Finite-dimensional quasi-Hopf algebras of Cartan type
aa r X i v : . [ m a t h . QA ] J un FINITE-DIMENSIONAL QUASI-HOPF ALGEBRAS OF CARTAN TYPE
YUPING YANG AND YINHUO ZHANG
Abstract.
In this paper, we present a general method for constructing finite-dimensionalquasi-Hopf algebras from finite abelian groups and braided vector spaces of Cartan type.The study of such quasi-Hopf algebras leads to the classification of finite-dimensional rad-ically graded basic quasi-Hopf algebras over abelian groups with dimensions not divisi-ble by 2 , , , introduction Quasi-Hopf algebras are generalizations of Hopf algebras, and are fundamental in the studyof finite integral tensor categories [14]. Recall that a tensor category is called integral ifthe Frobinus-Perron dimension of each object is an integer. According to [15], any finiteintegral tensor category over an algebraically closed filed is equivalent to the representationcategory of some finite-dimensional quasi-Hopf algebra. Pointed tensor categories are specialexamples of integral tensor categories, and the corresponding quasi-Hopf algebras are calledbasic quasi-Hopf algebras.In the past and half decades, the classification of finite-dimensional basic quasi-Hopf algebrashave attracted lots of attention. Since the dual of a finite-dimensional pointed Hopf algebrais a basic Hopf algebra, the duals of the finite dimensional pointed Hopf algebras over abeliangroups classified in[1, 4, 17, 6] provide a big family of finite-dimensional basic quasi-Hopfalgebras. Since our ultimate goal is to classify the tensor categories, we are only interested inthose quasi-Hopf algebras whose representation categories do not arise from any Hopf algebra.Such quasi-Hopf algebras are said to be genuine. In [10, 11, 12], Etingof and Gelaki gave amethod for constructing basic genuine quasi-Hopf algebras from known basic Hopf algebras,and classified the finite-dimensional radically graded basic quasi-Hopf algebras over cyclicgroups of prime order. In [16], Gelaki constructed the finite-dimensional basic quasi-Hopfalgebras of dimension N over cyclic groups of order N . Utilizing the classification resultof [4], Angiono [5] classified the finite-dimensional radically graded basic quasi-Hopf algebrasover cyclic groups with dimensions not divisible by small prime divisors. In [19, 20, 21],the quasi-commutative finite-dimensional graded pointed Majid algebras of low ranks (dualbasic quasi-Hopf algebras) are classified by the first author and his cooperators. Althoughthe aforementioned classification work covered a lot of new finite dimensional quasi-Hopfalgebras, the most important family of finite dimensional pointed Hopf algebras of Cartantype is not yet covered by the above classifications of quasi-Hopf algebras. In particular, wehave not found a natural quasi-version of the (generalized) small quantum groups althougha very close quasi-version of the Frobenius-Lusztig kernel was constructed by means of the Mathematics Subject Classification.
Key words and phrases. quasi-Hopf algebras, small quantum groups, Cartan matrices. quasi-quantum double in [25]. This is because the classic construction of a small quantumgroup as a particular quotient of the quantum double works not for the quasi-Hopf algebracase. So we have to look for an alternative way to define the notion of a small quasi-quantumgroup. The fact that the classical small quantum groups form a special class of the finitedimensional pointed Hopf algebras of finite Cartan type, see [22, 23, 1, 4], inspires us: if wecould construct finite dimensional quasi-Hopf algebras from Cartan matrices, then the smallquasi-quantum groups must be the special cases of those quasi-Hopf algebras of Cartan type.This motivates us to study the finite-dimensional quasi-Hopf algebras of Cartan type. Themain work of this paper are threefold.First of all, we present a general method for constructing finite-dimensional quasi-Hopf al-gebras from finite Cartan matrices. Such a quasi-Hopf algebra is generated by an abeliangroup and a braided vector space of Cartan type. In more detail, let G be a finite abeliangroup and G a bigger abelian group uniquely determined by G , see (3.4). Let u ( D , λ, µ )([4], or see Theorem 2.14) be the generalized small quantum group generated by grouplikeelements G and skew-primitive elements { X , · · · , X n } . We then determine the 2-cochains J on G such that the subalgebra of the twist quasi-Hopf algebra u ( D , λ, µ ) J generated by G and { X , · · · , X n } is a quasi-Hopf subalgebra, denoted u ( D , λ, µ, Φ J ), see Theorem 3.4. Note thatif λ = 0 and µ = 0, then u ( D , ,
0) is a radically graded basic Hopf algebra. Moreover, when G is a cyclic group, the quasi-Hopf algebra u ( D , , , Φ J ) is the same as those constructed in[5, 10, 11, 12]. However, if G is not cyclic, or u ( D , λ, µ ) is not radically graded and basic,then the construction and the study of u ( D , λ, µ, Φ J ) are much more complicated. One of thedifficulties is to compute suitable 2-cochain J ’s on G such that u ( D , λ, µ, Φ J ) is a quasi-Hopfalgebra. Even if we can compute such a suitable cochain J , we still have no standard methodto determine whether u ( D , λ, µ, Φ J ) is genuine or not. While in the case of λ = 0 , µ = 0 and G is a cyclic group, this problem is trivial.Secondly, the obtained quasi-Hopf algebras of Cartan type deliver the classification of finite-dimensional radically graded basic quasi-Hopf algebras over abelian groups. Let H be afinite-dimensional basic quasi-Hopf algebra, and rad( H ) the Jacobson radical of H . Then wehave H/ rad( H ) ∼ = [ k G ] ∗ , where G is the Grothendieck group of the representation categoryof H . We say that the basic quasi-Hopf algebra H is over the group G . When G is abelian, itis obvious that H/ rad( H ) ∼ = k G . If H is a radically graded and basic quasi-Hopf algebra over G , then the associator of H is determined by a normalized 3-cocycle on G , see [5, 19, 21]. Weshow that a finite-dimensional radically graded and basic quasi-Hopf algebra H over an abeliangroup G with dimension not divisible by 2 , , ,
7, and the associator is given by an abelian3-cocycle of G , must be isomorphic to a quasi-Hopf algebra of Cartan type u ( D , λ, µ, Φ J ),where λ = 0 , µ = 0, see Theorem 4.4. Since each normalized 3-cocycle of a cyclic group oran abelian group of the form Z m × Z n is abelian, our classification extends the correspondingclassification results of [5, 19] to more general cases.Thirdly, we introduce the quasi-version of the small quantum groups, which form a class offinite dimensional quasi-Hopf algebras of Cartan type, namely, those H c , where c is a familyof parameters. When c approaches 0, the small quasi-quantum group H c will be the usualsmall quantum group, see Proposition 5.3. As mentioned before, the small quasi-quantumgroup defined in this paper is substantially different from the one defined in [25], where asmall quasi-quantum group is defined as the quantum double of a quasi-Hopf algebra A q ( g )constructed in [11], where g is a simple Lie algebra. Note that the quantum double D ( A q ( g ))is a quasi-Hopf algebra of Cartan type as well. Unlike the Hopf algebra case, the small quasi-quantum group H c is not a quotient of the double D ( A q ( g )) in general. For example, if the INITE-DIMENSIONAL QUASI-HOPF ALGEBRAS OF CARTAN TYPE 3 order of q is odd and not divisible by 3 in case g is of type G , then the double D ( A q ( g )) is not agenuine quasi-Hopf algebra, see [13]. Under the same conditions for q , we can show that thereare many genuine small quasi-quantum groups. This means that those small quasi-quantumgroups can not be the quotients of D ( A q ( g )).Beside the study of the small quasi-quantum groups, we will provide lots of other genuinequasi-Hopf algebras associated to finite Cartan matrices in Section 6. As a matter of fact,our method will not only systematically produce many nonsemisimple, nonradically gradedgenuine quasi-Hopf algebras, but also yield many new classes of finite integral and non-pointedtensor categories.The paper is organized as follows. In Section 2, we introduce some concepts and known resultsabout quasi-Hopf algebras, generalized small quantum groups and 3-cocycle of abelian groups.In Section 3, the quasi-Hopf algebras of Cartan type are constructed, and some low ranknonradically graded examples are provided. In Section 4, we classify the finite-dimensionalradically graded quasi-Hopf algebras which are basic over abelian groups, and show thatall the radically graded quasi-Hopf algebras of Cartan type are genuine. In Section 5, weintroduce the small quasi-quantum groups, which are special nonradically graded quasi-Hopfalgebras of Cartan type, and present explicitly examples of genuine quasi-Hopf algebras ofCartan type. Section 6 is devoted to new examples of nonradically graded genuine quasi-Hopfalgebras associated to connected finite Cartan matrices.Throughout this paper, k denotes an algebraically closed field of characteristic zero. All thealgebras, tensor categories and the unadorned tensor product ⊗ are over k .2. Preliminaries
In this section, we introduce some notations and basic facts about Quais-Hopf algebras, tensorcategories and some important results [4] about pointed Hopf algebras.2.1.
Quasi-Hopf algebras.
A qausi-bialgebra H = ( H, △ , ε, Φ) is an unital associative al-gebra with two algebra maps △ : H → H ⊗ H (the comultiplication) and ε : H → k (thecounit), and an invertible element Φ ∈ H ⊗ (the associator), subject to:( ε ⊗ id ) △ ( h ) = h = ( id ⊗ ε ) △ ( h ) , ( id ⊗ △ ) △ ( h ) = Φ · ( △ ⊗ id ) △ ( h ) · Φ − , ( id ⊗ id ⊗ △ )(Φ) · ( △ ⊗ id ⊗ id )(Φ) = (1 ⊗ Φ) · ( id ⊗ △ ⊗ id )(Φ) · (Φ ⊗ , ( id ⊗ ε ⊗ id )(Φ) = 1for all h ∈ H . Write Φ = Φ ⊗ Φ ⊗ Φ and Φ − = Φ ⊗ Φ ⊗ Φ . A quasi-Hopf algebra H = ( H, △ , ε, Φ , S, α, β ) is a quasi-bialgebra ( H, △ , ε, Φ) with an antipode (
S, α, β ) , where α, β ∈ H and S : H → H is an angebra anti-homomorphism satisfying P S ( a ) αa = ε ( a ) α, P a βS ( a ) = ε ( a ) β, Φ βS (Φ ) α Φ = 1 , S (Φ ) α Φ βS (Φ ) = 1for all a ∈ H. Here we use Sweedler’s notation △ ( a ) = P a ⊗ a . Definition 2.1.
A twist for a quasi-Hopf algebra H is an invertible element J ∈ H ⊗ H satisfying ( ε ⊗ id )( J ) = ( id ⊗ ε )( J ) = 1 . YUPING YANG AND YINHUO ZHANG
Suppose that J = P i f i ⊗ h i is a twist of H with inverse J − = P i f i ⊗ h i . Write(2.1) α J = X i S ( f i ) αg i , β J = X i f i βS ( g i ) . According to [8], if β J is invertible then one can define a new quasi-Hopf algebra structure H J = ( H, △ J , ε, Φ J , S J , β J α J ,
1) on the algebra H, where △ J ( h ) = J △ ( h ) J − , h ∈ H, (2.2) Φ J = (1 ⊗ J )( id ⊗ △ )( J )Φ( △ ⊗ id )( J − )( J ⊗ − (2.3) S J ( h ) = β J S ( h ) β J − , h ∈ H. (2.4)Two quasi-Hopf algebras H and H ′ are said to be twist equivalent if H ′ ∼ = H J for some twist J of H. Definition 2.2.
A quasi-Hopf algebra H is genuine if H is not twist (or gauge) equivalentto any Hopf algebra. The following theorem is useful in Section 5.
Theorem 2.3. [27, Theorem 2.2]
Let H and B be two finite-dimensional quasi-Hopf algebras.Then the two module categories H -mod and B -mod are tensor equivalent if and only if H isequal to B J for some twist J of B. Let H = ( H, △ , ε, Φ , S, α, β ) be a quasi-Hopf algebra. If H = ⊕ i ≥ H [ i ] is a graded algebrasuch that ( H [0] , ε, Φ , S, α, β ) is a quasi-Hopf subalgebra, and I = ⊕ i ≥ H [ i ] is the Jacobsonradical of H and I k = ⊕ i ≥ k H [ i ] for each k ≥ , then we call H a radically graded quasi-Hopfalgebra. Suppose that H = ( H, △ , ε, Φ , S, α, β ) is a quasi-Hopf algebra, I is the Jacobsonradical of H. If I is a quasi-Hopf ideal of H , i.e., △ ( I ) ⊂ H ⊗ I + I ⊗ H , S ( I ) = I and ε ( I ) = 0 , then we can construct a radically graded quasi-Hopf algebra associated to H. Let H [0] = H/I and π : H → H [0] is the canonical projection. Define H [ k ] = I k /I k +1 for k ≥ . then the graded algebra gr ( H ) = ⊕ i ≥ H [ i ] has a natural quasi-Hopf algebra structure,with the associator π ⊗ π ⊗ π (Φ), and the antipode ( π ◦ S, π ( α ) , π ( β )) . For radically gradedquasi-Hopf algebras, we have the following useful lemma.
