Examples of finite-dimensional pointed Hopf algebras in characteristic 2
Nicolás Andruskiewitsch, Dirceu Bagio, Saradia Della Flora, Daiana Flôres
aa r X i v : . [ m a t h . QA ] M a y EXAMPLES OF FINITE-DIMENSIONAL POINTED HOPFALGEBRAS IN CHARACTERISTIC NICOLÁS ANDRUSKIEWITSCH, DIRCEU BAGIO, SARADIA DELLA FLORA,DAIANA FLÔRES
Abstract.
We present new examples of finite-dimensional Nichols al-gebra over fields of characteristic 2 starting from braided vector spacesthat are not of diagonal type, admit realizations as Yetter-Drinfeld mod-ules over finite abelian groups and are analogous to braidings over fieldsof odd characteristic with finite-dimensional Nichols algebras presentedin arXiv:1905.03074. As these last ones, they are related to the Nicholsalgebras of finite Gelfand-Kirillov dimension in characteristic 0 describedin arXiv:1606.02521. New finite-dimensional pointed Hopf algebras overfields of characteristic 2 are obtained by bosonization with group alge-bras of suitable finite abelian groups. Introduction
The goal of this paper is to present new examples of finite-dimensionalHopf algebras in characteristic 2, which are pointed, non-commutative andnon-cocommutative. Following the usual guidelines of the lifting method,we focus on finite-dimensional Nichols algebras, then the Hopf algebras areobtained routinely by bosonization. The main result of [AAH1] (in character-istic 0) is the classification of the Nichols algebras with finite Gelfand-Kirillovdimension arising from braided vector spaces ( V, c ) that decompose as V = V ⊕ · · · ⊕ V t ⊕ V t +1 ⊕ · · · ⊕ V θ , c ( V i ⊗ V j ) = V j ⊗ V i , i, j ∈ I θ , where V , . . . , V t are blocks (see § 2.2); V t +1 , . . . , V θ are points (i.e. havedimension 1); and the braidings have a specific form, see e. g. (3.2), (5.2).This result relies on the classification in [H] and assumes a Conjecture treatedpartially in [AAH2], both about Nichols algebras of diagonal type. Howeverin positive characteristic the classification of finite-dimensional Nichols alge-bras of diagonal type is known only in rank ≤ [HW, W1, W2]. Inspired by[CLW] and by familiar phenomena in Lie theory in positive characteristic,examples of finite-dimensional Nichols algebras in odd characteristic wereconstructed in [AAH3] by analogy with the Nichols algebras in [AAH1]–that Mathematics Subject Classification. have infinite dimension. Here we extend these constructions assuming thatthe base field k is algebraically closed of characteristic . There are newfeatures as − now. For instance in characteristic 0, two main actorsare the Jordan and the super Jordan planes. Their restricted versions incharacteristic p > have dimensions p [CLW] and p [AAH3] respectively.When char k = 2 they merge in the restricted Jordan plane that has dimen-sion
16 = 4 × [CLW]. Other families of [AAH1] also merge. Finally thefact that x i = 0 for suitable x i in the braided vector space brings on moreexamples with finite dimension. Let us present the main result of this paper. Theorem. If V is a braided vector space as in Table 1, then the dimensionof the Nichols algebra B ( V ) is finite. Table 1.
Finite-dimensional Nichols algebra in characteristic 2 V B ( V ) dim K dim B ( V ) L ℘ (1 , Proposition 3.6 L ℘ (1 , a ) , a = 1 Proposition 3.7 P ( q , a ) , a ∈ ( k × ) t Proposition 5.4 |A| t + |A| E ℘ (1) Proposition 6.2 E ℘ ( ω ) , ω ∈ G ′ Proposition 6.3 See 2.3.2 for the meaning of K . The braided vector spaces L ℘ (1 , appearto be close to L ( − , G ) and L − ( − , G ) in [AAH3, Table 1], but B ( L ℘ (1 , a )) , a = 1 has no finite-dimensional analogue in char k = p > . Similarly, thealgebras B ( P ( q , a )) are finite-dimensional in odd characteristic only whenthe entries of a belong to the prime field, in contrast with characteristic2. Also E ℘ ( ω ) does not appear in the loc. cit. Albeit no classification isenvisageable yet as the knowledge of diagonal type is still incomplete, wepresent partial results in Theorems 3.1, 4.1 and 6.1.After spelling out some preliminaries in Section 2, we devote Sections 3,4, 5 and 6 to Nichols algebras of one block and one point, one block andseveral points, several blocks and one point, and one pale block and onepoint respectively. Our proofs rely on the splitting technique §2.3.2 andthe classifications in [HW, W1, W2]. Explicit examples of finite-dimensionalpointed Hopf algebras are discussed in §3.2, §5.2, §6.2. More examples bylifting will be presented in a future work.2.
