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,{Rn}) is a local quasitriangular Hopf algebra. Then (HMcf,
C{Rn}), (
HMdcf, C{Rn}) and (HMdf, C{Rn}) are braided tensor categories. Furthermore, if (M, α−) is an H-module with finite cycles and R
n+1 = Rn+Wn with Wn ∈ Hn+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 ((TB(M))cop ⊲⊳τ TBc∗(M∗),{Rn}) is a local quasitriangular Hopf algebra. Furthermore, if (kQa, kQc) and (kQs, kQsc) are arrow dual pairings with finite Hopf quiver Q, then both ((kQa)cop ⊲⊳
τ kQc,{Rn}) and((kQs)cop ⊲⊳τ kQsc,{Rn}) 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
by definingCV,W(x⊗y) =y⊗xfor anyx∈V,y∈W. DenoteP byPP′⊗P′′forP ∈V⊗W. If V is a finite-dimensional vector space over field k with V∗ = Hom
k(V, k). Define maps
bV :k→V ⊗V∗ and dV :V∗⊗V →k by
bV(1) =
X
i
vi⊗vi∗ and
X
i,j
dV(v∗i ⊗vj) =hvi∗, vji,
where{vi |i= 1,2, . . . , n} is any basis ofV and {v∗i |i= 1,2, . . . , n} is its dual basis inV∗. dV andbV are called evaluation and coevaluation ofV, respectively. It is clear (dV⊗idU)(idU⊗bV) =
idU and (idV ⊗dV)(bV ⊗idV) = idV. ξV 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) = Px(−1)⊗x(0), δ+(x) = Px(0) ⊗x(1). Let (H, µ, η,∆, ǫ) be
a bialgebra and let ∆cop:=CH,H∆ andµop :=µCH,H.We denote (H, µ, η,∆cop, ǫ) byHcopand (H, µop, η,∆, ǫ) byHop. Sometimes, we also denote the unit element of H by 1H.
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) : Xy2R′⊗y1R′′= X
R′y1⊗R′′y2 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) =Xτ(x1, y)τ(x2, z);
(SP2) : τ(xu, z) =Xτ(x, z2)τ(u, z1);
(SP3) : τ(x, η) =ǫH(x); (SP4) : τ(η, y) =ǫA(y).
P is called a copairing of Y ⊗X if for any x, u ∈ H, y, z ∈ A the following conditions are satisfied:
(CP1) : XP′⊗P1′′⊗P2′′=XP′Q′⊗Q′′⊗P′′ with P =Q;
(CP1) : XP1′⊗P2′ ⊗P′′=XP′⊗Q′⊗P′′Q′′ with P =Q;
(CP3) : XP′⊗ǫA(P′′) =ηH;
(CP4) : XǫH(P′)⊗P′′=ηA.
ForR∈H⊗Hand twoH-modulesU andV, define a linear mapCR
U,V fromU⊗V toV ⊗U by sending (x⊗y) to P
R′′y⊗R′x for anyx∈U,y∈V.
If V = ⊕∞i=0Vi is a graded vector space, let V>n and V≤n denote ⊕∞i=n+1Vi and ⊕ni=0Vi, respectively. We usually denote⊕n
i=0Vi byV(n). If dimVi <∞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 Rn 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(n), Rn) is almost cocommutative for any natural number n. In this case, {Rn} is called a local quasitriangular structure of H. Obviously, if (H, R) is a quasitriangular bialgebra, then (H,{Rn}) is a local quasitriangular bialgebra withR0=R,Ri= 0, H0 =H,Hi= 0 fori >0.
