Examples of finite-dimensional pointed Hopf algebras in positive characteristic
aa r X i v : . [ m a t h . QA ] S e p EXAMPLES OF FINITE-DIMENSIONAL POINTED HOPFALGEBRAS IN POSITIVE CHARACTERISTIC
NICOLÁS ANDRUSKIEWITSCH, IVÁN ANGIONO AND ISTVÁNHECKENBERGER
To Nikolai Reshetikhin on his 60th birthday with admiration.
Abstract.
We present new examples of finite-dimensional Nichols al-gebras over fields of positive characteristic. The corresponding braidedvector spaces are not of diagonal type, admit a realization as Yetter-Drinfeld modules over finite abelian groups and are analogous to braid-ings over fields of characteristic zero whose Nichols algebras have finiteGelfand-Kirillov dimension.We obtain new examples of finite-dimensional pointed Hopf algebrasby bosonization with group algebras of suitable finite abelian groups. Introduction
Overview.
This is a contribution to the classification of finite-dimen-sional pointed Hopf algebras in positive characteristic. Beyond the classicaltheme of cocommutative Hopf algebras–see for instance [CF] and referencestherein–the problem was considered in several recent works [CLW, HW, NW,NWW1, NWW2, W]. As in various of these papers, the focus of our work ison finite-dimensional Nichols algebras over finite abelian groups. Let k be analgebraically closed field of characteristic p ≥ . When p = 0 , such Nicholsalgebras are necessarily of diagonal type and their classification was achievedin [H2]. When p > , finite-dimensional Nichols algebras of diagonal type ofrank 2 and 3 were classified in [HW, W]. Notice that there are more examplesthan in characteristic 0: indeed, 1 in the diagonal is no longer excluded. Example 1.1.
Assume that p > . Given θ ∈ N , we set I θ = { , , . . . , θ } .Let q = ( q ij ) i,j ∈ I θ ∈ k θ × θ , be a matrix with q ii = 1 = q ij q ji for all i = j ∈ I θ .Let ( V, c ) be a braided vector space of dimension θ , of diagonal type withmatrix q with respect to a basis ( x i ) i ∈ I θ , that is c : V ⊗ V → V ⊗ V is givenby c ( x i ⊗ x j ) = q ij x j ⊗ x i . Then the corresponding Nichols algebra is B ( V ) = s q ( V ) := T ( V ) / h x pi , i ∈ I θ , x i x j − q ij x j x i , i < j ∈ I θ i . Mathematics Subject Classification.
Clearly, dim s q ( V ) = p θ .Furthermore, if p > , then there are finite-dimensional Nichols algebrasover abelian groups that are not of diagonal type, a remarkable examplebeing the Jordan plane that has dimension p [CLW] (it gives rise to pointedHopf algebras of order p , see [NW]), in contrast with characteristic 0, whereit has Gelfand-Kirillov dimension . In fact, Nichols algebras over abeliangroups with finite Gelfand-Kirillov dimension and assuming p = 0 were thesubject of the recent papers [AAH1, AAH2]. Succinctly, the main relevantresults in loc. cit. are: ◦ It was conjectured in [AAH1] that finite GK-dimensional Nichols algebrasof diagonal type have arithmetic root system; the conjecture is true inrank 2 and also in affine Cartan type [AAH2]. ◦ A class of braided vector spaces arising from abelian groups was intro-duced in [AAH1]; they are decomposable with components being pointsand blocks. Assuming the validity of the above Conjecture, the finite GK-dimensional Nichols algebras from this class were classified in [AAH1].Beware that there are finite GK-dimensional Nichols algebras over abeliangroups that do not belong to the referred class, see [AAH1, Appendix].The braided vector spaces in the class alluded to above can be labelledwith flourished Dynkin diagrams. The main result of [AAH1] says that theNichols algebra of a braided vector space in the class has finite Gelfand-Kirillov dimension if and only if its flourished Dynkin diagram is admissible.
From now on we assume that p > . (The case p = 2 has to be treated sep-arately). In the present paper, we show, adapting arguments from [AAH1],that the Nichols algebras of many braided vector spaces of admissible flour-ished Dynkin diagrams are finite-dimensional. This result extends Example1.1 and the Jordan plane [CLW] and is reminiscent of a familiar phenomenonin Lie algebras in positive characteristic. By bosonization we obtain manynew examples of finite-dimensional pointed Hopf algebras.1.2. The main result.
To describe more precisely our main Theorem weneed first to discuss blocks.For k < ℓ ∈ N , we set I k,ℓ = { k, k + 1 , . . . , ℓ } , I ℓ = I ,ℓ .A block V ( ǫ, ℓ ) , where ǫ ∈ k × and ℓ ∈ N ≥ , is a braided vector space witha basis ( x i ) i ∈ I ℓ such that for i, j ∈ I ℓ , < j : c ( x i ⊗ x ) = ǫx ⊗ x i , c ( x i ⊗ x j ) = ( ǫx j + x j − ) ⊗ x i . (1.1)In characteristic 0, the only Nichols algebras of blocks with finite GKdim are the Jordan plane B ( V (1 , and the super Jordan plane B ( V ( − , ;both have GKdim = 2 . In our context with p > , the Jordan plane B ( V (1 , has dimension p [CLW]; see Lemma 3.1. Our starting resultis that the super Jordan plane B ( V ( − , has dimension p , see Propo-sition 3.2. For simplicity a block V ( ǫ, of dimension 2 is called an ǫ -block. OINTED HOPF ALGEBRAS IN POSITIVE CHARACTERISTIC 3
We also prove that a block V ( ǫ, has finite-dimensional Nichols algebra onlywhen ǫ = ± , see Proposition 3.3.The braided vector spaces in this paper belong to the class analogous tothe one considered in [AAH1]. Briefly, ( V, c ) belongs to this class if V = V ⊕ · · · ⊕ V t ⊕ V t +1 ⊕ · · · ⊕ V θ , (1.2) c ( V i ⊗ V j ) = V j ⊗ V i , i, j ∈ I θ , (1.3)where V h is a ǫ h -block, with ǫ h = 1 , for h ∈ I t ; and dim V i = 1 with braidingdetermined by q ii ∈ k × (we say that i is a point), i ∈ I t +1 ,θ ; the braidingbetween points i and j is given by q ij ∈ k × while the braiding between apoint and block, respectively two blocks, should have the form as in (4.1),respectively (6.1). For convenience, we attach to ( V, c ) a flourished graph D with θ vertices, those corresponding to a -block decorated with ⊞ , thoseto − -block decorated with ⊟ and the point i with q ii ◦ . If i = j are points,and there is an edge between them decorated by e q ij := q ij q ji when this is = − , or no edge if e q ij = 1 . If h is a block and j is a point, then there is anedge between h and j decorated either by G hj if the interaction is weak and G hj = 0 is the ghost, cf. (4.2), or by ( − , G hj ) if the interaction is mild and G hj is the ghost; but no edge if the interaction is weak and G hj = 0 . Thereare no edges between blocks and we assume that the diagram is connectedby a well-known reduction argument.This class of braided vector spaces together with those of diagonal typedoes not exhaust that of Yetter-Drinfeld modules arising from abelian groups;there are still those containing a pale block as in [AAH1, Chapter 8]. Syn-thetically our main result is the following. Theorem 1.2.
Let V be a braided vector space as in the following list, then dim B ( V ) < ∞ . (a) V has braiding (4.1) and is listed in Table 1, or (b) V has braiding (5.1) and is listed in Table 2, or (c) V has braiding (6.1) , or (d) V has braiding (7.1) and is listed in Table 3.By bosonization with suitable abelian groups, we get examples of finite-dimensional pointed Hopf algebras in positive characteristic. Concrete examples of such Hopf algebras are described in §3.3, §4.5, §5.3,§6.1 and §7.3. We also give a presentation by generators and relations of theNichols algebras; references to this information and the dimensions are alsogiven in the Tables.All braided vector spaces in this Theorem belong to the class describedabove except those in (d) that contain a pale block.1.3.
Contents of the paper.
Section 2 is devoted to preliminaries. Thenext Sections contain the examples of finite-dimensional Nichols algebrasand some realizations over abelian groups; each Section describes a family of
ANDRUSKIEWITSCH, ANGIONO AND HECKENBERGER
Table 1.
Finite-dimensional Nichols algebras of a block and a point V diagram q G B ( V ) dim K dim B ( V ) L (1 , G ) ⊞ G • discrete §4.3.1 p r +1 p r +3 L ( − , G ) ⊞ G − • − discrete §4.3.2 r +1 r +1 p L ( ω, ⊞ ω • ∈ G ′ p L − (1 , G ) ⊟ G • discrete §4.3.3 r p r +1 r +2 p r +3 L − ( − , G ) ⊟ G − • − discrete §4.3.4 r +1 p r r +3 p r +2 C ⊟ ( − , − • −
16 64 p Table 2.