Preliminaries
Notations and Conventions.
We denote the natural numbers by N , N = N ∪ { } . We set I k,ℓ = { n ∈ N : k ≤ n ≤ ℓ } , I ℓ = I ,ℓ and N ≥ ℓ = N \ I ℓ , OINTED HOPF ALGEBRAS IN CHARACTERISTIC for k < ℓ ∈ N . The group of N -th roots of unity in k is denoted by G N ; G ′ N is the subset of the primitive roots of order N and G ∞ = S N ∈ N G N .Throughout H is a Hopf algebra with bijective antipode. We use thenotations G ( H ) = the group of grouplikes in H , P ( H ) = the space of prim-itive elements, b H = Hom alg ( H, k ) , HH YD = the category of Yetter-Drinfeldmodules over H ; see e. g. [R, 11.6].2.2. Yetter-Drinfeld modules.
Braided vector spaces.
A braided vector space V is a pair ( V, c ) where V is a vector space and c ∈ GL ( V ⊗ ) is a solution of the braid equation ( c ⊗ id)(id ⊗ c )( c ⊗ id) = (id ⊗ c )( c ⊗ id)(id ⊗ c ) . We are interested in two classes of braided vector spaces. First, ( V, c ) orsimply V is of diagonal type if there exist a basis ( x i ) i ∈ I θ of V and a matrix q = ( q ij ) i,j ∈ I θ such that q ij ∈ k × and c ( x i ⊗ x j ) = q ij x j ⊗ x i for all i, j ∈ I = I θ . We denote in T ( V ) , or any quotient braided Hopf algebra, x ij = (ad c x i ) x j , x i i ...i M = (ad c x i ) x i ...i M , i, j, i , . . . , i M ∈ I , M ≥ . Second, let ǫ ∈ k × and ℓ ∈ N ≥ . A block V ( ǫ, ℓ ) is a braided vector spacewith a 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 . (2.1)For simplicity a block V ( ǫ, of dimension is called an ǫ -block.2.2.2. Realizations.
Any Yetter-Drinfeld module V bears a structure of brai-ded vector space by c ( v ⊗ w ) = v ( − · w ⊗ v (0) , v, w ∈ V where δ ( v ) = v ( − ⊗ v (0) . The braided vector spaces above appear as Yetter-Drinfeldmodules in different ways called realizations . Let Γ be an abelian group andlet b Γ be the group of characters of Γ . The Yetter-Drinfeld modules over thegroup algebra k Γ are the Γ -graded Γ -modules, the Γ -grading being denotedby V = ⊕ g ∈ Γ V g ; thus h · V g = V g for g, h ∈ Γ . If g ∈ Γ and χ ∈ b Γ , then theone-dimensional vector space k χg , with action and coaction given by g and χ ,is in k Γ k Γ YD . Given V ∈ k Γ k Γ YD with a basis ( v i ) i ∈ I where v i is homogeneousof degree g i , there are skew derivations ∂ i , i ∈ I , of T ( V ) such that ∂ i ( v j ) = δ ij , i, j ∈ I, ∂ i ( xy ) = ∂ i ( x )( g i · y ) + x∂ i ( y ) , x, y ∈ T ( V ) . (2.2)More generally a YD-pair for H is a pair ( g, χ ) ∈ G ( H ) × b H such that χ ( h ) g = χ ( h (2) ) h (1) g S ( h (3) ) , h ∈ H. (2.3)Let k χg be a one-dimensional vector space with H -action and H -coactiongiven by χ and g respectively; then (2.3) says that k χg ∈ HH YD . Thus arealization of V of diagonal type with matrix q = ( q ij ) i,j ∈ I θ is just a collection ( g , χ ) , . . . , ( g θ , χ θ ) such that q ij = χ j ( g i ) for all i, j ∈ I θ . ANDRUSKIEWITSCH, BAGIO, DELLA FLORA, FLÔRES
Realizations of ǫ -blocks. For χ ∈ b H , 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 } . The realizations of ǫ -blocks are given by the notion of YD-triple for H [AAH3]; this is a collection ( g, χ, η ) where ( g, χ ) , is a YD-pair for H , η ∈ Der χ,χ ( H, k ) , χ ( g ) = ǫ , η ( g ) = 1 and η ( h ) g = η ( h (2) ) h (1) g S ( h (3) ) , h ∈ H. (2.4)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 . Then V g ( χ, η ) ≃ V ( ǫ, as braided vector spaces. Example 2.1.
Let ǫ = 1 and Γ = h g i be a cyclic group of order N . Let V be the vector space with a basis ( x i ) i ∈ I with grading deg x i = g , i ∈ I .Then the assignment g ! defines a representation of Γ (hence astructure of Yetter-Drinfeld module over k Γ ) if and only if N is even. Thusif dim H < ∞ and H admits a YD-triple (for ǫ = 1 ), then dim H is even.2.3. Nichols algebras.