The following facts are obvious: τ−1 =τ(idH⊗S) (or =τ(S−1⊗idA)) 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 algebraTA(M) ofM overA is a graded algebra with TA(M)0 =A,TA(M)1 =M and TA(M)n=⊗nAM for n >1. That is,
TA(M) =A⊕(Ln>0⊗nAM) (see [12]). LetDbe another algebra. Ifhis an algebra map fromA toDandf is anA-bimodule map fromMtoD, then by the universal property ofTA(M) (see [12, Proposition 1.4.1]) there is a unique algebra mapTA(h, f) :TA(M)→Dsuch thatTA(h, f)ι0 =h
and TA(h, f)ι1 = f. One can easily see that TA(h, f) = h+Pn>0µn−1Tn(f), where Tn(f) is the map from⊗n
AM to⊗nADgiven by Tn(f)(x1⊗x2⊗ · · · ⊗xn) =f(x1)⊗f(x2)⊗ · · · ⊗f(xn), 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 Tc
C(M) of M over C is a graded coalgebra with TCc(M)0 = C, TCc(M)1 = M and
Tc
C(M)n = ✷nCM for n > 1. That is, TCc(M) = C ⊕(
L
n>0✷nCM) (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 Tc
C(M) (see [12, Proposi-tion 1.4.2]) there is a unique coalgebra mapTc
C(h, f) fromDtoTCc(M) such thatπ0TCc(h, f) =h andπ1TCc(h, f) =f. It is not difficult to see thatTCc(h, f) =h+
P
n>0Tnc(f)∆n−1, whereTnc(f) is the map from✷nCDto✷nCM induced byTn(f)(x1⊗x2⊗· · ·⊗xn) =f(x1)⊗f(x2)⊗· · ·⊗f(xn), i.e., Tc
n(f) =f⊗f ⊗ · · · ⊗f.
Furthermore, ifB is a Hopf algebra and M is a B-Hopf bimodule, thenTB(M) and TBc(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= εBπ0 and the comultiplication ∆ = (ι0⊗ι0)∆B+
P
n>0µn−1Tn(∆M), where ∆M = (ι0⊗ι1)δ−M + (ι1⊗ι0)δ+M. Dually, TBc(M) is a graded Hopf algebra with multiplicationµ=µB(π0⊗π0) +Pn>0Tnc(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);
(LQT4′) : Rn+1=Rn+Wn with Wn∈Hn+1⊗Hn+1.
Then (H,{Rn}) 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 > nx. Let HMcf denote the category of all left H-modules with finite cycles.
Proof . For anyx∈U,y∈V,there exist two natural numbersnx andny, such thatH>nxx= 0
and H>nyy = 0. Set nx⊗y = 2nx+ 2ny. It is clear that H>nx⊗y(x⊗y) = 0. Indeed, for any
h∈Hi withi > nx⊗y, we see
h(x⊗y) =Xh1x⊗h2y= 0 (since H is graded coalgebra).
Lemma 2. Assume that(H,{Rn})is a local quasitriangular Hopf algebra. Then for any leftH -modules U andV with finite cycles, there exists an invertible linear mapC{Rn}
U,V :U⊗V →V⊗U such that
C{Rn}
U,V (x⊗y) :=C
Rn(x⊗y) =XR′′
ny⊗R′nx
with n >2nx+ 2ny, for x∈U, y∈V.
Proof . We first define a mapf from U×V toV ⊗U by sending (x, y) to P
R′′
ny⊗R′nx with
n >2nx+ 2ny for anyx∈U,y∈V. It is clear thatf is well defined. Indeed, if n >2nx+ 2ny, thenCRn+1(x⊗y) =CRn(x⊗y) sinceR
n+1 =Rn+WnwithWn∈H(n+1)⊗Hn+1+Hn+1⊗H(n+1).
f is a k-balanced function. Indeed, forx, y∈U,z, w∈V,α∈k, let n >2nx+ 2ny+ 2nz. See
f(x+y, z) =XR′′nz⊗R′n(x+y) =
X
R′′nz⊗R′nx+
X
R′′nz⊗R′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{Rn}
U,V :U ⊗V →V ⊗U such that
C{Rn}
U,V (x⊗y) =C
Rn(x⊗y)
with n >2nx+ 2ny, forx∈U,y∈V. The inverse (C{Rn}
U,V )−1 of C {Rn}
U,V is defined by sending (y⊗x) to
P
(R−1
n )′x⊗(R−n1)′′y with
n >2nx+ 2ny for anyx∈U,y∈V.
