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arXiv:math/0409599v3 [math.QA] 1 Apr 2005

YETTER-DRINFELD MODULES OVER WEAK BIALGEBRAS

S. CAENEPEEL, DINGGUO WANG, AND YANMIN YIN

Abstract. We discuss properties of Yetter-Drinfeld modules over weak bial-

gebras over commutative rings. The categories of left-left, left-right, right-left

and right-right Yetter-Drinfeld modules over a weak Hopf algebra are isomor-

phic as braided monoidal categories. Yetter-Drinfeld modules can be viewed

as weak Doi-Hopf modules, and, a fortiori, as weak entwined modules. If H

is finitely generated and projective, then we introduce the Drinfeld double

using duality results between entwining structures and smash product struc-

tures, and show that the category of Yetter-Drinfeld modules is isomorphic

to the category of modules over the Drinfeld double. The category of finitely

generated projective Yetter-Drinfeld modules over a weak Hopf algebra has

duality.

Introduction

Weak bialgebras and Hopf algebras are generalizations of ordinary bialgebras and

Hopf algebras in the following sense: the defining axioms are the same, but the mul-

tiplicativity of the counit and comultiplicativity of the unit are replaced by weaker

axioms. The easiest example of a weak Hopf algebra is a groupoid algebra; other

examples are face algebras [10], quantum groupoids [19], generalized Kac algebras

[25] and quantum transformation groupoids [18]. Temperley-Lieb algebras give rise

to weak Hopf algebras (see [18]). A purely algebraic study of weak Hopf algebras

has been presented in [2]. A survey of weak Hopf algebras and their applications

may be found in [18]. It has turned out that many results of classical Hopf algebra

theory can be generalized to weak Hopf algebras.

Yetter-Drinfeld modules over finite dimensional weak Hopf algebras over fields have

been introduced by Nenciu [16]. It is shown in [16] that the category of finite dimen-

sional Yetter-Drinfeld modules is isomorphic to the category of finite dimensional

modules over the Drinfeld double, as introduced in the appendix of [1]. It is also

shown that this category is braided isomorphic to the center of the category of fi-

nite dimensional H-modules. In this note, we discuss Yetter-Drinfeld modules over

weak bialgebras over commutative rings. The results in [16] are slightly generalized

and more properties are given.

In Section 2, we compute the weak center of the category of modules over a weak

bialgebra H, and show that it is isomorphic to the category of Yetter-Drinfeld mod-

ules. If H is a weak Hopf algebra, then the weak center equals the center. In this

situation, properties of the center construction can be applied to show that the four

categories of Yetter-Drinfeld modules, namely the left-left, left-right, right-left and

1991 Mathematics Subject Classification. 16W30.

Research supported by the projects G.0278.01 “Construction and applications of non-

commutative geometry: from algebra to physics” from FWO-Vlaanderen and “New computa-

tional, geometric and algebraic methods applied to quantum groups and differential operators”

from the Flemish and Chinese governments.

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2S. CAENEPEEL, DINGGUO WANG, AND YANMIN YIN

right-right versions, are isomorphic as braided monoidal categories. Here we apply

methods that have been used before in [5], in the case of quasi-Hopf algebras.

In [7], it was observed that Yetter-Drinfeld modules over a classical Hopf algebra

are special cases of Doi-Hopf modules, as introduced by Doi and Koppinen (see

[8, 13]). In Section 3, we will show that Yetter-Drinfeld modules over weak Hopf

algebras are weak Doi-Hopf modules, in the sense of B¨ ohm [1], and, a fortiori, weak

entwined modules [6], and comodules over a coring [4].

The advantage of this approach is that it leads easily to a new description of the

Drinfeld double of a finitely generated projective weak Hopf algebra, using meth-

ods developed in [6]: we define the Drinfeld double as a weak smash product of

H and its dual. We show that our Drinfeld double is equal to the Drinfeld double

of [1, 16] (see Proposition 4.3) and anti-isomorphic to the Drinfeld double of [17]

(see Proposition 4.5). In Section 5, we show that the category of finitely generated

projective Yetter-Drinfeld modules over a weak Hopf algebra has duality.

