2Preliminaries 1Introduction LocalQuasitriangularHopfAlgebras

Local Quasitriangular Hopf Algebras

Shouchuan ZHANG ^{† ‡}, Mark D. GOULD ^‡ and Yao-Zhong ZHANG^‡

† Department of Mathematics, Hunan University, Changsha 410082, P.R. China E-mail: [email protected]

‡ Department of Mathematics, University of Queensland, Brisbane 4072, Australia E-mail: [email protected], [email protected]

Received January 31, 2008, in final form April 30, 2008; Published online May 09, 2008 Original article is available athttp://www.emis.de/journals/SIGMA/2008/042/

Abstract. We find a new class of Hopf algebras, local quasitriangular Hopf algebras, which generalize quasitriangular Hopf algebras. Using these Hopf algebras, we obtain solutions of the Yang–Baxter equation in a systematic way. The category of modules with finite cycles over a local quasitriangular Hopf algebra is a braided tensor category.

Key words: Hopf algebra; braided category

2000 Mathematics Subject Classification: 16W30; 16G10

1 Introduction

The Yang–Baxter equation first came up in the paper by Yang as factorization condition of the scattering S-matrix in the many-body problem in one dimension and in the work by Baxter on exactly solvable models in statistical mechanics. It has been playing an important role in mathematics and physics (see [2,16]). Attempts to find solutions of the Yang–Baxter equation in a systematic way have led to the theory of quantum groups and quasitriangular Hopf algebras (see [6,9]).

Since the category of modules with finite cycles over a local quasitriangular Hopf algebra is a braided tensor category, we may also find solutions of the Yang–Baxter equation in a systematic way.

The main results in this paper are summarized in the following statement.

Theorem 1. (i) Assume that (H,{R_n}) is a local quasitriangular Hopf algebra. Then (HM^cf, C^{Rn}), (_HM^dcf, C^{Rn}) and (_HM^df, C^{Rn}) are braided tensor categories. Furthermore, if (M, α⁻) is an H-module with finite cycles and R_n+1 = R_n+W_n with W_n ∈ H_n+1⊗H_(n+1), then (M, α⁻, δ⁻) is a Yetter–Drinfeld H-module.

(ii) Assume that B is a finite dimensional Hopf algebra and M is a finite dimensional B- Hopf bimodule. Then ((T_B(M))^cop ./_τ T_B^c∗(M^∗),{R_n}) is a local quasitriangular Hopf algebra.

Furthermore, if (kQ^a, kQ^c) and (kQ^s, kQ^sc) are arrow dual pairings with finite Hopf quiver Q, then both ((kQ^a)^cop ./τ kQ^c,{R_n}) and((kQ^s)^cop ./τ kQ^sc,{R_n}) are local quasitriangular Hopf algebras.

2 Preliminaries

Throughout, we work over a fixed field k. All algebras, coalgebras, Hopf algebras, and so on, are defined over k. Books [7, 11, 15, 13] provide the necessary background for Hopf algebras and book [1] provides a nice description of the path algebra approach.

Let V and W be two vector spaces. σV denotes the map from V to V^∗∗ by defining hσ_V(x), fi = hf, xi for any f ∈ V^∗, x ∈ V. CV,W denotes the map from V ⊗W to W ⊗V

by definingCV,W(x⊗y) =y⊗xfor anyx∈V,y∈W. DenoteP byP

P⁰⊗P⁰⁰forP ∈V⊗W. If V is a finite-dimensional vector space over field k with V^∗ = Hom_k(V, k). Define maps b_V :k→V ⊗V^∗ and d_V :V^∗⊗V →k by

bV(1) =X

i

vi⊗v_i^∗ and X

i,j

dV(v^∗_i ⊗vj) =hv_i^∗, vji,

where{v_i |i= 1,2, . . . , n} is any basis ofV and {v^∗_i |i= 1,2, . . . , n} is its dual basis inV^∗. dV

andb_V are called evaluation and coevaluation ofV, respectively. It is clear (d_V⊗id_U)(id_U⊗b_V) = id_U and (id_V ⊗d_V)(b_V ⊗id_V) = id_V. ξ_V denotes the linear isomorphism from V to V^∗ by sendingvi tov^∗_i fori= 1,2, . . . , n. Note that we can define evaluation dV when V is infinite.

We will use µ to denote the multiplication map of an algebra and use ∆ to denote the comultiplication of a coalgebra. For a (left or right) module and a (left or right) comodule, denote by α⁻,α⁺,δ⁻ and δ⁺ the left module, right module, left comodule and right comodule structure maps, respectively. The Sweedler’s sigma notations for coalgebras and comodules are ∆(x) = P

x₁ ⊗x₂, δ⁻(x) = P

x₍₋₁₎⊗x₍₀₎, δ⁺(x) = P

x₍₀₎ ⊗x₍₁₎. Let (H, µ, η,∆, ) be a bialgebra and let ∆^cop:=CH,H∆ andµ^op :=µCH,H.We denote (H, µ, η,∆^cop, ) byH^copand (H, µ^op, η,∆, ) byH^op. Sometimes, we also denote the unit element of H by 1_H.

LetA andH be two bialgebras with ∅6=X ⊆A ,∅6=Y ⊆H and P ∈Y ⊗X,R∈Y ⊗Y.

Assume thatτ is a linear map fromH⊗A tok.We give the following notations.

(Y, R) is called almost cocommutative if the following condition satisfied:

(ACO) : X

y₂R⁰⊗y₁R⁰⁰=X

R⁰y₁⊗R⁰⁰y₂ for any y∈Y.