Finite-dimensional Nichols algebras of a block andseveral points, ω ∈ G ′ . V diagram B ( V ) dim B ( V ) L ( A θ − ) , ⊞ − • − − ◦ . . . − ◦ − − ◦ §5.2.7 p θ > θ − vertices p ( θ − θ − L ( A , ⊞ − • − − ◦ §5.2.6 p L ( A (1 | ; ω ) ⊞ − • ω − ◦ §5.2.2 p L ( A (1 | ; ω ) ⊞ − • ω ω ◦ §5.2.1 p L ( A (1 | ; ω ) ⊞ ω • ω − ◦ §5.2.3 p L ( A (1 | ; r ) ⊞ − • r − r ◦ , r ∈ G ′ N , N > §5.2.1 p N L ( A (2 | ; ω ) ⊞ − • ω ω ◦ ω ω ◦ §5.2.4 p L ( D (2 | ω ) ⊞ − • ω ω ◦ ω ω ◦ §5.2.5 p braided vector spaces with a certain decomposition as we describe now. InSection 3 we compute Nichols algebras of a block. In Section 4 we presentexamples of Nichols algebras corresponding to one block and one point, whilein Section 5 we consider the case one block and several points. Section 6 isdevoted to examples of several blocks and one point. Finally in Section 7we give examples of finite-dimensional Nichols algebras whose braided vectorspaces decompose as one pale block and one point.2. Preliminaries
OINTED HOPF ALGEBRAS IN POSITIVE CHARACTERISTIC 5
Table 3.
Finite-dimensional Nichols algebras of a pale block anda point.
V ǫ e q q B ( V ) dim K dim B ( V ) E p ( q ) 1 1 − §7.1 p p p E + ( q ) − §7.2 p p E − ( q ) − − §7.2 p p E ⋆ ( q ) − − − §7.2 p p Conventions.
The q -numbers are the polynomials ( n ) q = n − X j =0 q j , ( n ) ! q = n Y j =1 ( j ) q , (cid:18) ni (cid:19) q = ( n ) ! q ( n − i ) ! q ( i ) ! q ∈ Z [ q ] ,n ∈ N , ≤ i ≤ n . If q ∈ k , then ( n ) q , ( n ) ! q , (cid:0) ni (cid:1) q denote the evaluations of ( n ) q , ( n ) ! q , (cid:0) ni (cid:1) q at q = q .Let G N be the group of N -th roots of unity, and G ′ N the subset of primitiveroots of order N ; G ∞ = S N ∈ N G N . All the vector spaces, algebras and tensorproducts are over k .All Hopf algebras have bijective antipode.2.2. Yetter-Drinfeld modules.
Let Γ be an abelian group. We denoteby b Γ the group of characters of Γ . The category k Γ k Γ YD of Yetter-Drinfeldmodules over the group algebra k Γ consists of Γ -graded Γ -modules, the Γ -grading being denoted by V = ⊕ g ∈ Γ V g ; that is, hV g = V g for all g, h ∈ Γ . If g ∈ Γ and χ ∈ b Γ , then the one-dimensional vector space k χg , with action andcoaction given by g and χ , is in HH YD . Let W ∈ k Γ k Γ YD and ( w i ) i ∈ I a basis of W consisting of homogeneous elements of degree g i , i ∈ I , respectively. Thenthere are skew derivations ∂ i , i ∈ I , of T ( W ) such that for all x, y ∈ T ( W ) , i, j ∈ I ∂ i ( w j ) = δ ij , ∂ i ( xy ) = ∂ i ( x )( g i · y ) + x∂ i ( y ) . (2.1)For a definition of Yetter-Drinfeld modules over arbitrary Hopf algebraswe refer e.g. to [R, 11.6].2.3. Nichols algebras.
Nichols algebras are graded Hopf algebras B = ⊕ n ≥ B n in HH YD coradically graded and generated in degree one. They arecompletely determined by V := B ∈ HH YD and it is customary to denote B = B ( V ) . If W ∈ k Γ k Γ YD as in Subsection 2.2, then the skew-derivations ∂ i induce skew-derivations on B ( W ) . Moreover, an element w ∈ B k ( W ) , k ≥ , is zero if and only if ∂ i ( w ) = 0 in B ( W ) for all i ∈ I . A pre-Nicholsalgebra of V is a graded Hopf algebra in HH YD generated in degree one, withthe one-component isomorphic to V . ANDRUSKIEWITSCH, ANGIONO AND HECKENBERGER
Example 2.1.
Let V be of dimension 1 with braiding c = ǫ id . Let N bethe smallest natural number such that ( N ) ǫ = 0 . Then B ( V ) = k [ T ] / h T N i ,or B ( V ) = k [ T ] if such N does not exist.A braided vector space V is of diagonal type if there exist a basis ( x i ) i ∈ I θ of V and q = ( q ij ) i,j ∈ I θ ∈ k θ × θ such that q ij = 0 and c ( x i ⊗ x j ) = q ij x j ⊗ x i for all i, j ∈ I = I θ . Given a braided vector space V of diagonal type with abasis ( x i ) , we denote in T ( V ) , or B ( V ) , or any intermediate Hopf algebra, x ij = (ad c x i ) x j , x i i ...i M = (ad c x i ) x i ...i M , (2.2)for i, j, i , . . . , i M ∈ I , M ≥ . A braided vector space V of diagonal typeis of Cartan type if there exists a generalized Cartan matrix a = ( a ij ) suchthat q ij q ji = q a ij ii for all i = j . Theorem 2.2. If V is of Cartan type with matrix a that is not finite, then dim B ( V ) = ∞ .Proof. The argument in [AAH2, Proposition 3.1] is characteristic-free andapplies here because there are infinite real roots in the root system of a . (cid:3) Blocks
We consider braided vector spaces V ( ǫ, with braiding (1.1), ǫ = 1 .3.1. The Jordan plane.
Here we deal with V = V (1 , . In characteristic0, B ( V ) is the well-known algebra presented by x and x with the relation(3.1). In positive characteristic, B ( V ) is a truncated version of that algebra. Lemma 3.1. [CLW] B ( V ) is presented by generators x , x and relations x x − x x + 12 x , (3.1) x p , (3.2) x p . (3.3) Also dim B ( V ) = p and { x a x b : 0 ≤ a, b < p } is a basis of B ( V ) . (cid:3) In characteristic 2, the relations of B ( V ) are different.Let Γ = Z /p = h g i . We realize V in k Γ k Γ YD by g · x = x , g · x = x + x , deg x i = g , i ∈ I . Thus the Hopf algebra B ( V ) k Γ has dimension p .3.2. The super Jordan plane.
Let V = V ( − , be the braided vectorspace with basis x , x and braiding c ( x i ⊗ x ) = − x ⊗ x i , c ( x i ⊗ x ) = ( − x + x ) ⊗ x i , i ∈ I . (3.4)Let g be a generator of the cyclic group Z . We realize V ( − , in kZkZ YD by g · x = − x , g · x = − x + x , deg x i = g , i ∈ I . As in (2.2), x = (ad c x ) x = x x + x x . (3.5) OINTED HOPF ALGEBRAS IN POSITIVE CHARACTERISTIC 7
The Nichols algebra B ( V ( − , T ( V ( − , / J ( V ( − , (called thesuper Jordan plane) was studied in [AAH1, 3.3] over fields of characteristic0. Assuming p > , the basic features of B ( V ( − , are summarized here: Proposition 3.2.