In Sections 1.1 and 1.2, we recall some general properties of weak bialgebras and

Hopf algebras. Further detail can be found in [4, 2, 18]. In Section 1.3, we recall

the center construction, and in Section 1.4, we recall the notions of weak Doi-Hopf

modules, weak entwining structures and weak smash products.

1. Preliminary results

1.1. Weak bialgebras. Let k be a commutative ring.

bialgebra is a k-module with a k-algebra structure (µ,η) and a k-coalgebra structure

(∆,ε) such that ∆(hk) = ∆(h)∆(k), for all h,k ∈ H, and

Recall that a weak k-

∆2(1)=1(1)⊗ 1(2)1(1′)⊗ 1(2′)= 1(1)⊗ 1(1′)1(2)⊗ 1(2′),

ε(hk(1))ε(k(2)l) = ε(hk(2))ε(k(1)l),

(1)

ε(hkl)=(2)

for all h,k,l ∈ H. We use the Sweedler-Heyneman notation for the comultiplication,

namely

∆(h) = h(1)⊗ h(2)= h(1′)⊗ h(2′).

We summarize the elementary properties of weak bialgebras. The proofs are direct

applications of the defining axioms (see [2, 18]). We have idempotent maps εt, εs:

H → H defined by

εt(h) = ε(1(1)h)1(2);εs(h) = 1(1)ε(h1(2)).

εt and εs are called the target map and the source map, and their images Ht =

Im(εt) = Ker(H − εt) and Hs= Im(εs) = Ker(H − εs) are called the target and

source space. For all g,h ∈ H, we have

(3)h(1)⊗ εt(h(2)) = 1(1)h ⊗ 1(2)and εs(h(1)) ⊗ h(2)= 1(1)⊗ h1(2),

and

(4)hεt(g) = ε(h(1)g)h(2)and εs(g)h = h(1)ε(gh(2)).

From (4), it follows immediately that

(5)ε(hεt(g)) = ε(hg) and ε(εs(g)h) = ε(gh).

The source and target space can be described as follows:

Ht= {h ∈ H | ∆(h) = 1(1)h ⊗ 1(2)} = {φ(1(1))1(2)| φ ∈ H∗};

Hs= {h ∈ H | ∆(h) = 1(1)⊗ h1(2)} = {1(1)φ(1(2)) | φ ∈ H∗}.

(6)

(7)

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YETTER-DRINFELD MODULES OVER WEAK BIALGEBRAS3

We also have

(8)εt(h)εs(k) = εs(k)εt(h),

and its dual property

(9)εs(h(1)) ⊗ εt(h(2)) = εs(h(2)) ⊗ εt(h(1)).

Finally εs(1) = εt(1) = 1, and

(10)εt(h)εt(g) = εt(εt(h)g) and εs(h)εs(g) = εs(hεs(g)).

This implies that Hsand Htare subalgebras of H.

Lemma 1.1. Let H be a weak bialgebra over a commutative ring. Then ∆(1) ∈

Hs⊗ Ht.

Proof. Applying H⊗ε⊗H to (1), we find that 1(1)⊗1(2)= εs(1(1))⊗1(2)∈ Hs⊗H

and 1(1)⊗ 1(2)= 1(1)⊗ εt(1(2)) ∈ H ⊗ Ht. Now let Ks= Ker(εs), Kt= Ker(εt).

Then H = Hs⊕ Ks= Ht⊕ Kt, and

H ⊗ H = Hs⊗ Ht⊕ Hs⊗ Kt⊕ Ks⊗ Ht⊕ Ks⊗ Kt,

so it follows that Hs⊗ Ht= H ⊗ Ht∩ Hs⊗ H.

?

The target and source map for the weak bialgebra Hopare

(11)

εt(h) = ε(h1(1))1(2)∈ Htand εs(h) = ε(1(2)h)1(1)∈ Hs.

εtand εsare also projections.

