Page 1
arXiv:math/0409599v3 [math.QA] 1 Apr 2005
YETTERDRINFELD MODULES OVER WEAK BIALGEBRAS
S. CAENEPEEL, DINGGUO WANG, AND YANMIN YIN
Abstract. We discuss properties of YetterDrinfeld modules over weak bial
gebras over commutative rings. The categories of leftleft, leftright, rightleft
and rightright YetterDrinfeld modules over a weak Hopf algebra are isomor
phic as braided monoidal categories. YetterDrinfeld modules can be viewed
as weak DoiHopf 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 YetterDrinfeld modules is isomorphic
to the category of modules over the Drinfeld double. The category of finitely
generated projective YetterDrinfeld 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]. TemperleyLieb 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.
YetterDrinfeld 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 YetterDrinfeld 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 Hmodules. In this note, we discuss YetterDrinfeld 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 YetterDrinfeld 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 YetterDrinfeld modules, namely the leftleft, leftright, rightleft 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 FWOVlaanderen and “New computa
tional, geometric and algebraic methods applied to quantum groups and differential operators”
from the Flemish and Chinese governments.
1
Page 2
2 S. CAENEPEEL, DINGGUO WANG, AND YANMIN YIN
rightright versions, are isomorphic as braided monoidal categories. Here we apply
methods that have been used before in [5], in the case of quasiHopf algebras.
In [7], it was observed that YetterDrinfeld modules over a classical Hopf algebra
are special cases of DoiHopf modules, as introduced by Doi and Koppinen (see
[8, 13]). In Section 3, we will show that YetterDrinfeld modules over weak Hopf
algebras are weak DoiHopf 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 antiisomorphic to the Drinfeld double of [17]
(see Proposition 4.5). In Section 5, we show that the category of finitely generated
projective YetterDrinfeld 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 DoiHopf
modules, weak entwining structures and weak smash products.
1. Preliminary results
1.1. Weak bialgebras. Let k be a commutative ring.
bialgebra is a kmodule with a kalgebra structure (µ,η) and a kcoalgebra 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 SweedlerHeyneman 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)
Page 3
YETTERDRINFELD 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 antiisomorphic, 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 antialgebra
isomorphism Ht→ Hswith inverse εt, and εtrestricts to an antialgebra isomor
phism Hs→ Htwith inverse εs.
Proposition 1.3. [22] Let H be a weak bialgebra. Then Hsand Htare Frobenius
separable kalgebras. 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,εHt) and (es,εHs). 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 Hmodule. By restriction of scalars, M is a left Htmodule; M
becomes an Htbimodule, if we define a right Htaction by
m · z = εs(z)m.
Page 4
4 S. CAENEPEEL, DINGGUO WANG, AND YANMIN YIN
Let M,N ∈HM, the category of left Hmodules. We define
M ⊗tN = ∆(1)(M ⊗ N),
the ksubmodule of M ⊗N generated by elements of the form 1(1)⊗1(2). M ⊗tN
is a left Hmodule, 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 Haction h⇀z = εt(hz). The induced Htbimodule 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 Htmodule 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)
?
Page 5
YETTERDRINFELD 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 antialgebra and an
anticoalgebra 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 SHt=
(εs)Ht, and S−1
Hs= (εt)Hs, so S restricts to an antialgebra 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,
Page 6
6 S. CAENEPEEL, DINGGUO WANG, AND YANMIN YIN
The left center Zl(C) is the full subcategory of Wl(C) consisting of objects (M,σM,−)
with σM,−a natural isomorphism. Zl(C) is a braided monoidal category. The tensor
product is
(M,σM,−) ⊗ (M′,σM′,−) = (M ⊗ M′,σM⊗M′,−)
with
σM⊗M′,X= aX,M,M′ ◦ (σM,X⊗ M′) ◦ a−1
(27)
M,X,M′ ◦ (M ⊗ σM′,X) ◦ aM,M′,X,
and the unit is (I,σI,−), with
(28)σI,M= r−1
M◦ lM.
The braiding c on Zl(C) is given by
cM,M′ = σM,M′ : (M,σM,−) ⊗ (M′,σM′,−) → (M′,σM′,−) ⊗ (M,σM,−).
Zl(C)inwill be our notation for the monoidal category Zl(C), together with the
inverse braiding ˜ c given by ˜ cM,M′ = c−1
The right center Zr(C) is defined in a similar way. An object is a couple (M,τ−,M),
where M ∈ C and τ−,M : − ⊗ M → M ⊗ − is a family of natural isomorphisms
such that τ−,Iis the natural isomorphism and
a−1
(29)
M′,M= σ−1
M′,M.
(30)
M,X,Y◦ τX⊗Y,M◦ a−1
for all X,Y ∈ C. A morphism between (M,τ−,M) and (M′,τ−,M′) consists of
ϑ : M → M′in C such that
X,Y,M= (τX,M⊗ Y ) ◦ a−1
X,M,Y◦ (X ⊗ τY,M),
(ϑ ⊗ X) ◦ τX,M= τX,M′ ◦ (X ⊗ ϑ),
for all X ∈ C. Zr(C) is a braided monoidal category. The unit is (I,l−1
the tensor product is
(M,τ−,M) ⊗ (M′,τ−,M′) = (M ⊗ M′,τ−,M⊗M′)
−◦ r−) and
with
(31)τX,M⊗M′ = a−1
M,M′,X◦ (M ⊗ τX,M′) ◦ aM,X,M′ ◦ (τX,M⊗ M′) ◦ a−1
The braiding d is given by
dM,M′ = τM,M′ : (M,τ−,M) ⊗ (M′,τ−,M′) → (M′,τ−,M′) ⊗ (M,τ−,M).
Zr(C)inis the monoidal category Zr(C) with the inverse braiding˜d given by˜dM,M′ =
d−1
M′,M.
For details in the case where C is a strict monoidal category, we refer to [12, Theorem
XIII.4.2]. The results remain valid in the case of an arbitrary monoidal category,
since every monoidal category is equivalent to a strict one. Recall the following
result from [5].
X,M,M′.
(32)
M′,M= τ−1
Proposition 1.10. Let C be a monoidal category. Then we have an isomorphism
of braided monoidal categories F : Zl(C) → Zr(C)in, given by
F(M,σM,−) = (M,σ−1
M,−) and F(ϑ) = ϑ.
We have a second monoidal structure on C, defined as follows:
C = (C,⊗ = ⊗ ◦ τ,I,a,r,l)
with τ : C × C → C × C, τ(M,N) = (N,M) and a defined by aM,N,X= a−1
If c is a braiding on C, then c, given by cM,N= cN,Mis a braiding on C. In [5], the
following obvious result was stated.
X,N,M.
Page 7
YETTERDRINFELD MODULES OVER WEAK BIALGEBRAS7
Proposition 1.11. Let C be a monoidal category. Then
Zl(C)∼= Zr(C) ; Zr(C)∼= Zl(C)
as braided monoidal categories.
1.4. Weak entwining structures and weak smash products. The results in
this Section are taken from [6]. Let A be a ring without unit. e ∈ A is called a
preunit if ea = ae = ae2, for all a ∈ A. Then map p : A → A, p(a) = ae, satisfies
the following properties: p ◦ p = p and p(ab) = p(a)p(b). Then A = Coim(p) is a
ring with unit e and A = Im(p) is a ring with unit e2. p induces a ring isomorphism
A → A.
Let k be a commutative ring, A, B kalgebras with unit, and R : B ⊗ A → A ⊗ B
a klinear map. We use the notation
(33)R(b ⊗ a) = aR⊗ bR= ar⊗ br,
where the summation is implicitely understood. A#RB is the kalgebra A⊗B with
newly defined multiplication
(a#b)(c#d) = acR#bRd.
