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The Category of Partial Doi-Hopf Modules and Functors

Q.-G. CHEN(*) - D.-G. WANG(**)

ABSTRACT- Let (H;A;C), (H0;A0;C0) be two partial Doi-Hopf datums consisting of a Hopf algebraH, a partial rightH-comodule algebraAand a partial rightH- module coalgebra. Givena:H!H0,b:A!A0andg:C!C0, we define an induction functor between the categoryM(H)CAof all partial Doi-Hopf modules and the categoryM(H0)CA00, and we prove that this functor has a right adjoint.

Specially, we then give necessary and sufficient conditions for the functor F:M(H)CA! M(H)A(exactly the category of rightA-modules). This leads to a generalized notion of integrals. Moreover, from these results, we deduce a version of Maschke-type Theorems for partial Doi-Hopf modules. The applica- tions of our results are considered.

MATHEMATICSSUBJECTCLASSIFICATION(2010). 16W30.

KEYWORDS. Hopf algebras, partial Doi-Hopf modules, integral.

1. Introduction

Partial group actions were considered first by Exel [E1] in the context of operator algebras and they turned out to be a powerful tool in the study ofC-algebras generated by partial isometries on a Hilbert space [E2]. A treatment from a purely algebraic point of view was given recently in

(*) Indirizzo dell'A.: School of Mathematics and Statistics, Yili Normal College Yining 835000, China.

E-mail: cqg211@163.com

(**) Indirizzo dell'A.: School of Mathematical Sciences, Qufu Normal Univer- sity, Qufu, Shandong 273165, P.R.China (corresponding author).

E-mail: dingguo95@126.com

The first author was supported by the National Natural Science Foundation of China (No. 11261063) and the Fund of the Key Disciplines in the General Colleges and Universities of Xin Jiang Uygur Autonomous Region (No. 2012ZDXK03). The second anthor was supported by the National Natural Science Foundation of China (No. 11171183) and the Shandong Provincial Natural Science Foundation of China (No. ZR2011AM013).

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[DEP], [DFP], [DZ] and [DE]. Partial Hopf actions were motivated by an attempt to generalize the notion of partial Galois extensions of commu- tative rings in [DFP] to a broader context. The definition of partial Hopf actions and co-actions were introduced by using the notions of partial entwining structures in [CJ].

The category M(H)CA of Doi-Hopf modules was introduced in [D], whereHis a Hopf algebra,Aa rightH-comodule algebra andCa rightH- module coalgebra. It is the category of the modules over the algebra A which are also comodules over the coalgebra C and satisfy certain com- patibility condition involvingH. The study ofM(H)CAturned out to be very useful: it was shown in [D] that many categories such as the module and comodule categories over bialgebras, the Hopf modules category [S], Ta- keuchi's relative Hopf modules, graded modules, modules graded by G-sets, Long dimodules and the Yetter-Drinfeld category ([CMI], [RT], [Y]) are special cases ofM(H)CA.

As a general version of the categoryM(H)CAof Doi-Hopf modules, we shall consider a partial Doi-Hopf datum (H;A;C), and the categoryM(H)CAof so- called partial Doi-Hopf modules. The starting point of this paper is an attempt to discuss the results of [CR] in the partial case. This would have meant in particulargivinggeneralizationsoftheinducedand thecoinduced functors.In the paper, we give a generalization of the induction functor and try to char- acterize the separability of the functorF:M(H)CA! M(H)Awhich leads to the generalized integral of the partial Doi-Hopf datum (H;A;C). Also, a version of Maschke-type Theorems for partial Doi-Hopf modules is proved.

The paper is organized as follows.

In Section 2, we recall definitions and basic results related to Hopf partial action, and introduce the induction functor: given maps a:H!H0,b:A!A0andg:C!C0, we have a functor (calledinduction functor) Fˆ AA0:M(H)CA! M(H0)CA00. This functor have a right adjoint G. In Section 3, we discuss the separability of the functor F:M(H)CA! M(H)A that forgets the C-coaction,which leads to a generalized integral of a partial Doi-Hopf datum (H;A;C). The applica- tions of our results are considered in Section 4.