Let V ∈ HH YD . The Nichols algebra of V is theunique graded connected Hopf algebra B ( V ) = ⊕ n ≥ B n ( V ) in HH YD suchthat V ≃ B ( V ) = P ( B ( V )) generates B ( V ) as algebra. See [A] for anexposition.The algebra and coalgebra underlying B ( V ) depend only on thebraiding. If V ∈ k Γ k Γ YD is as in § 2.2, then the ∂ i ’s induce skew-derivationson B ( V ) . Then w ∈ B k ( V ) , k ≥ , is if and only if ∂ i ( w ) = 0 in B ( V ) for all i ∈ I .2.3.1. The restricted Jordan plane.
This is the Nichols algebra of a -block. Theorem 2.2. [CLW]
The algebra B ( V (1 , is presented by generators x , x and relations x , x , x x + x x + x x x , x x x x + x x x x . (2.5) Let x := x x + x x . Then dim B ( V (1 , since B ( V ) has a basis { x m x m x n : m , m ∈ I , , n ∈ I , } . (cid:3) The splitting technique.
Let V = V ⊕ V be a direct sum of objectsin k Γ k Γ YD . Then B ( V ) ≃ K B ( V ) where K = B ( V ) co B ( V ) . By [HS,Proposition 8.6], K is the Nichols algebra of K = ad c B ( V )( V ) . (2.6)Here 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 . (2.7) OINTED HOPF ALGEBRAS IN CHARACTERISTIC One block and one point
Let ( q ij ) i,j ∈ I , q ij ∈ k × , i, j ∈ I , a ∈ k . In this Section we assume that q = 1 , q q = 1 . (3.1)Sometimes we use ℘ = q = q − . Let L ℘ ( q , a ) be the braided vector spacewith basis ( x i ) i ∈ I and braiding given by ( c ( x i ⊗ x j )) i,j ∈ I = x ⊗ x ( x + x ) ⊗ x q x ⊗ x x ⊗ x ( x + x ) ⊗ x q x ⊗ x q x ⊗ x q ( x + ax ) ⊗ x q x ⊗ x . (3.2)Let V = h x , x i ≃ V (1 , (the block) and V = h x i (the point); then L ℘ ( q , a ) = V ⊕ V . For simplicity, V = L ℘ ( q , a ) . Let Γ = Z withcanonical basis g , g . Observe that ( V, c ) can be realized in k Γ k Γ YD via: g · x = x , g · x = x + x , g · x = q x ,g · x = q x , g · x = q ( x + ax ) , g · x = q x , deg x = g , deg x = g , deg x = g . (3.3)If a = 0 , then B ( L ℘ ( q , ≃ B ( V ) ⊗ B ( V ) , where ⊗ is the braidedtensor product. Since dim B ( V ) = 2 , dim B ( L ℘ ( q , < ∞ ⇐⇒ dim B ( k x ) < ∞ ⇐⇒ q ∈ G ∞ . Thus we can assume that a ∈ k × .Our main goal in this Section is to prove the following result. Theorem 3.1.
Assume (3.1) and that a = 0 . Then dim B ( L ℘ ( q , a )) < ∞ if and only if q = 1 . Precisely, dim B ( L ℘ (1 , a )) = ( if a = 1 , if a ∈ k \ { , } . We shall apply the splitting technique cf. §2.3.2. To describe K , we set z n := (ad c x ) n x , n ∈ N . (3.4)We establish first a series of useful formulae. Lemma 3.2.
The following formulae hold in B ( V ) for all n ∈ N : g · z n = q z n , x z n = q z n x , x z n = q z n x , (3.5) g · z n = q n q z n , x z n = q z n x + z n +1 . (3.6) Proof.
Note that (3.5) holds for n = 0 . Indeed, g · z = g · x = q z and using derivations is easy to check that x z = q z x and x z = q z x . Now suppose that (3.5) holds for n . Then, z n +1 = (ad c x ) n +1 x =(ad c x ) z n = x z n + ( g · z n ) x = x z n + q z n x . So we compute g · z n +1 = g · ( x z n + q z n x ) = q ( x + x ) z n + q z n ( x + x )= q [( x z n + q z n x ) + ( x z n + q z n x )] = q z n +1 . Similarly, x z n +1 = x ( x z n + q z n x ) = ( x + x x ) z n + q z n x x ANDRUSKIEWITSCH, BAGIO, DELLA FLORA, FLÔRES = q z n x + q x z n x + q z n ( x + x x )= q ( x z n + q z n x ) x = q z n +1 x . Also, since x x = x x + x x we have that x z n +1 = x ( x z n + q z n x ) = x x z n + q z n x x = x x z n + x x z n + q z n x x = q x z n x + q x z n x + q z n ( x x + x x )= q x z n x + q z n x x + q z n x x + q z n x x = q ( x z n + q z n x ) x = q z n +1 x . Finally, the first equation in (3.6) follows by induction. For n = 0 , g · z = q z . Suppose that g · z n = q n q z n . Then, g · z n +1 = g · ( x z n + q z n x )= q ( x + ax )( q n q z n ) + q ( q n q z n ) q ( x + ax )= q n +121 q ( x z n + q z n x ) + aq n +121 q ( x z n + q z n x )= q n +121 q z n +1 . (cid:3) We define µ = 1 , µ = a, µ = a, µ = a ( a + 1) ,y = 1 , y = x , y = x , y = x x . Lemma 3.3.