(A,B,R) is called a weak smash product structure if A#RB is an associative k
algebra with preunit 1A#1B. The multiplication is associative if and only if
R(bd ⊗ a) = aRr⊗ brdRand R(b ⊗ ac) = aRcr⊗ bRr,
for all a,c ∈ A and b,d ∈ B. 1A#1Bis a preunit if and only if
R(1B⊗ a) = a(1A)R⊗ (1B)Rand R(b ⊗ 1A) = (1A)R⊗ (1B)Rb.
A leftright weak entwining structure is a triple (A,C,ψ), where A is an algebra, C
is a coalgebra, and ψ : A ⊗ C → A ⊗ C is a klinear map satisfying the conditions
aψ⊗ ∆(cψ) = aψΨ⊗ cΨ
(1)⊗ cψ
(2); (ab)ψ⊗ cψ= aψbΨ⊗ cΨψ;
1ψ⊗ cψ= ε(cψ
(1))1ψ⊗ c(2); aψε(cψ) = ε(cψ)a1ψ.
Here we use the notation (with summation implicitely understood):
ψ(a ⊗ c) = aψ⊗ cψ.
An entwined module is a kmodule M with a left Aaction and a right Ccoaction
such that
ρ(am) = aψm[0]⊗ mψ
The category of entwined modules and left Alinear right Ccolinear maps is de
noted byAM(ψ)C.
Let H be a weak bialgebra, and A a right Hcomodule, which is also an alge
bra with unit. A is called a right Hcomodule algebra if ρ(a)ρ(b) = ρ(ab) and
1[0]⊗ εt(1[1]) = ρ(1).
[1].
From [1], we recall the following definitions. Let C be a left Hmodule which is
also a coalgebra with counit. C is called a left Hcomodule algebra if ∆C(hc) =
∆H(h)∆C(c) and
(34)εC(hkc) = εH(hk(2))εC(k(1)c),
for all c ∈ C and h,k ∈ H. Several equivalent definitions are given in [6, Sec.
4]. We then call (H,A,C) a leftright weak DoiHopf datum. A weak DoiHopf
Page 8
8 S. CAENEPEEL, DINGGUO WANG, AND YANMIN YIN
module over (H,A,C) is a kmodule M with a left Aaction and a right Ccoaction,
satisfying the following compatibility relation, for all m ∈ M and a ∈ A:
(35)ρ(am) = a[0]m[0]⊗ a[1]m[1].
The category of weak DoiHopf modules over (H,A,C) and left Alinear right C
colinear maps is denoted byAM(H)C.
Let (H,A,C) be a weak leftright DoiHopf datum, and consider the map
ψ : A ⊗ C → A ⊗ C, ψ(a ⊗ c) = a[0]⊗ a[1]c.
Then (A,C,ψ) is a weak leftright entwining structure, and we have an isomorphism
of categoriesAM(H)C ∼=AM(ψ)C.
Let (A,C,ψ) be a weak leftright entwining structure, and assume that C is finitely
generated projective as a kmodule, with finite dual basis {(ci,c∗
Then we have a weak smash product structure (A,C∗,R), with R : C∗⊗A → A⊗C∗
given by
i)  i = 1,··· ,n}.
(36)R(c∗⊗ a) =?
i?c∗,cψ
i?aψ⊗ c∗
i.
We have an isomorphism of categories
(37)F : AM(ψ)C→A#RC∗M,
defined also follows: F(M) = M as a kmodule, with action [a#c∗]·m = ?c∗,m[1]?am[0].
Details can be found in [6, Theorem 3.4].
2. YetterDrinfeld modules over weak Hopf algebras
Let H be a weak bialgebra. A leftleft YetterDrinfeld module is a kmodule with
a left Haction and a left Hcoaction such that the following conditions hold, for
all m ∈ M and h ∈ H:
λ(m) = m[−1]⊗ m[0]∈ H ⊗tM;
h(1)m[−1]⊗ h(2)m[0]= (h(1)m)[−1]h(2)⊗ (h(1)m)[0].
(38)
(39)
We will now state some equivalent definitions. First we will rewrite the counit
property for YetterDrinfeld modules.
Lemma 2.1. Let H be a weak bialgebra, and λ : M → H⊗tM, ρ(m) = m[−1]⊗m[0]
a klinear map. Then
(40)ε(m[−1])m[0]= εt(m[−1])m[0].
Consequently, in the definition of a YetterDrinfeld module, the counit property
ε(m[−1])m[0]= m can be replaced by εt(m[−1])m[0]= m.
Proof.
εt(m[−1])m[0]= ε(1(1)m[−1])1(2)m[0]= ε(m[−1])m[0].
?
In the case of a weak Hopf algebra, the compatibility relation (39) can also be
restated:
Page 9
YETTERDRINFELD MODULES OVER WEAK BIALGEBRAS9
Proposition 2.2. (cf. [16, Remark 2.6]) Let H be a weak Hopf algebra, and M
a kmodule, with a left Haction and a left Hcoaction. M is a YetterDrinfeld
module if and only if
(41)λ(hm) = h(1)m[−1]S(h(3)) ⊗ h(2)m[0].
Proof. Let M be a YetterDrinfeld module. Then we compute
h(1)m[−1]S(h(3)) ⊗ h(2)m[0]= (h(1)m)[−1]h(2)S(h(3)) ⊗ (h(1)m)[0]
=(h(1)m)[−1]εt(h(2)) ⊗ (h(1)m)[0]
(39)
=1(1)(hm)[−1]⊗ 1(2)(hm)[0]
(3)
=(1(1)hm)[−1]1(2)⊗ (1(1)hm)[0]
= (hm)[−1]⊗ (hm)[0]= λ(hm).
(38)
Conversely, assume that (41) holds for all h ∈ H and m ∈ M. Taking h = 1 in
(41), we find
λ(m)=1(1)m[−1]S(1(3)) ⊗ 1(2)m[0]
1(1)m[−1]S(1(2′)) ⊗ 1(2)1(1′)m[0]∈ H ⊗tM=
and
λ(m)=1(1)m[−1]S(1(3)) ⊗ 1(2)m[0]= 1(1)m[−1]S(1(2′)) ⊗ 1(1′)1(2)m[0]
m[−1]S(1(2′)) ⊗ 1(1′)m[0].= (42)
Now
(h(1)m)[−1]h(2)⊗ (h(1)m)[0]
=h(1)m[−1]εs(h(3)) ⊗ h(2)m[0]
(42)
=h(1)m[−1]⊗ h(2)m[0],
(41)
= h(1)m[−1]S(h(3))h(4)⊗ h(2)m[0]
(21)
= h(1)m[−1]S(1(2)) ⊗ h(2)1(1)m[0]
as needed.
?
Corollary 2.3. Let M be a leftleft YetterDrinfeld module. For all y ∈ Hs, z ∈ Ht
and m ∈ M, we have
(43)λ(zm) = zm[−1]⊗ m[0]; λ(ym) = m[−1]S(y) ⊗ m[0].
Proof.
λ(zm)
(6,41)
=
(8)
=
1(1)zm[−1]S(1(3)) ⊗ 1(2)m[0]
z1(1)m[−1]S(1(3)) ⊗ 1(2)m[0]
(41)
= zm[−1]⊗ m[0].
The other assertion is proved in a similar way.
?
Corollary 2.4. Let M be a leftleft YetterDrinfeld module over a weak Hopf alge
bra with bijective antipode. Then we have the following identities, for all m ∈ M:
(44)1(1)m[0]⊗ 1(2)S−1(m[−1]) = m[0]⊗ S−1(m[−1]);
(45)εs(S−2(m[−1]))m[0]= m.
Proof. Apply S−1to the first factor of (42), and then switch the two tensor factors.
Then we obtain (44). (45) is proved as follows:
(40)
= ε(m[−1])m[0]= ε(S−1(m[−1]))m[0]
(44)
=ε(1(2)S−1(m[−1]))1(1)m[0]
m = εt(m[−1])m[0]
(20)
= εs(S−2(m[−1]))m[0].
?
Page 10
10 S. CAENEPEEL, DINGGUO WANG, AND YANMIN YIN
The category of leftleft YetterDrinfeld modules and left Hlinear, left Hcolinear
maps will be denoted byH
HYD.