2. Partial Module Coalgebras, Partial Comodule Algebras, Partial Doi-Hopf Modules

Throughtout this paper,kwill be a field. Unless specified otherwise, all modules, algebras, coalgebras, bialgebras(or Hopf algebras), tensor pro-

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ducts and homomorphisms are overk.idenotes the identity mapping.H will be a Hopf algebra with an invertible antipode S and we will use Sweedler's sigma-notation extensively. For example, if (C;DC;eH) is a coalgebra, then for allc2C, we write

DC(c)ˆc1c2 2CC:

DEFINITION2.1. Let Hbe a Hopf algebra. A k-algebra A is called a partial right H-comodule algebra, if there exists a k-linear map rA:A!AH, rA(a)ˆa[0] a[1] such that the following conditions satisfy:

rA(ab)ˆrA(a)rA(b);

(2:1)

rA(a[0])a[1]ˆa[0]1A[0]a[1]11A[1]a[1]2; (2:2)

e(a[1])(a[0])ˆa;

(2:3)

for alla;b2A.

EXAMPLE 2.2. Let e2H be an idempotent such that eeˆ D(e)(e1H) and e(e)ˆ1. Then we can define the following partial right H-coaction onAˆk:r(x)ˆxe2kH.

DEFINITION2.3. Let Hbe a Hopf algebra. A k-coalgebra Cis called a partial right H-module coalgebra, if there exists a k-linear map f:CH!C, f(ch)ˆch such that the following conditions sat- isfy:

(ch)gˆchg;

(2:4)

(ch)11H(ch)2ˆc1h1c2h2; (2:5)

e(ch)ˆe(c)e(h);

(2:6)

for allc2Candg;h2H.

EXAMPLE 2.4. Let e2H be a central idempotent such that eeˆ D(e)(e1H) ande(e)ˆ1. Then we can define the partial rightH-action on CˆH:ghˆegh.

A partial Doi-Hopf datum is a threetuple (H;A;C), where H is a Hopf algebra,Aa partial rightH-comodule algebra andCa partial right H-module coalgebra. Given a partial Doi-Hopf datum (H;A;C). Apartial

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Doi-Hopf module Mis a rightA-module and there exists ak-linear map r:M!MC such that

r2M(m)ˆm[0]1A[0][0]m[1]11A[0][1]m[1]21A[1]; (2:7)

wherer2Mˆ(rMi)rM,

r(ma)ˆm[0]a[0]m[1]a[1]; (2:8)

e(m[1])m[0]ˆm;

(2:9)

for allm2Manda2A.

EXAMPLE 2.5. Let e2H be a central idempotent such that eeˆ D(e)(e1H) ande(e)ˆ1. Then (H;k;H) is a partial Doi-Hopf datum.

M(H)CAwill be the category of partial Doi-Hopf modules andA-linear, C-colinear maps. Now we can give the following result for the category of partial Doi-Hopf modules which is analogous of [Theorem 1.1, CR].

THEOREM2.6. Consider two partial Doi-Hopf datums(H;A;C)and (H0;A0;C0), and suppose that we have maps a:H!H0, b:A!A0 and g:C!C0 which are respectively Hopf algebra, algebra and coalgebra maps satisfying

g(ch)ˆg(c)a(h);

(2:10)

rA(b(a))ˆb(a[0])a(a[1]);

(2:11)

for all c2C, h2H and a2A. Then we have a functor F:M(H)CA! M(H0)CA00, defined as follows:

F(M)ˆMAA0;

where A0is a left A-module viaband with structure maps defined by (mAa0)b0ˆmAa0b0;

(2:12)

rF(M)(mAa0)ˆm[0]Aa0[0]g(m[1])a0[1]; (2:13)

for all a0;b02A0and m2M.