For all k ∈ N , ∂ ( z k ) = ∂ ( z k ) = 0 , and ∂ ( z k ) = µ k y k , k ∈ I , , ∂ ( z k ) = 0 , k ≥ . Proof.
Clearly, ∂ ( z ) = ∂ ( z ) = 0 , ∂ ( z ) = 1 . Recursively, ∂ ( z k ) = 0 forall k . If ∂ ( z k ) = 0 , then ∂ ( z k +1 ) = ∂ ( x z k + q z k x ) = g · z k + q z k (3.5) = 0 .Next, ∂ ( z ) = ∂ ( x x + q x x ) = x + q ( q ( x + ax )) = ax = µ y ,∂ ( z ) = ∂ ( x z + q z x ) = ax x + q ax q ( x + ax ) = ax = µ y ,∂ ( z ) = ∂ ( x z + q z x ) = ax x + q ax q ( x + ax )= ax ( x x + x x ) + a ( x x + x x )( x + ax )= ax x + ax x + a x x x (2.5) = ( a + a ) x x x = ( a + a ) x x = µ y ,∂ ( z ) = ∂ ( x z + q z x )= ( a + a ) x x x x + q ( a + a ) x x x ( g · x )= ( a + a ) x x x x + q ( a + a ) x x x ( q ( x + ax ))= ( a + a )( x x x x + x x x x ) (2.5) = 0 . (cid:3) OINTED HOPF ALGEBRAS IN CHARACTERISTIC Lemma 3.4.
Let B := { z i : i ∈ I , } and B := { z i : i ∈ I , } . If a = 1 (resp. a = 1 ), then B (resp. B ) is a basis of K .Proof. Notice that (ad c x ) z n = 0 and (ad c x ) z n = 0 . By Theorem 2.2 andLemma 3.3, if a = 1 (resp. a = 1 ), then B (resp. B ) generates K . Sincethe elements of B i ( i ∈ I , ) are homogeneous of distinct degrees and arenon-zero, it follows that B i ( i ∈ I , ) is a linearly independent set. (cid:3) Let i ∈ N . We define recursively the scalars ν i,j , for j > i , by ν i,i = 1 , ν i,j = ( a + ( j − ν i,j − . Lemma 3.5.
The coaction (2.7) on z i , i ∈ I , , is given, (for n = 0 , ) by δ ( z n ) = n X k =1 ν k,n x x n − k g k − g ⊗ z k − + n X k =0 ν k,n x n − k g k g ⊗ z k ,δ ( z n +1 ) = n X k =0 ν k,n +1 x x n − k g k g ⊗ z k + n X k =0 ν k +1 ,n +1 x n − k g k +11 g ⊗ z k +1 . Proof.
Similar to the proof of [AAH1, Lemma 4.2.5]. (cid:3)
Lemma 3.5 implies that K is of diagonal type with braiding given by c ( z i ⊗ z j ) = q j − i q z j ⊗ z i , ∀ i, j. (3.7)Now we are ready for to prove the main result of this Section. Proof of Theorem 3.1. If q = 1 , then the Dynkin diagram of K istotally disconnected with vertices labelled with . Thus, if a = 1 then dim B ( K ) = 2 and dim B ( L ℘ (1 , ; if a = 1 , then dim B ( K ) = 2 and dim B ( L ℘ (1 , a )) = 2 . If q = 1 , then the Dynkin diagram of K is a = 1 : ◦ q q ❋❋❋❋❋❋❋❋❋ ◦ q q ②②②②②②②② q ◦ q ; a = 1 : ◦ q q ①①①①①①①① q ●●●●●●●● ◦ q q q ◦ q q q ◦ q . By inspection of the lists in [W1, W2] we conclude that dim B ( K ) = ∞ . (cid:3) The presentation by generators and relations.