τ is called a skew pairing onH⊗A if for anyx, u∈H,y, z ∈A the following conditions are satisfied:

(SP1) : τ(x, yz) =X

τ(x1, y)τ(x2, z);

(SP2) : τ(xu, z) =X

τ(x, z2)τ(u, z1);

(SP3) : τ(x, η) =_H(x);

(SP4) : τ(η, y) =A(y).

P is called a copairing of Y ⊗X if for any x, u ∈ H, y, z ∈ A the following conditions are satisfied:

(CP1) : X

P⁰⊗P₁⁰⁰⊗P₂⁰⁰=X

P⁰Q⁰⊗Q⁰⁰⊗P⁰⁰ with P =Q;

(CP1) : X

P₁⁰⊗P₂⁰ ⊗P⁰⁰=X

P⁰⊗Q⁰⊗P⁰⁰Q⁰⁰ with P =Q;

(CP3) : X

P⁰⊗_A(P⁰⁰) =η_H; (CP4) : X

_H(P⁰)⊗P⁰⁰=η_A.

ForR∈H⊗Hand twoH-modulesU andV, define a linear mapC_U,V^R fromU⊗V toV ⊗U by sending (x⊗y) to P

R⁰⁰y⊗R⁰x for any x∈U,y∈V.

If V = ⊕^∞_i=0Vi is a graded vector space, let V>n and V≤n denote ⊕^∞_i=n+1Vi and ⊕ⁿ_i=0Vi, respectively. We usually denote⊕ⁿ_i=0V_i byV_(n). If dimV_i <∞for any natural numberi, thenV is called a local finite graded vector space. We denote by ιi the natural injection from Vi toV and by πi the corresponding projection fromV toVi.

LetH be a bialgebra and a graded coalgebra with an invertible element R_n in H_(n)⊗H_(n) for any natural n. Assume Rn+1 = Rn +Wn with Wn ∈ H_(n+1) ⊗Hn+1 +Hn+1 ⊗H_(n+1). (H,{R_n}) is called a local quasitriangular bialgebra if Rn is a copairing on H_(n) ⊗H_(n), and

(H_(n), Rn) is almost cocommutative for any natural number n. In this case, {R_n} is called a local quasitriangular structure of H. Obviously, if (H, R) is a quasitriangular bialgebra, then (H,{R_n}) is a local quasitriangular bialgebra withR₀=R,R_i= 0, H₀ =H,H_i= 0 for i >0.

The following facts are obvious: τ⁻¹ =τ(id_H⊗S) (or =τ(S⁻¹⊗id_A)) ifAis a Hopf algebra (or H is a Hopf algebra with invertible antipode) and τ is a skew pairing; P⁻¹ = (S⊗id_A)P (or = (idH ⊗S⁻¹)P) if H is a Hopf algebra (or A is a Hopf algebra with invertible antipode) and P is a copairing.

LetA be an algebra andM be anA-bimodule. Then the tensor algebraT_A(M) ofM overA is a graded algebra with TA(M)0 =A,TA(M)1 =M and TA(M)n=⊗ⁿ_AM for n >1. That is, T_A(M) =A⊕(L

n>0⊗ⁿ_AM) (see [12]). LetDbe another algebra. Ifhis an algebra map fromA toDandf is anA-bimodule map fromMtoD, then by the universal property ofT_A(M) (see [12, Proposition 1.4.1]) there is a unique algebra mapTA(h, f) :TA(M)→Dsuch thatTA(h, f)ι0 =h and T_A(h, f)ι₁ = f. One can easily see that T_A(h, f) = h+P

n>0µⁿ⁻¹T_n(f), where T_n(f) is the map from⊗ⁿ_AM to⊗ⁿ_ADgiven by T_n(f)(x₁⊗x₂⊗ · · · ⊗x_n) =f(x₁)⊗f(x₂)⊗ · · · ⊗f(x_n), i.e., Tn(f) =f ⊗_Af⊗_A· · · ⊗_Af. Note thatµ can be viewed as a map fromD⊗_ADtoD. For the details, the reader is directed to [12, Section 1.4].

Dually, let C be a coalgebra and let M be a C-bicomodule. Then the cotensor coal- gebra T_C^c(M) of M over C is a graded coalgebra with T_C^c(M)0 = C, T_C^c(M)1 = M and T_C^c(M)_n = 2ⁿ_CM for n > 1. That is, T_C^c(M) = C ⊕(L

n>02ⁿ_CM) (see [12]). Let D be another coalgebra. Ifh is a coalgebra map fromD toC and f is aC-bicomodule map from D to M such that f(corad(D)) = 0, then by the universal property of T_C^c(M) (see [12, Proposi- tion 1.4.2]) there is a unique coalgebra mapT_C^c(h, f) fromDtoT_C^c(M) such thatπ₀T_C^c(h, f) =h andπ₁T_C^c(h, f) =f. It is not difficult to see thatT_C^c(h, f) =h+P

n>0T_n^c(f)∆n−1, whereT_n^c(f) is the map from2ⁿ_CDto2ⁿ_CM induced byTn(f)(x1⊗x₂⊗· · ·⊗xn) =f(x1)⊗f(x2)⊗· · ·⊗f(xn), i.e., T_n^c(f) =f⊗f ⊗ · · · ⊗f.

Furthermore, ifB is a Hopf algebra and M is a B-Hopf bimodule, thenT_B(M) and T_B^c(M) are two graded Hopf algebra. Indeed, by [12, Section 1.4] and [12, Proposition 1.5.1],TB(M) is a graded Hopf algebra with the counit ε= ε_Bπ₀ and the comultiplication ∆ = (ι₀⊗ι₀)∆_B+ P

n>0µⁿ⁻¹T_n(∆_M), where ∆_M = (ι₀⊗ι₁)δ⁻_M + (ι₁⊗ι₀)δ⁺_M. Dually, T_B^c(M) is a graded Hopf algebra with multiplicationµ=µB(π0⊗π0) +P

n>0T_n^c(µM)∆n−1, whereµM =α⁻_M(π0⊗π1) + α⁺_M(π₁⊗π₀).