PROOF. Let us show thatMAA0is an object ofM(H0)CA00. For this, we need to show that MAA0 satisfies conditions (2.7)-(2.9). Notice that

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MAA0 satisfies (2.9) obviously. We restrict here to check thatMAA0 satisfies conditions (2.7) and (2.8). Takem2Manda0;b02A0. Then

rF(M)((mAa0)b0)ˆm[0]Aa0[0]b0[0]g(m[1])a0[1]b0[1]

ˆ(m[0]Aa0[0])b0[0](g(m[1])a0[1])b0[1]; i.e., (2.8) holds. For (2.7), for allm2Manda02A0, we have

r2F(M)(mAa0)

ˆm[0][0]Aa0[0][0]g(m[0][1])a0[0][1]g(m[1])a0[1]

(2:7)ˆ m[0]1A[0]Aa0[0][0]g(m[1]11A[1])a0[0][1]g(m[1]2)a0[1]

ˆm[0]Ab(1A[0])a0[0][0]g(m[1]1)a(1A[1])a0[0][1]g(m[1]2)a0[1]

(2:11)ˆ m[0]A1A0[0]a0[0][0]g(m[1]1)1A0[1]a0[0][1]g(m[1]2)a0[1]

(2:1)ˆ m[0]Aa0[0][0]g(m[1]1)a0[0][1]g(m[1]2)a0[1]

ˆm[0]Aa0[0]1A0[0][0]g(m[1]1)a0[1]11A0[0][1]g(m[1]2)a0[1]21A0[1]

(2:5)ˆ (m[0]Aa0[0])1A0[0][0](g(m[1])a0[1])11A0[0][1](g(m[1])a0[1])21A0[1]:

This is exactly what we have to show. p

THEOREM 2.7. Under the assumptions of Theorem 2.6, we have a functor G:M(H0)CA00 ! M(H)CAwhich is right adjoint to F. G is defined by

G(M0)ˆM0pC0Cˆ fm0b(1A[0])c1A[1]g;

where m0c2M0C satisfies the following condition:

m0[0]b(1A[0][0])m0[1]a(1A[0][1])c1A[1]

(2:14)

ˆm0b(1A[0][0])g(c1)a(1A[0][1])c21A[1]; for all M02 M(H0)CA00, and with structure maps

rG(M0)(m0b(1A[0])c1A[1])ˆm0b(1A[0][0])c11A[0][1]c21A[1]; (2:15)

(m0b(1A[0])c1A[1])aˆm0b(a[0])ca[1]; (2:16)

for all a2A.

PROOF. Let us first show thatG(M0) is an object ofM(H)CA. It is routine to check thatG(M0) is a rightC-comodule. In order to prove thatMis a right A-module, we need to show that m0b(a[0])ca[1]2M0pC0C, for all

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a2A. Indeed,

(m0b(a[0]))[0]b(1A[0][0])(m0b(a[0]))[1]a(1A[0][1])ca[1]1A[1]

(2:8)ˆ m0[0]b(a[0])[0]b(1A[0][0])m0[1]b(a[0])[1]a(1A[0][1])ca[1]1A[1]

(2:11)ˆ m0[0]b(a[0][0])b(1A[0][0])m0[1]a(a[0][1])a(1A[0][1])ca[1]1A[1]

(2:1)ˆ m0[0]b(1A[0][0])b(a[0][0])m0[1]a(1A[0][1])a(a[0][1])c1A[1]a[1]

(2:14)ˆ m0b(1A[0][0])b(a[0][0])g(c1)a(1A[0][1])a(a[0][1])c21A[1]a[1]

(2:1)ˆ m0b(a[0][0])b(1A[0][0])g(c1)a(a[0][1])a(1A[0][1])c2a[1]1A[1]

(2:2;2:1)ˆ m0b(a[0])b(1A[0][0])g(c1)a(a[1]1)a(1A[0][1])c2a[1]21A[1]

(2:10)ˆ m0b(a[0])b(1A[0][0])g(c1a[1]1)a(1A[0][1])c2a[1]21A[1]

(2:5)ˆ m0b(a[0])b(1A[0][0])g((ca[1])1)a(1A[0][1])(ca[1])21A[1]: This is exactly what we have to show.

G(M0)2 M(H)CA and the functorial properties are checked in a straightforward way. Let us finally show that G is a right adjoint toF.