Theorem 2. Assume that (H,{Rn}) is a local quasitriangular Hopf algebra. Then (HMcf,
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 (HMcf,⊗, I, a, r, l) is a tensor subcategory of (HM,⊗, I, a, r, l). C{Rn} is a braiding of HMcf, 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
HiMj ⊆Mi+j for any natural number iand j.
Lemma 3. Assume that H is a local finite graded coalgebra and bialgebra. If M =⊕∞ i=0Mi 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 > nx. Since H/(0 : x)H ∼= Hx, where (0 : x)H := {h ∈ H |h·x = 0}, we have that Hx is finite dimensional.
(iii)⇒ (i). We first show that, for any homogeneous element x∈Mi, there exists a natural number nx such that Hjx = 0 with j > nx. In fact, if the above does not hold, then there exists hj ∈ Hnj such that hjx 6= 0 with n1 < n2 < · · ·. Considering hjx ∈ Mnj+i we have
that {hjx | j = 1,2, . . .} is linear independent in Hx, which contradicts to that Hx is finite dimensional.
For anyx∈M, thenx=Pl
i=1xi andxi 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 HMgf and HMgcf 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 HMcf. Therefore we have
Theorem 3. Assume that (H,{Rn}) is a local quasitriangular Hopf algebra. Then (HMgf,
C{Rn}) and (
HMgcf, 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,{Rn}), then CM,M{Rn} is a solution of Yang–Baxter equation on M.
It is easy to prove the following.
Theorem 4. Assume that(H,{Rn})is a local quasitriangular Hopf algebra andRn+1=Rn+Wn 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
Rn′′⊗R′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, δM−, δM+) and (N, δN−, δ+N) be respectively a finite dimensional A-B-bicomodule and a finite dimensional B-C -bicomodule. Then
(i) (M∗, δM−∗, δM+∗) is a finite dimensional A∗-B∗-bimodule; (ii)(M✷BN, δ−M✷BN, δ
+
M✷BN)is anA-C-bicomodule with structure mapsδ
−
M✷BN =δ
− M⊗idN and δM+✷
BN =idM ⊗δ +
N; (iii) M∗⊗
B∗N∗ ∼= (M✷BN)∗ (as A∗-C∗-bimodules).
Lemma 5 (See [3, 7]). Let A, B and C be finite dimensional algebras, (M, αM−, 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∗, αM+∗) 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⊗idN and α+M⊗
BN =idM ⊗α +
N; (iii) M∗✷
B∗N∗ ∼= (M ⊗BN)∗ (as A∗-C∗-bicomodules).
Proof . (i) and (ii) are easy. (iii) Consider
M∗✷B∗N∗ ∼= (M∗✷B∗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
Theorem 5. If A is a finite dimensional algebra and M is a finite dimensional A-bimodule, thenTA(M) is isomorphic to subalgebra
P∞ an A-bimodule isomorphism fromN toL. See
⊗nAM =N⊗AM
Lemma 6. Assume that B is a finite dimensional Hopf algebra and M is a finite dimensional
Proof . (i) We first show that (✷nB∗M∗)∗⊆(TBc∗(M∗))0. For anyf ∈(✷nB∗M∗)∗, P∞
i=n+1✷iB∗M∗
⊆ kerf and P∞
i=n+1✷iB∗M∗ is a finite codimensional ideal of TBc∗(M∗). Consequently, f ∈
(TBc∗(M∗))0.