3 Yang–Baxter equations

Assume thatHis a bialgebra and a graded coalgebra with an invertible elementRninH_(n)⊗H_(n) for any natural n. For convenience, let (LQT1), (LQT2) and (LQT3) denote (CP1), (CP2) and (ACO), respectively;

(LQT4) : Rn+1=Rn+Wn with Wn∈H_(n+1)⊗Hn+1+Hn+1⊗H_(n+1); (LQT4⁰) : R_n+1=R_n+W_n with W_n∈H_n+1⊗H_n+1.

Then (H,{R_n}) is a local quasitriangular bialgebra if and only if (LQT1), (LQT2), (LQT3) and (LQT4) hold for any natural numbern.

Let H be a graded coalgebra and a bialgebra. A left H-module M is called an H-module with finite cycles if, for any x ∈M, there exists a natural number nx such that Hix = 0 when i > n_x. Let _HM^cf denote the category of all left H-modules with finite cycles.

Lemma 1. Let H be a graded coalgebra and a bialgebra. If U and V are left H-modules with finite cycles, so is U ⊗V.

Proof . For anyx∈U,y∈V,there exist two natural numbersnx andny, such thatH>nxx= 0 and H_>nyy = 0. Set nx⊗y = 2n_x+ 2n_y. It is clear that H_>nx⊗y(x⊗y) = 0. Indeed, for any h∈H_i withi > nx⊗y, we see

h(x⊗y) =X

h1x⊗h2y= 0 (since H is graded coalgebra).

Lemma 2. Assume that(H,{R_n})is a local quasitriangular Hopf algebra. Then for any left H- modules U andV with finite cycles, there exists an invertible linear mapC_U,V^{Rn} :U⊗V →V⊗U such that

C_U,V^{Rn}(x⊗y) :=C^Rn(x⊗y) =X

R⁰⁰_ny⊗R⁰_nx with n >2n_x+ 2n_y, for x∈U, y∈V.

Proof . We first define a mapf from U×V toV ⊗U by sending (x, y) to P

R⁰⁰_ny⊗R⁰_nx with n >2nx+ 2ny for anyx∈U,y∈V. It is clear thatf is well defined. Indeed, if n >2nx+ 2ny, thenC^Rn+1(x⊗y) =C^Rn(x⊗y) sinceR_n+1 =R_n+W_nwithW_n∈H_(n+1)⊗H_n+1+H_n+1⊗H_(n+1). f is a k-balanced function. Indeed, forx, y∈U,z, w∈V,α∈k, let n >2n_x+ 2n_y+ 2n_z. See

f(x+y, z) =X

R⁰⁰_nz⊗R⁰_n(x+y) =X

R⁰⁰_nz⊗R⁰_nx+X

R⁰⁰_nz⊗R⁰_ny

=f(x, z) +f(y, z).

Similarly, we can show that f(x, z+w) =f(x, z) +f(x, w),f(xα, z) =f(x, αz). Consequently, there exists a linear map C_U,V^{Rn} :U ⊗V →V ⊗U such that

C_U,V^{Rn}(x⊗y) =C^Rn(x⊗y) with n >2n_x+ 2n_y, forx∈U,y∈V.

The inverse (C_U,V^{Rn})⁻¹ of C_U,V^{Rn} is defined by sending (y⊗x) toP

(R⁻¹_n )⁰x⊗(R⁻¹_n )⁰⁰y with

n >2n_x+ 2n_y for any x∈U,y∈V.

Theorem 2. Assume that (H,{R_n}) is a local quasitriangular Hopf algebra. Then (HM^cf, C^{Rn}) is a braided tensor category.

Proof . Since H is a bialgebra, we have that (_HM,⊗, I, a, r, l) is a tensor category by [14, Proposition XI.3.1]. It follows from Lemma 1 that (HM^cf,⊗, I, a, r, l) is a tensor subcategory of (_HM,⊗, I, a, r, l). C^{Rn} is a braiding of _HM^cf, which can be shown by the way similar to the proof of [14, Proposition VIII.3.1, Proposition XIII.1.4].

An H-moduleM is called a graded H-module if M =⊕^∞_i=0Mi is a graded vector space and H_iM_j ⊆M_i+j for any natural number iand j.

Lemma 3. Assume that H is a local finite graded coalgebra and bialgebra. If M =⊕^∞_i=0M_i is a graded H-module, then the following conditions are equivalent:

(i) M is an H-module with finite cycles;

(ii) Hx is finite dimensional for anyx∈M;

(iii) Hx is finite dimensional for any homogeneous elementx in M.

Proof . (i) ⇒ (ii). For any x ∈ M, there exists a natural number nx such thatHix = 0 with i > n_x. Since H/(0 : x)_H ∼= Hx, where (0 : x)_H := {h ∈ H |h·x = 0}, we have that Hx is finite dimensional.

(ii)⇒ (iii). It is clear.

(iii)⇒ (i). We first show that, for any homogeneous element x∈Mi, there exists a natural number n_x such that H_jx = 0 with j > n_x. In fact, if the above does not hold, then there exists h_j ∈ H_nj such that h_jx 6= 0 with n₁ < n₂ < · · ·. Considering h_jx ∈ M_nj+i we have that {h_jx | j = 1,2, . . .} is linear independent in Hx, which contradicts to that Hx is finite dimensional.