Let c be the braid-ing of K as in (3.7). Then q = 1 if and only if c = id . Hence, for a = 1 (resp. a = 1 ), B ( K ) is the algebra generated by z , z , z (resp. z , z , z , z ) with relations z i = 0 , z i z j = q j − i z j z i , i = j. Thus, we have the following results.
ANDRUSKIEWITSCH, BAGIO, DELLA FLORA, FLÔRES
Proposition 3.6.
The algebra B ( L ℘ (1 , is presented by generators x , x , x with defining relations (2.5) and x z j = q z j x , z j +1 = x z j + q z j x , j ∈ N , (3.8) z i z j = q j − i z j z i , z j = 0 , z k = 0 , i, j ∈ I , , k ≥ . (3.9) The dimension of B ( L ℘ (1 , is , since it has a PBW-basis { x m x m x m z n z n z n : m , m , n i ∈ I , , m ∈ I , } . (cid:3) Proposition 3.7.
The algebra B ( L ℘ (1 , a )) , a = 1 , is presented by genera-tors x , x , x with defining relations (2.5) and x z j = q z j x , z j +1 = x z j + q z j x , j ∈ N , (3.10) z i z j = q j − i z j z i , z j = 0 , z k = 0 , i, j ∈ I , , k ≥ . (3.11) The dimension of B ( L ℘ (1 , a )) is , since it has a PBW-basis { x m x m x m z n z n z n z n : m , m , n i ∈ I , , m ∈ I , } . (cid:3) Realizations.
Let ( g , χ , η ) be a YD-triple and ( g , χ ) a YD-pair for H , see §2.2.3. Let ( V, c ) be a braided vector space with braiding (3.2). 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 is finite-dimensional and ( V, c ) ≃ L ℘ (1 , a ) , a = 0 , then B (cid:0) V g ( χ , η ) ⊕ k χ g (cid:1) H is afinite-dimensional Hopf algebra. Observe that the existence of a YD-triplefor H finite-dimensional is not granted; for instance, ℘ = q should be a rootof 1, otherwise there is no such triple. Suppose that ord ℘ = M ∈ N . Here aresome explicit examples of finite-dimensional pointed Hopf algebras like this:take Γ = h g i × h g i where both g and g have order N := lcm(2 , M ) . Then ( V, c ) is realized in k Γ k Γ YD with structure as in (3.3) and dim B ( V ) k Γ =2 N (if a = 1 ) or N (if a = 1 ).4. One block and several points
Let θ ∈ N ≥ , I † θ = I θ ∪ { } ; as usual ⌊ x ⌋ is the integral part of x ∈ R . Wefix a matrix q = ( q ij ) i,j ∈ I θ with entries in k × and a = (1 , a , . . . , a θ ) ∈ k θ .We assume that q = 1 , q j q j = 1 , for all j ∈ I ,θ , a = (1 , , . . . , . (4.1)Let ( V, c ) be the braided vector space of dimension 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 = . (4.2) OINTED HOPF ALGEBRAS IN CHARACTERISTIC Then V = V ⊕ V where V = h x , x i ≃ V (1 , (the block) and V = h x , . . . , x θ i (the points). If Γ = Z θ with basis ( g h ) h ∈ I , then V can berealized in k Γ k Γ YD as in (3.3). Here is the main result of this Section. Theorem 4.1.
Assume (4.1) . Then dim B ( V ) = ∞ . We shall use the material from the previous Section with replacing 2for instance x = x x + x x . We shall apply the splitting technique cf.§2.3.2. To describe K , we introduce the elements z i,n := ( ad c x ) n x i , i ∈ I ,θ , n ∈ N . (4.3)Let i ∈ I ,θ , n ∈ N . By Lemma 3.2, we have that g · z i,n = q i z i,n , z i,n +1 = x z i,n + q i z i,n x , x z i,n = q ,i z i,n x . (4.4)Consequently, g h · z i,n = q nh q hi z i,n , h ∈ I ,θ . (4.5)In fact, g h · z i, = g h · x i = q hi x i . Suppose that g h · z i,n = q nh q hi z i,n . Thus, g h · z i,n +1 = g h · ( x z i,n + q i z i,n x )= q h ( x + a h x ) q nh q hi z i,n + q i q n +1 h q hi z i,n ( x + a h x )= q n +1 h q hi ( x z i,n + q i z i,n x ) = q n +1 h q hi z i,n +1 . As in Lemma 3.3, we define for i ∈ I ,θ , µ ( i )0 = 1 , µ ( i )1 = a i , µ ( i )2 = a i , µ ( i )3 = a i ( a i + 1) ,y = 1 , y = x , y = x , y = x x . Hence ∂ h ( z i,n ) = 0 for i ∈ I ,θ , n ∈ N , i = h ∈ I † θ and ∂ i ( z i,n ) = µ ( i ) n y n , n ∈ I , , ∂ i ( z i,n ) = 0 , n ≥ . For i ∈ I ,θ , we define J i = { ( i, } , a i = 0 , { ( i, , ( i, , ( i, } , a i = 1 , { ( i, , ( i, , ( i, , ( i, } , a i / ∈ { , } , J = [ i ∈ I ,θ J i . Lemma 4.2.