Lemma 2.4. [10, Lemma 2.1]
Let H = ⊕ i ≥ H i be a radically graded quasi-Hopf algebra.Then H is generated by H [0] and H [1] . Datum of Cartan type, root system and Wyle group.
For a finite group G, by b G we mean the character group of G. We give the definition of a datum of Cartan type accordingto [4].
Definition 2.5.
A datum of Cartan type (2.5) D = D ( G, ( h i ) ≤ i ≤ θ , ( χ i ) ≤ i ≤ θ , A ) consists of an abelian group G, elements h i ∈ G, characters χ i ∈ b G, ≤ i ≤ θ, and ageneralized Cartan matrix A = ( a ij ) ≤ i,j ≤ θ of size θ satisfying (2.6) q ij q ji = q a ij ii , where q ij = χ j ( h i ) , for all 1 ≤ i, j ≤ θ. We call θ the rank of D . A datum of Cartan type D is called finite Cartan type if the associatedCartan matrix A is finite; D is said to be connected if A is a connected Cartan matrix. INITE-DIMENSIONAL QUASI-HOPF ALGEBRAS OF CARTAN TYPE 5
Fix a datum D = D ( G, ( h i ) ≤ i ≤ n , ( χ i ) ≤ i ≤ θ , A ) of finite Cartan type. Let { α i | ≤ i ≤ θ } be the set of free generators of Z θ and s i : Z θ → Z θ the reflection s i ( α j ) = α j − a ij α i for 1 ≤ i, j ≤ θ. The Weyl group W of A is generated by { s i | ≤ i ≤ θ } and the rootsystem R = ∪ θi =1 W ( α i ) . Let R + be the set of positive roots with respect to the simple roots { α i | ≤ i ≤ θ } . For each α = P θi =1 k i α i ∈ Z θ , denote by ht ( α ) = P θi =1 k i , the height of α, and h α = h k h k · · · h k θ θ , (2.7) χ α = χ k χ k · · · χ k θ θ . (2.8)If α = P θi =1 k i α i ∈ R + , it is obvious that k i ≥ , ≤ i ≤ θ, and ht ( α ) > . For a datum ofCartan type D , we can define a Yetter-Drinfeld module V ( D ) in GG YD by(2.9) V ( D ) = θ X i =1 V χ i h i where V χ i h i = k { X i } is the 1-dimensional Yetter-Drinfeld module such that the module andthe comodule structures are given by(2.10) δ ( X i ) = X i ⊗ h i , g ⊲ X i = χ i ( h ) X i for all g ∈ G. A basis { X , · · · , X θ } of Yetter-Drinfeld module V ( D ) satisfying (2.10) is calleda canonical basis. It is well-known that GG YD is a braided tensor category. The naturalbraiding on V ( D ) is given by(2.11) c V,V : V ⊗ V −→ V ⊗ V, X i ⊗ X j → q ij X j ⊗ X i , ≤ i, j ≤ θ. Braided Hopf algebras.
Let (
V, c ) be a braided vector space with a basis { X , · · · , X n } such that c ( X i ⊗ X j ) = q ij X j ⊗ X i , q ij ∈ k , ≤ i, j ≤ n. Then we call (
V, c ) a braided vector space of diagonal type, { X , · · · , X n } a canonical basisof V, and ( q ij ) ≤ i,j ≤ n the braiding constants of V. Moreover, if q ij q ji = q a ij ii = q a ji jj , ≤ i, j ≤ n, where A = ( a ij ) ≤ i,j ≤ n is a Cartan matrix, then ( V, c ) is called braided vector space ofCartan type . For a datum of Cartan type D , it is obvious that V ( D ) is a braided vectorspace of Cartan type.Note that the braiding matrix ( q ij ) ≤ i,j ≤ θ of ( V, c ) defines a braided commutator on T ( V ) asfollows:(2.12) [ X, Y ] c = XY − ( Y ≤ k ≤ s, ≤ l ≤ t q x k y l i k j l ) Y X, where X = X x i X x i · · · X x s i s and Y = Y y j Y y j · · · Y y s j s . The braided adjoint action of an element X ∈ T ( V ) is defined by(2.13) ad c ( X )( Y ) = [ X, Y ] c for any Y ∈ T ( V ) . In the rest of this subsection, we let D = D ( G, ( h i ) ≤ i ≤ n , ( χ i ) ≤ i ≤ θ , A ) be a connected datumof finite Cartan type. In addition, we assume for 1 ≤ i ≤ θ,q ii has odd order , (2.14) the order of q ii is prime to 3 , if A is of type G , (2.15) YUPING YANG AND YINHUO ZHANG where q ij = χ j ( g i ) for 1 ≤ i, j ≤ θ. With these assumptions, we have the following:
Lemma 2.6. [4, Lemma 2.3]
There exists a root of unit q of odd order and integers d i ∈{ , , } , ≤ i ≤ θ, such that for ≤ i, j ≤ θ,q ii = q d i , d i a ij = d j a ji . Moreover, if A is of type G . Then the order of q is prime to . An immediate consequence of Lemma 2.6 is that the elements q ii , ≤ i ≤ θ, have the sameorder, hence we define(2.16) N = | q ii | , ≤ i ≤ θ. Let V = V ( D ) and { X , · · · , X θ } a canonical basis of V . Then the tensor algebra T ( V ) is abraided Hopf algebra in GG YD with comultiplication determined by △ ( X i ) = X i ⊗ ⊗ X i , ≤ i ≤ θ. Since ( ad c X i ) − a ij ( X j ) , ≤ i = j ≤ θ, are primitive elements in T ( V ), they generate abraided Hopf idea of T ( V ), denoted I . So we have a quotient braided Hopf algebra: R ( D ) = T ( V ) /I in GG YD . For convenience, we still denote by X i , 1 ≤ i ≤ θ , the image of the element X i in R ( D ).Now let w = s i s i · · · s i P be a fixed reduced presentation of the longest element of W interms of simple reflections. Then(2.17) β l = s i · · · s i l − ( α i l ) | ≤ l ≤ P is a convex order of positive roots. The root vectors { X α | α ∈ R + } can be defined as iter-ated braided commutators of the elements X , · · · , X θ with respect to the braiding given by( q ij ) ≤ i,j ≤ n such that X α i = X i , ≤ i ≤ θ, see [3, 4, 23] for detailed definition. Denote by K ( D ) the subalgebra of R ( D ) generated by the elements Y l = X Nβ l , ≤ l ≤ P. The followingdescription of K ( D ) comes from [4]. Theorem 2.7. [4, Theorem 2.6](1)
The elements X a β X a β · · · X a P β P , a , · · · a P ≥ , form a basis of R ( D ) . (2) K ( D ) is a braided Hopf subalgebra of R ( D ) . (2) For all α, β ∈ R + , [ X α , X Nβ ] C = 0 , that is, X α X Nβ − χ Nβ ( g α ) X Nβ X α = 0 . Let e l = ( δ kl ) ≤ k ≤ P ∈ N P , where δ kl is the Kronecker sign. For each a = ( a , a , · · · , a P ) ∈ N P , define Y a = Y a Y a · · · Y a P p , (2.18) h a = h Na β h Na β · h Na P β P , (2.19) a = a β + a β + · · · + a P β P . (2.20)Let △ R ( D ) be the comultiplication of R ( D ) , then we have the following lemma. INITE-DIMENSIONAL QUASI-HOPF ALGEBRAS OF CARTAN TYPE 7
Lemma 2.8. [4, Lemma 2.8]
For any nonzero a ∈ N P , there are uniquely determined scalars t ab,c ∈ k , = b, c ∈ N P , such that (2.21) △ R ( D ) ( Y a ) = Y a ⊗ ⊗ Y a + X b,c =0 ,b + c = a t ab,c Y b ⊗ Y c . Definition 2.9.
Let ( µ a ) a ∈ N P be a family of elements in k such that for all a, h a = 1 implies µ a = 0 . Then we can define u a ∈ k G inductively on ht ( a ) by (2.22) u a = µ a (1 − h a ) + X b,c =0 ,b + c = a t ab,c µ b u c . Proposition 2.10.
Let ( µ l ) ≤ l ≤ P be a family of elements in k such that: g Nβ l = 1 or χ Nβ l = ε implies µ l = 0 . Then there exists a unique family ( µ a ) a ∈ N P satisfying µ e l = µ l for ≤ l ≤ P such that (2.23) u a = u a − e l u e l , if a = ( a , · · · , a l , , · · · , , a l ≥ , ≤ l ≤ P, and a = e l . Proof.
It follows from [4, Lemma 2.10, Theorem 2.13] (cid:3)
Definition 2.11.
Suppose that µ = ( µ l ) ≤ l ≤ P is a family of elements in k satisfying thecondition of Proposition 2.10. For a root α ∈ R + , then there exists ≤ l ≤ P such that α = β l and define u α ( µ ) = u e l . Andruskiewitsch-Schneider’s Hopf algebras.
Fix a datum of finite Cartan type D = D ( G, ( h i ) ≤ i ≤ θ , ( χ i ) ≤ i ≤ θ , A ) , where A may not be a connected Cartan matrix. For1 ≤ i, j ≤ θ we define i ∼ j if i and j are in the same connected component of the Dynkindiagram of Cartan matrix A, and i, j are said to be connected if i ∼ j. Let Ω = { I , · · · , I t } be the set of the connected components of I = { , , · · · , θ } . Here we also assume that theconditions (2.14)-(2.15) hold for each connected component of I. For J ∈ Ω , let R J be theroot system of A J = ( a ij ) i,j ∈ J and N J the corresponding number defined by (2.16). Let R + J be the set of positive roots of A J with respect to the simple roots { α i | i ∈ J } . The followingpartitions are obvious: R = [ J ∈ Ω R J , R + = [ J ∈ Ω R + J . Definition 2.12.
A family λ = ( λ ij ) ≤ i,j ≤ n,i ≁ j of elements in k is called a family of linkingparameters for D if h i h j = 1 or χ i χ j = ε implies λ ij = 0 for all ≤ i, j ≤ θ, i ≁ j. Vertices ≤ i, j ≤ θ are called linkable if i ≁ j, h i h j = 1 and χ i χ j = ε. Definition 2.13.
A family µ = ( µ α ) α ∈ R + of elements in k is called a family of root vectorparameters for D if h N J α = 1 or χ N J α = ε implies µ α = 0 for all α ∈ R + J , J ∈ Ω . With these definitions and notations, we can give one of the main results of [4].