Next we show thatφ:=σB+Pn>0µn−1Tn(σM) (see the proof of Theorem5) is a coalgebra homomorphism from TB(M) to P∞i=0(✷cB∗M∗)∗. For any x ∈ ⊗nBM, f, g ∈ TBc∗(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=Xh(φ(x))1, fih(φ(x))2, gi,
since φ(x) ∈ (Tc
B∗(M∗))0. Considering TBc∗(M∗) = ⊕n≥0✷nB∗M∗ ∼= ⊕n≥0(⊗nBM)∗ as vec-tor spaces, we have that TB∗(M∗) is dense in (TB(M))∗. Consequently, Pφ(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 productAα ⊲⊳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 ǫD 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),
ǫD =ǫA⊗ǫH, ηD =ηA⊗ηH for anya, b∈A,h, g∈H.We denote (A⊗H, µD, ηD,∆D, ǫD) by
Aα⊲⊳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) =Pτ(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 Aα⊲⊳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
Aα⊲⊳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:Acop→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.
Proof . (i) It is clear that Hop is a graded bialgebra with (Hop)0 = (H0)(op). Thus Hop has
an antipode by [12, Proposition 1.5.1]. However, the antipode of Hop 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 [Pn] = 1A⊗Pn⊗1H. Then (D,{[Pn]}) is almost cocommutative on D(n) if and only if
(ACO1) : XP′y1⊗P′′⊗y2 = X
y4P′⊗P2′′⊗y2τ(y1, P1′′)τ−1(y3, P3′′) for any y∈H(n);
(ACO2) : Xx2⊗P′⊗x1P′′= X
x2⊗P2′⊗P′′x4τ(P1′, x1)τ−1(P3′, x3) for any x∈A(n).
Proof . It is clear that (D,{[Pn]}) is almost cocommutative on D(n) if and only if the following
holds:
X
x2⊗y4P′⊗x1P2′′⊗y2τ(y1, P1′′)τ−1(y3, P3′′)
=Xx2⊗P2′y1⊗P′′x4⊗y2τ(P1′, x1)τ−1(P3′, x3) (1)
for anyx∈A(n),y∈H(n).
Assume that both (ACO1) and (ACO2) hold. See that
the left hand of (1)by (ACO1)= Xx2⊗P′y1⊗x1P′′⊗y2
by (ACO2)
= the right hand of (1)
for anyx∈A(n),y∈H(n). That is, (1) holds.
Conversely, assume that (1) holds. Thus we have that
X
x2⊗P′⊗x1P2′′⊗ǫH(1H)τ(1H, P1′′)τ−1(1H, P3′′)
=Xx2⊗P2′ ⊗P′′x4⊗ǫH(1H)τ(P1′, x1)τ−1(P3′, x3)
and
X
ǫA(1A)⊗y4P′⊗P2′′⊗y2τ(y1, P1′′)τ−1(y3, P3′′)
=XǫA(1A)⊗P2′y1⊗P′′⊗y2τ(P1′,1A)τ−1(P3′,1A)
for anyx∈A(n),y∈H(n). Consequently, (ACO1) and (ACO2) hold.
Lemma 11. 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)) with
Pn+1 = Pn+Wn and Wn ∈ H(n+1)⊗An+1+Hn+1⊗A(n+1) for any natural number n. Set
D = A ⊲⊳τ H. If τ(Pn′, x)Pn′′ = x and τ(y, Pn′′)Pn′ = y for any x ∈ A(n), y ∈ H(n), then
(D,{[Pn]}) is a local quasitriangular Hopf algebra.
Proof . It follows from Lemma7thatD=⊕∞
n=0Dnis a Hopf algebra. LetDn=Pi+j=nAi⊗Hj. It is clear that D =⊕∞n=0Dn is a graded coalgebra. We only need to show that (D,{[Pn]}) is almost cocommutative on D(n). Now fix n. For convenience, we denote Pn by P and Qin the following formulae. For anyx∈Ai,y∈Hj with i+j≤n,
by assumption
= Xy4Q′y1⊗Q′′1 ⊗y2τ−1(y3, Q′′2)
by (CP1)
= Xy4Q′P′y1⊗P′′⊗y2τ−1(y3, Q′′)
=Xy4Q′P′y1⊗P′′⊗y2τ(S−1(y3), Q′′)
by assumption
= Xy4S−1(y3)P′y1⊗P′′⊗y2
=XP′y1⊗P′′⊗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 := TB(M)cop, H := TBc∗(M∗) and D = A ⊲⊳τ H with τ := dH(φ⊗
id)CH,A. Then ((TB(M))cop ⊲⊳τ TBc∗(M∗),{Rn}) is a local quasitriangular Hopf algebra. Here
Pn= (id⊗ψn)bH(n), Rn= [Pn] = 1B⊗(id⊗ψn)bH(n)⊗1B∗, φand ψn are defined in Lemma 6.