For anyx∈M, thenx=Pl

i=1x_i andx_i is a homogeneous element fori= 1,2, . . . , l. There exists a natural number nxi such that Hjxi = 0 with j > nxi. Set nx = Pl

s=1nxs. Then Hjx= 0 with j > nx. Consequently,M is anH-module with finite cycles.

Note that if M =⊕^∞_i=0Mi is a graded H-module, then both (ii)⇒ (iii) and (iii)⇒ (i) hold in Lemma 3.

Let _HM^gf and _HM^gcf denote the category of all finite dimensional graded left H-modules and the category of all graded leftH-modules with finite cycles. Obviously, they are two tensor subcategories of _HM^cf. Therefore we have

Theorem 3. Assume that (H,{R_n}) is a local quasitriangular Hopf algebra. Then (_HM^gf, C^{Rn}) and (HM^gcf, C^{Rn}) are two braided tensor categories.

Therefore, if M is a finite dimensional graded H-module (or H-module with finite cycles) over local quasitriangular Hopf algebra (H,{R_n}), then C_M,M^{Rn} is a solution of Yang–Baxter equation on M.

It is easy to prove the following.

Theorem 4. Assume that(H,{R_n})is a local quasitriangular Hopf algebra andR_n+1=R_n+W_n with Wn ∈ Hn+1 ⊗H_(n+1). If (M, α⁻) is an H-modules with finite cycles then (M, α⁻, δ⁻) is a Yetter–Drinfeld H-module, where δ⁻(x) =P

R_n⁰⁰⊗R⁰_nx for any x∈M and n≥nx.

4 Relation between tensor algebras and co-tensor coalgebras

Lemma 4 (See [3, 7]). Let A, B and C be finite dimensional coalgebras, (M, δ_M⁻, δ_M⁺) and (N, δ_N⁻, δ⁺_N) be respectively a finite dimensional A-B-bicomodule and a finite dimensional B-C- bicomodule. Then

(i) (M^∗, δ_M^−∗, δ_M^+∗) is a finite dimensional A^∗-B^∗-bimodule;

(ii)(M2BN, δ⁻_M2

BN, δ_M⁺₂

BN)is anA-C-bicomodule with structure mapsδ_M⁻₂

BN =δ⁻_M⊗id_N and δ_M⁺₂

BN =id_M ⊗δ_N⁺;

(iii) M^∗⊗_B^∗N^∗ ∼= (M2BN)^∗ (as A^∗-C^∗-bimodules).

Lemma 5 (See [3, 7]). Let A, B and C be finite dimensional algebras, (M, α_M⁻, α⁺_M) and (N, α⁻_N, α⁺_N) be respectively a finite dimensional A-B-bimodule and a finite dimensional B-C- bimodule. Then

(i) (M^∗, α⁻_M^∗, α⁺_M^∗) is a finite dimensional A^∗-B^∗-bicomodule;

(ii)(M⊗_BN, α⁻_M⊗

BN, α⁺_M⊗

BN)is anA-C-bimodule with structure mapsα⁻_M⊗

BN =α⁻_M⊗id_N and α⁺_M⊗

BN =idM ⊗α⁺_N;

(iii) M^∗2B^∗N^∗ ∼= (M ⊗_BN)^∗ (as A^∗-C^∗-bicomodules).

Proof . (i) and (ii) are easy.

(iii) Consider

M^∗2B^∗N^∗ ∼= (M^∗2B^∗N^∗)^∗∗∼= (M⊗_BN)^∗ (by Lemma 4).

Of course, we can also prove it in the dual way of the proof of Lemma 4, by sending f ⊗_kg to f ⊗_B^∗g for any f ∈M^∗,g∈N^∗ withf⊗g∈M^∗2B^∗N^∗.

Theorem 5. If A is a finite dimensional algebra and M is a finite dimensional A-bimodule, thenT_A(M) is isomorphic to subalgebraP∞

n=0(2ⁿ_A^∗M^∗)^∗ of (T_A^c∗(M^∗))⁰ under mapσ_TA(M) and σ_TA(M) =σ_A+P

n>0µⁿ⁻¹T_n(σ_M) with µⁿ⁻¹T_n(σ_M) =σ⊗ⁿ_AM.

Proof . We view ⊕^∞_n=0(2ⁿ_A^∗M^∗)^∗ as inner direct sum of vector spaces. It is clear that σ_A is algebra homomorphism from A toA^∗∗ ⊆(T_A^c∗(M^∗))^∗ and σM is a A-bimodule homomorphism from M to M^∗∗ ⊆ (T_A^c∗(M^∗))^∗. Thus it follows from [12, Proposition 1.4.1] that φ = σ_A+ P

n>0µⁿ⁻¹T_n(σ_M) is an algebra homomorphism from T_A(M) to (T_A^c∗(M^∗))^∗.

It follows from Lemma4 (iii) thatµⁿ⁻¹Tn(σM) =σ⊗ⁿ_AM. Indeed, we use induction onn >0.

Obviously, the conclusion holds when n = 1. Let n > 1, N = ⊗ⁿ⁻¹_A M, L = (2ⁿ⁻¹_A^∗ M^∗)^∗ and ζ =µⁿ⁻²Tn−1(σM). Obviously, µⁿ⁻¹Tn(σM) =µ(ζ⊗σM). By inductive assumption, ζ =σN is an A-bimodule isomorphism fromN toL. See

⊗ⁿ_AM =N⊗_AM ^ν∼=¹ L⊗_AM^∗∗ (by inductive assumption)

ν2

∼= (2ⁿ⁻¹_A^∗ M^∗)^∗⊗_A^∗∗M^∗∗

ν3

∼= ((2ⁿ⁻¹_A^∗ M^∗)2A^∗M^∗)^∗ (by Lemma4 (iii))