Take M2 M(H)CA, we define hM:M!GF(M)ˆ(MAA0)pC0C as fol- lows: for allm2M,

hM(m)ˆm[0]Ab(1A[0])m[1]1A[1]: For alla2A, we have

hM(ma)ˆ(ma)[0]Ab(1A[0])(ma)[1]1A[1]

(2:8)ˆ m[0]a[0]Ab(1A[0])m[1]a[1]1A[1]

ˆm[0]Ab(a[0])b(1A[0])m[1]a[1]1A[1]

(2:1)ˆ m[0]Ab(1A[0])b(a[0])m[1]1A[1]a[1]

ˆ(m[0]Ab(1A[0])m[1]1A[1])a and

(hMi)rM(m)ˆm[0][0]Ab(1A[0])m[0][1]1A[1]m[1]

(2:7)ˆ m[0]10A[0]Ab(1A[0])m[1]110A[1]1A[1]m[1]2

ˆm[0]Ab(10A[0]1A[0])m[1]110A[1]1A[1]m[1]2 (2:1)ˆ m[0]Ab(1A[0])m[1]11A[1]m[1]2

ˆrGF(M)hM(m):

SohM2 M(H)CA.

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TakeM02 M(H0)CA00. Then we definedM0:FG(M0)!M0, where dM0((m0b(1A[0])c1A[1])Aa0)ˆeC(c)m0a0:

Notice thatdM0 isA0-linear. ThatdNisC0-colinear is proved as follows:

(dM0i)(rFG(M0))(m0b(1A[0])c1A[1])Aa0)

ˆdM0((m0b(1A[0][0])c11A[0][1])Aa0[0])g(c21A[1])a0[1]

ˆm0b(1A[0])a0[0]g(c1A[1])a0[1]: ApplyingiieCto (2.14), this yields

m0b(1A[0])g(c1A[1])ˆe(c)m0[0]b(1A[0])m0[1]a(1A[1])

ˆe(c)m0[0]b(1A[0])m0[1]a(1A[1]):

Using the identity above, it follows that

m0b(1A[0])a0[0]g(c1A[1])a0[1]ˆe(c)m0[0]b(1A[0])a0[0]m0[1]a(1A[1])a0[1]

(2:11)ˆ e(c)m0[0]1A0[0]a0[0]m0[1]1A0[1]a0[1]

(2:1)ˆ e(c)m0[0]a0[0]m0[1]a0[1]

ˆrM0dM0((m0b(1A[0])c1A[1])Aa0):

This is what we need to show. We can checkhandddefined above are all natural transformations and they satisfy

G(dM0)hG(M0)ˆI; dF(M)F(hM)ˆI;

for allM2 M(H)CAandM02 M(H0)CA00.

The proof of Theorem is completed. p

REMARK 2.8. We consider (H;A;k) and the map iH, iB and eC:C!k. NowM(H)A(orUA), the category of rightA-modules, andF is the functor which forgets the C-comodule structures. From Theorem 2.7,G(M0)ˆ fm01A[0]c1A[1]g ˆM0C with structure maps

rG(M0)(m0A[0]c1A[1])ˆm01A[0][0]c11A[0][1]c21A[1]; (2:17)

(m01A[0]c1A[1])aˆm0a[0]ca[1]; (2:18)

for alla2AandM02 M(H)A. The unithof the adjunction (F;G) is given by hM0:M0!G(M0),

hM0(m0)ˆm0[0]1A[0]m0[1]1A[1]:

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3. Integral of partial Doi-Hopf datum

DEFINITION3.1. Let (H;A;C) be a partial Doi-Hopf datum. Ak-linear

map u:CC!A

is called anormalized A-integral, ifusatisfies the following conditions:

c21A[1]31A[0]u(d1A[1]1c11A[1]2) (3:1)

ˆc21A[1]1A[0][0][0]u(d1A[0][0][1]c11A[0][1])

ˆd11A[0][1]u(d21A[1]c)A[1]1A[0][0]u(d21A[1]c)[0]; u(c1c2)ˆ1Ae(c);

(3:2)

a[0][0]u(ca[0][1]da[1])ˆu(cd)a;

(3:3)

for alla2Aandc;d2C.

THEOREM3.2. For any partial Doi-Hopf datum(H;A;C), the follow- ing assertions are equivalent,

(1) hin Remark2.8is a split natural monomorphism.

(2) The forgetful functor F is separable.