The defining ideal J ( V ( − , is generated by x , (3.6) x x − x x − x x , (3.7) x p , (3.8) x p . (3.9) The set B = { x a x b x c : a ∈ I , , b ∈ I ,p − , c ∈ I , p − } is a basis of B ( V ) and dim B ( V ) = 4 p .Proof. Since ∂ ( x ) = 0 , we have that ∂ ( x n ) = 0 for every n ∈ N . Also ∂ ( x ) = x and g · x = x . Both (3.6) and (3.7) are in B ( V ) beingannihilated by ∂ and ∂ , cf. (2.1). From (3.6) and (3.7) we see that in B ( V ) x x = x ( x + x ) , (3.10) x x = x x x = x x . (3.11)By the preceding, we have ∂ ( x n ) = X ≤ i ≤ n x i − ∂ ( x ) x n − i = nx x n − . Hence x p = 0 , i.e. (3.8) holds. Next we prove (3.9). Clearly ∂ ( x n ) = 0 forevery n ∈ N . We observe that g · x = ( − x + x ) = x − x , ∂ ( x ) = g · x + x = x . Setting for simplicity a := x and b := x , we have for any n ∈ N : ∂ ( x n ) = X ≤ i ≤ n x i − ∂ ( x ) (cid:16) g · x n − i )2 (cid:17) = X ≤ i ≤ n b i − x ( b − a ) n − i . By (3.7), (3.10) and (3.11) we have ax = x a, bx = x ( b + a ) , ba = a ( a + b ) , (3.12)hence ∂ ( x n ) = x X ≤ i ≤ n ( b + a ) i − ( b − a ) n − i . We prove recursively that for all n ∈ N ( b − a ) n = b n − nab n − , A n := X i ∈ I n ( b + a ) i − ( b − a ) n − i = nb n − . (3.13)The case n = 1 is evident. We start with the first identity: ( b − a ) n +1 = ( b − a ) ( b − a ) n = ( b − a )( b n − nab n − )= b n +1 − nbab n − − ab n + na b n − = b n +1 − ( n + 1) ab n ANDRUSKIEWITSCH, ANGIONO AND HECKENBERGER as desired. For the second identity we use the first: A n +1 = ( b − a ) n + ( b + a ) A n = b n − nab n − + ( b + a ) nb n − = ( n + 1) b n . The claim is proved; summarizing we have ∂ ( x n ) = nx x n − . (3.14)In particular, this implies (3.9).We now argue as in [AAH1, 3.3.1]. The quotient e B of T ( V ) by (3.6), (3.7),(3.8) and (3.9) projects onto B ( V ) and the subspace I spanned by B is a leftideal of e B , by (3.7), (3.11). Since ∈ I , e B is spanned by B . To prove that e B ≃ B ( V ) , we just need to show that B is linearly independent in B ( V ) .We claim that this is equivalent to prove that B ′ = { x c x b x a : a ∈ { , } , b ∈ I ,p − , c ∈ I , p − } is linearly independent. Indeed, e B is spanned by B ′ sincethe subspace spanned B ′ is also a left ideal; if B ′ is linearly independent,then the dimension of e B is p , so B should be linearly independent and viceversa. Suppose that there is a non-trivial linear combination of elements of B ′ in B ( V ) of minimal degree. As ∂ ( x c x b ) = b x c x b − x , ∂ ( x c x b x ) = x c x b , (3.15)such linear combination does not have terms with a or b greater than 0. Weclaim that the elements x c , c ∈ I , p − , are linearly independent, yielding acontradiction. By homogeneity it is enough to prove that they are = 0 . If c is even this follows from (3.14). If c = 2 n + 1 with n < p , then ∂ ( x n +12 ) = ∂ ( x n ) g · x + x n = − nx x n − + x n . Again a degree argument gives the desired claim. (cid:3)
Let
Γ = Z / p . We may realize V ( − , in k Γ k Γ YD by the same formulas asabove; thus B ( V ) k Γ is a pointed Hopf algebra of dimension p .3.3. Realizations.
Let H be a Hopf algebra. A YD-pair for H is a pair ( g, χ ) ∈ G ( H ) × Hom alg ( H, k ) such that χ ( h ) g = χ ( h (2) ) h (1) g S ( h (3) ) , h ∈ H. (3.16)Let k χg be a one-dimensional vector space with H -action and H -coactiongiven by χ and g respectively; then (3.16) says that k χg ∈ HH YD .If χ ∈ Hom alg ( H, k ) , then the space of ( χ, χ ) -derivations is Der χ,χ ( H, k ) = { η ∈ H ∗ : η ( hℓ ) = χ ( h ) η ( ℓ ) + χ ( ℓ ) η ( h ) ∀ h, ℓ ∈ H } . A YD-triple for H is a collection ( g, χ, η ) where ( g, χ ) , is a YD-pair for H , η ∈ Der χ,χ ( H, k ) , η ( g ) = 1 and η ( h ) g = η ( h (2) ) h (1) g S ( h (3) ) , h ∈ H. (3.17)Given a YD-triple ( g, χ, η ) we define V g ( χ, η ) ∈ HH YD as the vector spacewith a basis ( x i ) i ∈ I , whose H -action and H -coaction are given by h · x = χ ( h ) x , h · x = χ ( h ) x + η ( h ) x , δ ( x i ) = g ⊗ x i , h ∈ H, i ∈ I ; OINTED HOPF ALGEBRAS IN POSITIVE CHARACTERISTIC 9 the compatibility is granted by (3.16), (3.17). As a braided vector space, V g ( χ, η ) ≃ V ( ǫ, , ǫ = χ ( g ) .Consequently, if H is finite-dimensional and ǫ = 1 , then B ( V g ( χ, η )) H is a Hopf algebra satisfying dim (cid:0) B ( V g ( χ, η )) H (cid:1) = ( p dim H, when ǫ = 1 , p dim H, when ǫ = − . (3.18)3.4. Exhaustion in rank . We recall some facts from [AAH1, §3.4].Let H be a Hopf algebra with bijective antipode and V ∈ HH YD . Let V ( V · · · ( V d = V be a flag of Yetter-Drinfeld submodules with dim V i = dim V i − + 1 for all i . Then V diag := gr V is of diagonal type. If B is a pre-Nichols algebra of V , then it is a graded filtered Hopf in HH YD and B diag := gr B is a pre-Nichols algebra of V diag . Proposition 3.3.
Let ǫ ∈ k × . If dim B ( V ( ǫ, < ∞ , then ǫ = 1 .Proof. Let V = V ( ǫ, ; it has a flag as above and V diag is the braided vectorspace of diagonal type with matrix ( q ij ) i,j ∈ I , q ij = ǫ for all i, j ∈ I . Hence dim B ( V diag ) ≤ dim B ( V ( ǫ, . (3.19) Step . If ǫ / ∈ G ∞ , then dim B ( V ( ǫ, ∞ . Proof.
Here dim B ( V diag ) = ∞ by Example 2.1 and (3.19) applies. (cid:3) Step . If ǫ ∈ G ′ N , N ≥ , then dim B ( V ( ǫ, ℓ )) = ∞ for all ℓ ≥ . Proof.
Here V diag is of Cartan type with Cartan matrix − N − N ! .Thus Theorem 2.2 and (3.19) apply. (cid:3) Step . Let ǫ ∈ G ′ . Then dim B ( V ( ǫ, ∞ . Proof.
The proof of [AAH1, §3.5 – Step 3] holds verbatim. (cid:3)
The Proposition is proved. (cid:3) One block and one point
The setting and the statement.
Let ( q ij ) ≤ i,j ≤ be a matrix of in-vertible scalars and a ∈ k . We assume that ǫ := q satisfies ǫ = 1 . Let V be a braided vector space with a basis ( x i ) i ∈ I and a braiding given by ( c ( x i ⊗ x j )) i,j ∈ I = ǫx ⊗ x ( ǫx + x ) ⊗ x q x ⊗ x ǫx ⊗ x ( ǫx + x ) ⊗ x q x ⊗ x q x ⊗ x q ( x + ax ) ⊗ x q x ⊗ x . (4.1)Let V = h x , x i (the block) and V = h x i (the point). Let Γ = Z with canonical basis g , g . We realize ( V, c ) in k Γ k Γ YD as V g ( χ , η ) ⊕ k χ g with suitable χ , χ and η , where V = V g ( χ , η ) , while V = k χ g . Thus V ≃ V ( ǫ, ; thus we use the notations and results from §3.2.The interaction between the block and the point is q q ; it isweak if q q = 1 , mild if q q = − , strong if q q / ∈ {± } . In characteristic 0, we introduced a normalized version of a called theghost, which is discrete when it belongs to N . In our context, p > , we needa variant of this notion. First we say that V has discrete ghost if a ∈ F × p .When this is the case, we pick a representative r ∈ Z of a by imposing r ∈ ( { − p, . . . , − } , ǫ = 1 , { , . . . , p − } ∩ Z , ǫ = − set G := ( − r , ǫ = 1 , r , ǫ = − . (4.2)Then G is called the ghost . In this Section we consider the following braidedvectors spaces with braiding (4.1), where the ghost is discrete and q ∈ G ∞ : L ( q , G ) : weak interaction, ǫ = 1; L − ( q , G ) : weak interaction, ǫ = − C : mild interaction, ǫ = q = − , G = 1 . In this Section, we shall prove part (a) of Theorem 1.2.
Theorem 4.1.
Let V be a braided vector space with braiding (4.1) . If V isas in Table 1, then dim B ( V ) < ∞ . To prove the Theorem, we consider K = B ( V ) co B ( V ) . By [HS, Proposi-tion 8.6], B ( V ) ≃ K B ( V ) and K is the Nichols algebra of K = ad c B ( V )( V ) . (4.3)Now K ∈ B ( V ) k Γ B ( V ) k Γ YD with the adjoint action and the coaction given by δ = ( π B ( V ) k Γ ⊗ id)∆ B ( V ) k Γ . (4.4)In order to describe K , we set z n := (ad c x ) n x , n ∈ N . (4.5)4.2. Weak interaction.
Here q q = 1 . In general, c | V ⊗ V = id ⇐⇒ q q = 1 and a = 0 . (4.6)If a = 0 , then B ( V ) ≃ B ( V ( ǫ, ⊗ B ( k x ) . (4.7)Here ⊗ denotes the braided tensor product of Hopf algebras (the structureof Hopf algebra in k G k G YD ) From now on we assume that the ghost is discrete, in particular = 0 . Wefollow the exposition in [AAH1, §4.2]. OINTED HOPF ALGEBRAS IN POSITIVE CHARACTERISTIC 11
Lemma 4.2.
The following formulae hold in B ( V ) for all n ∈ N : g · z n = ǫ n q z n , x z n = ǫ n q z n x , x n x = ( nǫx + x ) x n ,g · z n = q n q z n , x z n = q z n x , x z n = ǫ n q z n x + z n +1 . (4.8) Proof.