Proof . By Lemma 9 (ii), A and H have invertible antipodes. Assume that e1(i), e(2i), . . . , e(nii) is
a basis of Hi and e(1i)∗, e2(i)∗, . . . , en(ii)∗ is an its dual basis in (Hi)∗. Then{e (i)
j |i= 0,1,2, . . . , n;
j = 1,2, . . . , ni} is a basis ofH(n) and {ej(i)∗ |i= 0,1,2, . . . , n; j= 1,2, . . . , ni} 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(ji)⊗ψn+1(e(ji)∗) + nn+1
X
j=1
e(jn+1)⊗ψn+1(e(jn+1)∗)
= Pn+ nn+1
X
j=1
e(jn+1)⊗ψn+1(e(jn+1)∗).
Obviously, Pnn+1
j=1 e (n+1)
j ⊗ψn+1(e(jn+1)∗) ∈Hn+1⊗An+1. It is clear that Pn is a copairing on
H(n)⊗A(n) andτ is a skew pairing onH⊗A withPτ(Pn′, x)Pn′′ =xand
P
τ(y, Pn′′)Pn′ =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 lengthnin 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 yQxn denote the set of alln-paths fromxtoy,x, y∈Q0. That is,yQxn={p∈Qn|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)rCC 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
Kr(G) :={C∈ K(G)|rC 6= 0}={C∈ K(G)|IC(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 yQx
1 is equal
Hopf quiver of G with respect to r. edenotes the unit element ofG. {pg |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 kQc
1. In this case, the path coalgebra kQc is exactly
isomorphic to the cotensor coalgebra Tc
If Qis finite, then the above statements are also equivalent to the following: (iii) Arrow modulekQa
1 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 (kQc
1, δ−, δ+)
such that (kQc
1, α−, α+, δ−, δ+) becomes akG-Hopf bimodule, called a co-arrow Hopf bimodule.
We obtain two graded Hopf algebras TkG(kQc1) and TkGc (kQc1), called semi-path Hopf algebra
and co-path Hopf algebra, written askQs and kQc, 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
(kQa
1, α−, α+) such that (kQa1, α−, α+, δ−, δ+) becomes a (kG)∗-Hopf bimodule, called an arrow
Hopf bimodule. We obtain two graded Hopf algebras T(kG)∗(kQa1) andTc
(kG)∗(kQa1), called path
Hopf algebra and semi-co-path Hopf algebra, written as kQa and kQsc, respectively. From now on, we assume thatQis a finite Hopf quiver on finite group G. LetξkQa
1 becomes a kG-Hopf bimodule isomorphism from(kQ
c
∗-Hopf bimodule isomorphism from(kQa
1, α−kQa
1 becomes a kG-Hopf bimodule isomorphism from
LetB be a Hopf algebra and B
BMBB denote the category of B-Hopf bimodules. Let GHopf denote the category of graded Hopf algebras. Define TB(ψ) =: TB(ι0, ι1ψ) and TBc(ψ) :=
Obviously, the first equation holds. For the second equation, see
∆TB′(M′)Φι1 = ∆TB′(M′)ι1ψ= (ι0⊗ι1)δ
(ii) There exist τ and {Rn} such that ((kQs)cop ⊲⊳τ kQsc,{Rn}) becomes a local quasitrian-gular Hopf algebra.