Theorem 2.14. [4, Theorem 4.5]
Let D = D ( G, ( h i ) ≤ i ≤ θ , ( χ i ) ≤ i ≤ θ , A ) be a datum of finiteCartan type such that each connected component of I = { , , · · · , θ } satisfies the conditions (2.14) - (2.15) . Let λ and µ be a family of linking parameters and a family of root vector param-eters for D respectively. Then we have a finite-dimensional pointed Hopf algebra u ( D , λ, µ ) YUPING YANG AND YINHUO ZHANG generated by the group G and the skew-primitive elements { X i | ≤ i ≤ θ } subject to thefollowing relations: (Action of the group) gX i g − = χ i ( g ) X i , for all 1 ≤ i ≤ θ, g ∈ G, (Serre relations) ad c ( X i ) − a ij ( X j ) = 0 , for all i = j, i ∼ j, (Linking relations) ad c ( X i )( X j ) = λ ij (1 − h i h j ) , for all i < j, i ≁ j, (Root vector relations) X N J α = u α ( µ ) , for all α ∈ R + J , J ∈ Ω . The coalgebra structure is determined by △ ( X i ) = X i ⊗ h i ⊗ X i , △ ( g ) = g ⊗ g, for all 1 ≤ i ≤ θ, g ∈ G. The Hopf algebras constructed in Theorem 2.14 can be viewed as an axiomatic descriptionof generalized the small quantum groups, and Lusztig’s small quantum groups are specialexamples of such Hopf algebras. Another main result of [4] says that any finite-dimensionalpointed Hopf algebra over an abelian group G with dimension not divided by 2 , , , , is ofthe form u ( D , λ, µ ) for some D , λ, µ. In the sequel, we call such a Hopf algebra u ( D , λ, µ ) anAndruskiewitsch-Schneider Hopf algebra (or AS-Hopf algebra for short) following [12].2.5. Normalized -cocycles on finite groups. Let G be an arbitrary abelian group. So G ∼ = Z m × · · · × Z m n with m j ∈ N for 1 ≤ j ≤ n. A function φ : G × G × G k ∗ is called a3-cocycle on G if(2.24) φ ( ef, g, h ) φ ( e, f, gh ) = φ ( e, f, g ) φ ( e, f g, h ) φ ( f, g, h )for all e, f, g, h ∈ G . A 3-cocycle is called normalized if φ ( f, , g ) = 1 . Denote by A the set ofall sequences(2.25) ( c , . . . , c l , . . . , c n , c , . . . , c ij , . . . , c n − ,n , c , . . . , c rst , . . . , c n − ,n − ,n )such that 0 ≤ c l < m l , ≤ c ij < ( m i , m j ) , ≤ c rst < ( m r , m s , m t ) for 1 ≤ l ≤ n, ≤ i To see if the map is well-defined, we just need to show that σ ( φ ) is a normalized 3-cocycle on G for each φ ∈ Z ( G, k ∗ ) . Indeed, for any e, f, g, h ∈ G and φ ∈ Z ( G, k ∗ ) , we have ∂ ( σ ( φ ))( e, f, g, h ) = σ ( φ )( e, f, g ) σ ( φ )( e, f g, h ) σ ( φ )( f, g, h ) σ ( φ )( ef, g, h ) σ ( φ )( e, f, gh )= φ ( g, f, e ) φ ( h, f g, e ) φ ( h, g, f ) φ ( h, g, ef ) φ ( gh, f, e )= ∂ ( φ )( h, g, f, e )= 1 . This implies that σ ( φ ) is a 3-cocycle on G . The fact that σ ( φ ) is normalized follows from theequation: σ ( φ )( f, , g ) = φ ( g, , f ) = 1 , for all f, g ∈ G. It is obvious that σ is bijective since σ = id. Moreover, we have the following. Lemma 2.16. The map σ induces an involution of H ( G, k ∗ ) . Proof. It suffices show that σ preserves 3-coboundaries. Suppose that φ is a 3-coboundary.There exists a 2-cochain J : G × G → k ∗ such that φ = ∂ ( J ) . Define J ′ : G × G → k ∗ by(2.28) J ′ ( f, g ) = J − ( g, f ) , for all f, g ∈ G. Then we have: σ ( φ )( f, g, h ) = J ( h, g ) J ( gh, f ) J ( g, f ) J ( h, f g ) = J ′− ( g, h ) J ′− ( f, gh ) J ′− ( f, g ) J ′− ( f g, h ) = J ′ ( f, g ) J ′ ( f g, h ) J ′ ( g, h ) J ′ ( f, gh ) = ∂ ( J ′ )( f, g, h )for all f, g, h ∈ G. This implies that σ preserves 3-coboundaries. Thus, we have completedthe proof. (cid:3) For each c ∈ A , define φ c : G × G × G −→ k ∗ , [ g i · · · g i n n , g j · · · g j n n , g k · · · g k n n ] (2.29) n Y l =1 ζ c l k l [ il + jlml ] m l Y ≤ s Proposition 2.17. { φ c | c ∈ A} forms a complete set of representatives of the normalized -cocycles on G up to -cohomology. The original definition of an abelian cocycle was given in [9], and an equivalent definition viathe twisted quantum double appeared in [26]. Let φ be a 3-cocycle on G , and D φ ( G ) thetwisted quantum double of ( k G, φ ) (see [19] for the detail). φ is called an abelian -cocycle if D φ ( G ) is commutative. Using Proposition 2.17, one can easily determine all the abelian3-cocycles on G . A straightforward computation shows that φ c is an abelian 3-cocycle if andonly if c rst = 0 for all 1 ≤ r < s < t ≤ n . We point out that the twisted Yetter-Drinfeldcategory GG YD φ c is a pointed fusion category in case the 3-cocycle φ c is abelian. Denote by Vec G the category of G -graded vector spaces. Let ω be a 3-cocycle on G . Wedefine a tensor category Vec ωG . As a category, Vec ωG = Vec G . The tensor product V ⊗ W oftwo graded modules is endowed with the canonical grading:( V ⊗ W ) g = ⊕ ef = g V e ⊗ V f , ∀ g ∈ G. The associator a is given by a U,V,W : ( U ⊗ V ) ⊗ W −→ U ⊗ ( V ⊗ W )( x ⊗ y ) ⊗ z ω − ( e, f, g ) x ⊗ ( y ⊗ z ) , where x ∈ U e , y ∈ V f , z ∈ W g . According to [14, Proposition 2.6.1], Vec ωG is tensor equivalentto the representation category of some Hopf algebra if and only if φ is a 3-coboundary on G .3. Finite-dimensional Quasi-Hopf algebras General setup. In this subsection, we fix some notations on abelian groups, whichwill be used throughout this paper. Suppose that G is a finite abelian group, say, G = h g i × · · · × h g n i such that | g i | = m i for 1 ≤ i ≤ n. Let b G be the character group of G over k . For each g = Q ni g α i i , define a character χ g : G → k ∗ by(3.1) χ g ( h ) = n Y i ζ α i β i m i , where h = Q ni g β i i ∈ G. From the definition of χ g , it is obvious that χ − g ( h ) = χ g − ( h ) = χ g ( h − ) . So χ : G −→ b G, g → χ g is an group isomorphism. Let k [ G ] be the group algebra of G over field k . One can verify that(3.2) { g = 1 | G | X h ∈ G χ g ( h ) h | g ∈ G } forms a complete set of the orthogonal primitive idempotents of the algebra k [ G ] . Lemma 3.1. For g, h ∈ G, we have g h = h g = χ − g ( h )1 g . Proof. g h = | G | P f ∈ G χ g ( f ) f h = | G | P f ∈ G χ g ( f h ) χ g ( h − ) f h = χ − g ( h )1 g . (cid:3) Now let G be an abelian group. We can define a bigger abelian group G associated to G inthe following way: assume(3.3) G = h g i × · · · × h g n i , | g i | = m i , ≤ i ≤ n ;define the group G as follows:(3.4) G = h g i × · · · × h g n i , | g i | = m i = m i , ≤ i ≤ n. It is obvious that there is a group injection:(3.5) ι : G → G , ι ( g i ) = g m i i , ≤ i ≤ n. Let ζ m i be an m i -th primitive root of unity such that ζ m i m i = ζ m i for 1 ≤ i ≤ n. For each g = Q ni =1 g s i i ∈ G , define χ g : G → k ∗ by χ g ( h ) = n Y i =1 ζ s i t i m i , h = n Y i =1 g t i i . INITE-DIMENSIONAL QUASI-HOPF ALGEBRAS OF CARTAN TYPE 11 Similar to (3.2), one has a complete set { = | G | P h ∈ G χ g ( h ) h |{ | g ∈ G } ∈ G } oforthogonal primitive idempotents of the algebra k [ G ] . We have the following equality. Lemma 3.2. The following holds for all ≤ s i ≤ m i − , ≤ i ≤ n : (3.6) X ≤ k j ≤ m j − , ≤ j ≤ n ( Q ni =1 g miki + sii ) = 1 ( Q ni =1 g sii ) . Proof. By definition we have X ≤ k j ≤ m j − , ≤ j ≤ n ( Q ni =1 g miki + sii ) = 1 | G | X ≤ k j ≤ m j − , ≤ j ≤ n X h ∈ G χ ( Q ni =1 g miki + sii ) ( h ) h. Suppose h = g r g r · · · g r n n . Then we have the equation: X ≤ k j ≤ m j − , ≤ j ≤ n χ ( Q ni =1 g miki + sii ) ( h )= X ≤ k j ≤ m j − , ≤ j ≤ n n Y i ζ ( m i k i + s i ) r i m i = n Y i =1 ( X ≤ k i ≤ m i − ζ ( m i k i + s i ) r i m i ) . Note that P ≤ k l ≤ m l − ζ m l k l r l m l = 0 if r l = tm l for some integer 0 ≤ t ≤ m l − 1. Hence X ≤ k j ≤ m j − , ≤ j ≤ n χ ( Q ni =1 g miki + sii ) ( h ) = 0if and only if r i = t i m i for 0 ≤ t i ≤ m i − , ≤ i ≤ n, i.e., h is contained in the subgroup G. If r i = t i m i for 1 ≤ i ≤ n, then h = g t m g t m · · · g t n m n n and we have: X ≤ k j ≤ m j − , ≤ j ≤ n χ ( Q ni =1 g miki + sii ) ( h ) = n Y i =1 ( X ≤ k i ≤ m i − ζ ( m i k i + s i ) t i m i m i )= n Y i =1 ( X ≤ k i ≤ m i − ζ s i t i m i m i )= n Y i =1 ( m i ζ s i t i m i )= | G | n Y i =1 ζ s i t i m i . It follows that X ≤ k j ≤ m j − , ≤ j ≤ n ( Q ni =1 g miki + sii ) = 1 | G | X ≤ k j ≤ m j − , ≤ j ≤ n X h ∈ G χ ( Q ni =1 g miki + sii ) ( h ) h. = | G || G | X ≤ t j ≤ m j − , ≤ j ≤ n n Y i =1 ζ s i t i m i ( n Y i =1 g t i m i i )= 1 | G | X ≤ t j ≤ m j − , ≤ j ≤ n χ ( Q ni =1 g sii ) ( n Y i =1 g t i i )( n Y i =1 g t i i ) = 1 | G | X g ∈ G χ ( Q ni =1 g sii ) ( g ) g = 1 ( Q ni =1 g sii ) . Thus, the claimed equality holds. (cid:3) Finite dimensional quasi-Hopf algebras and Cartan matrices. Keep the nota-tions of the last subsection. Let D = D ( G , ( h i ) ≤ i ≤ θ , ( χ i ) ≤ i ≤ θ , A ) be a datum of finiteCartan type, where A = ( a ij ) ≤ i,j ≤ θ is a finite Cartan matrix. Let Ω be the set of the con-nected components of I = { , , · · · , θ } . Denote by ( s ij ) ≤ i ≤ θ, ≤ j ≤ n and ( r ij ) ≤ i ≤ θ, ≤ j ≤ n thetwo families of integers satisfying: h i = n Y j =1 g s ij j , χ i ( g j ) = ζ r ij m j , (3.7) 0 ≤ s ij , r ij < m j for all 1 ≤ i ≤ θ, ≤ j ≤ n. (3.8)It is obvious that ( s ij ) ≤ i ≤ θ, ≤ j ≤ n and ( r ij ) ≤ i ≤ θ, ≤ j ≤ n are uniquely determined by D . LetΓ( D ) be the subset of A such that for each c ∈ Γ( D ) , c rst = 0 for all 0 ≤ r < s < t ≤ n and s ij ≡ c j r ij mod m j , ≤ i ≤ θ, ≤ j ≤ n, (3.9) c ij r lj ≡ m j , ≤ l ≤ θ, ≤ i < j ≤ n, (3.10) c ij m i ≡ m j , ≤ i < j ≤ n. (3.11)For each c ∈ Γ( D ) , define on G the functions Θ , Ψ l , Υ and F i as follows:Θ l ( g ) = n Y i =1 ζ − t i s li m i , g = n Y i =1 g t i i ∈ G, ≤ l ≤ θ. (3.12) Ψ l ( f, h ) = n Y i =1 ζ c i q i ϕ li ( f ) m i Y ≤ i Definition 3.3. A family of linking parameters λ = ( λ ij ) ≤ i,j ≤ θ for D is said to be modifiedif (3.17) h i h j / ∈ G implies λ ij = 0 for all 1 ≤ i < j ≤ θ, i ≁ j. A family of root vector parameters µ = ( λ α ) α ∈ R for D is said to be modified if (3.18) h N J α / ∈ G implies µ α = 0 for all α ∈ R + J , J ∈ Ω . INITE-DIMENSIONAL QUASI-HOPF ALGEBRAS OF CARTAN TYPE 13 Now we can give the main result of this paper. Theorem 3.4. Let λ = ( λ ij ) ≤ i,j ≤ θ and µ = ( λ α ) α ∈ R be two families of modified link-ing parameters