The source and target space are anti-isomorphic, and they are separable Frobenius

algebras over k. This was first proved for weak Hopf algebras (see [2]), and then

generalized to weak bialgebras (see [22]).

Lemma 1.2. [22] Let H be a weak bialgebra. Then εsrestricts to an anti-algebra

isomorphism Ht→ Hswith inverse εt, and εtrestricts to an anti-algebra isomor-

phism Hs→ Htwith inverse εs.

Proposition 1.3. [22] Let H be a weak bialgebra. Then Hsand Htare Frobenius

separable k-algebras. The separability idempotents of Htand Hsare

et= εt(1(1)) ⊗ 1(2)= 1(2)⊗ εt(1(1));

es= 1(1)⊗ εs(1(2)) = εs(1(2)) ⊗ 1(1).

The Frobenius systems for Htand Hs are respectively (et,ε|Ht) and (es,ε|Hs). In

particular, we have for all z ∈ Htthat

(12)zεt(1(1)) ⊗ 1(2)= εt(1(1)) ⊗ 1(2)z.

It was shown in [17] that the category of modules over a weak Hopf algebra is

monoidal; it follows from the results of [22] that this property can be generalized

to weak bialgebras. We explain now how this can be done directly.

Let M be a left H-module. By restriction of scalars, M is a left Ht-module; M

becomes an Ht-bimodule, if we define a right Ht-action by

m · z = εs(z)m.

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4S. CAENEPEEL, DINGGUO WANG, AND YANMIN YIN

Let M,N ∈HM, the category of left H-modules. We define

M ⊗tN = ∆(1)(M ⊗ N),

the k-submodule of M ⊗N generated by elements of the form 1(1)⊗1(2). M ⊗tN

is a left H-module, with left diagonal action h·(m⊗n) = h(1)m⊗h(2)n. It follows

from (1) that the tensor product ⊗tis associative. Observe that

M ⊗tN ⊗tP = ∆2(1)(M ⊗ N ⊗ P).

Ht∈HM, with left H-action h⇀z = εt(hz). The induced Ht-bimodule structure

is given by left and right multiplication by elements of Ht.

For M,N ∈HM, consider the projection

π : M ⊗ N → M ⊗tN, π(m ⊗ n) = 1(1)m ⊗ 1(2)n.

Applying εs⊗ Htto (12), we find

εs(zεt(1(1))) ⊗ 1(2)= 1(1)εs(z) ⊗ 1(2)= 1(1)⊗ 1(2)z,

hence

π(mz ⊗ n) = π(εs(z)m ⊗ n) = 1(1)εs(z)m ⊗ 1(2)n = 1(1)m ⊗ 1(2)zn = π(m ⊗ zn).

So π induces a map π : M⊗HtN → M⊗tN, which is a left Ht-module isomorphism

with inverse given by

π−1(1(1)m ⊗ 1(2)n) = 1(1)m ⊗Ht1(2)n = m ⊗Htn.

Proposition 1.4. Let H be a weak bialgebra. Then we have a monoidal category

(HM,⊗t,Ht,a,l,r). The associativity constraints are the natural ones. The left

and right unit constraints lM: Ht⊗tM → M and rM: M ⊗tHt→ M and their

inverses are given by the formulas

lM(1(1)⇀z ⊗ 1(2)m) = zm ; l−1

M(m) = εt(1(1)) ⊗ 1(2)m;

rM(1(1)m ⊗ 1(2)⇀z) = εs(z)m ; r−1

M(m) = 1(1)m ⊗ 1(2).

Proof. This is a direct consequence of the observations made above. Let us check

that

l−1

M(lM(1(1)⇀z ⊗ 1(2)m)) = l−1

=zεt(1(1)) ⊗ 1(2)m

=εt(1(1)z) ⊗ 1(2)m = 1(1)⇀z ⊗ 1(2)m

lM(l−1

M(m)) = lM(εt(1(1)) ⊗ 1(2)m) = m

r−1

M(zm) = εt(1(1)) ⊗ 1(2)zm

(10)

= εt(z1(1)) ⊗ 1(2)m

M(rM(1(1)m ⊗ 1(2)⇀z)) = r−1

=1(1)m ⊗ 1(2)z = 1(1)m ⊗ 1(2)⇀z

rM(r−1

M(m)) = rM(1(1)m ⊗ 1(2)) = εs(1)m = m.