(3) There exists a normalized A-integralu:CC!A.

PROOF. (1)()(2) follows by Rafael Theorem ([R]).

(3)ˆ)(1). For any partial Doi-Hopf moduleM, we define nM:MC!M; nM(m1A[0]c1A[1])ˆm[0]u(m[1]c);

for allm2Mandc2C.

Now, we shall check thatnM2 M(H)CA. In fact, for allm2M,c2Cand a2A,

nM((m1A[0]c1A[1])a)ˆnM(ma[0]ca[1])

ˆ(ma[0])[0]u((ma[0])[1]ca[1])

(2:8)ˆ m[0]a[0][0]u(m[1]a[0][1]ca[1])

(3:3)ˆ m[0]u(m[1]c)a

ˆnM(m1A[0]c1A[1])a:

Hence it is a morphism of A-module. Next, we shall check that n is a

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morphism ofC-comodule. It is sufficient to check that rMnMˆ(nMi)rG(M) holds. For allm2Mandc2C, we have

rMnM(m1A[0]c1A[1])

ˆrM(m[0]u(m[1]c))

ˆ(m[0]u(m[1]c))[0](m[0]u(m[1]c))[1]

(2:8)ˆ m[0][0]u(m[1]c)[0]m[0][1]u(m[1]c)[1]

(2:7)ˆ m[0]1[0][0]u(m[1]21[1]c)[0]m[1]11[0][1]u(m[1]21[1]c)[1]

and

(nMi)rG(M)(m1A[0]c1A[1])

ˆ(nMi)(m1A[0][0]c11A[0][1]c21A[1])

ˆnM(m1A[0][0]c11A[0][1])c21A[1]

ˆ(m1A[0][0])[0]u((m1A[0][0])[1]c11A[0][1])c21A[1]

ˆm[0]10A[0]1A[0][0]u(m[1]1A[0][1]c11A[1])c210A[1]

ˆm[0]1A[0]u(m[1]c1)c21A[1]:

Using (3.1), we can get the desired result. For allm2M, Since nMhM(m)ˆnM(m[0]m[1])

ˆm[0][0]u(m[0][1]m[1])

(2:7)ˆ m[0]1A[0][0]u(m[1]11[0][1]m[1]21A[1])

ˆm[0]u(m[1]1m[1]2)

(3:2)ˆ m[0]e(m[1])ˆm:

So it follows thatnsplitsh. It is evidently natural.

(1)ˆ)(3). We consider the following partial Doi-Hopf module G(A).

Evaluating at this object, the retractionnof the unithyields a morphism nG(A) :ACC!AC;

whereACCˆ5a1[0][0]c1A[0][1]d1A[1];a2A;c;d2C> :From nhˆI, we have

nG(A)(a1A[0][0]c11A[0][1]c21A[1])ˆac:

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It can be used to constructuas follows:

u:CC!A;

u(cd)ˆ(ieC)nG(A)(1A[0][0]c1A[0][1]d1A[1]):

For allc2C, since

u(c1c2)ˆ(ieC)nG(A)(1A[0][0]c11A[0][1]c21A[1])

ˆ(idAeC)(1Ac)ˆ1AeC(c):

Hence (3.2) holds. It can be seen to obey (3.3) by naturality and the A- module map property ofn.

The verification of (3.1) is more involved. For any C-comoduleM, we consider the partial Doi-Hopf moduleMA. TheA-actions andC-coaction are defined as follows:

(ma)bˆmab;

rMA(ma)ˆm[0]a[0]m[1]a[1]; (

for all m2M;a;b2A: For C-comodule C, there is a partial Doi-Hopf moduleCAand the map

j:CA!AC; j(ca)ˆa[0]ca[1]

induces a morphism of partial Doi-Hopf modulesCA!AC. Thus by naturality ofn, we have the following commutative diagram

Explicitly, for alla2A;c;d2C, we have

(3:4) jnCA(ca1A[0]d1A[1])ˆ(nAC)(a[0]1A[0][0]ca[1]1A[0][1]d1A[1]):

We consider next the following partial Doi-Hopf moduleCCwith partial C-coaction given by comultiplication in the second factor. Then

xˆDi:CA!CCA; x(ca)ˆc1c2a

induces a morphism of partial Doi-Hopf modulesCA!CCA. Thus

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by naturality ofn, the following diagram

The commutative diagram above is equivalent to

xnCA(ca1A[0]d1A[1])ˆnCCA(c1c2a1A[0]d1A[1]);

(3:5)

for allc;d2Canda2A. Finally, for anyc2C, the map fc :CA!CCA; da7!cda

induces a morphism of partial Doi-Hopf modules CA!CCA.