Proof . (i) Let B := (kG)∗ and M :=kQa1.Thus TB(M) =kQa. Since (kQa1, kQc1) is an arrow
dual pairing, ξkQc
1 is akG-Hopf bimodule isomorphism fromkQ
c
1 to (kQa1)∗ by Lemma13. See
that
TBc∗(M∗) =T(ckG)∗∗((kQa1)∗) ν1
∼
=TkGc ((kQa1)∗) (by Lemma 15) ν2
∼
=TkGc (kQc1) (by Lemma 13and Lemma 14) =kQc,
where ν1=TkGc (σkG−1π0, π1),ν2=T(ckG)∗∗(π0,(ξkQc1)−1π1).
Let H =Tc
B∗(M∗) and A =TB(M)cop. By Theorem 6, ((kQa)cop ⊲⊳τ kQc,{Rn}) is a local quasitriangular Hopf algebra. Here τ =dH(φ⊗id)CH,A((ν2ν1)−1⊗idA); Rn = (idA⊗ν2ν1⊗
idA⊗ν2ν1)(1B⊗(id⊗ψn)bH(n) ⊗1B∗), φ and ψn are defined in Lemma 6. We have to show
that they are the same as in this theorem. That is,
dH(φ⊗id)CH,A((ν2ν1)−1⊗idA)(a⊗b) =δa,b (4)
(1B⊗bH(n) ⊗1B∗) = (idA⊗(ν2ν1)
−1⊗φ
n⊗(ν2ν1)−1)
×
X
g∈G
pe⊗g⊗pg⊗e+
X
q∈Q(n),q6∈G
pe⊗q⊗q⊗e
. (5)
If b=bnbn−1· · ·b1 is an-path inkQc withbi ∈Q1 fori= 1,2, . . . , n,then
(ν2ν1)−1(b) =bn∗ ⊗b∗n−1⊗ · · · ⊗b∗1. (6)
Ifb∈G, then (ν2ν1)−1(b) =σkG(b). Consequently, for anya, b∈Q∞withb∈kQcanda∈kQa, we have
dH(φ⊗id)CH,A((ν2ν1)−1⊗idA)(a⊗b) =dH(φ(a)⊗(ν2ν1)−1(b))
=
(
hφ(a), σkG(b)i=hσkG(b), ai=δa,b, when b∈Q0,
h(ν2ν1)−1, ai by (6)
= δa,b, when b6∈Q0.
Thus (4) holds. Note thatQn is a basis of not only (kQa)n but also (kQc)n forn >0.By (6),
{φ(q) | q ∈ Qn} is the dual basis of {(ν2ν1)−1(q) | q ∈ Qn} for any n > 0. Consequently, (5) holds.
(ii) Let B := kG and M := kQc
1. Thus TB(M) = kQs. Since (kQa1, kQc1) is an arrow dual pairing,ξkQa
1 is a (kG)
∗-Hopf bimodule isomorphism fromkQa
1 to (kQc1)∗ by Lemma13. Thus
TBc∗(M∗) =T(ckG)∗((kQc1)∗) ν3
∼
=T(ckG)∗(kQa1) (by Lemma13 and Lemma14)
=kQsc,
where ν3 =T(ckG)∗(π0,(ξkQa 1)
−1π
1). By Theorem 6, the double cross product ofTB(M)cop and
TB∗(M∗) is a local quasitriangular Hopf algebra. Consequently, so is the double cross product
Note (LQT4’) holds in the above theorem.
Example 1. LetG=Z2 = (g) ={e, g} be the group of order 2 with chark6= 2,X and Y be
respectively the set of arrows fromg0 tog0 and the set of arrows fromgtog, and|X|=|Y|= 3. The quiverQis a Hopf quiver with respect to ramificationr=r{e}{e}withr{e} = 3. Letχ(ei) ∈
Z2 and a(y,xi) denote the arrow from xtoy fori= 1,2,3. Defineδ−(a(x,xi)) =x⊗ax,x(i),δ+(a(x,xi)) =
ax,x(i) ⊗x, g ·a(x,xi) = agx,gx(i) , a(e,ei) ·g = χe(i)(g)a(xg,xgi) for x ∈ G, i = 1,2,3. By [4], kQ1 is a kG
-Hopf bialgebra. Therefore, it follows from Theorem 7 that ((kQa)cop ⊲⊳
τ kQc,{Rn}) is a local quasitriangular Hopf algebra and for every finite dimensional (kQa)cop ⊲⊳
τ kQc-module M,
C{Rn} is a solution of Yang–Baxter equations onM.
