�h� = u��t�h��� v ∗ u�h� = u��s�h���

× ∃v ∈ Hom�H�A��∀h ∈ H��

If u ∈ WC�H�A�, then we say u is weak convolution invertible.

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Example 3.5. Let H be a weak Hopf algebra with an antipode SH . Then we havethe identity map idH ∈ WC�H�H� with weak convolution inverse SH .

Next we will refer Böhm and Brzezinski [12] and give the necessary andsufficient condition such that � is invertible.

Proposition 3.6. Let A#�H be a weak crossed product. Then � H → A#�H definedby �h� = 1#�h is weak convolution invertible if and only if � is invertible.

Proof. We define the inverse of is −1�h� = ��S�h2�� h3�#�S�h1�. Here, � is theinverse of �. The other proof is similar to [12]. �

The following lemma given in [10] is needed in the sequel.

Lemma 3.7. The following conclusions on a weak Hopf algebra H are equivalent:

(1) H is semisimple;(2) There exists a normalized right integral x ∈ H; that is, for all h ∈ H , xh = x�s�h�,

and �s�x� = 1.

Lemma 3.8. Let H be a finite dimensional weak Hopf algebra and A#�H be a weakcrossed product with � invertible, and let x be a nonzero right integral in H . Let V�Wbe left A#�H-modules, and let � ∈ HomA#�1H

�V�W�. Define � � V → W by

��v� = −1�x1� · �� �x2�v�� (3.9)

for all v ∈ V and is given in Proposition 3.6. Then � is a left A#�H-linear map.

Proof. Note that −1 exists by Proposition 3.6 so that (3.9) makes sense. Weidentity A with A#�1H and write a#�h = �a#�1H��1A#�h� = a �h�� First, we show �is left A-linear. Since

�h�a = �1A#�h��a#�1H� = �h1 · a���h2� 11�#�h312�2�6�= �h1 · a���h211� 1H�#�h312 = �h1 · a���h2� 1H�#�h3

�2�9�= �h1 · a���t�h2� · 1A�#�h3 = �h1 · a��S�11� · 1A�#�12h2

= �h1 · a��S�11� · 1A��12 · 1A�#�h2 = ��h1 · a�#�1H��1A#�h2�

= �h1 · a� �h2��

we get

�h1�a −1�h2� = �h1 · a� �h2�

−1�h3�

= �h1 · a� ��t�h2�� = �11h · a� �12�= ��11h · a�#�1H��1A#�12�= �11h · a�#�12 = �11h · a��12 · 1A�#�1H�2�5�= h · a#�1H = h · a�

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Now we prove

��av� = −1�x1� · �� �x2�av�= −1�x1� · ���x2 · a� �x3�v�= −1�x1��x2 · a� · �� �x3�v�= � −1�x1� �x2�a

−1�x3�� · �� �x4�v�= � ��s�x1��a

−1�x2�� · �� �x3�v�= ��1#�11��a#�1����S�x213�� x314�#�S�x112��� · �� �x4�v�= ��1#�11��a��S�x2�� x3�#�S�12�S�x1��� · �� �x4�v�= ��11 · �a��S�x3�� x4�����12� S�15�S�x2��#�13S�14�S�x1�� · �� �x5�v�= ��11 · �a��S�x3�� x4�����12� S�14�S�x2����t�13� · 1A�#�S�x1�� · �� �x5�v��2�7�= ��11 · �a��S�x3�� x4�����S−1��t�13��12� S�14�S�x2��#�S�x1�� · �� �x5�v��2�6�= ��11 · �a��S�x3�� x4�����12� S�x2��#�S�x1�� · �� �x5�v��2�7�= ��11 · �a��S�x3�� x4����12 · 1A���1H� S�x2��#�S�x1�� · �� �x5�v��2�9�= �a��S�x3�� x4����t�S�x2�� · 1A�#�S�x1�� · �� �x5�v�= �a��S�x2�� x3��S�11� · 1A�#�12S�x1�� · �� �x4�v�= �a��S�x2�� x3�#�S�x1�� · �� �x4�v�= �a#�1����S�x2�� x3�#�S�x1�� · �� �x4�v�= a · � −1�x1� · �� �x2�v�� = a · ��v��

To check that �� �h�v� = �h���v�, we need the fact that x is a right integral. ByEq. (1.16) of Vecsernyés [25], that is, x1h⊗ x2 = x1 ⊗ x2S�h�, for any h ∈ H , we have

h11 ⊗ ��x12� = h11 ⊗ x112 ⊗ x213

= h11 ⊗ x1 ⊗ x2S�12�13 = h11 ⊗ x1 ⊗ x2S�12�

= h1 ⊗ x1 ⊗ x2�s�h2� = h1 ⊗ x1 ⊗ x2S�h2�h3

= h1 ⊗ x1h2 ⊗ x2h3�

According to Eq. (4.18) in [12], we have the equation ��h� l� = �h1� �l1� −1�h2l2�,

so we get

−1�h1���h2� l� = �l1� −1�hl2�� (3.10)

Then

�� �h�v� = −1�x1� · �� �x2� �h�v�= −1�x1� · �����x2� h1�#�x3h2�v�

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= −1�x1� · ����x2� h1� �x3h2�v�

= −1�x1���x2� h1� · �� �x3h2�v�

�3�10�= �h1� −1�x1h2� · �� �x2h3�v�

= �h11� −1�x112� · �� �x213�v�

= �h11� −1�x112� · �� �x2�v�

= ��1#�h11����S�x213�� x314�#�S�x112��� · �� �x4�v�= ��1#�h��1#�11����S�x2�� x3�#�S�x112��� · �� �x4�v�= �1#�h� · ����S�x2�� x3�#�S�x1�� · �� �x4�v��= �h� · � −1�x1� · �� �x2�v��= �h� · ��v��

We obtain � is a left A#�H-linear map. �

Theorem 3.9. Let H be a finite dimensional weak Hopf algebra with a normalizedright integral x and A#�H be a weak crossed product with � invertible. If W ⊆ V are(left) A#�H-modules so that W has an A-complement in V , then W has an A#�H-complement in V .

Proof. Since W has an A-module complement in V , there exists an A-moduleprojection � � V → W . From Proposition 3.6, we get is weak convolutioninvertible. By Lemma 3.8, the map

��v� = −1�x1� · �� �x2�v�

is a left A#�H-module map.We claim that � is also projection of V onto W . For if w ∈ W� then ��w� = w,

and thus

��w� = −1�x1� · �� �x2�w� = −1�x1� · � �x2�w�= −1�x1� �x2� · w = ��s�x�� · w= �1� · w = �1#�1� · w = w�

Hence W has an A#�H-complement ker �. �

We can now obtain our version of Maschke-type Theorem.

Theorem 3.10 (The Maschke-type Theorem). Let H be a finite dimensional weakHopf algebra with a normalized right integral x and A#�H be a weak crossed productwith � invertible.

(1) Let V be an A#�H-module. If V is completely reducible as an A-module, then V iscompletely reducible as an A#�H-module.

(2) If A is semisimple, so is A#�H .

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1812 SHEN

Proof. (1) is immediate from Theorem 3.9.(2) follows from (1), using the fact that an algebra is semisimple if and only if

every module is completely reducible. �

Remark 3.11.

(1) Let H be a finite dimensional semisimple Hopf algebra. Then A#�H is theordinary crossed product (see Blattner et al. [6]), and we get the Maschke-typetheorem in Blattner and Montgomery [8].

(2) If A is a left weak H-module algebra (see Nikshych [19]) and ��h� g� = hg · 1A,we define ��h� g� = hg · 1A. It is easy to check � satisfies (2.6)–(2.12) and � is theinverse of �. Then A#�H = A#H is the weak smash product (see Nikshych [19]).The multiplication is turned into �a#h��b#l� = a�h1 · b�#h2l, and the Maschke-type theorem for the weak smash product can be obtained immediately (seeZhang [27]).

4. THE DUALITY THEOREM FOR WEAK CROSSED PRODUCTS

In this section, we will prove an analogue of the Blattner–Cohen–Montgomery’s duality theorem for weak crossed products, which extends the mainresult given by Nikshych [19].