The proof of [AAH1, Lemma 4.2.1] is valid in any characteristic. (cid:3)
Let ( µ n ) n ∈ N be the family of elements of k defined recursively by µ = 1 , µ k +1 = − ( a + kǫ ) µ k , µ k = ( a + k + ǫ ( a + k − µ k − . This can be reformulated as µ n +1 = ( (2 a + n ) µ n if n is odd, − a + n µ n if n is even, when ǫ = 1; (4.9) µ n +1 = ( µ n if n is odd, − ( a − n ) µ n if n is even, when ǫ = − . (4.10)Thus µ n = 0 ⇐⇒ n > | r | . Lemma 4.3.
For all k ∈ { , . . . , p − } , ∂ ( z k ) = ∂ ( z k ) = 0 , ∂ ( z k ) = µ k x k , ∂ ( z k +1 ) = µ k +1 x x k . (4.11) Therefore, z n = 0 ⇐⇒ n > | r | .Proof. The proof of [AAH1, Lemma 4.2.2] is valid in any characteristic, tak-ing into the account the conventions on r . (cid:3) If ǫ = 1 , then (4.11) says that ∂ ( z k ) = ( − k k µ k x k , ∂ ( z k +1 ) = ( − k k µ k +1 x k +11 . (4.12)Recall that K = B ( V ) co B ( V ) ≃ B ( K ) , K = ad B ( V )( V ) . Lemma 4.4.
The family ( z n ) ≤ n ≤| r | is a basis of K .Proof. The family is linearly independent, because the z n ’s are homogeneousof distinct degrees, and are = 0 by (4.11). We have for all n ∈ N ad c x ( z n ) = x z n − g · z n x (4.8) = ǫ n q z n x − ǫ n q z n x = 0 , (4.13) ad c x ( z n ) = ad c x ad c x ( z n ) − ǫ ad c x ad c x ( z n ) = 0 . (4.14)The Lemma follows. (cid:3) If ǫ = 1 , then we define recursively ν k,n as follows: ν n,n = 1 , ν ,n +1 = − (cid:0) n a (cid:1) ν ,n , ν k,n +1 = ν k − ,n − (cid:18) n + k a (cid:19) ν k,n , ≤ k ≤ n. Lemma 4.5.
The coaction (4.4) on z n , ≤ n ≤ | r | , is given by (4.15) ,when ǫ = 1 , and by (4.16) , (4.17) , when ǫ = − : δ ( z n ) = n X k =0 ν k,n x n − k g k g ⊗ z k . (4.15) δ ( z n ) = n X k =1 k (cid:18) nk (cid:19) µ k,n x x n − k g k − g ⊗ z k − (4.16) + n X k =0 (cid:18) nk (cid:19) µ k,n x n − k g k g ⊗ z k ,δ ( z n +1 ) = n X k =0 (cid:18) nk (cid:19) µ k,n +1 x x n − k g k g ⊗ z k (4.17) + n X k =0 (cid:18) nk (cid:19) µ k +1 ,n +1 x n − k g k +11 g ⊗ z k +1 . Proof.
The proof of [AAH1, Lemma 4.2.1] can be adapted since (cid:0) nk (cid:1) = 0 byassumption. (cid:3) We are ready to prove the finite-dimensionality in the case of weak inter-action. We claim that the braided vector space K is of diagonal type withbraiding matrix ( p ij ) ≤ i,j ≤| r | = ( ǫ ij q i q j q ) ≤ i,j ≤| r | . Hence, the corresponding generalized Dynkin diagram has labels p ii = ǫ i q , p ij p ji = q , i = j ∈ I , | r | . Indeed, by Lemma 4.4 it is enough to compute c ( z i ⊗ z j ) = g i g · z j ⊗ z i = ǫ ij q i q j q z j ⊗ z i , by Lemmas 4.5 and 4.2, (4.13) and (4.14). We proceed then case by case. Case . q = 1 .Here the Dynkin diagram of K is totally disconnected with vertices i ∈ I , | r | labelled with ǫ i q . The vertices with label , respectively − ,contribute with p , respectively , to dim B ( K ) . Case . ǫ = 1 , q ∈ G ′ , | r | = 1 (provided that p > ).The Dynkin diagram is of Cartan type A , so dim B ( K ) < ∞ . OINTED HOPF ALGEBRAS IN POSITIVE CHARACTERISTIC 13
The presentation by generators and relations.
We still assumethat the interaction is weak. We start by some general Remarks that areproved exactly as in [AAH1, §4.3].
Remark . Let y k = x k , y k +1 = x x k , k ∈ N By Lemma 4.2 ∂ ( z t ) = µ t y t , z t y n = ǫ nt q n y n z t , t, n ∈ N . (4.18) Lemma 4.7.
Assume that ǫ = q = 1 . In B ( L ( q , G )) , or correspondingly B − ( L ( q , G )) , z | r | +1 = 0 , (4.19) z t z t +1 = q q z t +1 z t t ∈ N , t < | r | , (4.20) z t = 0 t ∈ N , ǫ t q = − . (4.21) ∂ ( z n +1 t ) = µ t q nt q n n y t z nt , n, t ∈ N , ǫ t q = 1 . (4.22) Lemma 4.8.
Let B be a quotient algebra of T ( V ) . Assume that x x = q x x , and either(a) (3.1) , or else(b) (3.7) , x x = q x x hold in B . Then for all n ∈ N , x z n = ǫ n q z n x (4.23) x z n = q z n x . (4.24) Lemma 4.9.
Let B be a quotient algebra of T ( V ) , ǫ = q = 1 . (i) Assume that (4.20) and (4.21) hold in B . Then for ≤ t < k ≤ | r | , z t z k = ǫ tk q k − t q z k z t . (4.25) (ii) Assume that z t = 0 in B for t ∈ N such that ǫ t q = − . Then z t z t +1 = q q z t +1 z t in B . In other words, (ii) says that (4.21) for a specific t implies (4.20) for t .4.3.1. The Nichols algebra B ( L (1 , G )) . Proposition 4.10.
Let G ∈ I p − . The algebra B ( L (1 , G )) is presented bygenerators x , x , x and relations (3.1) , (3.2) , (3.3) , together with x x = q x x , (4.26) z G = 0 , (4.27) z t z t +1 = q − z t +1 z t , ≤ t < G , (4.28) z pt = 0 , ≤ t ≤ G , (4.29) The dimension of B ( L (1 , G )) is p G +3 , since it has a PBW-basis B = { x m x m z n G G . . . z n z n : m i , n j ∈ I ,p } . . (cid:3) The Nichols algebra B ( L ( − , G )) . Proposition 4.11.
Let G ∈ I p − . The algebra B ( L ( − , G )) is presented bygenerators x , x , x and relations (3.1) , (3.2) , (3.3) , (4.26) , (4.27) and z t = 0 , ≤ t ≤ G . (4.30) The dimension of B ( L (1 , G )) is G +1 p , since it has a PBW-basis B = { x m x m z n G G . . . z n z n : n i ∈ { , } , m j ∈ I ,p − } . (cid:3) The Nichols algebra B ( L − (1 , G )) . Proposition 4.12.
Let G ∈ I p − ∩ Z . The algebra B ( L − (1 , G )) is pre-sented by generators x , x , x and relations (3.6) , (3.7) , (3.8) , (3.9) , (4.26) and z G = 0 , (4.31) x z = q z x , (4.32) z k +1 = 0 , ≤ k < G / , (4.33) z k z k +1 = q − z k +1 z k , ≤ k < G / . (4.34) The dimension of B ( L (1 , G )) is G +2 p G +3 , since it has a PBW-basis B = { x m x m x m z n G G . . . z n z n : m , n k +1 ∈ { , } ,m ∈ I , p − , m , n k ∈ I ,p − } . (cid:3) The Nichols algebra B ( L − ( − , G )) . Proposition 4.13.