The family B = ( z i,n ) ( i,n ) ∈ J is a basis of the braided vectorspace K , which is of diagonal type with braiding c ( z i,m ⊗ z j,n ) = q ni q m j q ij z j,n ⊗ z i,m , ( i, m ) , ( j, n ) ∈ J. (4.6) Proof.
Arguing as in Lemma 3.4, we see that B is a basis. We compute thecoaction (2.7) on z j,n as in Lemma 3.5 and then (4.6) follows. (cid:3) Proof of Theorem 4.1.
It is enough to show that dim B ( K ) = ∞ . By(4.6) we may assume that the Dynkin diagram of the matrix ( q ij ) i,j ∈ I ,θ is connected. We then may assume that θ = 3 by taking a suitable sub-diagram; thus e q := q q = 0 . We distinguish then three cases. Firstassume a = (1 , a, with a = 0 . By Theorem 3.1 applied to V ⊕ h x i , q = 1 . By Lemma 4.2, K is of diagonal type. If a = 1 , then its Dynkindiagram is ◦ ◦ e q ◦ q e q ①①①①①①① e q ◦ which does not appear in the list in [W2]. If a = 1 , then the diagram aboveappears a sub-diagram. The case a = (1 , , b ) with b = 0 is similar. Aswell, if a = (1 , a, b ) with a, b = 0 , then the diagram above also appears asub-diagram of that of K . (cid:3) Several blocks and one point
Let t ≥ and θ = t + 1 . As in [AAH1, AAH3] we use the notation: I ‡ k = { k, k + } , k ∈ I t ; I ‡ = I ‡ ∪ · · · ∪ I ‡ t ∪ { θ } . We fix a matrix q = ( q ij ) i,j ∈ I θ with entries in k × and a = ( a , . . . , a t ) ∈ k t .We assume that q ii = 1 , q ij q ji = 1 , for all i = j ∈ I θ ; a j = 0 , j ∈ I t . (5.1)Let P ( q , a ) be the braided vector space with basis ( x i ) i ∈ I ‡ and braiding c ( x i ⊗ x j ) = q ⌊ i ⌋⌊ j ⌋ x j ⊗ x i , ⌊ i ⌋ ≤ t, ⌊ i ⌋ 6 = ⌊ j ⌋ ,x j ⊗ x i , ⌊ i ⌋ = j ≤ t, ( x j + x ⌊ j ⌋ ) ⊗ x i , ⌊ i ⌋ ≤ t, j = ⌊ i ⌋ + ,q θj x j ⊗ x θ , i = θ, j ∈ I θ ,q θ ⌊ j ⌋ ( x j + a ⌊ j ⌋ x ⌊ j ⌋ ) ⊗ x θ , i = θ, j / ∈ I θ . (5.2)Let V = W ⊕ . . . ⊕ W t where W k = h x k , x k + i ≃ V (1 , (the blocks); andlet V = h x θ i (the point). Then P ( q , a ) = V ⊕ V . If Γ = Z θ with basis ( g i ) i ∈ I θ , then there is an action of Γ on V determined by c ( x i ⊗ x j ) = g i · x j ⊗ x i , i ∈ I θ , j ∈ I ‡ . (5.3)Thus V is realized in k Γ k Γ YD with the grading deg( x i ) = g ⌊ i ⌋ , i ∈ I ‡ .Here is the main result of this Section; see (5.9) for the explicit formulaof the dimension. Theorem 5.1.
Assume (5.1) . Then dim B ( P ( q , a )) < ∞ . OINTED HOPF ALGEBRAS IN CHARACTERISTIC Let j ∈ I t . We set x j + j = x j + x j + x j x j + and define µ ( j )0 = 1 , µ ( j )1 = a j , µ ( j )2 = a j , µ ( j )3 = a j ( a j + 1) , µ ( j ) n = 0 if n ≥ ,y j, = 1 , y j, = x j , y j, = x j + j , y j, = x j x j + j , y j,n = 0 if n ≥ . To apply the splitting technique, see §2.3.2, we introduce the elementsщ n := (ad c x ) n . . . (ad c x t + ) n t x θ , n = ( n , . . . , n t ) ∈ N t . (5.4)We start establishing some useful formulas. Lemma 5.2.