Let H be a finite dimensional weak Hopf algebra and A#�H be a weak crossedproduct. Then A#�H is a left H∗-module algebra via the formula

� · �a#�h� = a#��� ⇀ h� =< �� h2 > a#�h1�

for all � ∈ H∗� h ∈ H , and a ∈ A.Moreover, we can also define a right H∗

t -module on A#�H by

�a#�h� · �′ = a#��S−1H∗ ��

′� ⇀ h� =< S−1H∗ ��

′�� h2 > a#�h1 =< �′� S−1�h2� > a#�h1�

for all �′ ∈ H∗t �

Recall from Blattner and Montgomery [8] that if H is an ordinary finitedimensional Hopf algebra, then �A#�H�#H∗ Mn�A�, where n is the dimensionof H . Now we will construct a canonical isomorphism between the weaksmash product algebra �A#�H�#H∗ and the endomorphism algebra End�A#�H�A,where the right A-module action on A#�H is the multiplication, i.e., �a#�h� · b =�a#�h��b#�1H�.

Lemma 4.1. Let H be a finite dimensional weak Hopf algebra and A#�H be a weakcrossed product with � invertible. Then the following equations hold:

��h1� l1��h2 · �l2 · a�� = �h1l1 · a���h2� l2�� (4.1)

��h1� l1��h2 · ��l2�m�� = ��h1l1�m1���h2� l2m2�� (4.2)

for all h� l�m ∈ H� a ∈ A.

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Proof. Since A#�H is a weak crossed product, Eqs. (2.10) and (2.12) hold. FromEq. (2.12), we have

��h1� l1��h2 · �l2 · a����h3� l3��2�12�= ��h1� l1���h2� l2��h3l3 · a��3�3�= �h1l1 · 1A��h2l2 · a� = hl · a�

and

�h1l1 · a���h2� l2� = ��h1� l1��h2 · �l2 · a����h3� l3���h4� l4�

= ��h1� l1��h2 · �l2 · a���h3 · �l3 · 1A��= ��h1� l1��h2 · ��l2 · a��l3 · 1A���= ��h1� l1��h2 · �l2 · a���

Next from Eq. (2.10), we get

��h1� l1��h2 · ��l2�m1����h3� l3m2��2�10�= ��h1� l1���h2� l2���h3l3�m�

�3�3�= �h1l1 · 1A���h2l2�m�

= �S�11� · 1A���12hl�m�

�2�7�= �11 · 1A��12 · 1A���hl�m� = ��hl�m��

and

��h1l1�m1���h2� l2m2� = ��h1� l1��h2 · ��l2�m1����h3� l3m2���h4� l4m3�

�3�3�= ��h1� l1��h2 · ��l2�m1���h3 · �l3m2 · 1A��= ��h1� l1��h2 · ���l2�m1��l3m2 · 1A���= ��h1� l1��h2 · ���l2�m1���t�l3m2� · 1A���= ��h1� l1��h2 · ���l2�m1���t�l3�t�m2�� · 1A���= ��h1� l1��h2 · ���l2� 11m���t�l312� · 1A����2�6�= ��h1� l1��h2 · ���l2�m���t�l3� · 1A����2�7�= ��h1� l1��h2 · ��S−1��t�l3��l2�m��

= ��h1� l1��h2 · ��l2�m���

Hence the two equations hold. �

Lemma 4.2. The map � � �A#�H�#H∗ → End�A#�H�A defined by

���x#�h�#���y#�g� = �x#�h��y#��� ⇀ g�� = �x#�h��y#� < �� g2 > g1��

for all x� y ∈ A� h� g ∈ H�� ∈ H∗, is a homomorphism of algebras.