= (2ⁿ_A^∗M^∗)^∗,

whereν₁ =σ_N⊗_Aσ_M,ν₂(f^∗∗⊗_Ag^∗∗) =f^∗∗⊗_A^∗∗g^∗∗ and ν₃(f^∗∗⊗_A^∗∗g^∗∗) =f^∗∗⊗_kg^∗∗for any f^∗∗ ∈(2ⁿ⁻¹_A^∗ M^∗)^∗,g^∗∗ ∈M^∗∗. Now we have to show ν₃ν₂ν₁ =σ⊗ⁿ_AM =µⁿ⁻¹T_n(σ_M). For any f^∗ ∈2ⁿ⁻¹_A^∗ M^∗,g^∗ ∈M^∗,x∈ ⊗ⁿ⁻¹_A M,y ∈M, on the one hand

hσ_⊗ⁿ

AM(x⊗_Ay), f^∗⊗_kg^∗i=hf^∗, xihg^∗, yi.

On the other hand,

hν₃ν₂ν₁(x⊗_Ay), f^∗⊗_kg^∗i=hν₃ν₂(σ_N(x)⊗_Aσ_M(y)), f^∗⊗_kg^∗i

=hσ_M(x)⊗_kσN(y), f^∗⊗_kg^∗i=hf^∗, xihg^∗, yi.

Thus ν3ν2ν1 =σ⊗ⁿ_AM. See

hµ(ζ⊗_Aσ_M)(x⊗_Ay), f^∗⊗_kg^∗i=hζ(x)⊗_Aσ_M(y),∆(f^∗⊗_kg^∗)i

=hζ(x), f^∗ihσ_M(y), g^∗i=hf^∗, xihg^∗, yi.

Thus σ⊗ⁿ_AM =µⁿ⁻¹Tn(σM).

Finally, for anyx∈TA(M) withx=x⁽¹⁾+x⁽²⁾+· · ·+x⁽ⁿ⁾ and x⁽ⁱ⁾∈ ⊗ⁱ_AM, φ(x) =

n

X

i=1

φ(x⁽ⁱ⁾) =

n

X

i=1

σ_⊗ⁱ

AM(x⁽ⁱ⁾) =σ_TA(M)(x).

5 (Co-)tensor Hopf algebras

Lemma 6. Assume that B is a finite dimensional Hopf algebra and M is a finite dimensional B-Hopf bimodule. Let A:=TB(M)^cop,H :=T_B^c∗(M^∗). Then

(i) φ := σ_B +P

i>0µⁱ⁻¹T_i(σ_M) is a Hopf algebra isomorphism from T_B(M) to the Hopf subalgebra P∞

i=0(2ⁱ_B^∗M^∗)^∗ of (T_B^c∗(M^∗))⁰;

(ii) Let φn:= φ|_A(n) for any natural number n≥0. Then there exists ψn: (H_(n))^∗ → A_(n) such that φ_nψ_n = id_(H(n))^∗ and ψ_nφ_n = id_A(n), and ψ_n+1(x) = ψ_n(x) for any x ∈ (H_(n))^∗. Furthermore, φ_n and ψ_n preserve the (co)multiplication operations of T_B(M) and (T_B^c∗(M^∗))⁰, respectively.

Proof . (i) We first show that (2ⁿ_B^∗M^∗)^∗⊆(T_B^c∗(M^∗))⁰. For anyf ∈(2ⁿ_B^∗M^∗)^∗,P∞

i=n+12ⁱ_B^∗M^∗

⊆ kerf and P∞

i=n+12ⁱ_B^∗M^∗ is a finite codimensional ideal of T_B^c∗(M^∗). Consequently, f ∈ (T_B^c∗(M^∗))⁰.

Next we show thatφ:=σB+P

n>0µⁿ⁻¹Tn(σM) (see the proof of Theorem5) is a coalgebra homomorphism from T_B(M) to P∞

i=0(2^c_B^∗M^∗)^∗. For any x ∈ ⊗ⁿ_BM, f, g ∈ T_B^c∗(M^∗), on the one hand

hφ(x), f ∗gi=hf∗g, xi (by Theorem5)

=X

x

hf, x₁ihg, x₂i=X

x

hφ(x₁), fihφ(x₂), gi.

On the other hand hφ(x), f ∗gi=X

h(φ(x))₁, fih(φ(x))₂, gi,

since φ(x) ∈ (T_B^c∗(M^∗))⁰. Considering T_B^c∗(M^∗) = ⊕_n≥02ⁿ_B^∗M^∗ ∼= ⊕_n≥0(⊗ⁿ_BM)^∗ as vec- tor spaces, we have that T_B^∗(M^∗) is dense in (T_B(M))^∗. Consequently, P

φ(x₁)⊗φ(x₂) = P(φ(x))1⊗(φ(x))2, i.e.φis a coalgebra homomorphism.

(ii) It follows from Theorem5.

Recall the double cross productA_α ./_β H, defined in [17, p. 36]) and [14, Definition IX.2.2].

Assume that H and A are two bialgebras; (A, α) is a left H-module coalgebra and (H, β) is a right A-module coalgebra. We define the multiplication mD, unit ηD, comultiplication ∆D

and counit _D onA⊗H as follows:

µD((a⊗h)⊗(b⊗g)) =X

aα(h1, b1)⊗β(h2, b2)g,

∆_D(a⊗h) =X

(a1⊗h1⊗a2⊗h2),

_D =_A⊗_H,η_D =η_A⊗η_H for anya, b∈A,h, g∈H.We denote (A⊗H, µ_D, η_D,∆_D, _D) by A_α./_βH, which is called the double cross product of A and H.