Let j ∈ I t and n = ( n , . . . , n t ) ∈ N t . Then ad c x j ( щ n ) = ad c x j + j ( щ n ) = 0 , (5.5) ad c x j + ( щ n ) = Y i Similar to the proof of [AAH1, Lemma 7.2.3]. (cid:3) Let us set b j := 2 , if a j = 1 , b j := 3 , if a j = 1 , and b = ( b , . . . , b t ) ∈ N t . Arguing as in [AAH1, § . ], we conclude from Lemma 5.2: Lemma 5.3. Let A = { n ∈ N t : n ≤ b } ordered lexicographically. (i) The elements ( щ n ) n ∈A form a basis of K . (ii) The coaction (2.7) on щ n is given by δ ( щ n ) = X ≤ k ≤ n ν nk y ,n − k . . . y t,n t − k t g k . . . g k t t g θ ⊗ щ k for some scalars ν nk , ≤ k ≤ n , with ν nn = 1 . (iii) The braided vector space K is of diagonal type with respect to the basis ( щ n ) n ∈A with matrix braiding ( p m , n ) m , n ∈A , where p m , n = t Y i,j =1 q m i n j ij q m i iθ q n j θj . Hence, the corresponding generalized Dynkin diagram has labels p m , m = 1 p m , n p n , m = 1 , m = n . Proof of Theorem 5.1. By Lemma 5.3, dim B ( K ) = 2 |A| . Now the blocks W i and W j , i = j , commute in the braided sense by definition, therefore B ( V ) ≃ B ( W ) ⊗ B ( W ) . . . ⊗ B ( W ) . Hence dim B ( P ( q , a )) = 2 t + |A| . (5.9) (cid:3) The presentation by generators and relations.Proposition 5.4. The algebra B ( P ( q , a )) is presented by generators x i , i ∈ I ‡ , and relations x i = 0 , x i + = 0 , i ∈ I t , (5.10) x i + x i + x i x i + + x i x i + x i = 0 , i ∈ I t , (5.11) x i x i + x i x i + + x i + x i x i + x i = 0 , i ∈ I t , (5.12) x i x j = q ⌊ i ⌋⌊ j ⌋ x j x i , ⌊ i ⌋ 6 = ⌊ j ⌋ ∈ I t , (5.13) x i x θ = q iθ x θ x i , i ∈ I t , (5.14) (cid:16) ad c x i + (cid:17) b i ( x θ ) = 0 , i ∈ I t , (5.15) щ m щ n = p m , n щ n щ m , m = n ∈ A , (5.16) щ n = 0 , n ∈ A . (5.17) A basis of B ( P ( q , a )) is given by B = { y ,m x m . . . y t,m t − x m t t + Y n ∈A щ b n n : 0 ≤ b n < , ≤ m i < } . Hence dim B ( P ( q , a )) = 2 t + |A| . (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 §2.2.3. Let ( V, c ) be a braidedvector space with braiding (5.2). Then V := (cid:16) ⊕ i ∈ I t V g i ( χ i , η i ) (cid:17) ⊕ k χ θ g θ ∈ HH YD (5.18)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 j ( g j ) , j ∈ I t . Consequently, if H is finite-dimensional, then so is B ( V ) H . But the ex-istence of such H requires that all q ij ’s are roots of 1. In this case, let Γ = ( Z /N ) t where N is even and divisible by ord q ij for all i, j . Then ( V, c ) is realized in k Γ k Γ YD with action (5.3). Thus B ( P ( q , a )) k Γ is a pointedHopf algebra of dimension t + |A| N t . OINTED HOPF ALGEBRAS IN CHARACTERISTIC One pale block and one point An indecomposable Yetter-Drinfeld module which is decomposable as brai-ded vector space is called a pale block [AAM]; the simplest examples werestudied in [AAH1, AAH3]. We extend the analysis there to characteristic 2.Let ( q ij ) i,j ∈ I be a matrix with non-zero entries; we assume that q = 1 and q q = 1 ; we set ℘ = q = q − . Let V = E ℘ ( q ) be the braidedvector space of dimension 3 with basis ( x i ) i ∈ I and braiding given by ( c ( x i ⊗ x j )) i,j ∈ I = x ⊗ x x ⊗ x q x ⊗ x x ⊗ x x ⊗ x q x ⊗ x q x ⊗ x q ( x + x ) ⊗ x q x ⊗ x . (6.1)Let V = h x , x i (the pale block), V = h x i (the point) and Γ = Z with a basis g , g . Notice that B ( V ) is a truncated symmetric algebra ofdimension 4. We realize V in k Γ k Γ YD by deg x = deg x = g , deg x = g , g · x = x , g · x = x , g · x = q x ,g · x = q x , g · x = q ( x + x ) , g · x = q x . (6.2) Theorem 6.1. The Nichols algebra B ( E ℘ ( q )) is finite-dimensional if andonly if q = 1 or q = ω , with ω ∈ G ′ . To apply the splitting technique, see §2.3.2, we introduce the elementsш