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1814 SHEN

Proof. First we need to check that � is well defined. We check

���x#�zh�#���y#�g� = �x#�zh��y#� < �� g2 > g1�

= x�zh1 · y���h2� g1�#�h3g2 < �� g3 >

�2�4�= x�z · 1A��h1 · y���h2� g1�#�h3g2 < �� g3 >

= �x�z · 1A�#�h��y#� < �� g2 > g1�

= ���x�z · 1A�#�h�#���y#�g��and

���x#�h�#���y#�zg� = �x#�h��y#��� ⇀ zg�� = �x#�h��y#�zg1 < �� g2 >�

= x�h1 · y���h2� 11zg1�#�h312g2 < �� g3 >

�2�6�= x�h1 · y���h2� zg1�#�h3g2 < �� g3 >

�2�8�= x�h1 · y��h2 · �z · 1A����h3� g1�#�h4g2 < �� g3 >

= x�h1 · y�z · 1A����h2� g1�#�h3g2 < �� g3 >

= �x#�h��y�z · 1A�#�� ⇀ g�

= ���x#�h�#���y�z · 1A�#�g��for all x� y ∈ A� h� g ∈ H and z ∈ Ht. For � ∈ H∗

t , we compute

���x#�h�#����y#�g� = x�h1 · y���h2� g1�#�h3��� ⇀ g2�

= x�h111 · y���h212� g1�#�h313�� ⇀ 1H��� ⇀ g2�

= x�h111 · y���h212�� ⇀ 1H�1� g1�#�h3�� ⇀ 1H�2�� ⇀ g2�

= x�h1�� ⇀ 1H�1 · y���h2�� ⇀ 1H�2� g1�#�h3�� ⇀ 1H�3�� ⇀ g2�

= ���x#�h�� ⇀ 1H��#���y#�g�

= ���x#�S−1H∗ ��� ⇀ h�#���y#�g�

= ���x#�h� · �#���y#�g��The following computation shows that � commutes with the right action ofall w ∈ A.

���x#�h�#����y#�g� · w� = ���x#�h�#���y�g1 · w���g2� 11�#�g312��2�9�= ���x#�h�#���y�g1 · w�#�g2�= �x#�h��y�g1 · w�#�� ⇀ g2�

= �x#�h��y�g1 · w�#� < �� g3 > g2�

= �x#�h��y#��� ⇀ g���w#�1H�

= ����x#�h�#���y#�g�� · w�

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DUALITY AND GLOBAL DIMENSION FOR CROSSED PRODUCTS 1815

Finally,

����x#�h�#����x′#�h

′�#�′���y#�g� = ���x#�h��x′#���1 ⇀ h′��#�2�

′��y#�g�

= �x#�h��x′#���1 ⇀ h′���y#���2�

′ ⇀ g��

= �x#�h��� · ��x′#�h′��y#���′ ⇀ g����

= ���x#�h�#�� � ���x′#�h′�#�′��y#�g��

for all x� x′� y ∈ A� h� h′� g ∈ H��� �′ ∈ H∗; therefore, � is a homomorphism ofalgebras. �

Let fi� be a basis of H and �i� be the dual basis of H∗, i.e., such that <fi� �j >= �ij for all i� j. Then we have identities

∑

i

fi < h��i >= h�∑

i

< fi� � > �i = ��

for all h ∈ H and � ∈ H∗; moreover, the element of∑

i fi ⊗ �i ∈ H ⊗H∗ does notdepend on the choice of fi�.

Let us define a linear map � � End�A#�H�A → �A#�H�#H∗ by

T → ∑

i

�T���fi3� S−1�fi2��#�fi4��1A#�S

−1�fi1���#�i�

Lemma 4.3. The maps � and � are inverse of each other.