Lemma 7 (See [8]). LetH andAbe two bialgebras. Assume thatτ is an invertible skew pairing onH⊗A. If we defineα(h, a) =P

τ(h1, a1)a2τ⁻¹(h2, a3)andβ(h, a) =P

τ(h1, a1)h2τ⁻¹(h3, a2) then the double cross product Aα ./_β H of A and H is a bialgebra. Furthermore, if A and H are two Hopf algebras, then so is A_α./_β H.

Proof . We can check that (A, α) is an H-module coalgebra and (H, β) is an A-module coal- gebra step by step. We can also check that (M1)–(M4) in [17, pp. 36–37] hold step by step.

Consequently, it follows from [17, Corollary 1.8, Theorem 1.5] or [14, Theorem IX.2.3] that

A_α./_β H is a Hopf algebra.

In this case,Aα ./β H can be written as A ./τ H.

Lemma 8. LetHandAbe two Hopf algebra. Assume that there exists a Hopf algebra monomor- phism φ:A^cop→H⁰. Setτ =dH(φ⊗idH)CH,A. Then A ./τ H is Hopf algebra.

Proof . Using [10, Proposition 2.4] or the definition of the evaluation and coevaluation on tensor product, we can obtain thatτ is a skew pairing onH⊗A. Considering Lemma7, we complete

the proof.

Lemma 9. (i) If H=⊕^∞_n=0Hnis a graded bialgebra and H0 has an invertible antipode, then H has an invertible antipode.

(ii) Assume thatB is a finite dimensional Hopf algebra andM is aB-Hopf bimodule. Then both TB(M) andT_B^c(M) have invertible antipodes.

Proof . (i) It is clear that H^op is a graded bialgebra with (H^op)0 = (H0)^(op). Thus H^op has an antipode by [12, Proposition 1.5.1]. However, the antipode of H^op is the inverse of antipode of H.

(ii) It follows from (i).

Lemma 10. Let A = ⊕^∞_n=0An and H = ⊕^∞_n=0Hn be two graded Hopf algebras with invertible antipodes. Let τ be a skew pairing on (H⊗A) and Pn be a copairing of H_(n)⊗A_(n) for any natural number n. Set D = A ./_τ H and [P_n] = 1_A⊗P_n⊗1_H. Then (D,{[P_n]}) is almost cocommutative on D_(n) if and only if

(ACO1) : X

P⁰y1⊗P⁰⁰⊗y2 =X

y4P⁰⊗P₂⁰⁰⊗y2τ(y1, P₁⁰⁰)τ⁻¹(y3, P₃⁰⁰) for any y∈H_(n); (ACO2) : X

x2⊗P⁰⊗x1P⁰⁰=X

x2⊗P₂⁰⊗P⁰⁰x4τ(P₁⁰, x1)τ⁻¹(P₃⁰, x3) for any x∈A_(n). Proof . It is clear that (D,{[P_n]}) is almost cocommutative on D_(n) if and only if the following holds:

Xx2⊗y4P⁰⊗x1P₂⁰⁰⊗y2τ(y1, P₁⁰⁰)τ⁻¹(y3, P₃⁰⁰)

=X

x2⊗P₂⁰y1⊗P⁰⁰x4⊗y2τ(P₁⁰, x1)τ⁻¹(P₃⁰, x3) (1) for any x∈A_(n),y∈H_(n).

Assume that both (ACO1) and (ACO2) hold. See that the left hand of (1)^{by (ACO1)}= X

x₂⊗P⁰y₁⊗x₁P⁰⁰⊗y₂^{by (ACO2)}= the right hand of (1) for any x∈A_(n),y∈H_(n). That is, (1) holds.

Conversely, assume that (1) holds. Thus we have that Xx₂⊗P⁰⊗x₁P₂⁰⁰⊗_H(1_H)τ(1_H, P₁⁰⁰)τ⁻¹(1_H, P₃⁰⁰)

=X

x2⊗P₂⁰ ⊗P⁰⁰x4⊗H(1H)τ(P₁⁰, x1)τ⁻¹(P₃⁰, x3) and

X_A(1_A)⊗y4P⁰⊗P₂⁰⁰⊗y2τ(y1, P₁⁰⁰)τ⁻¹(y3, P₃⁰⁰)

=X

_A(1_A)⊗P₂⁰y₁⊗P⁰⁰⊗y₂τ(P₁⁰,1_A)τ⁻¹(P₃⁰,1_A)

for any x∈A_(n),y∈H_(n). Consequently, (ACO1) and (ACO2) hold.

Lemma 11. Let A = ⊕^∞_n=0A_n and H = ⊕^∞_n=0H_n be two graded Hopf algebras with invertible antipodes. Let τ be a skew pairing on (H ⊗A) and Pn be a copairing of (H_(n)⊗A_(n)) with P_n+1 = P_n+W_n and W_n ∈ H_(n+1)⊗A_n+1+H_n+1⊗A_(n+1) for any natural number n. Set D = A ./_τ H. If τ(P_n⁰, x)P_n⁰⁰ = x and τ(y, P_n⁰⁰)P_n⁰ = y for any x ∈ A_(n), y ∈ H_(n), then (D,{[P_n]}) is a local quasitriangular Hopf algebra.