Let G ∈ I p − ∩ Z . The algebra B ( L − ( − , G )) is pre-sented by generators x , x , x and relations (3.6) , (3.7) , (3.8) , (3.9) , (4.26) , (4.31) , (4.32) and z k = 0 , ≤ k ≤ G / , (4.35) z k − z k = − q − z k z k − , The Nichols algebra B ( L ( ω, . Proposition 4.14. Let ω ∈ G ′ . The algebra B ( L ( ω, is presented bygenerators x , x , x and relations (3.1) , (3.2) , (3.3) , (4.26) , z = 0 , (4.37) z = 0 , (4.38) z = 0 , (4.39) z , = 0 . (4.40) The dimension of B ( L ( ω, is p , since it has a PBW-basis B = { x m x m z n z n , z n : m i ∈ I ,p − , n j ∈ I , } . (cid:3) Mild interaction. We assume in this Subsection that q q = − ǫ , a = 1 , q = − . The corresponding braided vector space is denoted C , asabove. We proceed as above but now the elements z n = (ad c x ) n x are notenough to describe K and we need f n = ad c x ( z n ) , n = 0 , . Then x z = f + q z x ,x z = f − q z x + q f x ,x z = z + q z x . (4.41) Proposition 4.15. The Nichols algebra B ( C ) is presented by generators x , x , x and relations (3.6) , (3.7) , (3.8) , (3.9) , x z + q z x = 12 f + q f x ,x f = q f x , x f + q f x = − f ,z = 0 , f = 0 , z = 0 , f = 0 . (4.42) The dimension of B ( C ) is p , since it has a PBW-basis B = { x m x m x m f n f n z n z n : m , n i ∈ { , } , m , m ∈ I p } . (cid:3) Realizations. Let H be a Hopf algebra, ( g , χ , η ) a YD-triple and ( g , χ ) a YD-pair for H , see §3.3. Let ( V, c ) be a braided vector space withbraiding (4.1). Then V g ( χ , η ) ⊕ k χ g ∈ HH YD is a principal realization of ( V, c ) over H if q ij = χ j ( g i ) , i, j ∈ I ; a = q − η ( g ) . Thus ( V, c ) ≃ V g ( χ , η ) ⊕ k χ g as braided vector space. Hence, if H isfinite-dimensional and ( V, c ) is as in Table 1, then B (cid:0) V g ( χ , η ) ⊕ k χ g (cid:1) H is a finite-dimensional Hopf algebra. Examples of finite-dimensional pointedHopf algebras A = B (cid:0) V g ( χ , η ) ⊕ k χ g (cid:1) k Γ like this are listed in Table 4.In all cases Γ is a product of two cyclic groups, g = (1 , , g = (0 , and χ j ( g i ) = 1 if i = j ; hence it remains to fix the value of q . Table 4. Pointed Hopf algebras K from a block and a point V diagram Γ q dim A L (1 , G ) ⊞ G • Z /p × Z /p p r +5 L ( − , G ) ⊞ G − • Z /p × Z / p r +2 p L ( ω, ⊞ ω • Z /p × Z / p p L − (1 , G ) ⊟ G • Z / p × Z /p r +3 p r +5 L − ( − , G ) ⊟ G − • Z / p × Z / p ± r +5 p r +4 C ⊟ ( − , − • Z / p × Z / p ± p One block and several points The setting and the main result. Let θ ∈ N ≥ , I ,θ = I θ − { } , I † θ = I θ ∪ { } . Let ⌊ i ⌋ be the largest integer ≤ i . We start from the data ( q ij ) i,j ∈ I θ ∈ ( k × ) θ × θ , q = 1; ( a , . . . , a θ ) ∈ k I ,θ . We assume that q = 1 =: a . Let ( V, c ) be the braided vector space ofdimension θ + 1 , with a basis ( x i ) i ∈ I † θ and braiding given by c ( x i ⊗ x j ) = ( q ⌊ i ⌋ j x j ⊗ x i , i ∈ I † θ , j ∈ I θ ; q ⌊ i ⌋ ( x + a ⌊ i ⌋ x ) ⊗ x i , i ∈ I † θ , j = . (5.1)We say that the block and the points have discrete ghost if a j ∈ F I ,θ p , ( a j ) = 0 . When this is the case, we pick the representative r j ∈ Z of a j byimposing r j ∈ { − p, . . . , − , } , and set G j = − r j . The ghost between theblock and the points is the vector G = ( G j ) j ∈ I ,θ given by G = − ( r j ) j ∈ I ,θ ∈ N I ,θ . (5.2)The braided subspace V spanned by x , x is ≃ V (1 , , while V diag spanned by ( x i ) i ∈ I ,θ is of diagonal type. Obviously, V = V ⊕ V diag . (5.3)Let X be the set of connected components of the Dynkin diagram of thematrix q = ( q ij ) i,j ∈ I ,θ . If J ∈ X , then we set J ′ = I ,θ − J , V J = X j ∈ J k χ j g j , G J = ( G j ) j ∈ J . OINTED HOPF ALGEBRAS IN POSITIVE CHARACTERISTIC 17 We shall use the results and notations from the preceding Sections, butwith replacing 2 when appropriate, e. g. x = x x − x x . Let K = B ( V ) co B ( V ) and K = ad c B ( V )( V diag ) ∈ B ( V ) k Γ B ( V ) k Γ YD , so that B ( V ) ≃ K B ( V ) , K ≃ B ( K ) Let z j,n := ( ad c x ) n x j , j ∈ I ,θ , n ∈ N . (5.4)For all i, j ∈ I ,θ , n ∈ N , we have as in [AAH1, §5.2.1] that g · z j,n = q j z j,n , (by Lemma 4.2)(5.5) g i · z j,n = q ni q ij z j,n , (5.6) Lemma 5.1. The braided vector space K is of diagonal type in the basis ( z j,n ) j ∈ I ,θ , ≤ n ≤ G j (5.7) with braiding matrix ( p im,jn ) i,j ∈ I ,θ , ≤ m ≤ G i , ≤ n ≤ G j = ( q ni q m j q ij ) i,j ∈ I ,θ , ≤ m ≤ G i , ≤ n ≤ G j . Hence, the corresponding generalized Dynkin diagram has labels p im,im = q ii , p im,jn p jn,im = q ij q ji , ( i, m ) = ( j, n ) . Proof. The proof in [AAH1, Lemma 7.2.5] applies as the combinatorial num-bers appearing there are not zero. (cid:3) Let K J be the braided vector subspace of K spanned by ( z j,n ) j ∈ J, ≤ n ≤ G j . Corollary 5.2. The braided subspaces corresponding to the connected com-ponents of the Dynkin diagram of K are K J , J ∈ X . Hence dim K = dim B ( K ) = Y J ∈X dim B ( K J ) . (cid:3) (5.8)Observe that if G J = 0 , then K J = V J .In this Section, we shall prove part (b) of Theorem 1.2. Theorem 5.3. Let V be a braided vector space with braiding (5.1) . Assumethat for every J ∈ X , either G J = 0 , or else dim V J = 1 and V ⊕ V j is as inTable 1, or else V J is as in Table 2. Then dim B ( V ) = p Y J ∈X dim B ( K J ) < ∞ . (5.9) Proof. By Corollary 5.2 we reduce to connected components in X . If J ∈ X has weak interaction and G J = 0 , then B ( V ) ≃ B ( V ⊕ V J ′ ) ⊗ B ( V J ) , hence dim B ( V ) = dim B ( V ⊕ V J ′ ) dim B ( V J ) .If J ∈ X is a point, then Theorem 4.1 applies. We need to analyze those J with | J | ≥ and G J = 0 .Below we denote ı = √− . L ( A ) , G J = (1 , : Here K J is of Cartan type A and dim B ( K J ) = 2 . L ( A j ) , j > , G J = (1 , : Here K J is of Cartan type D j − and dim B ( K J ) =2 j ( j +1) . L ( A , : Here K J is of Cartan type D and dim B ( K J ) = 2 . L ( A (1 | ; ω ) , provided that p > : Here K J is of diagonal type in the basis z i, , z i, , z j, with diagram − ◦ ω ✾✾✾✾✾✾✾✾ − ◦ ω − ◦ Then dim B ( K J ) < ∞ by [H2, Table 2, row 15]; it has the same root systemas g (2 , and dimension , see [AA, 8.3.4]. L ( A (1 | ; ω ) and L ( A (1 | ; ω ) , provided that p > : Here K J is of diagonaltype in the basis z i, , z j, , z j, , respectively z i, , z i, , z j, , and its diagram is − ◦ ω ✞✞✞✞✞✞✞✞✞ ω ◦ ω − ◦ ω ◦ ω ω ✼✼✼✼✼✼✼✼ ω ◦ ω − ◦ Then dim B ( K J ) < ∞ by [H2, Table 2, row 8], respectively [H2, Table 2, row15]. In the first case it has the same root system as sl (2 | and dimension , see [AA, 5.1.8]. In the second case it has the same root system as g (2 , and dimension , see [AA, 8.3.4].R L ( A (1 | ; r ) : analogous to L ( A (1 | ; ω ) ; same root system as sl (2 | anddimension N , see [AA, 5.1.8]. L ( A (2 | ; ω ) and L ( D (2 | ω ) , p > : In both cases, dim B ( K J ) = 2 ; ithas the same root system as g (3 , , see [H2, Table 3, row 18], [AA, 8.4.5]. (cid:3) The presentation of the Nichols algebras. We give defining re-lations and an explicit PBW basis of B ( V ) , for all V as in Theorem 5.3,assuming that the Dynkin diagram of V diag is connected, i.e. V diag = V J ,where J = I ,θ . Essentially the relations are the same as in [AAH1] up toadding the suitable p -powers; we omit the proofs as they are minor variationsof those in loc. cit. The passage from connected V diag to the general caseis standard, just add the quantum commutators between points in differentcomponents. Since the case | J | = 1 was treated in §4, we also suppose that OINTED HOPF ALGEBRAS IN POSITIVE CHARACTERISTIC 19 | J | > . These braided vector spaces have names given in [AAH1], see Table2. The braided vector subspace V ⊕ k x of such V is of type • L ( − , when V is of type L ( A , , • L ( ω, when V is of type L ( A (1 | ; ω ) , or • L ( − , for all the other cases.Thus the subalgebra generated by V ⊕ k x is a Nichols algebra. We recallits relations up to the change of index with respect to §4; the and thereare now and . As in (2.2), we set x i i ...i M = ad c x i x i ...i M . Also, wehave now z n = (ad c x ) n x , n ∈ N .First, the defining relations of B ( L ( − , are x x − x x + 12 x , (5.10) x p , (3.2) x p , (5.11) x x − q x x , (5.12) (ad c x ) x , (5.13) x , x . (5.14)Second, the defining relations of B ( L ( − , are (5.10), (3.2), (5.11), (5.12),(5.14) and (ad c x ) x , (5.15) x 32 32 . (5.16)Third, the defining relations of B ( L ( ω, are (5.10), (3.2), (5.11), (5.12),(5.13) and x , (5.17) x , (5.18) [ x , x ] c . (5.19)We also observe that, since q j q j = 1 and G j = 0 , we have x x j = q j x j x , x x j = q j x j x , j ∈ I ,θ . (5.20)5.2.1. The Nichols algebra B ( L ( A (1 | ; r )) , r ∈ G ′ N , N ≥ . Letц = x , ц = [ x , x ] c . (5.21) Proposition 5.4. The algebra B ( L ( A (1 | ; r )) is presented by generators x i , i ∈ I † , and relations (5.10) , (3.2) , (5.11) , (5.12) , (5.13) , (5.14) (5.20) ,and (ad c x ) x = 0 , x N = 0 , (5.22) ц N = 0 , (5.23) The set B = (cid:8) x m x m x n ц n ц n x n x n x n :0 ≤ n , n , n , n < , ≤ n , n < N, ≤ m , m < p (cid:9) is a basis of B ( L ( A (1 | ; r )) and dim B ( L ( A (1 | ; r )) = p N . (cid:3) The Nichols algebra B ( L ( A (1 | ; ω )) . Proposition 5.5. The algebra B ( L ( A (1 | ; ω )) is presented by generators x i , i ∈ I † , and relations (5.10) , (3.2) , (5.11) , (5.12) , (5.13) , (5.14) , (5.20) ,and x = 0 , x = 0 , x = 0 , (5.24) ц = 0 , ц = 0 , [ ц , x ] c = 0 , (5.25) The set B = (cid:8) x m x m x n ц n [ ц , [ ц , ц ] c ] n c [ ц , ц ] n c ц n [ ц , x ] n c [ ц , x ] n c x n x n x n : 0 ≤ m , m , < p, ≤ n , n , n < , ≤ n < , ≤ n , n , n , n , n , n < (cid:9) is a basis of B ( L ( A (1 | ; ω )) and dim B ( L ( A (1 | ; ω )) = p . (cid:3) The Nichols algebra B ( L ( A (1 | ; ω )) . Proposition 5.6. The algebra B ( L ( A (1 | ; ω )) is presented by generators x i , i ∈ I † , and relations (5.10) , (3.2) , (5.11) , (5.12) , (5.13) , (5.17) , (5.18) , (5.19) (5.20) , x = 0 , x = 0 , x = 0 , (5.26) x x + q q x x = 0 , ц = 0 . (5.27) The set B = (cid:8) x m x m x n ц n [ ц , ц ] n c ц n [ ц , ц ] n c [ ц , x ] n c ц n x n x n x n : 0 ≤ m , m < p, ≤ n , n , n , n , n , n < , ≤ n , n , n < , ≤ n < (cid:9) is a basis of B ( L ( A (1 | ; ω )) and dim B ( L ( A (1 | ; ω )) = p . (cid:3) The Nichols algebra B ( L ( A (2 | ; ω )) . Proposition 5.7. The algebra B ( L ( A (2 | ; ω )) is presented by generators x i , i ∈ I † , and relations (5.10) , (3.2) , (5.11) , (5.12) , (5.13) , (5.14) , (5.20) ,and x = 0 , x = 0 , x = 0 , x = 0 , (5.28) OINTED HOPF ALGEBRAS IN POSITIVE CHARACTERISTIC 21 x = 0 , x = 0 , x = 0 , x = 0 , (5.29) [ x , x ] c = 0 , [[ x , x ] c , [ x , x ] c ] c = 0 , [ x , x ] c = 0 , [[ x , x ] c , [[ x , x ] c , x ] c ] c = 0 , [[ x , x ] c , x ] c = 0 , [[ x , x ] c , [[ x , x ] c , x ] c ] c = 0 . (5.30) The set B = (cid:8) x m x m x n ц n [ ц , ц ] n c ц n [ ц , ц ] n c [ ц , [ ц , x ] c ] n c ц n [ ц , [ ц , x ] c ] n c [ ц , x ] n c [ ц , x ] n c x n x n x n x n x n x n x n : 0 ≤ n , n , n , n , n , n , n , n < , ≤ m , m < p, ≤ n , n , n , n , n , n , n , n , n < (cid:9) is a basis of B ( L ( A (2 | ; ω )) and dim B ( L ( A (2 | ; ω )) = p . (cid:3) The Nichols algebra B ( L ( D (2 | ω )) . Proposition 5.8. The algebra B ( L ( D (2 | ω )) is presented by generators x i , i ∈ I † , and relations (5.10) , (3.2) , (5.11) , (5.12) , (5.13) , (5.14) , (5.20) ,and [[[ x , x ] c , x ] c , x ] c = 0 , [[ x , x ] c , x ] c = 0 , [[ x , x ] c , x ] c = 0 , [ x , x ] c = 0 , [ x , x ] c = 0 , (5.31) x = 0 , x = 0 , x = 0 , [[ x , x ] c , x ] c = 0 , (5.32) x = 0 , x = 0 , x = 0 , x = 0 , x = 0 . (5.33) The set B = { x m x m x n ц n ц n ц n [ ц , x ] n c [[ ц , x ] c , x ] n c [ ц , x ] n c x n [ x , x ] n c x n x n x n [ x , x ] n c x n x n x n x n :0 ≤ m , m < p, ≤ n , n , n , n , n , n , n , n < , ≤ n , n , n , n , n , n , n , n , n < } . is a basis of B ( L ( D (2 | ω )) and dim B ( L ( D (2 | ω )) = p . (cid:3) The Nichols algebra B ( L ( A , . Proposition 5.9. The algebra B ( L ( A , is presented by generators x i , i ∈ I † , and relations (5.10) , (3.2) , (5.11) , (5.12) , (5.14) , (5.15) , (5.16) , (5.20) ,and x = 0 , x = 0 , x = 0 . (5.34) [ x 32 32 , x ] c , x ] c = 0 , x 32 32 = 0 , [ x 32 32 , x ] c = 0 , [ x 32 32 , x ] c = 0 , x = 0 , [ x , x ] c = 0 , [[ x 32 32 , x ] c , x ] c , x ] c = 0 . (5.35) The set B = { x m x m x n 132 32 x n 232 32 [ x 32 32 , x ] n c [[ x 32 32 , x ] c , x ] n c [[[ x 32 32 , x ] c , x ] c , x ] n c [ x 32 32 , x ] n c x n x n [ x , x ] n x n x n x n : m , m ∈ I ,p − , n i ∈ I , } . is a basis of B ( L ( A , and dim B ( L ( A , p . (cid:3) The Nichols algebra B ( L ( A θ − )) . For details of the following resultwe refer to [AAH1, §5.3.8]. In particular, one defines x ij , ≤ i ≤ j , as usual,я ℓ = [ x , x ℓ ] c and я kℓ = [ я ℓ , x k ] , k > . Proposition 5.10. The algebra B ( L ( A θ − )) is presented by generators x i , i ∈ I † θ , and relations (5.10) , (3.2) , (5.11) , (5.12) , (5.13) , (5.14) , (5.20) , and x ij = 0 , ≤ i ≤ j ≤ θ, (5.36) [ x k − k k +1 , x k ] = 0 , ≤ k < θ. (5.37) x j , я kℓ , j, k, ℓ ∈ I θ ,k < ℓ, (5.38) Furthermore there is a PBW-basis in terms of the positive roots of the rootsystem of type D θ and dim B ( L ( A θ − )) = p θ ( θ − . (cid:3) Realizations. Let H be a Hopf algebra, ( g , χ , η ) a YD-triple and ( g j , χ j ) , j ∈ I ,θ , a family of YD-pairs for H , see §3.3. Let ( V, c ) be abraided vector space with braiding (5.1). Then V := V g ( χ , η ) ⊕ (cid:16) ⊕ j ∈ I ,θ k χ j g j (cid:17) ∈ HH YD (5.39)is a principal realization of ( V, c ) over H if q ij = χ j ( g i ) , i, j ∈ I θ ; a j = q − j η ( g j ) , j ∈ I ,θ . Thus ( V, c ) ≃ V as braided vector space. Consequently, if H is finite-dimensional and ( V, c ) is as in Table 2, then B ( V ) H is a finite-dimensionalHopf algebra. Examples of finite-dimensional pointed Hopf algebras A = B ( V ) k Γ with Γ abelian like this are listed in Table 5 where the interac-tion is weak, ǫ = 1 and ω ∈ G ′ . As in §4.5, Γ is a product of θ cyclic groups, g i is the i -th canonical generator, χ j ( g i ) = 1 if i = j ; we set q ij = 1 , i < j .6. Several blocks, one point Let t ≥ and θ = t + 1 . We shall use the following notation: I ‡ k = { k, k + } , k ∈ I t ; I ‡ = I ‡ ∪ · · · ∪ I ‡ t ∪ { θ } ; The Poseidon braided vector space P ( q , G ) depends on a datum • q ∈ k θ × θ such that ǫ i := q ii = ± , q ij q ji = 1 for all i = j ∈ I θ . OINTED HOPF ALGEBRAS IN POSITIVE CHARACTERISTIC 23 Table 5. Pointed Hopf algebras from a block and several points V J G J Γ dimA − ◦ − − ◦ . . . − ◦ − − ◦ (1 , , . . . , Z /p × Z / p × Z / p Z /p × Z / p × (cid:0) Z / (cid:1) θ − ( θ − p − ◦ − − ◦ (2 , Z /p × Z / p × Z / p − ◦ ω − ◦ (1 , Z /p × Z / p × Z / p − ◦ ω ω ◦ (1 , Z /p × Z / p × Z / p (0 , Z /p × Z / × Z / p p − ◦ r − r ◦ , r ∈ G ′ N , N > , Z /p × Z / p × Z /N N p − ◦ ω ω ◦ ω ω ◦ (1 , , Z /p × Z / p × Z / × Z / p − ◦ ω ω ◦ ω ω ◦ (1 , , Z /p × Z / p × Z / × Z / p • G = ( G j ) ∈ N t , ( a j ) such that G j = ( − r j , ǫ j = 1 , r j , ǫ j = − cf. (4.2);it has a basis ( x i ) i ∈ I ‡ and braiding c ( x i ⊗ x j ) = q ij x j ⊗ x i , ⌊ i ⌋ ≤ t, ⌊ i ⌋ 6 = ⌊ j ⌋ ,ǫ j x j ⊗ x i , ⌊ i ⌋ = j ≤ t, ( ǫ j x j + x ⌊ j ⌋ ) ⊗ x i , ⌊ i ⌋ ≤ t, j = ⌊ i ⌋ + ,q θj x j ⊗ x θ , i = θ, j ∈ I θ ,q θj ( x j + a j x ⌊ j ⌋ ) ⊗ x θ , i = θ, j / ∈ I θ . (6.1)We shall use the elements y h n i j := x j x mj + j , n = 2 m + 1 odd ; x mj + j , n = 2 m even . j ∈ I t , n ∈ N ; (6.2) щ n := (ad c x ) n . . . (ad c x t + ) n t x θ , n = ( n , . . . , n t ) ∈ N t . (6.3)Let A := { n ∈ N t : 0 ≤ n ≤ a = ( | r | , . . . , | r t | ) } , ordered lexicographi-cally. For m , n ∈ A we set p m , n := ǫ θ Y i,j ∈ I t q m i n j ij q m i iθ q n j θj ǫ n := p n , n We also need the following notation: t + := { i ∈ I t : ǫ i = 1 } , t − = t − t + ,M + := { m ∈ A : ǫ m = 1 } , M − = |A| − M + . The main result of this Section is the following. Proposition 6.1. The algebra B ( P ( q , G )) is presented by generators x i , i ∈ I ‡ , and relations x pi + = 0 , x pi = 0 ,x i + x i − x i x i + + 12 x i = 0 , i ∈ I t , ǫ i = 1; (6.4) x i = 0 , x pi + i = 0 , x pi + = 0 ,x i + x i + i − x i + i x i + − x i x i + i = 0 , i ∈ I t , ǫ i = − (6.5) x i x j = q ij x j x i , ⌊ i ⌋ 6 = ⌊ j ⌋ ∈ I t ; (6.6) x i x θ = q iθ x θ x i , i ∈ I t ; (6.7) (ad c x i + ) | r i | ( x θ ) = 0 , i ∈ I t , (6.8) щ m щ n = p m , n щ n щ m m = n ∈ A ; (6.9) щ n = 0 , n ∈ A , ǫ n = − , (6.10) щ p n = 0 , n ∈ A , ǫ n = 1 . (6.11) A basis of B ( P ( q , G )) is given by B = (cid:8) y h m i x m . . . y h m t − i t x m t t + Y n ∈A щ b n n : 0 ≤ b n < if ǫ n = − , ≤ b n , m i < p if ǫ n = 1 , i ∈ I t (cid:9) . Hence dim B ( P ( q , G )) = 2 t − + M − p t + M + . (cid:3) Realizations. Let H be a Hopf algebra, ( g i , χ i , η i ) , i ∈ I t , a family ofYD-triples and ( g θ , χ θ ) a YD-pair for H , see §3.3. Let ( V, c ) be a braidedvector space with braiding (6.1). Then V := (cid:16) ⊕ i ∈ I t V g i ( χ i , η i ) (cid:17) ⊕ k χ θ g θ ∈ HH YD (6.12)is a principal realization of ( V, c ) over H if q ij = χ j ( g i ) , i, j ∈ I θ ; a j = q − j η ( g j ) , j ∈ I t . Thus ( V, c ) ≃ V as braided vector space. Consequently, if H is finite-dimensional, then B ( V ) H is a finite-dimensional Hopf algebra.If q ij = 1 for i = j , then we may choose H = k Γ , where Γ = Z /n × . . . Z /n θ , n i = ( p if q ii = 1 , p if q ii = − . Thus B ( P ( q , G )) k Γ is a Hopf algebra of dimension t − + M − + δ ǫθ, − p t + M + +1 . OINTED HOPF ALGEBRAS IN POSITIVE CHARACTERISTIC 25 A pale block and a point Let V be a braided vector space of dimension 3 with braiding given in thebasis ( x i ) i ∈ I by ( c ( x i ⊗ x j )) i,j ∈ I = ǫx ⊗ x ǫx ⊗ x q x ⊗ x ǫx ⊗ x ǫx ⊗ x q x ⊗ x q x ⊗ x q ( x + x ) ⊗ x q x ⊗ x . (7.1)Let V = h x , x i , V = h x i . Let Γ = Z with a basis g , g . We realize V in k Γ k Γ YD by V = V g , V = V g , g · x = ǫx , g · x = q x , g · x = ǫx , g · x = q ( x + x ) , g i · x = q i x .As usual, let e q = q q ; in particular the Dynkin diagram of the braidedsubspace h x , x i is ǫ ◦ e q q ◦ .As for other cases, we consider K = B ( V ) co B ( V ) ; then K = ⊕ n ≥ K n inherits the grading of B ( V ) ; B ( V ) ≃ K B ( V ) and K is the Nicholsalgebra of K = ad c B ( V )( V ) . Now K ∈ B ( V ) k Γ B ( V ) k Γ YD with the adjointaction and the coaction given by (4.4), i.e. δ = ( π B ( V ) k Γ ⊗ id)∆ B ( V ) k Γ .Next we introduce ш m,n = (ad c x ) m (ad c x ) n x ; we distinguish two cases: w m = (ad c x ) m x = ш m, , z n = (ad c x ) n x = ш ,n . By direct computation, g · ш m,n = q ǫ m + n ш m,n , g · w m = q m q w m , (7.2) z n +1 = x z n − q ǫ n z n x , ш m +1 ,n = x ш m,n − q ǫ m + n ш m,n x , (7.3) ∂ ( ш m,n ) = 0 , ∂ ( ш m,n ) = 0 , (7.4) ∂ ( w m ) = Y ≤ j ≤ m − (1 − ǫ j e q ) x m . (7.5)7.1. The block has ǫ = 1 . Here B ( V ) ≃ S ( V ) is a polynomial algebra,so that x and x commute, and (ad c x ) s ш m,n = ш m,n + s for all m, n, s ∈ N . (7.6)Thus ш m,n , m, n ∈ N generate K . As in [AAH1, §8.1], we have that g · ш m,n = q m + n q X ≤ j ≤ n (cid:18) nj (cid:19) ш m + j,n − j , (7.7)For q ∈ k × , let E p ( q ) = V be the braided vector space as in (7.1) underthe assumptions that ǫ = 1 , q = q = q − , q = − . We call B ( E p ( q )) and the Nichols algebras B ( E ± ( q )) , B ( E ⋆ ( q )) studied in Propositions 7.2,7.3 and 7.4 the Endymion algebras . Proposition 7.1. The algebra B ( E p ( q )) is presented by generators x , x , x and relations x p = 0 , x p = 0 , x x = x x , (7.8) x x = qx x , (7.9) z t = 0 , t ∈ I ,p − . (7.10) The dimension of B ( E p ( q )) is p p , since it has a PBW-basis B = { x m x m z n p − p − . . . z n : n i ∈ { , } , m j ∈ I ,p − } . Proof. We claim that dim K = p and K ≃ Λ( K ) . In fact, by (7.4), (7.5)and the hypothesis e q = 1 , w m = 0 for all m > ; thus