Proof. We need to check that

� � � = id�A#�H�#H∗ and � � � = idEnd�A#�H�A�

For all a ∈ A� h ∈ H and � ∈ H∗, we compute

� � ���a#�h�#�� = ∑

i

����a#�h�#�����fi3� S−1�fi2��#�fi4��1A#�S

−1�fi1���#�i

= ∑

i

��a#�h����fi3� S−1�fi2��#� < �� fi5 > fi4��1A#�S

−1�fi1���#�i

= ∑

i

��a�h1 · ��fi3� S−1�fi2�����h2� fi4�#�h3

< �� fi6 > fi5��1A#�S−1�fi1���#�i

= ∑

i

�a�h1 · ��fi4� S−1�fi3�����h2� fi5���h3fi6� S−1�fi2��

#�h4 < �� fi8 > fi7S−1�fi1��#�i

�2�10�= ∑

i

�a�h1 · ��fi5� S−1�fi4����h2 · ��fi6� S−1�fi3�����h3� fi7S−1�fi2��

#�h4fi8S−1�fi1��#�i < �� fi9 >

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1816 SHEN

= ∑

i

�a�h1 · ���fi5� S−1�fi4����fi6� S−1�fi3������h2� fi7S

−1�fi2��

#�h3fi8S−1�fi1��#�i < �� fi9 >

�3�3�= ∑

i

�a�h1 · �fi4S−1�fi3� · 1A����h2� fi5S−1�fi2��

#�h3fi6S−1�fi1��#�i < �� fi7 >

= ∑

i

�a�h1 · ��s�fi3� · 1A����h2� fi4S−1�fi2��

#�h3fi5S−1�fi1��#�i < �� fi6 >

= ∑

i

�a�h1 · �11 · 1A����h2� 12fi3S−1�fi2��

#�h3fi4S−1�fi1��#�i < �� fi5 >

�2�8�= ∑

i

�a�h1 · �11 · 1A���h2 · �12 · 1A����h3� fi3S−1�fi2��

#�h4fi4S−1�fi1��#�i < �� fi5 >

= ∑

i

�a�h1 · 1A���h2� �s�fi2��#�h3fi3S−1�fi1��#�i < �� fi4 >

= ∑

i

�a�h1 · 1A���h2� 11�#�h312fi2S−1�fi1��#�i < �� fi3 >

= ∑

i

�a�h1 · 1A�#�h2�s�fi1��#�i < �� fi2 >

= ∑

i

�a�S�1′1� · 1A�#�1′2h11�#�i < �� 12fi >

= ∑

i

�a#�h11�#�i < �� 12fi >

= �a#�h11�#�i < �1� 12 >< �2� fi >

= �a#�h��1 ⇀ 1H��#�2

= �a#���∗s ��1� ⇀ h��#�2

= �S∗−1�∗t ��1� · �a#�h��#�2

= �a#�h� · �∗t ��1�#�2

= �a#�h�#��

Also, for every T ∈ End�A#�H�A� y ∈ A� g ∈ H , we have

� � ��T��y#�g� = ∑

i

��T���fi3� S−1�fi2��#�fi4��1A#�S

−1�fi1��#�i��y#�g�

= ∑

i

T���fi3� S−1�fi2��#�fi4��1A#�S

−1�fi1���y#� < �i� g2 > g1�

= T���g4� S−1�g3��#�g5��1A#�S

−1�g2���y#�g1�

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DUALITY AND GLOBAL DIMENSION FOR CROSSED PRODUCTS 1817

= T���g7� S−1�g6��#�g8���S

−1�g5� · y���S−1�g4�� g1�#�S−1�g3�g2�

= T���g5� S−1�g4��#�g6���S

−1�g3� · y���11S−1�g2�� g1�#�12��2�7�= T���g5� S

−1�g4��#�g6���S−1�g3� · y���S−1�g2�� g1�

×�S�11� · 1A��12 · 1A�#�1H�= T���g5� S

−1�g4��#�g6���S−1�g3� · y���S−1�g2�� g1�#�1H�

= T����g5� S−1�g4��#�g6���S

−1�g3� · y���S−1�g2�� g1�#�1H��

�T is right A-linear�

= T���g5� S−1�g4���g6 · �S−1�g3� · y���g7 · ��S−1�g2�� g1��#�g8�

�4�1�= T��g5S−1�g4� · y���g6� S−1�g3���g7 · ��S−1�g2�� g1��#�g8�

= T��11 · y���12g4� S−1�g3���g5 · ��S−1�g2�� g1��#�g6��3�1�= T�y��g4� S

−1�g3���g5 · ��S−1�g2�� g1��#�g6��4�2�= T�y��g5S

−1�g4�� g1���g6� S−1�g3�g2��11 · 1A�#�12g7�

= T�y���s�g3�� g1���g4� �t�g2��#�g5�

= T�y��11� g1���12g3� �t�g2��#�g4�

= T�y��11� g1���12g3�t�g2�� 1H�#�g4�

= T�y��11� g1���12g2� 1H�#�g3�

= T�y�g1 · 1A��g2 · 1A�#�g3� �by Eqs. (3.1) and (2.7))

= T�y�g1 · 1A�#�g2� = T�y�S�11� · 1A�#�12g�= T�y#�g��

So we get � and � are inverse of each other. �

We now obtain the main result of this section as follows.

Theorem 4.4 (The Duality Theorem). Let H be a finite dimensional weak Hopfalgebra measuring an algebra A and A#�H be a weak crossed product with �

invertible. Then there is a canonical isomorphism between the algebras �A#�H�#H∗ andEnd�A#�H�A.