Hence by naturality ofn, we have

cnCA(ea1A[0]d1A[1])ˆnCCA(cea1A[0]d1A[1]);

(3:6)

for allc;e;d2Canda2A. From (3.5) and (3.6),

xnCA(ca1A[0]d1A[1])ˆc1nCA(c2a1A[0]d1A[1]);

(3:7)

for allc;d2Canda2A.

FromnG(A)beingC-colinear, it follows that rG(A)nG(A)(1A[0][0]c1A[0][1]d1A[1])

ˆnG(A)(1A[0][0][0]c1A[0][0][1]d11A[0][1])d21A[1]; for allc;d2C.

For allc;d2C, since

c21A[1]1A[0][0][0]u(d1A[0][0][1]c11A[0][1])

ˆc21A[1]31A[0]u(d1A[1]1c11A[1]2)

ˆtA;C(ieCi)(nG(A)(1A[0]10A[0][0]d1A[1]110[0][1]c11A[1]210A[1])c21A[1]3)

ˆtA;C(ieCi)(nG(A)(1A[0][0][0]d1A[0][0][1]c11A[0][1])c21A[1])

ˆtA;C(ieCi)rG(A)nG(A)(1A[0][0]d1A[0][1]c1A[1])

ˆtA;CnG(A)(1A[0][0]d1A[0][1]c1A[1]);

where the second equals is followed fromnbeing a leftA-module. In fact, for

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alla2A, the map

fa:AC!AC; fa(b1A[0]c1A[1])ˆab1A[0]c1A[1]

is a morphism in the categoryUCA. Hence by naturality ofn, we have thatnis a leftA-module. Since

d11A[0][1]u(d21A[1]c)[1]1A[0][0]u(d21A[1]c)[0]

ˆd1(1A[0]u(d21A[1]c))[1](1A[0]u(d21A[1]c))[0]

ˆd1(1A[0](ieC)nG(A)(10A[0][0]d21A[1]10A[0][1]c10A[1]))[1]

(1A[0](ieC)nG(A)(10A[0][0]d21A[1]10A[0][1]c10A[1]))[0]

ˆd1((ieC)nG(A)(1A[0]10A[0][0]d21A[1]10A[0][1]c10A[1]))[1]

((ieC)nG(A)(1A[0]10A[0][0]d21A[1]10A[0][1]c10A[1]))[0]

(3:4)ˆ d1((ieC)jnCA(d2 1A[1]1A[0]c1A[1]))[1]

((ieC)jnCA(d2 1A[0]c1A[1]))[0]

LetnCA(da1A[0]c1A[1])ˆciai. Then

d1((ieC)jnCA(d2 1A[1]1A[0]c1A[1]))[1]

((ieC)jnCA(d2 1A[0]c1A[1]))[0]

(3:7)ˆ ci1((ieC)j(ci2ai))[1]((ieC)j(ci2ai))[0]

ˆci1ai[1]ai[0]ˆX

tA;Cj(ciai)

ˆ X

tA;CjnCA(da1A[0]c1A[1]):

Hence we can get (3.1) by using (3.4). p

4. Applications

4.1 ±Maschke-type Theorems for partial Doi-Hopf modules

Since separable functors reflect well the semisimplicity of the objects of a categogy, by Theroem 3.2, we will prove the Maschke-type theorems for partial Doi-Hopf modules.

COROLLARY 4.1. Let (H;A;C) be a partial Doi-Hopf datum, and M;N2 UCA. Suppose that there exists a total integralu:CC!A. Then a monomorphism (resp. epimorphism) f :M!N splits in UCA, if the monomorphism (resp. epimorphism) f splits as an A-module morphism.