and root vector parameters respectively for a datum of Cartan type D = D ( G , ( h i ) ≤ i ≤ θ , ( χ i ) ≤ i ≤ θ , A ) , and c a nonzero element in Γ( D ) . Then we have a finite-dimensional quasi-Hopf algebra u ( D , λ, µ, Φ c ) generated by G and { X , · · · , X θ } subject tothe relations: gX i g − = χ i ( g ) X i , for all 1 ≤ i ≤ θ, g ∈ G, (3.19) ad C ( X i ) − a ij ( X j ) = 0 , for all i = j, i ∼ j, (3.20) ad C ( X i )( X j ) = λ ij (1 − h i h j ) , for all i < j, i ≁ j, (3.21) X N J α = u α ( µ ) , for all α ∈ R + J , J ∈ Ω . (3.22) The coalgebra structure of u ( D , λ, µ, Φ c ) is given by (3.23) △ ( X l ) = X f,g ∈ G Ψ l ( f, g ) X l f ⊗ g + X f ∈ G Θ l ( f )1 f ⊗ X l , △ ( g ) = g ⊗ g, ≤ l ≤ θ, g ∈ G. The associator of u ( D , λ, µ, Φ c ) is determined by (3.24) Φ c = X f,g,h ∈ G φ c ( f, g, h )(1 f ⊗ g ⊗ h ) . The antipode ( S , α, is defined by (3.25) α = X g ∈ G Υ( g )1 g , S ( X i ) = X g ∈ G F i ( g ) X i g . The proof of Theorem 3.4 will be delivered in the next subsection. Since the quasi-Hopfalgebra u ( D , λ, µ, Φ c ) is generated by the abelian group G and the braided vector space ofCartan type V = k { X , · · · , X θ } , we shall call it a quasi-Hopf algebra of Cartan type in thesequel. We will say that u ( D , λ, µ, Φ c ) is associated to the Cartan matrix A , and call θ the rank of u ( D , λ, µ, Φ c ) , as well as the rank of A . Remark 3.5. The relations (3.20)-(3.23) for u ( D , λ, µ, Φ c ) are similar to those of AS-Hopf algebras u ( D , λ, µ ) , but the generators of the two algebras are essential different. Infact, u ( D , λ, µ ) is generated by G and { X , · · · , X θ } , and u ( D , λ, µ, Φ c ) is the subalgebra of u ( D , λ, µ ) generated by subgroup G and { X , · · · , X θ } . Moreover, we will prove that u ( D , λ, µ, Φ c ) is a quasi-Hopf subalgebra of u ( D , λ, µ ) J for some twist J of u ( D , λ, µ ) . It is obvious that u ( D , λ, µ, Φ c ) is radically graded if and only if u α ( µ ) = 0 , λ ij (1 − h i h j ) = 0 , for all α ∈ R + , ≤ i, j ≤ θ, and the radical is the ideal generated by X , · · · , X θ . Notethat if µ = 0 , then u α ( µ ) = 0 by (2.19). It follows the definitions of a familly of modifiedlinking parameters and a family of modified root vector parameters thta the quasi-hopf algebra u ( D , λ, µ, Φ c ) is radically graded if and only if both λ = 0 and µ = 0. Remark 3.6. (1). Let G be a cyclic group, H the AS-Hopf algebra u ( D , , . The quasi-Hopf algebra u ( D , , , Φ c ) is nothing but the basic quasi-Hopf algebra A ( H, c ) (overthe cyclic group G ) classified in [5] . (2). Suppose that A is a connected Cartan Matrix of rank n , and g is the simple Liealgebra associated to A . Let G = Z nm for some positive odd integer m, which is notdivisible by if A is of type G . Let h i = Q nj =1 g a ij j , χ i ( g l ) = ζ δ il m for ≤ i, l ≤ n. Let c i = a ii , c j,k = a jk for ≤ i ≤ n, ≤ j < k ≤ n. Then u ( D , , , Φ c ) is the halfsmall quasi-quantum groups A q ( g ) given in [11] , where q = ζ m . (2). Suppose that A is the diagonal Cartan matrix A × A × · · · A . Then the dual of thequasi-Hopf algebras u ( D , , , Φ c ) is a quasi-quantum linear space, see [21] . The proof of Theorem 3.4. Let H = u ( D , λ, µ ) be the AS-Hopf algebra given inTheorem 2.14, which is generated by G and { X , · · · , X θ } . By [4, Theorem 4.5], the groupof group-like elements of H is G . According to Subsection 3.1, we know that χ : G → b G isa group isomorphism. Let { η , · · · , η θ } be the set of elements in G such that χ η i = χ i for1 ≤ i ≤ θ. By (3.7), it is obvious that(3.26) η i = n Y j =1 g r ij j , ≤ i ≤ θ. Moreover we have the following. Lemma 3.7. (3.27) X i = X i η i , for all g ∈ G , ≤ i ≤ θ. Proof. Follows from the following equations: X i = 1 | G | X f ∈ G χ g ( f ) f X i = 1 | G | X f ∈ G χ g ( f ) χ η i ( f ) X i f = 1 | G | X i X f ∈ G χ g η i ( f ) f = X i η i . (cid:3) Given c ∈ Γ( D ) , we define:(3.28) J c : G × G → k ∗ ; ( g x · · · g x n n , g y · · · g y n n ) n Y l =1 ζ c l y l ( x ′ l − x l ) m l Y ≤ s It is obvious that J c is invertible with inverse J − c = P f , g ∈ G J − c ( f , g ) ⊗ . Nextwe verify that ( ε ⊗ id )( J c ) = ( id ⊗ ε )( J c ) = 1 holds. Suppose g = Q ni =1 g k i i for some0 ≤ k i ≤ m i , ≤ i ≤ n. Then the following equations hold: ε ( ) = 1 | G | ε ( X h ∈ G χ g ( h ) h ) = 1 | G | X h ∈ G χ g ( h )= 1 | G | X ≤ l j ≤ m j − , ≤ j ≤ n ( n Y i =1 ζ k i l i m i )= 1 | G | n Y i =1 ( X ≤ l i ≤ m i − ζ k i l i m i ) . It follows that ε ( ) = (cid:26) , if h = 1;1 , if h = 1.Hence, ( ε ⊗ id )( J c ) = P g ∈ G J c (1 , g ) = P g ∈ G 1g = 1 . Similarly, the equation: ( id ⊗ ε )( J c ) = 1 holds. (cid:3) INITE-DIMENSIONAL QUASI-HOPF ALGEBRAS OF CARTAN TYPE 15 Since H is a Hopf algebra, we can view it as a quasi-Hopf algebras with the trivial associatorΦ = 1 ⊗ ⊗ S, , H J c = ( H J c , △ J c , ε, Φ J c , S J c , β J c α J c , . The associator of H J c can beexplicitly described as follows: Lemma 3.9. (3.29) Φ J c = X f,g,h ∈ G φ c ( f, g, h )1 f ⊗ g ⊗ h Proof. First of all, we need to verify the comultiplication of the element for every g ∈ G :(3.30) △ ( ) = X fh = g 1f ⊗ . Indeed, we have: X fh = g 1f ⊗ = 1 | G | X fh = g X x ∈ G χ f ( x ) x ⊗ X y ∈ G χ h ( y ) y = 1 | G | X x , y ∈ G [ X fh = g χ f ( x ) χ h ( y ) x ⊗ y ]= 1 | G | X x ∈ G χ g ( x ) x ⊗ x = △ ( ) , where the third identity follows from the equation: X fh = g χ f ( x ) χ h ( y ) = (cid:26) , if x = y ; | G | χ g ( x ) , if x = y .Hence, it yields: Φ J c = (1 ⊗ J c )( id ⊗ △ )( J c )( △ ⊗ id )( J − c )( J c ⊗ − = X f , g , h ∈ G J c ( g , h ) J c ( f , gh ) J c ( f , g ) J c ( fg , h ) ⊗ ⊗ = X f , g , h ∈ G ∂ ( J c )( f , g , h ) ⊗ ⊗ . Now suppose f = Q ni =1 g x i m i + r i i , g = Q ni =1 g y i m i + s i i , h = Q ni =1 g z i m i + t i i for 0 ≤ x i , y i , z i , r i , s i , t i ≤ m i − , ≤ i ≤ n. Let f = Q ni =1 g r i i , g = Q ni =1 g s i i , h = Q ni =1 g t i i . We compute theelement ∂ ( J c )( f , g , h ): ∂ ( J c )( f , g , h ) = n Y i =1 ζ c i t i [ ri + simi ] m i m i Y ≤ j Since R + = ∪ J ∈ Ω R + J , it suffices to show u α ( µ ) ∈ k G for any α ∈ R + J with a fixed J ∈ Ω . Suppose that J = { i , · · · , i η } ⊂ I, and { α i , · · · , α i η } is the set of the simple rootscorresponding to the vertexes of J. Let w J = s j s j · · · s j PJ be the reduced presentation of thelongest element of the Weyl group W J in terms of simple reflections. Define(3.31) β j l = s j s j · · · s j l − i ( α j l ) for 1 ≤ l ≤ P J and(3.32) a = a β j i + a β j + · · · + a P J β j PJ , a ∈ N P J . We show that u a ∈ k G for each a ∈ N P J . Consequently, it leads to u α ( µ ) ∈ k G for all α ∈ R + J . We will prove it by induction on ht ( a ) . In case ht ( a ) = 1 , then a is a simple root contained in { α i , · · · , α i η } , say, α i k . By (2.22), wehave u a = µ a (1 − h a ) = h N J a ik . It follows from 3.18 that u a = µ a (1 − h a ) ∈ k G. Now assume that u a ∈ k G holds for all a ∈ N P J such that ht ( a ) < l . Let a ∈ N P J such that ht ( a ) = l. If a = β j s for some 1 ≤ s ≤ P J , then u a = µ a (1 − h a ) + X b,c =0 ,b + c = a t ab,c µ b u c , and h a = h N J β js . From 3.18 we see that the part µ a (1 − h a ) ∈ k G . The fact that second part P b,c =0 ,b + c = a t ab,c µ b u c belongs to k G follows from the induction assumption. If a = β j s forany 1 ≤ s ≤ P J , then a = ( a , a , · · · , a s , · · · , 0) with a s > ≤ s ≤ P J . Let e s = (0 , · · · , | {z } s , , · · · , . By Proposition 2.10, we have u a = u a − e s u e s . Since both heights of a − e s and e s are lessthan l , the elements u a − e s , u e s belong to k G by the induction assumption. This implies that u a ∈ k G. (cid:3) Now we denote by A ( H, c ) the subalgebra of H J c generated by G and { X , · · · , X θ } . Weare going to show that A ( H, c ) is the desired quasi-Hopf algebra if we choose an approriateelement c . We first describe the defining relations of the generators of A ( H, c ). INITE-DIMENSIONAL QUASI-HOPF ALGEBRAS OF CARTAN TYPE 17 Proposition 3.11. The algebra A ( H, c ) can be presented by the generators G and { X , · · · , X θ } and the following relations: gX i g − = χ i ( g ) X i , for all 1 ≤ i ≤ θ, g ∈ G, (3.33) ad C ( X i ) − a ij ( X j ) = 0 , for all i = j, i ∼ j, (3.34) ad C ( X i )( X j ) = λ ij (1 − h i h j ) , for all i < j, i ≁ j, (3.35) X N J α = u α ( µ ) , for all α ∈ R + J , J ∈ Ω . (3.36) Moreover, A ( H, c ) has a basis of the form X x β X x β · · · X x P β P g, g ∈ G, ≤ x i ≤ N J , β i ∈ R + J . Proof. Since the relations (3.34)-(3.35) hold in H for the group G and the generators X , · · · X θ ,they hold as well for the subgroup G and X , · · · , X θ . Relation (3.37) follows from Lemma3.10. For the relation (3.36), it is enought to show that the elements λ ij (1 − h i h j ) fall in k G .But this is true because of (3.17). The last part of the proposition follows from [4, Theorem3.3] and Relation (3.29). (cid:3) The algebra A ( H, c ) is apparently not a Hopf subalgebra of H . However, it is a quasi-Hopfsubalgebra of some twist of H . Proposition 3.12. A ( H, c ) is a quasi-Hopf subalgebra of H J c if and only if c ∈ Γ( D ) . Proof. ⇐ . First of all, we show that A ( H, c ) is closed under the comultiplication △ J c of H J c . It is obvious that △ J c ( g ) = J c ( g ⊗ g ) J − c = g ⊗ g for any g ∈ G ⊂ G since G is abelian. Itremains to show that △ J c ( X i ) ∈ A ( H, c ) ⊗ A ( H, c ) for 1 ≤ i ≤ θ. By Lemma 3.1 and 3.7, wehave △ J c ( X i ) = J c ( X i ⊗ h i ⊗ X i ) J − c = ( X f , g ∈ G J c ( f , g ) ⊗ )( X i ⊗ h i ⊗ X i )( X f , g ∈ G J − c ( f , g ) ⊗ )= X f , g ∈ G [ J c ( f η − i , g ) J c ( f , g ) X i ⊗ + J c ( f , g η − i ) J c ( f , g ) χ − f ( h i ) ⊗ X i ] . Suppose f = Q nj =1 g x j m j + k j j , g = Q nj =1 g y j m j + l j j for 0 ≤ x j , y j , k j , l j ≤ m j − , ≤ j ≤ n. Let( k j − r ij ) ′ be the remainder of ( k j − r ij ) divided by m j for 1 ≤ ≤ j ≤ n, and define(3.37) ψ ij = (cid:26) ( k j − r ij ) ′ − ( x j m j + k j − r ij ) , if x j m j + k j − r ij ≥ k j − r ij ) ′ − m j − ( x j m j + k j − r ij ) , if x j m j + k j − r ij < J c ( f η − i , g ) J c ( f , g ) = Q nj =1 ζ c j l j ψ ij m j Q ≤ s 1) is a antipode of A ( H, c ) . For all g ∈ G , we have S ( ) = S ( 1 | G | X h ∈ G χ g ( h ) h ) = 1 | G | X h ∈ G χ g ( h ) h − = − . So we obtain α J c = X f , g ∈ G J − c ( f , g ) − = X g ∈ G J − c ( g − , g ) , (3.41) β J c = X f , g ∈ G J c ( f , g ) 1f 1g − = X g ∈ G J c ( g , g − ) . (3.42) INITE-DIMENSIONAL QUASI-HOPF ALGEBRAS OF CARTAN TYPE 19 It is obvious that β J c is invertible with inverse P g ∈ G J − c ( g , g − ) , and we have: β J c α J c = X g ∈ G J c ( g , g − ) J c ( g − , g ) (3.43) = X ≤ k i ,l i ≤ m i − , ≤ i ≤ n Q ni =1 ζ c i l i k i m i m i Q ≤ s 1) is determined by (3.43) and (3.44), and the associator Φis given by (3.29). Therefore, we have proved the theorem. ✷ Examples of quasi-Hopf algebras of Cartan type. In this subsection, we will givesome examples of quasi-Hopf algebras of Cartan type. We make a convention that the comul-tiplications, the associators, and the antipodes of the quasi-Hopf algebras in those examplesbelow can be written in the forms as listed in (3.23)-(3.25), and hence will be omitted. Example 3.13. Basic quasi-Hopf algebras over cyclic groups. Let G = Z m = h g i .In this case, G = Z m = h g i , and G is identical to the subgroup h g m i of G , see (3.5) . Let D = D ( G , ( h i ) ≤ i ≤ θ , ( χ i ) ≤ i ≤ θ , A ) be a datum of finite Cartan type, where h i = g s i , χ i ( g ) = ζ r i m for some < s i , r i < m , ≤ i ≤ θ. Denote by H the AS-Hopf algebra u ( D , , , and let s bea number satisfying < s < m and sr i ≡ s i mod m for all ≤ i ≤ θ, and c = { s } . Then we geta quasi-Hopf algebra u ( D , , , Φ c ) = A ( H, s ) , see [5, Proposition 3.1.1] for the definition of A ( H, s ) . According to [5, Theorem 3.4.1] , any nonsemisimple, genuine basic graded quasi-Hopfalgebra over a cyclic group with dimension not divisible by and must be twist equivalentto some u ( D , , , Φ c ) .Let m = p and D = D ( G , g , χ, A ) such that χ ( g ) = ζ p . Then it is obvious that s = 1 , c = { } , and we get a quasi-Hopf algebra u ( D , , , Φ c ) generated by G and X with therelations gXg − = ζ p X, X p = 0 . According to the classification of pointed Hopf algebras of dimension p in [2] , we know that u ( D , , , Φ c ) does not admit a pointed Hopf algebra structure. Next we will construct a few more non-radically graded quasi-Hopf algebras of rank 2 . Firstwe give an example of a quasi-Hopf algebra associated to A × A , and then present someexamples of quasi-Hopf algebras associated to A , B , G . Example 3.14. The quasi-version of u q ( sl ) . Let N > and d be two positive oddnumbers, and G = Z m = h g i , G = Z m = h g i , where m = N d. As usual, G is viewed as thesubgroup of G . Let D = D ( G , ( h , h ) , ( χ , χ ) , A × A ) , where h = h = g m , χ ( g ) = ζ dm , χ ( g ) = ζ − dm . It is easy to verify that D is a datum of Cartan type. Since Γ( D ) is the set of numbers ≤ c ≤ m − satisfying m ≡ cd mod m,m ≡ − cd mod m. Both equations are equivalent with N | c since m = N d and N is odd.In this case, it is clear that Γ( D ) = { c = kN | ≤ k < d } . Let q = ζ mdm . By H c we denote thealgebra generated by g, X , X subject to the relations as follows: gX g − = q X , gX g − = q − X ,X X − q − X X = λ (1 − g ) ,X N = X N = 0 . INITE-DIMENSIONAL QUASI-HOPF ALGEBRAS OF CARTAN TYPE 21 Let E = X , F = X g − , and λ = q − − q, then we can see that H c is generated by g, E, F satisfying the relations: gEg − = q X , qF g − = q − X ,EF − F E = g − g − q − q − ,E N = F N = 0 . When d = 1 , we have c = 0 and H c ∼ = u q ( sl ) . When d > , we know that Γ( D ) has nonzeroelements. If c = 0 , we have H c ∼ = u ( D ′ , λ, , where D ′ = D ( G, ( g, g ) , ( χ ′ , χ ′ ) , A × A ) and χ ′ ( g ) = q , χ ( g ) = q − . If c ∈ Γ( D ) is nonzero, then we have H c = u ( D , λ, , Φ c ) . Becauseof this fact, u ( D , λ, , Φ c ) can be viewed as the quasi-version of u q ( sl ) , and is called a smallquasi-quantum group . More small quasi-quantum groups will be studied in Section 5. The above example provides us some non-radically graded quasi-Hopf algebras associated to A × A . We point out that there is a similar notion of a quasi-version of u q ( sl ) in [24], whereLiu defined a quasi-Hopf analogue of u q ( sl ) as a quantum double of a quasi-Hopf algebraassociated to A . It is obvious that these two definitions are different, since the dimensionof a quasi-version of u q ( sl ) is not a square in general. Hence, it should not be a quantumdouble. In order to construct non-radically graded examples of type A , B , G , we need thefollowing well-known proposition from number theory. Proposition 3.15. Let a, b, n be nonzero integers. Then the equation ax ≡ b mod n hassolutions if and only if ( a, n ) | b. Moreover, if there exists a solution, then it is unique up tomodulo n ( a,n ) . Example 3.16. Quasi-Hopf algebras associated to A , B , G . Let A be a Cartan matrixof type A , B or G . Suppose that m, n, d are positive odd numbers such that ( m, n ) =( m, d ) = ( n, d ) = 1 . In addition, in case A is of type G , we will assume that the threenumbers m, n and d are not divisible by . Let G = h g i × h g i × h g i × h g i , where | g | = md, | g | = nd, | g | = md , and | g | = nd . The group G and the generators g i , ≤ i ≤ aredefined in a similar manner as before. Let a, b, k be the numbers listed in the Table 1. Define D = D ( G , ( h , h ) , ( χ , χ ) , A ) , where h = ( g g ) mn , h = ( g g g g ) mn χ ( g ) = ζ ad , χ ( g ) = ζ ad , χ ( g ) = ζ bd , χ ( g ) = ζ bd ,χ ( g ) = ζ ad , χ ( g ) = ζ ad , χ ( g ) = ζ d , χ ( g ) = ζ kd − d . Let A = ( a ij ) ≤ i,j ≤ such that a ≤ a . Then one can easily show that χ ( h ) χ ( h ) = χ ( h ) a = χ ( h ) a . Hence D is a datum of Cartan type.Next we will show that Γ( D ) contains nonzero elements. By definition, Γ( D ) is the set offamilies c = ( c i , c st ) ≤ i ≤ , ≤ s 1) mod nd . (3.48) Table 1. a, b, k associated to A , B , G . . a, b, k associated to A Cartan matrix A a = 1 , b = − , k = 0 A a = 1 , b = − , k = − B a = 3 , b = − , k = − G By Proposition 3.15, these equations have solutions. It is obvious that any solution ( c , c , c , c ) of Equations (3.45) - (3.48) should not be zero since ( n, d ) = 1 . Hence, Γ( D ) contains nonzeroelements.At last, we will show that there exists a family of nonzero modified root vector parameters µ for D . Note that A is connected, so λ must be zero. Since N = | χ i ( h i ) | for i = 1 , , (see (2.16) for definition), it is obvious that N = | ζ amnd | = d . Let α i be the simple root correspondingto X i , ≤ i ≤ . We have h N = ( g g ) d mn = g dn g dm ∈ G, h N = 1 , and χ N = χ d = 1 . So µ α is a nonzero parameter. Thus, there exists a family µ of nonzero modified root vectorparameters for D . The quasi-Hopf algebra u ( D , , µ, A ) is a nonradically graded quasi-Hopfalgebra associated to A. In Section 6, we will show that these quasi-Hopf algebras are genuine. Remark 3.17. Consider the subalgebra of u ( D , , µ, A ) generated by g , g , X in Example3.16. This algebra is a quasi-Hopf algebra of rank one with a nontrivial root vector relation.Hence, it is nonradically graded. We make a convention that if u ( D , λ, µ, A ) is a quasi-Hopfalgebra over a cyclic group, then the root vector relation must be trivial, i.e. µ = 0 . For thisreason, we can only construct quasi-Hopf algebras with nontrivial root vector relations overnoncyclic groups. Radically graded quasi-Hopf algebras of Cartan type In this section, we study radically graded quasi-Hopf algebras of Cartan type. We show thatall the radically graded quasi-Hopf algebras given in Theorem 3.4 are genuine quasi-Hopfalgebras, which leads to some interesting classification results.4.1. Radically graded quasi-Hopf algebras of Cartan type are genuine. In general,it is very difficult to determine whether a nonradically graded quasi-Hopf algebra is genuineor not. However, for radically graded quasi-Hopf algebras, we have the following proposition. Proposition 4.1. Suppose that H = ⊕ i ≥ H [ i ] is a finite-dimensional radically graded quasi-Hopf algebra. Then H is genuine if and only if H [0] is a genuine quasi-Hopf algebra.Proof. “ ⇐ ”: Suppose H = ( H, △ , ε, Φ , S, α, β ). By the definition of a radically graded quasi-Hopf algebra, H [0] = ( H [0] , △ , ε, Φ , S | H [0] , α, β ) is a quasi-Hopf subalgebra of H . If H is notgenuine, then there is a twist J of H , such that H J is a Hopf algebra, i.e.,Φ J = 1 ⊗ ⊗ , α J β J = 1 . Let π : H → H [0] = H/I be the natural projection, where I = ⊕ i ≥ H [ i ] is the Jacobsonradical of H. Define J = ( π ⊗ π )( J ). It is clear that J = J + J ≥ , where J ≥ ∈ H ⊗ I + I ⊗ H. Since ε ( I ) = 0, we have ( id ⊗ ε )( J ) = ( π ⊗ π )( id ⊗ ε )( J ) = 1 . Similarly, ( ε ⊗ id )( J ) = 1 . It INITE-DIMENSIONAL QUASI-HOPF ALGEBRAS OF CARTAN TYPE 23 is obvious that J has the inverse ( π ⊗ π )( J − ) because π is an algebra morphism. It followsthat J is a twist for H [0]. Now Φ ∈ H [0] ⊗ implies that (cid:26) Φ J ∈ H [0] ⊗ , Φ J ≥ ∈ H ⊗ H ⊗ I + H ⊗ I ⊗ H + I ⊗ H ⊗ H. Combining Φ J = 1 ⊗ ⊗ , we obtain Φ J = Φ J = 1 ⊗ ⊗ . Similarly, we have α J β J = 1 . Hence, H [0] J is a Hopf algebra, a contradiction to the fact that H [0] is genuine.