Remark 4.5. If A is a left weak H-module algebra and ��h� g� = hg · 1A, wedefine ��h� g� = hg · 1A. Then A#�H is the weak smash product and fromTheorem 4.4, we get the duality theorem for weak smash products: There is acanonical isomorphism between the algebras �A#H�#H∗ and End�A#H�A. Here themaps ���x#h�#���y#g� = �x#h��y#�� ⇀ g�� = �x#h��y# < �� g2 > g1� and ��T� =∑

i T���fi3� S−1�fi2��#fi4��1A#S

−1�fi1��#�i =∑

i T�1#fi2��1#S−1�fi1��#�i� We can find

the results in [19].

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1818 SHEN

5. THE GLOBAL DIMENSION FOR WEAK CROSSED PRODUCTS

Let H be a finite dimensional semisimple weak Hopf algebra and A#�H a weakcrossed product with � invertible. We consider the relationship of global dimensionbetween A and A#�H in this section.

Lemma 5.1. Let H be a finite dimensional semisimple weak Hopf algebra, and let Pbe an A#�H-module. Then P is a projective A#�H-module if and only if it is a projectiveA-module.

Proof. Necessity can be obtained directly since A#�H is a projective A-module.Next, let us consider the sufficiency. For A#�H-modules M , N , let g � M → N andh � P → N be two A#�H-module morphisms such that g is surjective. In order toprove P is projective as an A#�H-module, it is enough to find an f ∈ HomA#�H

�P� N�

satisfying h = gf . Since P is projective as an A-module, there is an f ∈ HomA�P�N�such that h = gf , where we consider A#�H-module as A-module in the naturalway. Define f �v� = −1�x1� · f� �x2�v�, where x is a normalized right integral. FromLemma 3.8, we get f is A#�H-linear. We check

gf �p� = g� −1�x1� · f� �x2�p��= h�� −1�x1� �x2�� · p�= h� ��s�x�� · p� = h� �1� · p�= h��1#�1� · p� = h�p��

for all p ∈ P. �

Theorem 5.2. If H and H∗ are semisimple, then gl.dim�A#�H� = gl.dim�A�.

Proof. First claim: gl.dim�A#�H� ≤ gl.dim�A�. Clearly, it is harmless to assume theleft global dimension of A is finite, say n. For any A#�H-module M , by Lemma 5.1,any projective resolution of M as A#�H-module is also a projective resolution ofM as an A-module. Hence, its nth syzygy is projective as an A-module and thus isprojective as an A#�H-module. This implies gl.dim�A#�H� ≤ gl.dim�A�.

Note that A#�H is an H∗-module algebra via � · �a#�h� = a#��� ⇀h� =< �� h2 > a#�h1, for a#�h ∈ A#�H and � ∈ H∗. Thus, by the above claim,gl.dim��A#�H�#H∗� ≤ gl.dim�A#�H�. By Theorem 4.4, we have �A#�H�#H∗ End�A#�H�A. Since A#�H is a right A-progenerator, we have �A#�H�#H∗ is Moritaequivalent to A. Therefore, gl.dim�A� = gl.dim��A#�H�#H∗�. Thus

gl.dim�A� = gl.dim��A#�H�#H∗� ≤ gl.dim�A#�H� ≤ gl.dim�A��

This means that gl.dim�A#�H� = gl.dim�A�. �

Remark 5.3.

(1) If H and H∗ are semisimple Hopf algebras, then gl.dim�A#�H� = gl.dim�A�.(2) (Liu [17]) If H and H∗ are semisimple (weak) Hopf algebras and A is a left

(weak) H-module algebra, then gl.dim�A#H� = gl.dim�A�.

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DUALITY AND GLOBAL DIMENSION FOR CROSSED PRODUCTS 1819

ACKNOWLEDGMENTS

The author sincerely thanks the referee for his or her valuable suggestions andcomments. This work was supported by the NSF of China (10725104) and STCSM(09XD1402500).

REFERENCES

[1] Abuhlail, J. Y. (2005). Duality theorems for crossed products over rings. J. Algebra288:212–240.

[2] Alonso, Álvarez, J. N., Fernández Vilaboa, J. M., González Rodríguez, R., RodríguezRaposo, A. B. (2010). Crossed Products in Weak Contexts. Applied CategoricalStructures 18(3):231–258.