Proof . It follows from Lemma7thatD=⊕^∞_n=0D_nis a Hopf algebra. LetD_n=P

i+j=nA_i⊗H_j. It is clear that D =⊕^∞_n=0Dn is a graded coalgebra. We only need to show that (D,{[P_n]}) is almost cocommutative on D_(n). Now fix n. For convenience, we denote P_n by P and Qin the following formulae. For anyx∈A_i,y∈H_j with i+j≤n,

the right hand of (ACO1)^{by (CP1)}= X

y₄Q⁰P⁰⊗Q⁰⁰₁ ⊗y₂τ(y₁, P⁰⁰)τ⁻¹(y₃, Q⁰⁰₂)

by assumption

= X

y4Q⁰y1⊗Q⁰⁰₁ ⊗y2τ⁻¹(y3, Q⁰⁰₂)^{by (CP1)}= X

y4Q⁰P⁰y1⊗P⁰⁰⊗y2τ⁻¹(y3, Q⁰⁰)

=X

y4Q⁰P⁰y1⊗P⁰⁰⊗y2τ(S⁻¹(y3), Q⁰⁰)by assumption

= X

y4S⁻¹(y3)P⁰y1⊗P⁰⁰⊗y2

=X

P⁰y₁⊗P⁰⁰⊗y₂ = the left hand of (ACO1).

Similarly, we can show that (ACO2) holds onA_(n).

Theorem 6. Assume thatB is a finite dimensional Hopf algebra andM is a finite dimenaional B-Hopf bimodule. Let A := T_B(M)^cop, H := T_B^c∗(M^∗) and D = A ./_τ H with τ := d_H(φ⊗ id)CH,A. Then ((TB(M))^cop ./τ T_B^c∗(M^∗),{R_n}) is a local quasitriangular Hopf algebra. Here P_n= (id⊗ψ_n)b_H(n), R_n= [P_n] = 1_B⊗(id⊗ψ_n)b_H(n)⊗1_B^∗, φand ψ_n are defined in Lemma 6.

Proof . By Lemma 9 (ii), A and H have invertible antipodes. Assume that e⁽ⁱ⁾₁ , e⁽ⁱ⁾₂ , . . . , e⁽ⁱ⁾_ni is a basis of H_i and e^(i)∗₁ , e^(i)∗₂ , . . . , e^(i)∗ni is an its dual basis in (H_i)^∗. Then{e⁽ⁱ⁾_j |i= 0,1,2, . . . , n;

j = 1,2, . . . , n_i} is a basis ofH_(n) and {e^(i)∗_j |i= 0,1,2, . . . , n; j= 1,2, . . . , n_i} is its dual basis in (H_(n))^∗. ThusbH_(n) =Pn

i=0

Pni

j=1e⁽ⁱ⁾_j ⊗e^(i)∗_j .See that Pn+1 =

n

X

i=0 ni

X

j=1

e⁽ⁱ⁾_j ⊗ψn+1(e^(i)∗_j ) +

nn+1

X

j=1

e⁽ⁿ⁺¹⁾_j ⊗ψn+1(e^(n+1)∗_j )

= P_n+

nn+1

X

j=1

e⁽ⁿ⁺¹⁾_j ⊗ψ_n+1(e^(n+1)∗_j ).

Obviously, Pnn+1

j=1 e⁽ⁿ⁺¹⁾_j ⊗ψ_n+1(e^(n+1)∗_j ) ∈H_n+1⊗A_n+1. It is clear that P_n is a copairing on H_(n)⊗A_(n) andτ is a skew pairing onH⊗A withP

τ(P_n⁰, x)P_n⁰⁰ =xand P

τ(y, P_n⁰⁰)P_n⁰ =y for any x∈A_(n),y∈H_(n). We complete the proof by Lemma11.

Note thatRn+1=Rn+Wn withWn∈Dn+1⊗Dn+1 in the above theorem.

6 Quiver Hopf algebras

A quiver Q= (Q0, Q1, s, t) is an oriented graph, whereQ0 and Q1 are the sets of vertices and arrows, respectively; sandt are two maps fromQ1 toQ0. For any arrow a∈Q1,s(a) andt(a) are called its start vertex and end vertex, respectively, andais called an arrow froms(a) tot(a).

For any n≥0, ann-path or a path of length nin the quiverQis an ordered sequence of arrows p=anan−1· · ·a1witht(ai) =s(ai+1) for all 1≤i≤n−1. Note that a 0-path is exactly a vertex and a 1-path is exactly an arrow. In this case, we defines(p) =s(a₁), the start vertex ofp, and t(p) =t(an), the end vertex of p. For a 0-path x, we have s(x) = t(x) = x. Let Qn be the set of n-paths, Q_(n) be the set of i-paths with i≤n and Q∞ be the set of all paths inQ. Let ^yQ^x_n denote the set of alln-paths fromxtoy,x, y∈Q₀. That is,^yQ^x_n={p∈Q_n|s(p) =x, t(p) =y}.

A quiver Qisfinite ifQ0 and Q1 are finite sets.

Let G be a group. Let K(G) denote the set of conjugate classes in G. r = P

C∈K(G)r_CC is called a ramification (or ramification data) of G, if rC is the cardinal number of a set for any C ∈ K(G). We always assume that the cardinal number of the set IC(r) is rC. Let K_r(G) :={C∈ K(G)|r_C 6= 0}={C∈ K(G)|I_C(r)6=∅}.

LetG be a group. A quiver Q is calleda quiver of G ifQ0 =G (i.e., Q= (G, Q1, s, t)). If, in addition, there exists a ramification r of G such that the cardinal number of ^yQ^x₁ is equal torC for anyx, y∈Gwithx⁻¹y∈C∈ K(G), thenQis called aHopf quiver with respect to the ramification data r. In this case, there is a bijection fromIC(r) to^yQ^x₁. Denote by (Q, G, r) the

Hopf quiver of G with respect to r. edenotes the unit element ofG. {p_g |g ∈G} denotes the dual basis of{g|g∈G} of finite group algebrakG.