M(εs(z)m) = 1(1)εs(z)m ⊗ 1(2)

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YETTER-DRINFELD MODULES OVER WEAK BIALGEBRAS5

1.2. Weak Hopf algebras. A weak Hopf algebra is a weak bialgebra together

with a map S : H → H, called the antipode, satisfying

(13)S ∗ H = εs, H ∗ S = εt, and S ∗ H ∗ S = S,

where ∗ is the convolution product. It follows immediately that

(14)S = εs∗ S = S ∗ εt.

If the antipode exists, then it is unique. We will always assume that S is bijective;

if H is a finite dimensional weak Hopf algebra over a field, then S is automatically

bijective (see [2, Theorem 2.10]).

Lemma 1.5. Let H be a weak Hopf algebra. Then S is an anti-algebra and an

anti-coalgebra morphism. For all h,g ∈ H, we have

εt(hg)

εs(hg)

∆(εt(h))

∆(εs(h))

=εt(hεt(g)) = h(1)εt(g)S(h(2));

εs(εs(h)g) = S(g(1))εs(h)g(2);

h(1)S(h(3)) ⊗ εt(h(2))

εs(h(2)) ⊗ S(h(1))h(3).

(15)

=(16)

=(17)

=(18)

Lemma 1.6. Let H be a weak Hopf algebra. For all h ∈ H, we have

εt(h)

εs(h)

=ε(S(h)1(1))1(2)= ε(1(2)h)S(1(1)) = S(εs(h))

1(1)ε(1(2)S(h)) = ε(h1(1))S(1(2)) = S(εt(h)).

(19)

=(20)

Corollary 1.7. Let H be a weak Hopf algebra. For all h ∈ H, we have

(21)εt(h(1)) ⊗ h(2)= S(1(1)) ⊗ 1(2)h ; h(1)⊗ εs(h(2)) = h1(1)⊗ S(1(2)).

Proposition 1.8. Let H be a weak Hopf algebra. Then

(22)εt◦ S = εt◦ εs= S ◦ εs; εs◦ S = εs◦ εt= S ◦ εt.

Corollary 1.9. Let H be a weak Hopf algebra with bijective antipode. Then S|Ht=

(εs)|Ht, and S−1

|Hs= (εt)|Hs, so S restricts to an anti-algebra isomorphism Ht→ Hs.

It follows that the separability idempotents of Htand Hs are et= S(1(1)) ⊗ 1(2)

and es= 1(1)⊗ S(1(2)). Consequently, we have the following formulas, for z ∈ Ht

and y ∈ Hs:

zS(1(1)) ⊗ 1(2)

=S(1(1)) ⊗ 1(2)z;

1(1)⊗ S−1(y)1(2).

(23)

y1(1)⊗ 1(2)

= (24)

Applying S−1⊗ H to (23), we find

(25)1(1)S−1(z) ⊗ 1(2)= 1(1)⊗ 1(2)z.

1.3. The center of a monoidal category. Let C = (C,⊗,I,a,l,r) be a monoidal

category. The weak left center Wl(C) is the category with the following objects and

morphisms. An object is a couple (M,σM,−), with M ∈ C and σM,−: M ⊗ − →

−⊗M a natural transformation, satisfying the following condition, for all X,Y ∈ C:

(26)

and such that σM,I is the composition of the natural isomorphisms M ⊗ I∼= M∼=

I ⊗ M. A morphism between (M,σM,−) and (M′,σM′,−) consists of ϑ : M → M′

in C such that

(X ⊗ ϑ) ◦ σM,X= σM′,X◦ (ϑ ⊗ X).

(X ⊗ σM,Y) ◦ aX,M,Y◦ (σM,X⊗ Y ) = aX,Y,M◦ σM,X⊗Y ◦ aM,X,Y,