Example 2.5. Let G be a groupoid, and kG the corresponding groupoid algebra.
Then kG is a weak Hopf algebra. Let M be a leftleft YetterDrinfeld module.
Then M is a kGcomodule, so M is graded by the set G, that is
M =
?
σ∈G1
Mσ,
and λ(m) = σ ⊗ m if and only if m ∈ Mσ, or deg(m) = σ.
Recall that the unit element of kG is 1 =?
of the object x ∈ G0. Take m ∈ Mσ. Using (41), we find
x∈G0x, where x is the identity morphism
λ(m) = λ(1m) =
?
x∈G
xσx ⊗ xm = 0,
unless s(σ) = τ(σ) = x. So we have
M =
?
σ∈G1
s(σ)=t(σ)
Mσ.
Take m ∈ Mσ, with s(σ) = τ(σ), and τ ∈ G1. It follows from (41) that λ(τm) =
τστ−1⊗ τm = 0, unless s(τ) = x. If s(τ) = x, then deg(τm) = τστ−1.
Theorem 2.6. Let H be a weak bialgebra. Then the categoryH
to the weak left center Wl(HM) of the category of left Hmodules. If H is a weak
Hopf algebra with bijective antipode, thenH
Zl(HM)
HYD is isomorphic
HYD is isomorphic to the left center
Proof. We will restrict to a brief description of the connecting functors; for more
detail (in the leftright case), we refer to [16, Lemma 4.3]. Take (M,σM,−) ∈
Wl(HM). For each left Hmodule V , we have a map σM,V : M ⊗tV → V ⊗tM
inHM. We will show that the map
λ : M → H ⊗tM, λ(m) = σM,H(1(1)m ⊗ 1(2)) = m[−1]⊗ m[0]
makes M into a YetterDrinfeld module. Conversely, let (M,λ) is a YetterDrinfeld
module; a natural transformation σ is then defined by the formula
(46)σM,V(1(1)m ⊗ 1(2)v) = m[−1]v ⊗ m[0].
Straightforward computations show that (M,σ) ∈ Wl(HM). If H is a Hopf algebra
with invertible antipode, then the inverse of σM,V is
(47)σ−1
M,V(1(1)v ⊗ 1(2)m) = m[0]⊗ S−1(m[−1])v.
?
From now on, we assume that H is a weak Hopf algebra with bijective antipode.
Since the left center of a monoidal category is a braided monoidal category, it follows
from Theorem 2.6 thatH
HYD is a braided monoidal category; a direct but long proof
can be given: see [16, Prop. 2.7]. The monoidal structure can be computed using
(27). Take M,N ∈H
HYD, the Hcoaction on M ⊗tN is given by the formula
λ(1(1)m ⊗ 1(2)n) = ((σM,H⊗ N) ◦ (M ⊗ σN,H))(1(1′)(1(1)m ⊗ 1(2)n) ⊗ 1(2′)).
Page 11
YETTERDRINFELD MODULES OVER WEAK BIALGEBRAS 11
Observe that
x=
=
1(1′)(1(1)m ⊗ 1(2)n) ⊗ 1(2′)
1(1′)1(1)m ⊗ 1(1′′)1(2′)1(2)n ⊗ 1(2′′)= 1(1)m ⊗ 1(1′′)1(2)n ⊗ 1(2′′),
so that
(M ⊗ σN,H)(x) = 1(1)m ⊗ (1(2)n)[−1]⊗ (1(2)n)[0]
=1(1)m ⊗ 1(2)n[−1]S(1(4)) ⊗ 1(3)n[0]
=1(1)m ⊗ 1(2)1(1′)n[−1]S(1(3′)) ⊗ 1(2′)n[0]
=1(1)m ⊗ 1(2)n[−1]⊗ n[0]
and
(48)λ(1(1)m ⊗ 1(2)n) = m[−1]n[−1]⊗ m[0]⊗ n[0].
We compute the left Hcoaction on Htusing (28) and (46). For any z ∈ Ht, this
gives
λ(z)=σHt,H((1(1)⇀z) ⊗ 1(2)) = r−1
r−1
M(z) = 1(1)z ⊗ 1(2)= ∆(z).
M(lM((1(1)⇀z) ⊗ 1(2)))
=
(49)
The braiding and its inverse are given by the formulas
σM,N(1(1)m⊗1(2)n) = m[−1]n⊗m[0]; σ−1
M,N(1(1)n⊗1(2)m) = m[0]⊗S−1(m[−1])n.
A leftright YetterDrinfeld module is a kmodule with a left Haction and a right
Hcoaction such that the following conditions hold, for all m ∈ M and h ∈ H:
ρ(m) = m[0]⊗ m[1]∈ M ⊗tH;
h(1)m[0]⊗ h(2)m[1]= (h(2)m)[0]⊗ (h(2)m)[1]h(1).
(50)
(51)
The category of leftright YetterDrinfeld modules and left Hlinear right Hcolinear
maps is denoted byHYDH.
Proposition 2.7. Let H be a weak Hopf algebra with bijective antipode. Then the
categoryHYDHis isomorphic to the right center Zr(HM).
Proof. Take (M,τ−,M) ∈ Zr(HM). We know from Proposition 1.10 that (M,σM,−=
τ−1
−,M) ∈ Zl(HM). Take the corresponding leftleft YetterDrinfeld (M,λ), as in
Theorem 2.6, and define ρ : M → M ⊗ H by
(52)ρ(m) = m[0]⊗ m[1]= m[0]⊗ S−1(m[−1]).
It follows from (44) that ρ(m) ∈ M ⊗tH. The coassociativity of ρ follows immedi
ately from the coassociativity of λ and the anticomultiplicativity of S−1. Also
ε(m[1])m[0]= ε(S−1(m[−1]))m[0]= ε(m[−1])m[0]= m.
From (47), it follows that
(53)τV,M(1(1)v ⊗ 1(2)m) = m[0]⊗ m[1]v.
In particular, τM,H(1(1)⊗ 1(2)m) = ρ(m), and the fact that τM,H is left Hlinear
implies (51). Hence (M,ρ) is a leftright YetterDrinfeld module.
Conversely, if (M,ρ) is a leftright YetterDrinfeld module, then (M,τ−,M), with τ
defined by (53) is an object of Zr(HM).
?
Page 12
12 S. CAENEPEEL, DINGGUO WANG, AND YANMIN YIN
Corollary 2.8. Let M be a kmodule with a left Haction and a right Hcoaction.
Then M is a leftright YetterDrinfeld module if and only if
ρ(hm) = h(2)m[0]⊗ h(3)m[1]S−1(h(1)). (54)
Corollary 2.9. Let M be a leftright YetterDrinfeld module. For all y ∈ Hs,
z ∈ Htand m ∈ M, we have that
ρ(ym) = m[0]⊗ ym[1]; ρ(zm) = m[0]⊗ m[1]S−1(z). (55)
Corollary 2.10. Let M be a leftright YetterDrinfeld module. Then
1(2)m[0]⊗ m[1]S−1(1(1)) = ρ(m), (56)
for all m ∈ M.
Proof. Apply S−1⊗ M to λ(m) = 1(1)S(m[1]) ⊗ 1(2)m[0].
?
Corollary 2.11. The categoryHYDHis a braided monoidal category, isomorphic
toH
HYDin.
In a similar way, we can introduce rightright and rightleft YetterDrinfeld mod
ules. The categories YDH
modules are isomorphic to the right and left center of MH. Let us summarize the
results.
A rightright YetterDrinfeld module is a kmodule M with a right Haction and a
right Hcoaction such that
HandHYDH of rightright and rightleft YetterDrinfeld
ρ(m) = m[0]⊗ m[1]∈ M ⊗sH;
m[0]h(1)⊗ m[1]h(2)= (mh(2))[0]⊗ h(1)(mh(2))[1];
(57)
(58)
or, equivalently,
(59)ρ(mh) = m[0]h(2)⊗ S(h(1))m[1]h(3).
The counit condition m = ε(m[1])m[0]is equivalent to
m = m[0]ε(m[1]).
The natural isomorphism τ−,Mcorresponding to (M,ρ) ∈ YDH
given by the formulas
(60) τM,V(v1(1)⊗m1(2)) = m[0]⊗mv[1]; τ−1
Hand its inverse are
M,V(m1(1)⊗v1(2)) = vS−1(m[1])⊗m[0].
Furthermore
m[0]εt(S−2(m[1])) = m,
and S−1(m[1]) ⊗ m[0]∈ H ⊗sM.
The monoidal structure on YDH
His given by the formula
ρ(m1(1)⊗ n1(2)) = m[0]⊗ n[0]⊗ m[1]n[1].
The category YDH
The braiding is given by (60).
monoidal category to Zr(MH).
His isomorphic as a braided
Let M be a right Hmodule and a left Hcomodule. M is a rightleft YetterDrinfeld
module if one of the three following equivalent conditions is satisfied, for all m ∈ M
and h ∈ H:
1) λ(m) ∈ H ⊗sM and
h(2)(mh(1))[0]⊗ (mh(1))[1]= m[−1]h(1)⊗ m[0]h(2),
Page 13
YETTERDRINFELD MODULES OVER WEAK BIALGEBRAS13
2) λ(mh) = S−1(h(3))m[−1]h(1)⊗ m[0]h(2);
3) (M,ρ), with ρ(m) = m[0]⊗ S(m[−1]) is a rightright YetterDrinfeld module.
The category of rightleft YetterDrinfeld modules,HYDH, is a braided monoidal
category. The monoidal structure and the braiding are given by
λ(m1(1)⊗ n1(2)) = m[−1]n[−1]⊗ m[0]⊗ n[0];
σM,N(m1(1)⊗ n1(2)) = nm[−1]⊗ m[0].
As a braided monoidal category,HYDHis isomorphic to Zl(MH) and (YDH
H)in.
The antipode S : H → Hop,copis an isomorphism of weak Hopf algebras. Observe
that the target map of Hop,copis εs, and that its source map is εt. Thus S induces
an isomorphism between the monoidal categoriesHM andHop,copM. We also have
a monoidal isomorphism F :
Hop,copM → MH, given by
F(M) = M, mh = hop,copm.
indeed, inHop,copM, M⊗tN is generated by elements of the form 1(2)m⊗1(1)n, and
F(M ⊗tN) is generated by elements of the form m1(2)⊗ n1(1). F(N) ⊗sF(M) is
generated by elements of the form n1(1)⊗m1(2), and it follows that the switch map is
an isomorphism F(M ⊗tN) → F(N)⊗sF(M). We conclude from Proposition 1.11
that we have isomorphisms of braided monoidal categories
H
HYD∼= Zl(HM)∼= Zl(Hop,copM)∼= Zl(MH)∼= Zr(MH)∼= YD
H
H.
This isomorphism can be described explicitely as follows:
F :
H
HYD → YD
H
H, F(M) = M,
with
m · h = S−1(h)m ; ρ(m) = m[0]⊗ S(m[−1]).
We summarize our results as follows:
Theorem 2.12. Let H be a weak Hopf algebra with bijective antipode. Then we
have the following isomorphisms of braided monoidal categories:
H
HYD∼=HYDHin∼= YDH
H∼=HYDH
in.
3. YetterDrinfeld modules are DoiHopf modules
It was shown in [7] that YetterDrinfeld modules (over a classical Hopf algebra)
can be considered as DoiHopf modules, and, a fortiori, as entwined modules, and
as comodules over a coring (see [4]). Weak DoiHopf modules were introduced by
B¨ ohm [1], and they are special cases of weak entwined modules (see [6]), and these
are in turn examples of comodules over a coring (see [4]). In this Section, we will
show that YetterDrinfeld modules over weak Hopf algebras are special cases of
weak DoiHopf modules. We will discuss the leftright case.
Proposition 3.1. Let H be a weak Hopf algebra with a bijective antipode. Then
H is a right H ⊗ Hopcomodule algebra, with Hcoaction
ρ(h) = h(2)⊗ S−1(h(1)) ⊗ h(3).
Page 14
14S. CAENEPEEL, DINGGUO WANG, AND YANMIN YIN
Proof. It is easy to verify that H is a right H ⊗ Hopcomodule and that ρ(hk) =
ρ(h)ρ(k). Recall that Ht= Im(εt) = Im(εt). The target map of Hop⊗H is εt⊗εt.
We now have
1[0]⊗ (εt⊗ εt)(1[1]) = 1(2)1(1′)⊗ εt(S−1(1(1))) ⊗ εt(1(2′))
1(2)1(1′)⊗ S−1(1(1)) ⊗ 1(2′)= ρ(1),=
where we used the fact that S−1(1(1)) ⊗ 1(2)∈ Ht⊗ Ht.
?
Proposition 3.2. Let H be a weak Hopf algebra with a bijective antipode. Then
H is a left Hop⊗ Hmodule coalgebra with left action
(k ⊗ h) ⊲ c = hck.
Proof. We easily compute that
ε((m ⊗ l)(k(2)⊗ h(2)))ε((k(1)⊗ h(1)) ⊲ c)
=ε(k(2)m)ε(lh(2))ε(h(1)ck(1))
=ε(lhckm) = ε(((m ⊗ l)(k ⊗ h)) ⊲ c).
The other conditions are easily verified.
?
Corollary 3.3. Let H be a weak Hopf algebra with bijective antipode. Then we
have a weak DoiHopf datum (Hop⊗H,H,H) and the categoriesHM(Hop⊗H)H
andHYDHare isomorphic.
Proof. The compatibility relation (35) reduces to (54).
?
As we have seen in Section 1.4, weak DoiHopf modules are special cases of entwined
modules. The entwining map ψ : H ⊗ H → H ⊗ H corresponding to the weak
DoiHopf datum (Hop⊗ H,H,H) is given by
(61)ψ(h ⊗ k) = h(2)⊗ h(3)kS−1(h(1)).
4. The Drinfeld double
Now we consider the particular case where H is finitely generated and projective
as a kmodule, with finite dual basis {(hi,h∗
weak Hopf algebra, in view of the selfduality of the axioms of a weak Hopf algebra.
Recall that the comultiplication is given by the formula ?∆(h∗),h ⊗ k? = ?h∗,hk?;
the counit is evaluation at 1. Also recall that H∗is an Hbimodule, with left and
right Haction
?h⇀h∗↼k,l? = ?h∗,klh?,
i)  i = 1,··· ,n}. Then H∗is also a
or
(62)h⇀h∗↼k = ?h∗
(1),k??h∗
(3),h?h∗
(2).
Using (36), we find a weak smash product structure (H,H∗,R), with R : H∗⊗H →
H ⊗ H∗given by
R(h∗⊗ h)=
?
?
h(2)⊗
i?h∗,h(3)hiS−1(h(1))?h(2)⊗ h∗
i?S−1(h(1))⇀h∗↼h(3),hi?h(2)⊗ h∗
?
i
=
i
=S−1(h(1))⇀h∗↼h(3)
?
.(63)
Page 15
YETTERDRINFELD MODULES OVER WEAK BIALGEBRAS15
From Section 1.4, we know that H#RH∗, which we will also denote by H ⊲⊳ H∗, is
an associative algebra with preunit 1#ε. Using (33), we compute the multiplication
rule on H ⊲⊳ H∗.
(h ⊲⊳ h∗)(k ⊲⊳ k∗) =?