ш m,n = 0 for all m > by (7.6). Using this fact and (7.7), g · z n = q n q z n , n ∈ N . (7.11)We have that for all n ∈ N : ∂ ( z n ) = ( − n x n , (7.12) δ ( z n ) = X ≤ j ≤ n ( − n + j (cid:18) nj (cid:19) x n − j g j g ⊗ z j . (7.13)Indeed the proof follows as in [AAH1, Lemma 8.1.4]. As x p = 0 , we have z p = 0 . Thus the set ( z n ) n ∈ I ,p − is a basis of K . The braiding is c ( z n ⊗ z s ) = X ≤ j ≤ n ( − n + j (cid:18) nj (cid:19) ad c ( x n − j g j g ) z s ⊗ z j = q n q s q z s ⊗ z n . That is, K is of diagonal type and the Dynkin diagram consists of discon-nected p points labeled with − . Hence B ( K ) = Λ( K ) . Moreover, B is abasis of B ( V ) since B ( V ) ≃ K B ( V ) .The presentation follows as in [AAH1, Proposition 4.3.7]. (cid:3) The block has ǫ = − . Here B ( V ) ≃ Λ( V ) is an exterior algebraand consequently ш m,n , m, n ∈ { , } generates K . By direct computation, g · z = q q ( z + w ) , ∂ ( z ) = (1 − e q ) x − e q x , (7.14) δ ( z ) = g g ⊗ z + (cid:0) (1 − e q ) x − e q x (cid:1) g ⊗ x . (7.15)7.2.1. Case 1: e q = 1 . Here w = 0 by (7.4) and (7.5), soш , = − (ad c x ) w = 0 . Thus z = x and z form a basis of K and the braiding of K is given by c ( x ⊗ x ) = q x ⊗ x , c ( x ⊗ z ) = q q z ⊗ x ,c ( z ⊗ x ) = q q x ⊗ z , c ( z ⊗ z ) = − q z ⊗ z . (7.16)That is, K is of diagonal type with Dynkin diagram q ◦ q − q ◦ . OINTED HOPF ALGEBRAS IN POSITIVE CHARACTERISTIC 27 For q ∈ k × , let E ± ( q ) = V be the braided vector space as in (7.1) underthe assumptions that ǫ = − , q = q = q − , q = ± . Proposition 7.2. The algebra B ( E + ( q )) is presented by generators x , x , x and relations x = 0 , x = 0 , x x = − x x , (7.17) ( x x − qx x ) = 0 , x p = 0 , (7.18) x ( x x − qx x ) = q − ( x x − qx x ) x , (7.19) x x = qx x . (7.20) Let z = x x − qx x . Then B ( E + ( q )) has a PBW-basis B = { x m x m x m z m : m , m , m ∈ { , } , m ∈ I ,p − } ; hence dim B ( E + ( q )) = 2 p .Proof. Notice that x p = 0 since x is a point labeled with q = 1 in K .Also, B is a basis thanks to the isomorphism B ( E + ( q )) ≃ B ( K ) B ( V ) .The rest of the proof follows as in [AAH1, Proposition 8.1.6]. (cid:3) Proposition 7.3. The algebra B ( E − ( q )) is presented by generators x , x , x and relations (7.17) , (7.20) , x = 0 , ( x x − qx x ) p = 0 , (7.21) x ( x x − qx x ) = − q − ( x x − qx x ) x . (7.22) Let z = x x − qx x . Then B ( E − ( q )) has a PBW-basis B = { x m x m x m z m : m , m , m ∈ { , } , m ∈ I ,p − } ; hence dim B ( E − ( q )) = 2 p .Proof. Notice that z p = 0 since z is a point labeled with in K . Also, B is a basis thanks to the isomorphism B ( E − ( q )) ≃ B ( K ) B ( V ) . The restof the proof follows as in [AAH1, Proposition 8.1.6]. (cid:3) Case 2: e q = − . We consider now a fixed choice of q and e q ,which is the corresponding one to the example of finite GKdim over a fieldof characteristic 0.For q ∈ k × , let E ⋆ ( q ) = V be the braided vector space as in (7.1) underthe assumptions that ǫ = − , q = − , q = q , q = − q − . Recall (2.2). Proposition 7.4. The algebra B ( E ⋆ ( q )) is presented by generators x , x , x and relations (7.17) , x = 0 , x = 0 , (7.23) x [ x , x ] c − q [ x , x ] c x = q x x , (7.24) x p = 0 , [ x , x ] pc = 0 , x = 0 . (7.25) Moreover B ( E ⋆ ( q )) has a PBW-basis B = { x m x m x m [ x , x ] m c x m x m x m : m , m , m , m , m ∈ { , } , m ∈ I , p − , m ∈ I ,p − } ; hence dim B ( E ⋆ ( q )) = 2 p .Proof. Relations (7.17) are 0 in B ( E ⋆ ( q )) because B ( V ) ≃ Λ( V ) ; (7.23) are0 since x , x generate a Nichols algebra of Cartan type A at − .Notice that [ x , x ] c = x x + x x . By (7.17) and (7.23), x x = − q x x , x x = q x x , (7.26) x x = − q x x , x x = − q − x x − q − x , (7.27) x x = q − x x , [ x , x ] c x = − q − x [ x , x ] c , (7.28) x x = − q x x , x x = q [ x , x ] c − q x x , (7.29) x x = − q x x , x [ x , x ] c = [ x , x ] c x , (7.30) x x = q x x , [ x , x ] c x = q x [ x , x ] c . (7.31)As in [AAH1, Proposition 8.1.8] we check that ∂ ( x ) = 4 x x = 0 , ∂ ([ x , x ] c ) = 2 q − x x = 0 . Now we prove that (7.25) holds in B ( V ) . We check that ∂ i annihilatesthese terms for i = 1 , , . To simplify the notation, let u = [ x , x ] c . As ∂ , ∂ annihilate x , x and u , it remains the case i = 3 . Using (7.26)-(7.31), ∂ ( x ) = 4 (cid:0) q − x x x + x x x (cid:1) = 0 ,∂ ( u p ) = 2 q − p − X k =0 ( − q ) k − p u k x x u p − − k = 2 q − p u p − x x = 0 . For the remaining relation, we check that ∂ ( x ) = q − x − x (2 x + x ) and the following equalities hold (using (7.26)-(7.31)): x x = q ( x + u ) x , x u = q ux ,x x = ux + x x , x u = ux ,x x = q x x − qx x , x u = q ux ,x x = q x x , x u = q ux + qx x , Using the previous computations and x = 0 , (cid:0) q − x − x x − x x (cid:1) ( x + u ) = q ( x + u ) x − q x x x − q x x x + q ux − q x ux − q x ux = qx x + q ux − q ( ux + x x ) x − q ( ux + x x ) x + q ux − q ux x − q ux x = q ( x + 2 u ) (cid:0) q − x − x x − x x (cid:1) . OINTED HOPF ALGEBRAS IN POSITIVE CHARACTERISTIC 29 We apply this equality to compute: ∂ ( x p ) = p − X k =0 q k +2 − p x k ∂ ( x )( x + x ) p − − k = p − X k =0 q k +2 − p x k (cid:0) q − x − x (2 x + x ) (cid:1) ( x + u ) p − − k = p − X k =0 x k ( x + 2 u ) p − − k (cid:0) q − x − x (2 x + x ) (cid:1) Using (7.24), (7.29) and (7.30) we get ux = ( u + x ) u . Hence a = x + u and b = u satisfy the last equation of (3.12), so (3.13) applies and we have ∂ ( x p ) = p ( x + u ) p − (cid:0) q − x − x (2 x + x ) (cid:1) = 0 . The rest of the proof follows as in [AAH1, Proposition 8.1.8]. (cid:3) Realizations. Here we present a realization of a braided vector spaceas in (7.1) over a group algebra H = k Γ , with Γ a finite abelian group.We consider V = h x , x i , V = h x i . We realize V in k Γ k Γ YD by V = V g , V = V g , g · x = ǫx , g · x = q x , g · x = ǫx , g · x = q ( x + x ) , g i · x = q i x . In all the cases Γ will be a product of two cyclic groups, g = (1 , , g = (0 , . Examples of finite-dimensional pointed Hopf algebras A = B (cid:0) V g ⊕ V g (cid:1) H are listed in Table 6. Table 6. Pointed Hopf algebras K from a pale block and a point V ( ǫ, e q , q ) Γ q dim A E p ( q ) (1 , , − Z /p × Z / p p +1 p E + ( q ) ( − , , Z / p × Z /p p E − ( q ) ( − , , − Z / p × Z / p p E ⋆ ( q ) ( − , − , − Z / p × Z / p ± p References [AA] N. 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E-mail address : (andrus|angiono)@famaf.unc.edu.ar Philipps-Universität Marburg, Fachbereich Mathematik und Informatik,Hans-Meerwein-Straße, D-35032 Marburg, Germany. E-mail address ::