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4.2 ±Partial relative modules

Let H be a Hopf algebra and A a partial right H-comodule algebra.

Then the threetuple (H;A;H) is a partial Doi-Hopf datum. The category M(H)HAis called a partial (H;A)-Hopf module category and denoted byUHA. COROLLARY 4.2. Let H be a Hopf algebra and A a partial right H- comodule algebra. Then the following statements are equivalent:

(1) The forgetful functor F:UHA ! UAis separable, (2) There exists a normalized A-integralu:HH!A.

We will now introduce the partial total integral for the partial rightH- comodule algebra, and investigate the difference between the partial total integral and the total integral in sense of Doi.

PROPOSITION4.3. Let H be a Hopf algebra and A a partial right H-comodule algebra. If u:HH!k is a normalized A-integral for (H;A;H), the k-linear map

W:H!A; W(h)ˆu(1Hh);

for all h2H, satisfies the relations:

W(h)[0]W(h)[1]ˆW(h1)1A[0]h21A[1]; (4:1)

W(1H)ˆ1A: (4:2)

PROOF. Notice first thatW(1H)ˆu(1H1H)ˆeH(1H)1Aˆ1A. Since h21A[1]u(gh1)1A[0]

ˆh21A[1]1A[0][0][0]u(g1A[0][0][1]h11A[0][1])

ˆg11A[0][1]u(g21A[1]h)[1]1A[0][0]u(g21A[1]h)[0]

ˆg1(1A[0]u(g21A[1]h))[1](1A[0]u(g21A[1]h))[0]

ˆg1(u(g2h))[1](u(g2h))[0]

It follows by takinggˆ1Hthat

h21A[1]u(1Hh1)1A[0]ˆu(1Hh)[1]u(1Hh)[0]: So (4.1) holds.

DEFINITION4.4. LetHbe a Hopf algebra andAa right partialH-co- module algebra. Ak-linear mapW:H!Ais call a partial total integral for (H;A), ifWsatisfies the conditions (4.1) and (4.2).

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REMARK4.5. If 1A[0]1A[1]ˆ1A1H, then the right partial H-co- module algebra A is just the ordinary right H-comodule algebra, and the partial total integral is the same with the total integral in sense of Doi in (D2).

LetW:H!Abe the total integral for the partial rightH-coalgebraA, and define

u:HH!A; u(hg)ˆ1A[0]W(gS 1(1A[1]h));

for allg;h2H.

THEOREM 4.6. Let A be a partial right H-comodule algebra and W:H!A be a partial total integral. If

gW(h)[1]W(h)[0]ˆW(h)[1]gW(h)[0];

W(h)2Z(A) (the center of A); 1A[0]W(S 1(1[1]))ˆ1A; Thenuis a normalized A-integral.

PROOF. For alla2Aandg;h2H, one has

a[0][0]u(ga[0][1]ha[1])ˆa[0][0]1A[0]W(ha[1]S 1(1A[1]ga[0][1]))

ˆa[0]1A[0]W(ha[1]2S 1(ga[1]11A[1]))

ˆ1A[0]W(hS 1(g1A[1]))a

ˆu(gh) and

g1(u(g2h))[1](u(g2h))[0]

ˆg11A[0][1]W(hS 1(1A[1]g2))[1]1A[0][0]W(hS 1(1A[1]g2))[0]

ˆW(hS 1(1A[1]g2))[1]g11A[0][1]1A[0][0]W(hS 1(1A[1]g2))[0]

ˆh2S 1(1A[1]g2)210A[1]g11A[0][1]1A[0][0]W(h1S 1(1A[1]g2)1)10A[0]

ˆh2S 1(1A[1]1)S 1(g2)g11A[0][1]1A[0][0]W(h1S 1(1A[1]2g3))

ˆh2S 1(1A[1]2)1A[1]110A[1]1A[0]10A[0]W(h1S 1(1A[1]3g))

ˆh210A[1]1A[0]10A[0]W(h1S 1(1A[1]g))

ˆh21A[1]u(gh1)1A[0]:

u(h1h2)ˆ1A[0]W(h2S 1(1A[1]h1))

ˆeH(h)1A[0]W(S 1(1A[1]))ˆeH(h)1A

Souis a normalizedA-integral. p

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4.3 ±Partial Doi-Hopf Datum(H;k;H)

COROLLARY 4.7. Under the assumptions of Example 2.5. Then the following statements are equivalent:

(1) The forgetful functor F:M(H)H! Mk(the category of all vector spaces) is separable.