“ ⇒ ”: If H [0] is not genuine, then there is a twist J of H [0] such that H [0] J is a Hopf algebra.It is easy to see that J is also a twist of H , and H J is a Hopf algebra. (cid:3) Theorem 4.2. Suppose that u ( D , λ, µ, Φ c ) is a quasi-Hopf algebra of Cartan type with λ =0 , µ = 0 . Then u ( D , λ, µ, Φ c ) is a genuine quasi-Hopf algebra.Proof. Keep the same notations as those in Theorem 3.4. Let H = u ( D , , , Φ c ) and theradically graded structure is given by H = ⊕ i ≥ H [ i ]. Then H [0] = k G, and the associatorΦ c = X e,f,g ∈ G φ c ( e, f, g )1 e ⊗ f ⊗ g for a nonzero c ∈ Γ( D ) . By Proposition 2.17, φ c is a 3-cocycle on G , but not a coboundary.So ( H [0] , Φ c ) is a genuine quasi-Hopf algebra. It follows from Proposition 4.1 that H = u ( D , , , Φ c ) is a genuine quasi-Hopf algebra. (cid:3) Some classification results. Let H = ⊕ i ≥ H [ i ] be a finite-dimensional radicallygraded quasi-Hopf algebra. The ideal I = ⊕ i ≥ H [ i ] is the radical of H . Note that H [ i ] = I i /I i +1 . In fact, we have the following relations:(4.1) H [ i ] = H [1] i , i ≥ . Assume that H = k [ G ] and G an abelian group. Then ( k [ G ] , Φ) is a quasi-Hopf subalgebra of H with the inherited associator Φ and the restricted antipode ( S | H , α, β ). Now we constructa new quasi-Hopf algebra b H . By ⊲ we denote the inner action of H [0] on H [1]: g ⊲ X = g · X · g − , g ∈ G, X ∈ H [1] , where · stands for the multiplication of H . Extend the action of H [0] on H [1] to the tensoralgebra T ( H [1]) naturally.Let b H be the smash product algebra T ( H [1]) ⋊ H [0]. The algebra b H has a natural comulti-plication given by △ b H ( X ) = △ H ( X ) , △ b H ( g ) = g ⊗ g, X ∈ H [1] , g ∈ G. Let S ′ : b H → b H ⊗ b H be a algebra antimorphism such that S ′ | H [0] ⊗ H [1] = S | H [0] ⊗ H [1] . Onemay verify straightforward that ( b H, △ b H , Φ , S ′ , α, β ) forms a quasi-Hopf algebra. It is obviousthat we have a canonical surjective homomorphism P : b H → H such that P restricts to theidentity on H [0] ⊕ H [1] . In what follows, the elements in H [ n ] will be said to be of degree n . In order to classifythe finite-dimensional radically graded quasi-Hopf algebras over abelian groups, we need thefollowing proposition, whose proof is parallel to [5, Proposition 3.3.2, Proposition 3.3.3], hencewill be omitted. Proposition 4.3. Let H be a finite-dimensional radically graded quasi-Hopf algebra, and π : H → u ( D , , , Φ c ) a quasi-Hopf algebra epimorphism such that the restriction of π to theparts of degree and is the identity. Then ad c ( X i ) − a ij ( X j ) = 0 , i = j and X N J α = 0 , α ∈ R + J , J ∈ Ω hold in H. Now we are able to give one of the main results of the paper. The notations of G, G , g i , g i , ≤ i ≤ n are the same as those in Subsection 3.1. Theorem 4.4. Suppose that H is a radically graded finite-dimensional genuine quasi-Hopfalgebra over an abelian group G with an associator Φ = P e,f,g ∈ G φ ( e, f, g )1 e ⊗ f ⊗ g , where φ an abelian -cocycle on G satisfying , , , ∤ dim( H ) . Then H ∼ = u ( D , , , Φ c ) for somedatum of finite Cartan type D and some nonzero c ∈ Γ( D ) . Proof. According to Subsection 2.5, we know that { φ c | c ∈ A , c r,s,t = 0 , ≤ r < s < t ≤ n } is a complete set of representatives of abelian 3-cocycles on G . Thus there exists a 2-cochain J on G such that φ∂J = φ c for some c in A satisfying c r,s,t = 0 for all 1 ≤ r < s < t ≤ n. Let J = P f,g J ( f, g )1 f ⊗ g . It is clear that the associator of H J is Φ c . Thus, without loss ofgenerality, we may assume that the associator of H is Φ c for some c ∈ { c ∈ A| c r,s,t = 0 , ≤ r < s < t ≤ n } . Since H = ⊕ i ≥ H [ i ] is a radically graded quasi-Hopf algebra, we can construct a new quasi-Hopf algebra b H, so that there is an epimorphism P : b H → H such that P restricts to theidentity on H [0] ⊕ H [1]. Denote by L the sum of all quasi-Hopf ideals of b H contained in P i ≥ H [ i ] . It is easy to see that KerP ∈ L. Let H be the quotient b H/L , and η : b H → H thecanonical projection. Thus, there exists an epimorphism π : H → H such that η = πP . Notethat π restricts to the identity as well on H [0] ⊕ H [1].Next we show that H ∼ = u ( D , , , Φ c ) for some D and some c ∈ Γ( D ) . Then, it follows fromProposition 4.3 that π must be an isomorphism, and the proof will be done. Decompose H [1] = ⊕ χ ∈ b G H χ [1] , where(4.2) H χ [1] = { X ∈ H [1] | gXg − = χ ( g ) X, ∀ g ∈ G } . For each χ ∈ b G, define a e χ ∈ b G by e χ ( g i ) = χ ( g i ) mi , ≤ i ≤ n. Denote by e H the quasi-Hopfalgebra generated by H and g i , ≤ i ≤ n, where g m i i = g i , and g X g − = e χ ( g ) X for all g ∈ G and X ∈ H χ [1] . It is obvious that e H is a radically graded quasi-Hopf algebra over G , and H is the quasi-Hopf subalgebra of e H generated by e H [1] and g i , ≤ i ≤ n. Consider A = e H J − c . Since A is a finite-dimensional radically graded Hopf algebra over G , and A is of the form R kG for some braided graded Hopf algebra in the category of Yeter-Drinfeldmodules over G . So A is also a finite-dimensional pointed Hopf algebra over G . By the classi-fication result of [4], there exists a datum of finite Cartan type D = D ( G , ( h i ) ≤ i ≤ θ , ( χ i ) ≤ i ≤ θ such that A = u ( D , , H is generated by A J c [1] and g i , ≤ i ≤ n, there is a nonzero c ∈ Γ( D ) such that H ∼ = u ( D , , , Φ c ) by Proposition 3.12. (cid:3) From Proposition 2.17 we know that every 3-cocycle on a cyclic group or on an abelian groupof the form Z m × Z m is abelian. So we have the following. Corollary 4.5. Suppose that H is a radically graded finite-dimensional genuine quasi-Hopf al-gebra over an abelian group G = Z m ⊗ Z n such that , , , ∤ dim( H ) . Then H ∼ = u ( D , , , Φ c ) for some datum of finite Cartan type D and some c ∈ Γ( D ) . INITE-DIMENSIONAL QUASI-HOPF ALGEBRAS OF CARTAN TYPE 25 Small quasi-quantum groups In this section, we will introduce small quasi-quantum groups. These algebras can be viewedas natural generalization of small quantum groups. We will present several examples of smallquasi-quantum groups which are genuine quasi-Hopf algebras. We fix a finite abelian group G with free generators { g i | ≤ i ≤ n } . The notations G and { g i | ≤ i ≤ n } are defined inthe the same way as those in Subsection 3.1.5.1. Small quasi-quantum groups. Suppose that A = ( a ij ) ≤ i,j ≤ n is a finite Cartan ma-trix and that D ( A ) is the Cartan matrix (cid:18) A A (cid:19) . Let G = Z nm = h g i × · · · × h g n i . Definition 5.1. Let D ( G , ( h i ) ≤ i ≤ n , ( χ i ) ≤ i ≤ n , D ( A )) be a datum of finite Cartan typesuch that h i = h n + i ∈ G, χ i = χ − i + n , ≤ i ≤ n. Suppose that c ∈ Γ( D ) is nonzero. We callthe quasi-Hopf algebra Qu ( D , λ, Φ c ) = u ( D , λ, µ, Φ c ) a quasi-small quantum group if µ = 0 and λ i,j = 0 if and only if j = i + n for ≤ i ≤ n. The quasi-Hopf algebras in Example 3.14 are examples of quasi-small quantum groups. Moreexamples will be given before we show that small quasi-quantum groups are natural gener-alization of small quantum groups. For 1 ≤ i ≤ n, define E i = X i , F i = X i + n h − i and X ′ i = X i + n . Let V = k { E , · · · , E n } , V ′ = k { X ′ , · · · , X ′ n } and U = k { F , · · · , F n } . It isobvious that V and V ′ are braided vector spaces of Cartan type with the braiding matrices( q ij ) ≤ i,j ≤ n and ( q − ij ) ≤ i,j ≤ n respectively, where q ij = χ j ( h i ) , ≤ i, j ≤ n. Note that Bydefinition de braided vector spaces of Cartan type (see Subsection 2.3), the associated Cartanmatrices of V and V ′ are both equal to A. Let q ′ ij = q − ji , ≤ i, j ≤ n . We define a braiding c on U as follows: c ( F i ⊗ F j ) = q ′ ij F j ⊗ F i , ≤ i, j ≤ n. For all 1 ≤ i, j ≤ n, we have q ′ ij q ′ ji = q − ji q − ij = q − a ij ii = q ′ a ij ii . So ( U, c ) is also a braided vectorspace of Cartan type, and the associated Cartan matrix is A as well. So we can define braidedcommutators, the braided adjoint action and the root vectors over T ( U ), see Subsection 2.3for details. Now let R + be the positive root system corresponding to the Cartan matrix A with respect to the simple roots α , · · · , α n , and F α , α ∈ R + the root vectors such that F α i = F i for 1 ≤ i ≤ n. Let Ω be the set of the connected components of I = { , · · · , n } , and R + J be the positive root system corresponding to J ∈ Ω . Proposition 5.2. Qu ( D , λ, Φ c ) is generated by g i , E i , F i , ≤ i ≤ n subject to the relations: g i E j g − i = χ j ( g i ) E j , g i F j g − i = χ − j ( g i ) F j , ≤ i, j ≤ n, (5.1) E i F j − F j E i = δ ij λ i,i + n ( h − i − h i ) , λ i,i + n = 0 , ≤ i, j ≤ n, (5.2) ad c ( E i ) − a ij ( E j ) = 0 , ad c ( F i ) − a ij ( F j ) = 0 , ≤ i = j ≤ n, (5.3) E N J α = 0 , F N J α = 0 , α ∈ R + J , J ∈ Ω . (5.4) The comultiplication is determined by △ ( E i ) = X f,g ∈ G Ψ i ( f, g ) E i f ⊗ g + h i ⊗ E i , (5.5) △ ( F i ) = X f,g ∈ G Ψ i + n ( f, g ) χ g ( h − i ) F i f ⊗ g + 1 ⊗ F i . (5.6) The associator is Φ c and the antipode ( S , α, is given by α = P g ∈ G Υ( g )1 g , (5.7) S ( F i ) = χ − i ( h i ) P g ∈ G χ g ( h i ) F i + n ( g ) F i g , S ( E i ) = P g ∈ G F i ( g ) E i g , (5.8) for ≤ i ≤ n. Proof. By Theorem 3.4, Qu ( D , λ, Φ c ) is generated by G and X j , ≤ i ≤ n, ≤ j ≤ n subjectto the relations (3.19)-(3.22). Since F i h i = X i , the algebra Qu ( D , λ, Φ c ) is also generated by g i , E i , F i , ≤ i ≤ n. Now we show that the relations (5.1)-(5.4) are equivalent to the relations(3.19)-(3.22). It is easy to see that (5.1) equals (3.19). For all 1 ≤ i, j ≤ n, we have: E i F j − F j E i = X i X j + n h − j − X j + n h − j X i = X i X j + n h − j − χ j + n ( h − j ) X i X j + n h − j = X i X j + n h − j − χ j ( h j ) X i X j + n h − j = ad c ( X i )( X j ) h − j . So Relation (5.2) is equivalent to Relation (3.21). For 1 ≤ i = j ≤ n , we have ad c ( F i )( F j ) = F i F j − q ′ ij F j F i = F i F j − χ − i ( h j ) F j F i = X i + n h − i X j + n h − j − χ − i ( h j ) X j + n h − j X i + n h − i = χ − j ( h − i ) X i + n X j + n h − i h − j − χ − i ( h j ) χ − i ( h − j ) X j + n X i + n h − i h − j = χ j ( h i )[ X i + n X j + n − χ − j ( h i ) X j + n X i + n ] h − i h − j = χ j ( h i )[ ad c ( X i + n )( X j + n )] h − i h − j . By induction on l ≥ , one can show that(5.9) ( ad c ( F i )) l ( F j ) = χ lj ( h i ) χ l ( l − i ( h i )[( ad c ( X i + n )) l ( X j + n )] h − li h − j . Hence, the relation (5.3) is equivalent to the relation (3.20). Similarly, for each α ∈ R + , onecan show that F α = λ α X ′ α h − α , where λ α is some nonzero number depend on α, and h α is defined by (2.7). Thus, for each α ∈ R + J , J ∈ Ω , we have F N J α = 0 if and only if X ′ N J α = 0. Therefore, the relation (5.4)equivalent to the relation (3.22) since µ = 0. We have proved that Qu ( D , λ, Φ c ) is generatedby g i , E i , F i , ≤ i ≤ n subject to the relations (5.1)-(5.4).Next we compute the comultiplication and the antipode of Qu ( D , λ, Φ c ) for the generators.Formula (5.5) follows from the fact h i = P f ∈ G Θ i ( f )1 f for h i ∈ G . Formula (5.6) holdsbecause of the following equations: △ ( F i ) = △ ( X i + n h − i )= ( X f,g ∈ G Ψ i + n ( f, g ) X i + n f ⊗ g + X f ∈ G Θ i + n ( f )1 f ⊗ X i + n )( h − i ⊗ h − i )= X f,g ∈ G Ψ i + n ( f, g ) χ g ( h − i ) F i f ⊗ g + X f ∈ G Θ i + n ( f ) χ f ( h − i )1 f ⊗ F i = X f,g ∈ G Ψ i + n ( f, g ) χ g ( h − i ) F i f ⊗ g + 1 ⊗ F i . INITE-DIMENSIONAL QUASI-HOPF ALGEBRAS OF CARTAN TYPE 27 Formula (5.7) is obvious. Formula (5.8) follows from S ( F i ) = S ( h − i ) S ( X i + n ) = h i X g ∈ G F i + n ( g ) X i + n g = χ i + n ( h i ) X g ∈ G χ g ( h i ) F i + n ( g ) F i g , where χ g , g ∈ G , is defined by (3.1). (cid:3) Now let D ( G , ( h i ) ≤ i ≤ n , ( χ i ) ≤ i ≤ n , D ( A )) be a datum of finite Cartan type such that h i = h n + i ∈ G, χ i = χ − i + n , ≤ i ≤ n . Note that Γ( D ) is not empty since 0 ∈ Γ( D ) . Take an element c ∈ Γ( D ). We define an algebra H c generated by G and E i , F i , ≤ i ≤ n subject to the Relations (5.1)-(5.4). Define an algebra morphism △ : H c → H c ⊗ H c by (5.5)-(5.6), and an algebra antimorphism S : H c → H c by (5.8). Let α be an element of H c definedby (5.7). Define an algebra morphism ε : H c → k such that ε ( g ) = 1 , ε ( E i ) = ε ( F i ) = 0 for g ∈ G, ≤ i ≤ n. We have the following identification of the algebra H c . Proposition 5.3. (1) If c = 0 , then ( H c , △ , ε, S ) is isomorphic to the AS-Hopf algebra u ( D ′ , λ, , where D ′ = D ( G, ( h i ) ≤ i ≤ n , ( χ ′ i ) ≤ i ≤ n , D ( A )) , and χ ′ i = χ i | G , ≤ i ≤ n. Each small quantum group is isomorphic to a H as Hopf algebras. (2) If c = 0 , then ( H c , △ , ε, Φ c , S, α, is isomorphic to the small quasi-quantum group Qu ( D , λ, Φ c ) . Proof. Observe that the functions defined by (3.13)-(3.15) are equivalent to the constantfunction 1 if c = 0. Hence, the algebra morphism Υ : H c → u ( D ′ , λ, 0) given byΥ( g ) = g, Υ( E i ) = X i , Υ( F i ) = X i + n h i , for all g ∈ G, ≤ i ≤ n, is a Hopf algebra isomorphism. The second part of the propositionfollows from Proposition 5.2. (cid:3) From this proposition, we can see that small quasi-quantum groups are natural generalizationsof small quantum groups.5.2. Triangular decomposition and half small quasi-quantum groups. In this subsec-tion, we study the triangular decomposition of small quasi-quantum groups. Fix an abeliangroup G = Z nm = h g i × · · · × h g n i and a finite Cartan matrix A = ( a ij ) ≤ i,j ≤ n . Assume that D = D ( G , ( h i ) ≤ i ≤ n , ( χ i ) ≤ i ≤ n , D ( A )) is a datum of finite Cartan type and Qu ( D , λ, Φ c )is a small quantum group. Denote by Qu + ( D , Φ c ) (resp. Qu − ( D , Φ c )) the subalgebra gen-erated by E i , ≤ i ≤ n (resp. F i , ≤ i ≤ n ), and Qu = k G. So we have a natural linearisomorphism ϕ : Qu + ( D , Φ c ) ⊗ Qu ⊗ Qu − ( D , Φ c ) −→ Qu ( D , λ, Φ c ) x ⊗ y ⊗ z → xyz, for all x ∈ Qu + ( D , Φ c ) , y ∈ Qu , z ∈ Qu − ( D , Φ c ) . This decomposition Qu ( D , λ, Φ c ) = Qu + ( D , Φ c ) ⊗ Qu ⊗ Qu − ( D , Φ c ) is called the triangular decomposition of Qu ( D , λ, Φ c ).Denote by Qu ≥ ( D , Φ c ) (resp. Qu ≤ ( D , Φ c )) the subalgebra of Qu ( D , λ, Φ c ) generated by G and E i , ≤ i ≤ n (resp. G and E i , ≤ i ≤ n ). The following isomorphisms are obvious. Qu ≥ ( D , Φ c ) ∼ = u ( D ′ , , , Φ c ) , (5.10) Qu ≤ ( D , Φ c ) ∼ = u ( D ′′ , , , Φ c ) , (5.11) where D ′ = D ( G , ( h i ) ≤ i ≤ n , ( χ i ) ≤ i ≤ n , A ) , D ′′ = D ( G , ( h i ) n +1 ≤ i ≤ n , ( χ i ) n +1 ≤ i ≤ n , A ) . Thetwo radically graded quasi-Hopf algebras are called the half small quasi-quantum groups .Keep the assumption of G and A as above, we have the following. Proposition 5.4. Let D = D ( G , ( h i ) ≤ i ≤ n , ( χ i ) ≤ i ≤ n , A ) be a datum of finite Cartan typesuch that h i ∈ G, ≤ i ≤ n, and χ i ( h j ) = χ j ( h i ) , ≤ i, j ≤ n. Then for any nonzero c ∈ Γ( D ) , u ( D , , , Φ c ) is a half small quasi-quantum group.Proof. We need to show that there exists a small quasi-quantum group Qu ( D ′ , λ, Φ c ) suchthat u ( D , , , Φ c ) ∼ = Qu ≥ ( D ′ , Φ c ) . Define h i + n = h i and χ i + n = χ − i for 1 ≤ i ≤ n. For1 ≤ i, j ≤ n, we have the following: χ i ( h j + n ) χ j + n ( h i ) = χ i ( h j ) χ − j ( h i ) = 1 . It is clear that D ′ = D ( G , ( h i ) ≤ i ≤ n , ( χ i ) ≤ i ≤ n , D ( A )) is a datum of finite Cartan type. Foreach 1 ≤ i ≤ n , let h i = Q nj =1 g s ij . Since h i ∈ G , we have s ij ≡ m. It follows thatΓ( D ) = Γ( D ′ ) . Moreover, h i h i + j = h i ∈ G and χ i χ i + n = χ i χ − i = ε . Thus, we obtain a familyof linking parameters λ = ( λ ij ) ≤ i Let A = ( a ij ) ≤ i,j ≤ n be afinite Cartan matrix, ( d , · · · , d n ) a vector with elements in { , , } such that ( d i a ij ) ≤ i,j ≤ n is symmetric. Let q be an N -th primitive root of unity, where N is an odd positive integer.Moreover, in case A has a connected component of type G , we will add one more assumptionthat N is prime to 3 . Let p be a positive odd integer and m = pN. Choose ζ m and ζ m suchthat ζ mm = ζ m , ζ pm = q. Let G = Z nm = h g i × · · · × h g n i , and l , 1 ≤ l < m , be an integer suchthat lp = 0 mod m . Define characters { χ i | ≤ i ≤ n } on G as follows:(5.12) χ i ( g j ) = ζ pd i m , if i = j ; ζ pd i a ij m , if i = j and a ij = 0;1 , if a ij = 0.With the above notations, it is easy to verify that D = D ( G , ( h i ) ≤ i ≤ n , ( χ i ) ≤ i ≤ n , D ( A )) isa datum of finite Cartan type, where h i = h i + n = g mli and χ i + n = χ − i for 1 ≤ i ≤ n. Lemma 5.5. There are nonzero elements c = ( c i , c jk ) ≤ i ≤ n, ≤ j It is obvious that (3.10)-(3.11) are solvable for ( c jk ) ≤ j INITE-DIMENSIONAL QUASI-HOPF ALGEBRAS OF CARTAN TYPE 29 We need the following lemma. Lemma 5.6. There exists a family of modified linking parameters λ for D satisfying condition: λ ij = 0 if and only if i + n = j. Proof. Note that for each 1 ≤ i ≤ n, we have χ i χ i + n = ε because χ i + n = χ − i . The fact that lp = 0 mod m, implies that l = 0 mod N . Hence 2 l = 0 mod N because N is odd. It followsthat h i h i + n = g mli = g li ∈ G and h i h i + n = 1 . By definition, there exists a family of modifiedlinking parameters λ for D such that λ ij = 0 if and only if i + n = j. (cid:3) From Lemmas 5.5-5.6 and the definition of small quasi-quantum groups, we obtain the fol-lowing. Proposition 5.7. Let c be a nonzero parameter in Γ( D ) , and λ a a family of modified linkingparameters such that λ ij = 0 if and only if i + n = j . Then u ( D , λ, , Φ c ) is a small quasi-quantum group. In what follows, we let Qu ( D , λ, Φ c ) = u ( D , λ, , Φ c ), c = 0, be a small quasi-quantum groupgiven in Proposition 5.7. We conjecture that Qu ( D , λ, Φ c ) is genuine. At this moment, weare not able to prove it in general. But we have the following partial result. Proposition 5.8. Suppose l > , l | m and l ∤ c i for ≤ i ≤ n . Then Qu ( D , λ, Φ c ) is genuine.Proof. Let I be the quasi-Hopf ideal of Qu ( D , λ, Φ c ) generated by { X i | ≤ i ≤ n } . Set e u = Qu ( D , λ, Φ c ) /I . It is evident that e u ∼ = k G ′ , where G ′ = G/ h g li − | ≤ i ≤ n i . Sincegcd(2 l, m ) = l, we have G ′ = G/ h g li − | ≤ i ≤ n i . For an element g ∈ G, we denote by g thecorresponding element in the quotient group G ′ . Let G ◦ be the subgroup of G generated by { g ml i | ≤ i ≤ n } . Define a group isomorphism ϕ : G ◦ → G ′ by ϕ ( g ml i ) = g i , ≤ i ≤ n. Let f = Q n g n i i . We have the following expression of the element 1 f in k G :1 f = 1 | G | X g ∈ G χ f ( g ) g = 1 | G | X ≤ i j Rep Φ ′ c ( k G ′ ) is a full tensor subcategory of M . Hence Rep φ ′ c ( k G ′ ) has afiber functor as well. Thus e u is twist equivalent to a Hopf algebra, a contradiction since e u isgenuine. (cid:3) It is easy to see that there are (infinitely) many numbers l, p, N to choose such that l, m, c i , ≤ i ≤ n satisfy the conditions in Proposition 5.8. Therefore, we obtain many examples of genuinesmall quasi-quantum groups associated to each finite Cartan matrix.6. Nonradically graded genuine Quasi-Hopf algebras associated toconnected Cartan matrices In this section, we construct many new examples of nonradically graded genuine quasi-Hopfalgebras associated to each finite connected Cartan matrix. Explicitly, for each finite con-nected Cartan matrix A , we will show that there exists a Cartan datum D associated to A , nontrivial modified root vector parameters µ for D and a nonzero c ∈ Γ( D ) such that u ( D , , µ, Φ c ) is a finite-dimensional genuine quasi-Hopf algebras. All these quasi-Hopf alge-bras have only trivial linking relations since the Cartan matrices are assumed to be connected.In the following, the groups G = h g i × · · · × h g n i and G = h g i × · · · × h g n i are defines thesame as those in Subsection 3.1. For each fixed Cartan matrix A , the corresponding rootsystem R, positive root system R + with simple roots { α i | ≤ i ≤ n } are defined to be thesame as those in Subsection 2.2. Throughout this section, p, q, d are positive odd numberssatisfying ( p, q ) = ( p, d ) = ( q, d ) = 1.6.1. Quasi-Hopf algebras of Cartan type A n , B n and C n . Notice that examples ofnon-radically graded quasi-Hopf algebras associated to A , B (= C ) , G have been given inSubsection 3.5, so in this subsection we always assume n ≥ 3. Let A = ( a ij ) ≤ i,j ≤ n be the INITE-DIMENSIONAL QUASI-HOPF ALGEBRAS OF CARTAN TYPE 31 Cartan matrix of type A n , B n or C n . Let G = h g i × h g i × · · · × h g n i be the abelian groupdetermined by(6.1) | g i | = pd, if i = 4 k + 1 for some k ; qd, if i = 4 k + 2 for some k ; pd , if i = 4 k + 3 for some k ; qd , if i = 4 k for some k .For each 1 ≤ i ≤ n, define an element in G as follows:(6.2) h i = ( g i − g i ) pq , if i is odd ;( g i − g i − g i − g i g i +1 g i +2 ) pq , if i is even and i = n ;( g i − g i − g i − g i ) pq , if i is even and i = n. In order to give a datum of Cartan type associated to Cartan matrix A, we need to introducesome characters on G . Let a, b, r be the numbers given in Table 2. Define(6.3) χ ( g j ) = ζ ad , j = 1 , ζ rd , j = 3 , , j ≥ χ ( g j ) = ζ ad , j = 1 , ζ d , j = 3, ζ − ad − d , j = 4, ζ bd , j = 5 , , j ≥ ≤ i < n, define(6.5) χ i ( g j ) = ζ − bd , if i is odd and j = 2 i − , i − , i + 1 , i + 2, ζ bd , if i is odd and j = 2 i − , i , ζ bd , if i is even and j = 2 i − , i − , i + 1 , i + 2, ζ d , if i is even and j = 2 i − ζ − bd − d , if i is even and j = 2 i ,1 , otherwise.When n is odd, define(6.6) χ n ( g j ) = ζ − bd , j = 2 n − , n − ζ bd , j = 2 n − , n ,1 , otherwise.When n is even, define(6.7) χ n ( g j ) = ζ bd , j = 2 n − , n − ζ d , j = 2 n − ζ − d , j = 2 n ,1 , otherwise.With these notations, we can give the following lemma. Lemma 6.1. D = D ( G , ( h i ) ≤ i ≤ n , ( χ i ) ≤ i ≤ n , A ) is a datum of Cartan type.Proof. Let q ij = χ j ( h i ) for all 1 ≤ i = j ≤ n . By definition of datum of Cartan type, we onlyneed to show(6.8) q ij q ji = q a ij ii = q a ji jj , ≤ i = j ≤ n, and it follows a direct verification. (cid:3) Table 2. a, b, r associated to A n , B n , C n . . a, b, r associated to A Cartan matrix A a = 1 , b = 1 , r = − A n a = 2 , b = 1 , r = − B n a = 1 , b = 2 , r = − C n Lemma 6.2. Γ( D ) is not empty. For each family c = ( c i , c jk ) ≤ i ≤ n, ≤ j Notice that Equations (3.10)-(3.11) always have solutions, for examples c ij = 0 for all1 ≤ i < j ≤ n. So we only need to prove that Equations (3.9) have solutions and c i = 0 for1 ≤ i ≤ n. Let m i = | g i | for each 1 ≤ i ≤ n. It is clear that Equations (3.9) on variables c , c , c , c are given by c ap ≡ pq mod pd, (6.9) c aq ≡ pq mod qd, (6.10) c p ≡ pq mod pd , (6.11) c rp d ≡ pd , (6.12) c ( − ad − q ≡ pq mod qd , (6.13) c rq d ≡ qd . (6.14)When 3 ≤ i ≤ n and i is odd, Equations (3.9) are c i − bp ≡ pq mod pd, (6.15) c i bq ≡ pq mod qd. (6.16)When 3 < i < n and i is even, Equations (3.9) become c i − p ≡ pq mod pd , (6.17) c i − ( − b ) p d ≡ pd (6.18) c i ( − bd − q ≡ pq mod qd , (6.19) c i ( − b ) q d ≡ qd . (6.20)When i = n is even, the equations (3.9) are given by c n − p ≡ pq mod pd , (6.21) c n − ( − b ) p d ≡ pd (6.22) c n ( − q ≡ pq mod qd , (6.23) c n ( − b ) q d ≡ qd . (6.24)It is obvious that any integers c i − , c i are solution of (6.12), (6.14), (6.18),(6.20), (6.22) and(6.24). Since ( ap , pd ) = p, ( aq , qd ) = q, ( p , pd ) = p, (( − ad − q , qd ) = q , so (6.9)-(6.14) have solutions by Proposition (3.15). Any solution c , c , c or c of (6.9)-(6.14) shouldnot be zero since ( p, d ) = ( q, d ) = 1. Similarly, one can show that (6.15)-(6.17), (6.19),(6.21)and (6.23) have nonzero solutions. (cid:3) Lemma 6.3. There exists modified root vector parameters µ for D satisfying the condition: (6.25) µ α is a nonzero parameter if and only if α = α i for some odd number 1 ≤ i ≤ n. INITE-DIMENSIONAL QUASI-HOPF ALGEBRAS OF CARTAN TYPE 33 Proof. Firstly, one can verify that | χ i ( h i ) | = d for 1 ≤ i ≤ n. So for all 1 ≤ i ≤ n and i isodd, we have h d i = ( g i − g i ) pqd = g qd i − g pd i ∈ G, h d i = 1 and χ d i = ε , hence µ α i = 0.We proved the lemma. (cid:3) Proposition 6.4. Let c ∈ Γ( D ) and µ a family of modified root vector parameters for D satisfying condition (6.25) . Then u ( D , , µ, Φ c ) is a finite-dimensional nonradically gradedgenuine quasi-Hopf algebra associated to A. Proof. By Lemma 6.2-6.3 and Theorem 3.4, u ( D , , µ, Φ c ) is a finite-dimensional nonradicallygraded quasi-Hopf algebra. So we only need to prove that u ( D , , µ, Φ c ) is a genuine quasi-Hopf algebra.Let α be a positive root in R + , then we have µ α = 0 if α = α i some odd number 1 ≤ i ≤ n. Let I be the ideal of u ( D , , µ, Φ c ) generated by { X i , − g j | ≤ i ≤ n, j = 1 or 2 mod 4 } . It is obvious that I is a quasi-Hopf ideal of u ( D , , µ, Φ c ) . Denote by G ′ = h g i | i = 0 or 3 mod 4 i . Then it is obvious that(6.26) k G ′ = u ( D , , µ, Φ c ) / I . Similar to the proof of Proposition 5.8, one can show that the associator of the k G ′ isΦ c = P e,f,g ∈ G ′ φ c | G ′ ( e, f, g )1 e ⊗ f ⊗ g . By lemma 6.2, φ c | G ′ is a not 3-coboundary on G ′ . Hence ( k G ′ , Φ c ) is a genuine quasi-Hopf algebra. This implies u ( D , , µ, Φ c ) is genuine,since otherwise ( k G ′ , Φ c ) = u ( D , , µ, Φ c ) / I should not be genuine, which is a contradiction. (cid:3) Quasi-Hopf algebras of Cartan type D n . In this subsection, we always assume that n ≥ . Let G = h g i × · · · × h g n i such that(6.27) | g i | = pd, i = 1 or 4 k − k ≥ qd, i = 2 or 4 k for k ≥ pd , i = 4 k + 1 for k ≥ qd , i = 4 k + 2 for k ≥ h i ) ≤ i ≤ n be a family of elements in G given by(6.28) h i = ( g i − g i ) pq , if i = 1 or 2 k for k ≥ Q j =1 g j ) pq , if i = 3;( g i − g i − g i − g i g i +1 g i +2 ) pq , if i = 2 k + 1 = n for k ≥ g n − g n − g n − g n ) pq , if i = n is odd .Now define a family ( χ i ) ≤ i ≤ n of characters on G as following.When i = 1 , , (6.29) χ i ( g j ) = ζ d , j = 2 i − , i ; ζ − d , j = 5 , , otherwise. (6.30) χ ( g j ) = ζ d , j = 1 , , , , , ζ d , j = 5; ζ − d − d , j = 6;1 , otherwise.When 3 < i < n ,(6.31) χ i ( g j ) = ζ − d , if i is even and j = 2 i − , i − , i + 1 , i + 2 ; ζ d , if i is even and j = 2 i − , i ; ζ d , if i is odd and j = 2 i − ζ − d − d , if i is odd and j = 2 i ; ζ d , if i is odd and j = 2 i − , i − , i + 1 , i + 2;1 , otherwise.If n is even,(6.32) χ n ( g j ) = ζ d , if j = 2 n − , n ; ζ − d , if j = 2 n − , n − , otherwise.If n is odd,(6.33) χ n ( g j ) = ζ d , if j = 2 n − , n − ζ d , if j = 2 n − ζ − d , if j = 2 n ;1 , otherwise;With these definitions, one can verify that D = D ( G , ( h i ) ≤ i ≤ n , ( χ i ) ≤ i ≤ n , D n ) is a datumof Cartan type. Moreover, we have the following two lemmas, and the proofs, omitted, aresimilar as the proofs of Lemma 6.2 and 6.3. Lemma 6.5. Γ( D ) is a nonempty set, and for each family c = ( c i , c jk ) ≤ i ≤ n, ≤ j There exists a family of modified root vector parameters µ for D satisfying thecondition: (6.34) µ α is a nonzero if and only if α = α or α i for some even number 1 ≤ i ≤ n. Proposition 6.7. Let c ∈ Γ( D ) , and µ a family of modified root vector parameters for D satisfying the condition (6.34) . Then u ( D , , µ, Φ c ) is a genuine quasi-Hopf algebra.Proof. Similar to the proof of Proposition 6.4. (cid:3) Quasi-Hopf algebras of Cartan type E , E and E . In this subsection, we alwaysassume n = 6 , G = h g i × h g i × · · · × h g i such that | g i − | = pd , | g i | = qd for i = 1 , , , | g i − | = pd, | g i | = qd for i = 2 , , , . (6.36) INITE-DIMENSIONAL QUASI-HOPF ALGEBRAS OF CARTAN TYPE 35 Table 3. Characters of G = h g i × · · · × h g i χ χ χ χ χ χ χ ′ χ χ g ζ d ζ − d g ζ − d ζ − d g ζ d ζ d ζ d g ζ d ζ d ζ d g ζ − d ζ d ζ − d ζ − d g ζ − d ζ − d − d ζ − d ζ − d g ζ d ζ d g ζ d ζ d g ζ d ζ d ζ d ζ d g ζ d ζ d ζ d ζ d g ζ − d ζ d ζ d ζ − d g ζ − d ζ − d ζ − d − d ζ − d g ζ d ζ d ζ d g ζ d ζ d ζ d g ζ − d ζ d g ζ − d ζ − d Let G = h g i × h g i × · · · × h g i and G = h g i × h g i × · · · × h g i . Define h i = ( g i − g i ) pq for i = 2 , , , , (6.37) h = ( g g g g ) pq , (6.38) h = ( Y i =3 g i ) pq , (6.39) h = ( g g g g ) pq , (6.40) h ′ = ( g g g g g g ) pq , (6.41) h = ( g g g g ) pq . (6.42)Let χ i , ≤ i ≤ χ ′ be the characters of G given in Table 3, then one can verify that D = D ( G , ( h , · · · , h ) , ( χ , · · · , χ ) , E ) , D = D ( G , ( h , · · · , h , h ′ , h ) , ( χ , · · · , χ , χ ′ , χ ) , E ) , D = D ( G , ( h , · · · , h , h ′ , h , h ) , ( χ , · · · , χ , χ ′ , χ , χ ) , E ) . are datums of Cartan type. And similar as Lemma 6.2 and 6.3 we have the following: Lemma 6.8. Γ( D n ) is a nonempty set. For any c = ( c i , c jk ) ≤ i ≤ n, ≤ j There exists nonzero modified root vector parameters µ for D n satisfying theconditions: if n = 6 , µ α = 0 if and only if α = α i for i = 2 , ; if n = 7 or 8 , µ α = 0 ifand only if α = α i for i = 2 , , . Proposition 6.10. Let c ∈ Γ( D n ) , and µ a family of modified root vector parameters for Γ( D n ) satisfying the conditions of Lemma 6.9. Then u ( D n , , µ, Φ c ) is a nonradically gradedgenuine quasi-Hopf algebra. Table 4. Characters of G = h g i × · · · × h g i g g g g g g g g χ ζ d ζ d ζ − d ζ − d χ ζ d ζ d ζ d ζ − d − d ζ d ζ d χ ζ − d ζ − d ζ d ζ d ζ − d ζ − d χ ζ d ζ d ζ d ζ − d Proof. Similar to the proof of Proposition 6.4. (cid:3) Quasi-Hopf algebras of Cartan type F . Let G = h g i × · · · × h g i such that(6.43) | g i | = pd, i = 1 , qd, i = 2 , pd , i = 3 , qd , i = 4 , h i ) ≤ i ≤ a family of elements in G through h = ( g g ) pq , h = ( g g g g g g ) pq , (6.44) h = ( g g ) pq , h = ( g g g g ) pq . Let ( χ i ) ≤ i ≤ be the characters of G given in Table 4, then one can easily verify that D = D ( G , ( h i ) ≤ i ≤ , ( χ i ) ≤ i ≤ , F )is a datum of Cartan type. Similar to Lemma 6.2-6.3, we have the following two lemmas. Lemma 6.11. Γ( D ) is a nonempty set. Let c = ( c i , c jk ) ≤ i ≤ , ≤ j There exists a family of modified root vector parameter µ for Γ( D ) satisfyingthe condition: µ α is nonzero if and only if α = α or α . Proposition 6.13. Let c ∈ Γ( D ) , and µ a family of modified root vector parameters satisfyingthe condition of Lemma 6.12. Then u ( D , , µ, Φ c ) is a nonradically graded genuine quasi-Hopfalgebra.Proof. Similar to the proof of Proposition 6.4. (cid:3) References [1] N. Andruskiewitsch, H-J. 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