By the way, we obtain the relation between path algebras and path coalgebras by Theorem5.
Corollary 1. Let Q be a finite quiver over finite group G. Then Path algebra kQa is algebra isomorphic to subalgebra P∞
n=0(✷nkGkQc1)∗ of (kQc)∗. Proof . Let A = (kG)∗ and M = kQc
1. It is clear that ξkQc
1 is a kG-bicomodule isomorphism
from kQc
1 to (kQa1)∗. See that
TBc∗(M∗) =T(ckG)∗∗((kQa1)∗) ν1
∼
=TkGc ((kQa1)∗)=∼ν2TkGc (kQc1) =kQc,
where ν1 = TkGc (σkG−1π0, π1), ν2 = TkGc (π0,(ξkQc 1)
−1π
1). Obviously, ν1 and ν2 are coalgebra
isomorphism. Consequently, it follows from Theorem 5 that kQa is algebra isomorphism to subalgebraP∞
n=0(✷nkGkQc1)∗ of (kQc)∗.
Acknowledgement
The authors were financially supported by the Australian Research Council. S.Z. thanks De-partment of Mathematics, University of Queensland for hospitality.
References
[1] Auslander M., Reiten I., Smalø S.O., Representation theory of Artin algebras, Cambridge University Press, 1995.
[2] Belavin A.A., Drinfel’d V.G., solutions of the classical Yang–Baxter equations for simple Lie algebras,
Functional Anal. Appl.16(1982), no. 3, 159–180.
[3] Chin W., Montgomery S., Basic coalgebras, in Modular Interfaces (1995, Riverside, CA),AMS/IP Stud. Adv. Math., Vol. 4, Amer. Math. Soc., Providence, RI, 1997, 41–47.
[4] Cibils C., Rosso M., Hopf quivers,J. Algebra254(2002), 241–251,math.QA/0009106. [5] Cibils C., Rosso M., Algebres des chemins quantiques,Adv. Math.125(1997), 171–199.
[6] Drinfel’d V.G., Quantum groups, in Proceedings International Congress of Mathematicians (August 3–11, 1986, Berkeley, CA), Amer. Math. Soc., Providence, RI, 1987, 798–820.
[7] Dascalescu S., Nastasescu C., Raianu S., Hopf algebras: an introduction, Marcel Dekker Inc., 2001.
[8] Doi Y., Takeuchi M., Multiplication alteration by two-cocycles – the quantum version,Comm. Algebra22 (1994), 5715–5732.
[9] Jantzen J.C., Lectures on quantum groups, Graduate Studies in Mathematics, Vol. 6, Amer. Math. Soc., Providence, RI, 1996.
[10] Majid S., Algebras and Hopf algebras in braided categories, in Advances in Hopf Algebras (1992, Chicago, IL),Lecture Notes in Pure and Appl. Math., Vol. 158, Marcel Dekker, New York, 1994, 55–105.
[12] Nichols W., Bialgebras of type one,Comm. Algebra6(1978), 1521–1552.
[13] Lusztig G., Introduction to quantum groups,Progress in Mathematics, Vol. 110, Birkh¨auser, Boston, MA, 1993.
[14] Kassel C., Quantum groups,Graduate Texts in Mathematics, Vol. 155, Springer-Verlag, New York, 1995.
[15] Sweedler M.E., Hopf algebras, Benjamin, New York, 1969.
[16] Yang C.N., Ge M.L. (Editors), Braid group, knot theory and statistical mechanics, World Scientific, Singa-pore, 1989.