(2) k-linear mapu:HH!k such that the following conditions are satisfied:

u(h1h2)ˆeH(h);

(4:3)

u(eheg)ˆu(hg);

(4:4)

eh2u(gh1)ˆeg1u(eg2h):

(4:5)

Takeeˆ1. Then the partial Doi-Hopf datum (H;k;H) is just the Doi- Hopf datum, the categoryM(H)His the categoryMHofH-Hopf module.

Suppose that Wis the right integral ofH, the mapu:HH!kis de- fined by

u(hg)ˆW(gS 1(h)):

By the properties of the right integralW, we can check thatusatisfies (4.3)- (4.4).

COROLLARY 4.8. Let H be a finite dimensional cosemisimple Hopf algebra. The forgetful functor F:MH! Mk is separable.

PROOF. SinceHis a finite dimensional cosemisimple Hopf algebra, it follows that there exists a right integralW2H such thatW(1H)ˆ1. The desired total integralucan be constructed by usingW. p Acknowledgment. The authors would like to thank the referee for the valuable suggestions and comments.

REFERENCES

[CJ] S. CAENEPEEL - K. JANSSEN, Partial (Co)Actions of hopf algebras and Partial Hopf-Galois Theory,Comm. Algebra,36(2008), pp. 2923-2946.

[CMI] S. CAENEPEEL- G. MILITARU- B. ION,Separable functors for the category of Doi-Hopf modules, Applications, Adv. Mathematics,145(1999), pp. 239-290.

[CR] S. CAENEPEEL - S. RAIANU, Induction functors for the Doi-Koppinen unified Hopf modules, in Abelian Groups and Modules, pp. 73-94, Kluwer Acad. Publ., Dordrecht, 1995.

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[D1] Y. DOI,Unifying Hopf modules, J. Algebra,153(2) (1992), pp. 373-385.

[D2] Y. DOI,Algebras with total integrals, Comm. Algebra,13(1986), pp. 2137- 2159.

[DE] M. DOKUCHAEV- R. EXEL,Associativity of Crossed Products by Partial Actions, Enveloping Actions and Partial Representations, Trans. Amer.

Math. Soc.,357(2005), pp. 1931-1952.

[DEP] M. DOKUCHAEV - R. EXEL - P. PICCIONE, Partial Representations and Partial Group Algebras,J. Algebra,226(2000), pp. 251-268.

[DFP] M. DOKUCHAEV - M. FERRERO - A. PACQUES, Partial Galois theory of commutative rings, J. Pure Appl. Algebra,208(2007), pp. 77-87.

[DZ] M. DOKUCHAEV - N. ZHUKAVETS, On Finite Degree Partial Partial

Representations of Groups,J. Algebra,274(2000), pp. 309-334.

[E1] R. EXEL, Twisted Partial Actions: A Classfication of Regular C-Alge- braic Bundles, Proc. London. Math. Soc.,74(1997), pp. 417-443.

[E2] R. EXEL, Circle Actions on C-Algebras, Partial Automorphisms and Generalized Pimsner-Voiculescu Exact Sequences, J. Funct. Anal., 122 (1994), pp. 361-401.

[R] M. D. RAFAEL, Separable functors revised, Comm. Algebra,18(1990), pp.

1445-1459.

[RT] D. E. RADFORD- J. TOWBER,Yetter-Drinfeld categories associated to an arbitrary bialgebra, J. Pure Appl. Algebra,87(3) (1993), pp. 259-279.

[S] M. E. SWEEDLER,Hopf Algebras, Benjamin, New York, 1969.

[Y] D. N. YETTER, Quantum groups and representations of monoidal cate- gories, Math. Proc. Cambridge Philos. Soc.,108(2) (1990), pp. 261-290.

Manoscritto pervenuto in redazione il 13 Dicembre 2011.

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