=
i?h∗,k(3)hiS−1(k(1))?hk(2)⊲⊳ h∗
hk(2)⊲⊳ (S−1(k(1))⇀h∗↼k(3)) ∗ k∗
hk(2)⊲⊳ ?h∗
i∗ k∗
(64)
=
(1),k(3)??h∗
(3),S−1(k(1))?h∗
(2)∗ k∗. (65)
We have a projection p : H ⊲⊳ H∗→ H ⊲⊳ H∗,
p(h ⊲⊳ h∗) = (1 ⊲⊳ ε)(h ⊲⊳ h∗) = (h ⊲⊳ h∗)(1 ⊲⊳ ε) = (h ⊲⊳ h∗)(1 ⊲⊳ ε)2,
and D(H) = H ⊲⊳ H∗= (H ⊲⊳ H∗)/Kerp is a kalgebra with unit [1 ⊲⊳ ε], which
we call the Drinfeld double of H. D(H) is also isomorphic to H ⊲⊳ H∗= Im(p),
which is a kalgebra with unit (1 ⊲⊳ ε)2. Observe that the multiplication rule (65)
is the same as in [1, 16]. We show that the ideal J that is divided out in [1, 16]
is equal to Kerp, and this will imply that D(H) is equal to the Drinfeld double
introduced in [1, 16]. We first need some Lemmas.
Lemma 4.1. Let H a weak bialgebra. For all h∗∈ H∗, y ∈ Hs and z ∈ Ht, we
have
h∗∗ (y⇀ε)
h∗∗ (ε↼y)
(z⇀ε) ∗ h∗
(ε↼z) ∗ h∗
=?h∗
?h∗
?h∗
?h∗
(2),y?h∗
(1),y?h∗
(2),z?h∗
(1),z?h∗
(1)= y⇀h∗
(2)= h∗↼y
(1)= z⇀h∗
(2)= h∗↼z
(66)
=
(67)
= (68)
=
(69)
Proof. We only prove (68). For all h ∈ H, we have
?(z⇀ε) ∗ h∗,h? = ?ε,h(1)z??h∗,h(2)? = ?ε,h(1)1(1)z??h∗,h(2)1(2)?
=?ε ∗ h∗,hz? = ?h∗,hz? = ?z⇀h∗,h? = ?h∗
(2),z??h∗
(1),h?.
?
Lemma 4.2. Let H be a weak Hopf algebra with bijective antipode. For all y ∈ Hs,
z ∈ Ht, we have
(70)S−1(z)⇀ε = z⇀ε and ε↼y = ε↼S−1(y).
Proof. For all h ∈ H, we have
?S−1(z)⇀ε,h? = ε(hS−1(z))
(21)
=ε(h(1))ε(zεs(h(2)))
(2)
=ε(h1(1))ε(1(2)S−1(z))1.5
(8)
=ε(εs(h)z)
=ε(h1(1))ε(zS(1(2)))
=ε(hz) = ?z⇀ε,h?.
(5)
The second statement can be proved in a similar way.
?
Proposition 4.3. Let H be a finitely generated projective weak Hopf algebra. Then
Ker(p) is the klinear span J of elements of the form
A = hz ⊲⊳ h∗− h ⊲⊳ (z⇀ε) ∗ h∗and B = hy ⊲⊳ h∗− h ⊲⊳ (ε↼y) ∗ h∗,
where h ∈ H, h∗∈ H∗, y ∈ Hsand z ∈ Ht.
Page 16
16 S. CAENEPEEL, DINGGUO WANG, AND YANMIN YIN
Proof. A ∈ Ker(p) since
(1 ⊲⊳ ε)(hz ⊲⊳ h∗)
(65)
= h(2)1(2)⊲⊳ ε(2)∗ h∗?ε(1),h(3)1(3)??ε(3),S−1(h(1)1(1)z)?
h(2)⊲⊳ ε(2)∗ h∗?ε(1),h(3)??ε(3),S−1(z)??ε(4),S−1(h(1))?
h(2)⊲⊳ (ε(2)∗ (S−1(z)⇀ε) ∗ h∗)?ε(1),h(3)??ε(3),S−1(h(1))?
(1 ⊲⊳ ε)?h ⊲⊳ ((S−1(z)⇀ε) ∗ h∗)?(70)
In a similar way, B ∈ Ker(p):
=
(66)
=
(65)
== (1 ⊲⊳ ε)(h ⊲⊳ (z⇀ε) ∗ h∗).
(1 ⊲⊳ ε)(hy ⊲⊳ h∗)
(65)
= (h(2)1(2)⊲⊳ ε∗
(h(2)⊲⊳ ε∗
(h(2)⊲⊳ (ε(2)↼y) ∗ h∗)?ε(1),h(3)??ε(3),S−1(h(1))?
(1 ⊲⊳ ε)(h ⊲⊳ (ε↼y) ∗ h∗).
(2)∗ h∗)?ε(1),h(3)1(3)y??ε(3),S−1(h(1)1(1))?
(3)∗ h∗)?ε(1),h(3)??ε(2),y??ε(4),S−1(h(1))?
=
(67)
=
(65)
=
This shows that J ⊂ Ker(p). We now compute for all h ∈ H and h∗∈ H∗that
(h ⊲⊳ h∗)(1 ⊲⊳ ε)
(65)
= (h1(2)1(1′)⊲⊳ h∗
(2))?h∗
(1),1(2′)??h∗
(3),S−1(1(1))?,
and?
h ⊲⊳ (S−1(1(2))⇀ε) ∗ (ε↼1(1′)) ∗ h∗
(2)
?
?h∗
(1),1(2′)??h∗
(3),S−1(1(1))?
(62)
=
?
h ⊲⊳ ε(1)∗ ε(2′)∗ h∗
(2)
?
?ε(2),S−1(1(2))??ε(1′),1(1′)?
?h∗
(1),1(2′)??h∗
(3),S−1(1(1))?
?
=
?
?
h ⊲⊳ ε(1)∗ ε(2′)∗ h∗
(2)
?ε(1′)∗ h∗
(1),1??ε(2)∗ h∗
(3),S−1(1)?
= h ⊲⊳ ε(1)∗ h∗
(1)
?
?ε(2)∗ h∗
(2),1? = h ⊲⊳ (ε ∗ h∗) = h ⊲⊳ h∗.
Observing that
hzy ⊲⊳ h∗− h ⊲⊳ ((S−1(z)⇀ε) ∗ (ε↼y) ∗ h∗)
hzy ⊲⊳ h∗− hz ⊲⊳ (ε↼y) ∗ h∗)
hz ⊲⊳ (ε↼y) ∗ h∗) − h ⊲⊳ ((S−1(z)⇀ε) ∗ (ε↼y) ∗ h∗) ∈ J,
=
+
it follows that (h ⊲⊳ h∗)(1 ⊲⊳ ε) − (h ⊲⊳ h∗) ∈ J, for all h ∈ H and h∗∈ H∗. If
x ∈ Ker(p), then x(1 ⊲⊳ ε) = 0, and x = x − x(1 ⊲⊳ ε) ∈ J. We conclude that
Ker(p) ⊂ J, finishing our proof.
?
We now recall the following results from [17]. On H∗⊗H, there exists an associative
multiplication
(h∗⊗ h)(k∗⊗ k)=k∗
(h(3)⇀k∗↼S(h(1))) ∗ h∗⊗ h(2)k.
(2)h∗⊗ h(2)k?S(h(1)),k∗
(1)??h(3),k∗
(3)?
=
The kmodule I generated by elements of the form
A′= h∗⊗ hz − (ε↼z)h∗⊗ h and B′= h∗⊗ yh − (y⇀ε)h∗⊗ h
is a twosided ideal of H∗⊗H. The quotient D′(H) = (H∗⊗H)/I is an algebra with
unit element ε ⊗ 1. It is a weak Hopf algebra, with the following comultiplication,
Page 17
YETTERDRINFELD MODULES OVER WEAK BIALGEBRAS17
counit and antipode:
∆[h∗⊗ h] = [h∗
ε[h∗⊗ h] = ?h∗,εt(h)?
S[h∗⊗ h] = [S−1(h∗
(1)⊗ h(1)] ⊗ [h∗
(2)⊗ h(2)] (71)
(72)
(2)) ⊗ S(h(2))]?h∗
(1),h(1)??h∗
(3),S(h(3))? (73)
Proposition 4.4. The klinear isomorphism
f : H ⊲⊳ H∗→ H∗⊗ H, f(h ⊲⊳ h∗) = h∗⊗ S−1(h)
is antimultiplicative, and induces an algebra isomorphism f : D(H) → D′(H)op.