[3] Alonso Álvarez, J. N., Fernández Vilaboa, J. M., González Rodríguez, R., RodríguezRaposo, A. B. (2006). Weak C-cleft extensions and weak Galois extensions. J. Algebra299:276–293.

[4] Alonso Álvarez, J. N., Fernández Vilaboa, J. M., González Rodríguez, R., RodríguezRaposo, A. B. (2005). Weak C-cleft extensions, weak entwining structures and weakHopf algebras. J. Algebra 284:679–704.

[5] Alonso Álvarez, J. N., González Rodríguez, R. (2004). Crossed products for weakHopf algebras with coalgebras splitting. J. Algebra 281:731–752.

[6] Blattner, R., Cohen, M., Montgomery, S. (1986). Crossed product and inner actionsof Hopf algebras. Trans. Amer. Math. Soc. 298:671–711.

[7] Blattner, R., Montgomery, S. (1985). A duality theorem for Hopf module algebras.J. Algebra 95:153–172.

[8] Blattner, R., Montgomery, S. (1989). Crossed product and Galois extensions of Hopfalgebras. Pacific Journal of Math. 137:37–53.

[9] Böhm, G., Szlachányi, K. (1996). A coassociative C∗-quantum group with nonintegraldimensions. Lett. in Math. Phys. 35:437–456.

[10] Böhm, G., Nill, F., Szlachányi, K. (1999). Weak Hopf algebras I. Integral theory andC∗-structure. J. Algebra 221:385–438.

[11] Böhm, G. (2005). Integral theory for Hopf algebroids. Algebra and RepresentationTheory 8:563–599.

[12] Böhm, G., Brzezinski, T. (2006). Cleft extensions of Hopf algebroids. Appl. Categor.Struct. 14:431–469.

[13] Böhm, G., Szlachányi, K. (2004). Hopf algebroids with bijective antipodes: axioms,integrals, and duals. J. Algebra 274:708–750.

[14] Brzezinski, T., Militaru, G. (2002). Bialgebroids, ×A-bialgebras and duality. J. Algebra251:279–294.

[15] Cohen, M., Fishman, D. (1986). Hopf algebra actions. J. Algebra 100:363–379.[16] Cohen, M., Montgomery, S. (1984). Group-graded rings, smash products, and group

actions. Trans. Amer. Math. Soc. 282:237–258.[17] Liu, G. X. (2005). A note on the global dimension of smash products. Comm. Algebra

33:2625–2627.[18] Montgomery, S. (1993). Hopf Algebras and Their Actions on Rings. CBMS, Lecture

Notes. Vol. 82. Providence, RI: Amer Math. Soc.[19] Nikshych, D. (2000). A duality theorem for quantum groupoids. Contemporary

Mathematics 267:237–243.[20] Nikshych, D. (2002). On the structure of weak Hopf algebras. Adv. in Math.

170:257–286.[21] Nikshych, D., Vainerman, L. (2002). Finite quantum groupoids and their applications.

In: New Directions in Hopf Algebras. In: MSRI Publications 43:211–262.

Dow

nloa

ded

by [

Uni

vers

ity o

f C

onne

ctic

ut]

at 1

3:57

11

Oct

ober

201

4

1820 SHEN

[22] Nikshych, D., Turaev, V., Vainerman, L. (2003). Invariants of knots and 3-manifoldsfrom quantum groupoids. Topology and its applications 127:91–123.

[23] Rodríguez Raposo, A. B. (2009). Crossed products for Weak Hopf algebras. Comm. inAlgebra 37(7):2274–2289.

[24] Sweedler, M. (1969). Hopf Algebras. New York: Benjamin.[25] Vecsernyés, P. (2003). Larson-Sweedler theorem and the role of grouplike elements in

weak Hopf algebras. J. Algebra 270:471–520.[26] Wang, S. H., Kim, Y. G. (2004). Quasitriangular structure for a class of Hopf algebras

of dimension p6. Comm. Algebra 32:1401–1423.[27] Zhang, L. Y. (2006). Maschke-type theorem and morita context over weak Hopf

algebras. Science in China: Series A Math. 49(5):587–598.

Dow

nloa

ded

by [

Uni

vers

ity o

f C

onne

ctic

ut]

at 1

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11

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ober

201

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