LetQ= (G, Q₁, s, t) be a quiver of a group G. Then kQ₁ becomes a kG-bicomodule under the natural comodule structures:

δ⁻(a) =t(a)⊗a, δ⁺(a) =a⊗s(a), a∈Q1, (2)

called an arrow comodule, written as kQ^c₁. In this case, the path coalgebra kQ^c is exactly isomorphic to the cotensor coalgebra T_kG^c (kQ^c₁) over kGin a natural way (see [3] and [4]). We will regard kQ^c = T_kG^c (kQ^c₁) in the following. Moreover, kQ₁ becomes a (kG)^∗-bimodule with the module structures defined by

p·a:=hp, t(a)ia, a·p:=hp, s(a)ia, p∈(kG)^∗, a∈Q1, (3) written as kQ^a₁, called an arrow module. Therefore, we have a tensor algebra T_(kG)^∗(kQ^a₁).

Note that the tensor algebra T_(kG)^∗(kQ^a₁) of kQ^a₁ over (kG)^∗ is exactly isomorphic to the path algebra kQ^a. We will regard kQ^a=T_(kG)^∗(kQ^a₁) in the following.

Lemma 12 (See [4], Theorem 3.3, and [5], Theorem 3.1). LetQbe a quiver over groupG.

Then the following statements are equivalent:

(i) Q is a Hopf quiver.

(ii) Arrow comodulekQ^c₁ admits a kG-Hopf bimodule structure.

If Qis finite, then the above statements are also equivalent to the following:

(iii) Arrow modulekQ^a₁ admits a (kG)^∗-Hopf bimodule structure.

Assume that Q is a Hopf quiver. It follows from Lemma 12 that there exist a left kG- module structure α⁻ and a right kG- module structure α⁺ on arrow comodule (kQ^c₁, δ⁻, δ⁺) such that (kQ^c₁, α⁻, α⁺, δ⁻, δ⁺) becomes akG-Hopf bimodule, called a co-arrow Hopf bimodule.

We obtain two graded Hopf algebras T_kG(kQ^c₁) and T_kG^c (kQ^c₁), called semi-path Hopf algebra and co-path Hopf algebra, written askQ^s and kQ^c, respectively.

Assume that Q is a finite Hopf quiver. Dually, it follows from Lemma 12 that there exist a left (kG)^∗-comodule structureδ⁻ and a right (kG)^∗-comodule structureδ⁺ on arrow module (kQ^a₁, α⁻, α⁺) such that (kQ^a₁, α⁻, α⁺, δ⁻, δ⁺) becomes a (kG)^∗-Hopf bimodule, called an arrow Hopf bimodule. We obtain two graded Hopf algebras T_(kG)^∗(kQ^a₁) andT_(kG)^c ∗(kQ^a₁), called path Hopf algebra and semi-co-path Hopf algebra, written as kQ^a and kQ^sc, respectively.

From now on, we assume thatQis a finite Hopf quiver on finite group G. LetξkQ^a₁ denote the linear map fromkQ^a₁ to (kQ^c₁)^∗ by sendingatoa^∗for anya∈Q₁ andξ_kQ^c

1 denote the linear map fromkQ^c₁ to (kQ^a₁)^∗ by sendinga toa^∗ for any a∈Q₁. It is easy to check the following.

Lemma 13. (i) If(M, α⁻, α⁺, δ⁻, δ⁺)is a finite dimensionalB-Hopf bimodule andB is a finite dimensional Hopf algebra, then (M^∗, δ^−∗, δ^+∗, α^−∗, α^+∗) is a B^∗-Hopf bimodule.

(ii) If(kQ^c₁, α⁻, α⁺, δ⁻, δ⁺) is a co-arrow Hopf bimodule, then there exist unique left (kG)^∗- comodule operation δ_kQ⁻ a

1 and right (kG)^∗-comodule operation δ⁺_kQa

1 such that (kQ^a₁, α⁻_kQa 1, α⁺_kQa

1, δ⁻_kQa

1, δ⁺_kQa

1) becomes a (kG)^∗-Hopf bimodule and ξkQ^a₁ becomes a (kG)^∗-Hopf bimodule isomor- phism from (kQ^a₁, α⁻_kQa

1, α⁺_kQa 1, δ_kQ⁻ a

1, δ_kQ⁺ a

1) to ((kQ^c₁)^∗, δ^−∗, δ^+∗, α^−∗, α^+∗).

(iii) If (kQ^a₁, α⁻, α⁺, δ⁻, δ⁺) is an arrow Hopf bimodule, then there exist unique left kG- module operationα⁻_kQc

1 and rightkG-moduleα⁺_kQc

1 such that(kQ^c₁, α⁻_kQc 1, α⁺_kQc

1, δ⁻_kQc 1, δ_kQ⁺ c

1)become a kG-Hopf bimodule andξ_kQ^c

1 becomes a kG-Hopf bimodule isomorphism from(kQ^c₁, α_kQ⁻ c 1, α⁺_kQc

1, δ⁻_kQc

1, δ_kQ⁺ c

1) to ((kQ^a₁)^∗, δ^−∗, δ^+∗, α^−∗, α^+∗).

(iv)ξkQ^a₁ is a(kG)^∗-Hopf bimodule isomorphism from(kQ^a₁, α⁻_kQa 1, α⁺_kQa

1, δ⁻_kQa 1, δ⁺_kQa

1)to((kQ^c₁)^∗, δ⁻_kQc

1

∗, δ_kQ⁺ c 1

∗, α⁻_kQc 1

∗, α⁺_kQc 1

∗) if and only if ξ_kQ^c

1 becomes a kG-Hopf bimodule isomorphism from (kQ^c₁, α⁻_kQc

1, α⁺_kQc 1, δ⁻_kQc

1, δ_kQ⁺ c

1) to ((kQ^a₁)^∗, δ_kQ⁻ a 1

∗, δ_kQ⁺ a 1

∗, α⁻_kQa 1

∗, α⁺_kQa 1

∗).