Proof. Let us first prove that f reverses the multiplication. Indeed,
f(k ⊲⊳ k∗)f(h ⊲⊳ h∗) = (k∗⊗ S−1(k))(h∗⊗ S−1(h))
(S−1(k(1))⇀h∗↼k(3)) ∗ k∗⊗ S−1(k(2))S−1(h)
=f((h ⊲⊳ h∗)(k ⊲⊳ k∗)).
=
Using Lemma 4.2, we easily compute that f(J) = I, and the result follows.
?
Let us now define a comultiplication, counit and antipode on D(H), in such a way
that f :D(H) → D′(H) is an isomorphism of Hopf algebras. Obviously, the
comultiplication is given by the formula
(74) ∆[h ⊲⊳ h∗] = [h(2)⊲⊳ h∗
(1)] ⊗ [h(1)⊲⊳ h∗
(2)].
The counit is computed as follows:
(75)ε[h ⊲⊳ h∗] = ε[h∗⊗ S−1(h)]
(72)
= ?h∗,εt(S−1(h))?
(15)
= ?h∗,1(2)??ε,h1(1)?.
Since the antipode of H is the inverse of the antipode of Hop, the antipode of D′(H)
is transported to the inverse of the antipode of D(H). We find
S−1[h ⊲⊳ h∗] = (f−1◦ S ◦ f)[h ⊲⊳ h∗] = f−1(S[h∗⊗ S−1(h)])
f−1[S−1(h∗
=
(2)) ⊗ h(2)]?h∗
(2))]?h∗
(1),S−1(h(3))??h∗
(1),S−1(h(3))??h∗
(3),h(1)?
=[S(h(2)) ⊲⊳ S−1(h∗
(3),h(1)?(76)
The antipode S is then given by the formula
(77)S[h ⊲⊳ h∗] = [S−1(h(2)) ⊲⊳ S(h∗
(2))]?h∗
(1),S−1(h(3))??h∗
(3),h(1)?
Indeed,
S(S−1[h ⊲⊳ h∗])
=[h(3)⊲⊳ h∗
[h(3)⊲⊳ h∗
[h(2)⊲⊳ h∗
ε([h ⊲⊳ h∗](1))[h ⊲⊳ h∗](2)ε([h ⊲⊳ h∗](3)) = [h ⊲⊳ h∗].
(3)]?h∗
(2)]?h∗
(2)]?h∗
(1),S−1(h(5))??h∗
(1),S−1(h(5))h(4)??h∗
(1),εt(S−1(h(3)))??h∗
(2),h(4)??h∗
(2),S−1(h(2))h(1)?
(3),εt(S−1(h(1)))?
(5),h(1)??h∗
(4),S−1(h(2))?
=
=
=
Similar arguments show that S−1(S[h ⊲⊳ h∗]) = [h ⊲⊳ h∗].
Proposition 4.5. Let H be a weak Hopf algebra with bijective antipode, which is
finitely generated and projective as a kmodule. Then D(H) is a weak Hopf algebra,
with comultiplication, counit and antipode given by the formulas (74,75,76). As a
weak Hopf algebra, D(H) is isomorphic to D′(H)op.
Page 18
18 S. CAENEPEEL, DINGGUO WANG, AND YANMIN YIN
Proposition 4.6. Let H be a weak Hopf algebra with bijective antipode, which is
finitely generated and projective as a kmodule. The functor
F :
HYDH→D(H)M, F(M) = M,
with
(h ⊲⊳ h∗)m = ?h∗,m[1]?hm[0],
for all h ∈ H, h∗∈ H∗and m ∈ M is an isomorphism of monoidal categories.
Proof. We already know (see (37)) that F is an isomorphism of categories, so we
only have to show that F preserves the product. Take M,N ∈HYDH. The right
Hcoaction on M ⊗tN is given by the formula (use (48) and (52)):
ρ(1(1)m ⊗ 1(2)n) = m[0]⊗ n[0]⊗ n[1]m[1],
hence the left D(H)action on F(M ⊗tN) is the following
(78)[h ⊲⊳ h∗](1(1)m ⊗ 1(2)n) = ?h∗,n[1]m[1]?h(1)m[0]⊗ h(2)n[0].
We now compute
F(N) ⊗tF(M) = {[1 ⊲⊳ ε]X  X ∈ F(N) ⊗ F(M)}.
Observe that
[1 ⊲⊳ ε](1)n ⊗ [1 ⊲⊳ ε](2)m = ?ε(1),n[1]?1(2)n[0]⊗ ?ε(2),m[1]?1(1)m[0]
=?ε,n[1]m[1]?1(2)n[0]⊗ 1(1)m[0].
We claim that the switch map τ : M ⊗ N → N ⊗ M induces an isomorphism τ :
F(M ⊗tN) → F(N)⊗tF(M) of kmodules. Indeed, take 1(1)m⊗1(2)n ∈ M ⊗tN.
Since M ⊗tN is a YetterDrinfeld module, we have that ε(n[1]m[1])m[0]⊗ n[0]=
1(1)m ⊗ 1(2)n, and
τ(1(1)m ⊗ 1(2)n) = 1(2)n ⊗ 1(1)m = 1(2′)1(2)n ⊗ 1(1′)1(1)m
=ε(n[1]m[1])1(2)n[0]⊗ 1(1)m[0]
= [1 ⊲⊳ ε](1)n ⊗ [1 ⊲⊳ ε](2)m ∈ F(N) ⊗tF(M).
Conversely,
τ([1 ⊲⊳ ε](1)n ⊗ [1 ⊲⊳ ε](2)m) = ε(n[1]m[1])1(1)m[0]⊗ 1(2)n[0]∈ F(M ⊗tN).
Let us now show that τ is left D(H)linear. To this end, we compute the left
D(H)action on F(N) ⊗tF(M).
[h ⊲⊳ h∗]τ(1(1)m ⊗ 1(2)n) = [h ⊲⊳ h∗](1(2)n ⊗ 1(1)m)
=[h(2)⊲⊳ h∗
(55)
=?h∗
(1)](1(2)n) ⊗ [h(1)⊲⊳ h∗
(1),n[1]S−1(1(2))?h(2)n[0]⊗ ?h∗
?h∗,n[1]S−1(1(2))1(1)m[1]?h(2)n[0]⊗ h(1)m[0]
τ?[h ⊲⊳ h∗](1(1)m ⊗ 1(2)n)?
It also follows that F(Ht) is a unit object inD(H)M. Since the unit object in a
monoidal category is unique up to automorphism, we concluce that the target space
of D(H)tis isomorphic to Ht. This can also be seen as follows: in [17], it is shown
that D′(H)t= [ε ⊗ Ht]∼= Ht. Since the target spaces of a weak Hopf algebra and
its opposite coincide, it follows that D(H)t∼= Ht.
(2)](1(1)m)
(2),1(1)m[1]?h(1)m[0]
=
(78)
=
?
Page 19
YETTERDRINFELD MODULES OVER WEAK BIALGEBRAS 19
5. Duality
Let H be a weak Hopf algebra with bijective antipode, andHRep the category
of left Hmodules M which are finitely generated projective as a kmodule. Let
M ∈HRep, and let {(ni,n∗
i)  i = 1,···n} be a finite dual basis of M. From [17],
we recall the following result. We refer to [12] for the definition of duality in a
monoidal category.
Proposition 5.1. The categoryHRep has left duality. The left dual of M ∈HRep
is M∗= Hom(M,k) with left Haction defined by
(79)?h · m∗,m? = ?m∗,S(h)m?,
for all h ∈ H, m ∈ M and m∗∈ M∗. The evaluation map evM: M∗⊗tM → Ht
and the coevaluation map coevM: Ht→ M ⊗tM∗are defined as follows:
evM(1(1)· m∗⊗ 1(2)m) = ?m∗,1(1)m?1(2);
coevM(z) = z · (?
ini⊗ n∗
i).