m,n = (ad c x ) m (ad c x ) n x , w m = ш m, , z n = ш ,n , m, n ∈ N . By direct computation g · ш m,n = q ш m,n , g · w m = q m q w m , (6.3) z n +1 = x z n + q z n x , ш m +1 ,n = x ш m,n + q ш m,n x , (6.4) ∂ ( ш m,n ) = ∂ ( ш m,n ) = 0 , ∂ ( w m ) = 0 , for all m > . (6.5)Since x and x commute, ш m,n = (ad c x ) n ( ш m, ) = (ad c x ) n ( w m ) . By(6.5) w m = 0 and thus ш m,n = 0 , for all m > . Hence { z n : n ∈ N } generates K . It is easy to check that g · z n = q n q z n , ∂ ( z n ) = x n , n ∈ N . (6.6)As x = 0 we conclude that { z , z } is a basis of K . The coaction is givenby δ ( z ) = g ⊗ z and δ ( z ) = x g ⊗ z + g g ⊗ z . From (6.6) follows that K is a braided vector space of diagonal type with braiding c ( z i ⊗ z j ) = q j − i q z j ⊗ z i , i, j ∈ I , . Proof of Theorem 6.1. If q = 1 , then the Dynkin diagram of K is totallydisconnected with vertices labelled with q . In this case z = z = 0 , dim B ( K ) = 4 and so dim B ( E ℘ (1)) = 2 . If q = 1 , the Dynkin diagramof K is ◦ q q ◦ q . By inspection in the list of [HW], dim B ( K ) < ∞ if and only if q = ω , with ω ∈ G ′ . In this case, dim B ( K ) = 3 and so dim B ( E ℘ ( ω )) = 2 . (cid:3) The presentation by generators and relations.Proposition 6.2. The algebra B ( E ℘ (1)) is presented by generators x , x , x with defining relations x = 0 , x = 0 , x x = x x , (6.7) x x = ℘x x , z = x x + ℘x x (6.8) x = 0 , z = 0 . (6.9) The dimension of B ( E ℘ (1)) is , since it has a PBW-basis { x m x m z n x n : m i , n i ∈ I , } . (cid:3) Proposition 6.3. Let z := ad c x ( z ) . The algebra B ( E ℘ ( ω )) is presentedby generators x , x , x with defining relations x = 0 , x = 0 , x x = x x , (6.10) x x = ℘x x , z = x x + ℘x x (6.11) x = 0 , z = 0 . (6.12) z = 0 , (ad c x ) ( z ) = 0 . (6.13) The dimension of B ( E ℘ ( ω )) is , since it has a PBW-basis { x m x m z n z n x n : m i ∈ I , , n i ∈ I , } . (cid:3) Realizations. Assume that ℘ is a root of 1 of order M . Take Γ = h g i × h g i where g has order M and g has order N := lcm(2 , M ) . Werealize E ℘ (1) in k Γ k Γ YD by deg x = deg x = g , deg x = g and action (6.2).Then B (cid:0) E ℘ (1) (cid:1) k Γ is a pointed Hopf algebra of dimension M N .Also, let Υ = h h i × h h i where h has order M and h have order P :=lcm(6 , M ) . We realize E ℘ ( ω ) in k Υ k Υ YD by deg x = deg x = h , deg x = h and action as in (6.2) with h i ’s instead of the g i ’s. Then B (cid:0) E ℘ ( ω ) (cid:1) k Υ isa pointed Hopf algebra of dimension M P . References [A] N. Andruskiewitsch. An Introduction to Nichols Algebras . In Quantization, Geom-etry and Noncommutative Structures in Mathematics and Physics. A. Cardona,P. Morales, H. Ocampo, S. Paycha, A. Reyes, eds., pp. 135–195, Springer (2017).[AAH1] N. Andruskiewitsch, I. Angiono, I. Heckenberger, On finite GK-dimensionalNichols algebras over abelian groups . Mem. Amer. Math. Soc., to appear.[AAH2] . On finite GK-dimensional Nichols algebras of diagonal type . 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Rank three Nichols algebras of diagonal type over arbitrary fields . Isr. J.Math. , 1–26 (2017).[W2] . Rank 4 finite-dimensional Nichols algebras of diagonal type in positivecharacteristic . arXiv:1911.03555 . J. Algebra, to appear.[R] D. E. Radford. Hopf algebras . Series on Knots and Everything 49. Hackensack,NJ: World Scientific. xxii, 559 p. (2012). FaMAF-Universidad Nacional de Córdoba, CIEM (CONICET),Medina Allende s/n, Ciudad Universitaria,(5000) Córdoba, República Argentina. E-mail address : [email protected] Departamento de Matemática, Universidade Federal de Santa Maria,97105-900, Santa Maria, RS, Brazil E-mail address ::