Let M be a finitely generated projective left Hcomodule. Then M∗is also a left
Hcomodule, with left Hcoaction λ : M∗→ H ⊗ M∗given by
λ(m∗) =?
i?m∗,ni[0]?S−1(ni[−1]) ⊗ n∗
i.
The definition of λ can also be stated as follows: λ(m∗) = m∗
if
[−1]⊗m∗
[0]if and only
(80)?m∗
[0],m?S(m∗
[−1]) = ?m∗,m[0]?m[−1],
for all m ∈ M.
Proposition 5.2. Let M be a finitely generated projective leftleft YetterDrinfeld
module over the weak Hopf algebra H. Then M∗with Haction and Hcoaction
given by (79) and (80) is also a leftleft YetterDrinfeld module.
Proof. We have to show that
λ(h · m∗) =?
i?m∗,S(h)ni[0]?S−1(ni[−1]) ⊗ n∗
i
equals
h(1)m∗
[−1]S(h(3)) ⊗ h[2]m∗
[−1]=?
i?m∗,ni[0]?h(1)S−1(ni[−1])S(h(3)) ⊗ (h(2)· n∗
i).
It suffices to show that both terms coincide after we evaluate the second tensor
factor at an arbitrary m ∈ M.
?
i?m∗,ni[0]?h(1)S−1(ni[−1])S(h(3))?n∗
?m∗,(S(h(2))m)[0]?h(1)S−1?
?m∗,S(h(3))m[0]?h(1)S−1?
?m∗,S(h(3))m[0]?h(1)S(h(2))S−1(m[−1])h(4)S(h(5))
?m∗,S(h(2))m[0]?εt(h(1))S−1(m[−1])εt(h(3))
i,S(h(2))m?
=
(S(h(2))m)[−1]
?
S(h(3))
(41)
=
S(h(4))m[−1]S2(h(2))
?
S(h(5))
=
=
Page 20
20 S. CAENEPEEL, DINGGUO WANG, AND YANMIN YIN
(21)
=?m∗,S(1(2)h(1))m[0]?S(1(1))S−1(m[−1])εt(h(2))
?m∗,S(h(1))1(1)m[0]?1(2)S−1(m[−1])εt(h(2))
?m∗,S(1(1)h)m[0]?S−1(m[−1])1(2)
?m∗,S(h)S(1(1))m[0]?S−1(S(1(2))m[−1])
?m∗,S(h)1(2)m[0]?S−1(1(1)m[−1])
?m∗,S(h)m[0]?S−1(m[−1])
?
=
(3,44)
=
=
=
(38)
=
=
i?m∗,S(h)ni[0]?S−1(n∗
i[−1])?n∗
i,m?
?
Proposition 5.3. The category of finitely generated projective leftleft Yetter
Drinfeld modules has left duality.
Proof. In view of the previous results, it suffices to show that the evaluation map
evM and the coevaluation map coevM are left Hcolinear, for every finitely gener
ated projective leftleft YetterDrinfeld module M. Let us first show that evM is
left Hcolinear.
(H ⊗ evM)(λ(1(1)· m∗⊗ 1(2)m))
=m∗
(80)
=?m∗,(1(1)m[0])[0]?S−1((1(1)m[0])[−1])m[−1]⊗ 1(2)
(43)
=?m∗,m[0]?1(1)S−1(m[−1])m[−2]⊗ 1(2)
?m∗,m[0]?1(1)εt(S−1(m[−1])) ⊗ 1(2)
(44)
=?m∗,1(1′)m[0]?1(1)εt(1(2′)S−1(m[−1])) ⊗ 1(2)
(10)
=?m∗,1(1′)m[0]?1(1)1(2′)εt(S−1(m[−1])) ⊗ 1(2)
(25)
=?m∗,1(1′)S−1(εt(S−1(m[−1])))m[0]?1(1)1(2′)⊗ 1(2)
(22)
=?m∗,1(1′)εs(S−2(m[−1]))m[0]?1(1)1(2′)⊗ 1(2)
(1,45)
=?m∗,1(1)m?1(2)⊗ 1(3)
=λ(evM(1(1)· m∗⊗ 1(2)m)).
[−1]m[−1]⊗ ?m∗
[0],1(1)m[0]?1(2)
=
(49)
= λ(?m∗,1(1)m?1(2))
To prove that coevM is left Hcolinear, we have to show that, for all z ∈ Ht,
λ(coevM(z)) =?
=
iλ(1(1)zni⊗ 1(2)· n∗
i(1(1)zni)[−1](1(2)· n∗
i)
?
i)[−1]⊗ (1(1)zni)[0]⊗ (1(2)· n∗
i)[0]
equals
(H ⊗ coevM)(λ(z)) = (H ⊗ coevM)(1(1)z ⊗ 1(2)) =?
It suffices to show that both terms coincide after we evaluate the third tensor factor
at an arbitrary m ∈ M. Indeed
i1(1)z ⊗ 1(2)ni⊗ 1(3)· n∗
i.
?
i(1(1)zni)[−1](1(2)· n∗
(80)
=
?
=
?
= (1(1)zS(1(2))m[0])[−1]S−1(m[−1]) ⊗ (1(1)zS(1(2))m[0])[0]
i)[−1]⊗ (1(1)zni)[0]?(1(2)· n∗
i(1(1)zni)[−1]?1(2)· n∗
i(1(1)zni)[−1]?n∗
i)[0],m?
i,m[0]?S−1(m[−1]) ⊗ (1(1)zni)[0]
i,S(1(2))m[0]?S−1(m[−1]) ⊗ (1(1)zni)[0]
Page 21
YETTERDRINFELD MODULES OVER WEAK BIALGEBRAS 21
(6,41)
=1(1)zm[−1]S(1(3))S−1(m[−2]) ⊗ 1(2)m[0]
1(1)zm[−1]S(1(2′))S−1(m[−2]) ⊗ 1(2)1(1′)m[0]
1(1)zm[−1]S−1(m[−2]) ⊗ 1(2)m[0]
1(1)zS−1(εt(m[−1])) ⊗ 1(2)m[0]
1(1)z ⊗ 1(2)εt(m[−1])m[0]
1(1)z ⊗ 1(2)S(1(3))m =?
?
=
(42)
=
=
(25)
=
(40)
= 1(1)z ⊗ 1(2)m
i1(1)z ⊗ 1(2)ni?n∗
i,m?,
=
i,S(1(3))m?
=
i1(1)z ⊗ 1(2)ni?1(3)· n∗
as needed.
?
6. Appendix. Weak bialgebras and bialgebroids
In [21], YetterDrinfeld modules over a ×Rbialgebra (see [24]) are introduced, and
it is shown that the weak center of the category of left modules is isomorphic to
the category of YetterDrinfeld modules. The notion of ×Rbialgebra is equivalent
to the notion of Rbialgebroid, we refer to [3] for a detailed discussion. So we can
consider YetterDrinfeld modules over bialgebroids.
A weak bialgebra H can be viewed as a bialgebroid over the target space Ht; this was
shown in [9] in the weak Hopf algebra case, and generalized to the weak bialgebra
case in [22]. The aim of this Section is to make clear that YetterDrinfeld modules
over H considered as a weak bialgebra coincide with YetterDrinfeld modules over
Hconsidered as a bialgebroid.
To this end, we first recall the definition of a bialgebroid, as introduced by Lu [14].
Let k be a commutative ring, and R a kalgebra. An R ⊗ Ropring is a pair (H,i),
with H a kalgebra and i : R ⊗ Rop→ H. Giving i is equivalent to giving algebra
maps sH: R → H and tH: R → Hopsatisfying sH(a)tH(b) = tH(b)sH(a), for all
a,b ∈ R. We then have that i(a⊗b) = sH(a)tH(b). Restriction of scalars makes H
into a left R ⊗ Ropmodule, and an Rbimodule:
a · h · b = sH(a)tH(b)h.
Consider
H ×RH = {
?
i
?
i
hi⊗Rki∈ H ⊗RH

hitH(a) ⊗Rki=
?
i
hi⊗RkisH(a), for all a ∈ R}
It is easy to show that H ×RH is a ksubalgebra of H ⊗RH.
Recall that an Rcoring is a triple (H,˜∆, ˜ ε), with H an Rbimodule and˜∆ : H →
H ⊗RH and ˜ ε : H → R Rbimodule maps satisfying the usual coassociativity and
counit properties; we refer to [4] for a detailed discussion of corings.
Definition 6.1. [14] A left Rbialgebroid is a fivetuple (H,sH,tH,˜∆, ˜ ε) satisfying
the following conditions.
(1) (H,˜∆, ˜ ε) is an Rcoring;
(2) (H,m ◦ (sH⊗ tH) = i) is an R ⊗ Ropring;
(3) Im(˜∆) ⊂ H ×RH;
(4)˜∆ : H → H ×RH is an algebra map, ˜ ε(1H) = 1Rand
˜ ε(gh) = ˜ ε(gsH(˜ ε(h))) = ˜ ε(gtH(˜ ε(h))),
Page 22
22 S. CAENEPEEL, DINGGUO WANG, AND YANMIN YIN
for all g,h ∈ H.
Take two left Hmodules M and N; then M and N are Rbimodules, by restriction
of scalars. M ⊗RN is a left Hmodule, with
h · (m ⊗Rn) = h(1)m ⊗Rh(2)n.
Also R is a left Hmodule, with
h · r = ˜ ε(hsH(r)) = ˜ ε(htH(r)).
(HM,⊗R,R) is a monoidal category, and the restriction of scalars functorHM →
RMRis strictly monoidal; this can be used to reformulate the definition of a bial
gebroid (see [3, 20, 23]).
In [21, Sec. 4], leftleft YetterDrinfeld modules over H are introduced, and it
is shown that Wl(HM) is isomorphic to the category of YetterDrinfeld modules.
According to [21], a leftleft YetterDrinfeld Hmodule is a left comodule M over
the coring H, together with a left Haction on M such that the underlying left
Ractions coincide, and such that
(81)h(1)m[−1]⊗Rh(2)· m[0]= (h(1)· m)[−1]h(2)⊗R(h(1)· m)[0]
holds in H ⊗RM, for all h ∈ H and m ∈ M.
Let H be a weak bialgebra, and consider the maps
sH: Ht
tH= εsHt: Ht→ Hs⊂ H;
⊂H;
˜∆ = can ◦ ∆ : H → H ⊗ H
canH ⊗HtH;
˜ ε = εt: H → Ht.
Then (H,sH,tH,˜∆, ˜ ε) is a left Htbialgebroid. The fact that Im(˜∆) ⊂ H ×HtH
follows from the separability of Htas a kalgebra (cf. Proposition 1.3).
We have seen in Section 1.1 that, for any two left Hmodules M and N, we have
an isomorphism π : M ⊗HtN → M ⊗tN. This entails that the monoidal cate
gories (HM,⊗t,Ht) and (HM,⊗Ht,Ht) are isomorphic, and a fortiori, their weak
left centers are isomorphic categories. Consequently, the two corresponding cate
gories of YetterDrinfeld modules are isomorphic. This can also be seen directly,
comparing the definitions in Section 2 and (81).
Acknowledgment
We thank Tomasz Brzezi´ nski and the referee for their useful comments, and Adriana
Nenciu for sending us her paper [16].
References
[1] G. B¨ ohm, DoiHopf modules over weak Hopf algebras, Comm. Algebra 28 (2000), 4687–4698.
[2] G. B¨ ohm, F. Nill, K. Szlach´ anyi, Weak Hopf algebras I. Integral theory and C∗structure, J.
Algebra 221 (1999), 385438.
[3] T. Brzezi´ nski and G. Militaru, Bialgebroids, ×Rbialgebras and duality, J. Algebra 251
(2002), 279–294.
[4] T. Brzezi´ nski and R. Wisbauer, “Corings and comodules”, London Math. Soc. Lect. Note
Ser. 309, Cambridge University Press, Cambridge, 2003.
[5] D. Bulacu, S. Caenepeel, F. Panaite, YetterDrinfeld categories for quasiHopf algebras,
Comm. Algebra, to appear.
Page 23
YETTERDRINFELD MODULES OVER WEAK BIALGEBRAS 23
[6] S. Caenepeel and E. De Groot, Modules over weak entwining structures, Contemp. Math.
267 (2000), 31–54.
[7] S. Caenepeel, G. Militaru and S. Zhu, Crossed modules and DoiHopf modules, Israel J.
Math. 100 (1997), 221–247.
[8] Y. Doi, Unifying Hopf modules, J. Algebra 153 (1992), 373385.
[9] P. Etingof, D. Nikshych, Dynamical quantum groups at roots of 1, Duke Math. J. 108 (2001),
135–168.
[10] T. Hayashi, Quantum group symmetry of partition functions of IRF models and its applica
tions to Jones’ index theory, Comm. Math. Phys. 157 (1993), 331345.
[11] A. Joyal, R. Street, Tortile YangBaxter operators in tensor categories, J. Pure Appl. Algebra
71 (1991), 43–51.
[12] C. Kassel, “Quantum groups”, Grad. Texts Math. 155, Springer Verlag, Berlin, 1995.
[13] M. Koppinen, Variations on the smash product with applications to groupgraded rings, J.
Pure Appl. Algebra 104 (1995), 6180.
[14] J. H. Lu, Hopf algebroids and quantum groupoids, Intern. J. Math. 7 (1996), 47–70.
[15] S. Majid, Representations, duals and quantum doubles of monoidal categories, Rend. Circ.
Mat. Palermo (2) Suppl. No. 26 (1991), 197–206.
[16] A. Nenciu, The center construction for weak Hopf algebras, Tsukuba J. Math. 26 (2002),
189–2004.
[17] D. Nikshych, V. Turaev, L. Vainerman, Invariants of knots and 3manifolds from quantum
groupoids, Topology Appl. 127 (2003), 91–123..
[18] D. Nikshych, L. Vainerman, Finite quantum groupoids and their applications, in “New di
rections in Hopf algebras”, S. Montgomery and H.J. Schneider (eds.), Math. Sci. Res. Inst.
Publ. 43, Cambridge Univ. Press, Cambridge, 2002, 211–262.
[19] A. Ocneanu, Quantum cohomology, quantum groupoids, and subfactors, unpublished talk
given at the First Caribbean School of Mathematics and Theoretical Physics, Guadeloupe,
1993.
[20] P. Schauenburg, Bialgebras over noncommutative rings and structure theorems for Hopf bi
modules, Appl. Categorical Structures 6 (1998), 193–222.
[21] P. Schauenburg, Duals and doubles of quantum groupoids (×Ralgebras), Contemp. Math.
267 (2000), 273–299.
[22] P. Schauenburg, Weak Hopf algebras and quantum groupoids, Banach Center Publ. 61 (2003),
171–181.
[23] K. Szlachanyi, Finite quantum groupoids and inclusion of finite type, Fields Institute Comm.
30 (2001), 393–407.
[24] M. Takeuchi, Groups of algebras over A ⊗ A, J. Math. Soc. Japan 29 (1977), 459–492.
[25] T. Yamanouchi, Duality for generalized Kac algebras and a characterization of finite groupoid
algebras, J. Algebra 163 (1994), 950.
Faculty of Engineering Sciences, Vrije Universiteit Brussel, VUB, B1050 Brussels,
Belgium
Email address: scaenepe@vub.ac.be
URL: http://homepages.vub.ac.be/~scaenepe/
Department of Mathematics, Qufu Normal University, Qufu, Shandong 273165, China
Email address: dgwang@qfnu.edu.cn
Department of Mathematics, Shandong Institute of Architecture and Engineering, Ji
nan, Shandong 250014, China
Email address: yanmin yin@163.com