Finite dimensional Nichols algebras over Suzuki algebra I: over simple Yetter-Drinfeld modules of A μλ N2n
aa r X i v : . [ m a t h . QA ] N ov FINITE DIMENSIONAL NICHOLS ALGEBRAS OVER SUZUKIALGEBRA A µλ N n I: OVER SIMPLE YETTER-DRINFELD MODULES
YUXING SHIA bstract . The Suzuki algebra A µλ Nn was introduced by Suzuki Satoshi in 1998,which is a class of cosemisimple Hopf algebras[32]. In this paper, the authorgives a complete set of simple Yetter-Drinfeld modules over Suzuki algebra A µλ N n and investigates the Nichols algebras over those simple Yetter-Drinfeld modules.The involved finite dimensional Nichols algebras of diagonal type are of Cartantype A , A × A , A , Super type A ( q ; I ) and the Nichols algebra ufo (8). Andthe 4 m and m -dimensional Nichols algebras which discovered in [6, section3.7] can be realized in the category of Yetter-Drinfeld modules over A µλ Nn .
1. I ntroduction
Let k be an algebraicaly closed field of characteristic 0. The motivation of thepaper is to make some contributions to the following classification program offinite dimensional Hopf algebras. Problem 1.1.
How to classify all finite dimensional Hopf algebras over Suzukialgebra A µλ Nn ?There are only a few works to deal with the problem. In 2004, Menini et al.studied the quantum lines over A m and B m , which are isomorphic to A ++ m and A + − m respectively[13]. In 2019, the author [29] investigated finite dimensional Hopfalgebras over the Kac-Paljutkin algebra A + − and Fantino et al. [16] classified finitedimensional Hopf algebras over the dual of dihedral group b D m , with m = a ≥ b D m is a 2-cocycle deformation of A ++ a [27].Why we are interested in Suzuki algebra? In [29], the author started a programto study finite dimensional Hopf algebras over non-trivial semisimple Hopf alge-bras. For the non-trivial semisimple Hopf algebras, we mean Hopf algebras thatare neither commutative nor cocommutative, and are not 2-cocyle deformation ofgroup algebras. According to [27], the 2-cocycle deformation of A + − m is trivial.When N is odd, n ≥ λ = − λ =
1, Wakui
Mathematics Subject Classification.
Keywords:
Nichols algebra; Hopf algebra; Suzuki algebra.This work was partially supported by Foundation of Jiangxi Educational Committee (No.12020447)and Natural Science Foundation of Jiangxi Normal University(No. 12018937) . [35] showed A + λ Nn has no triangle structure. This means that A + λ Nn can’t be obtainedfrom a group algebra by a Drinfeld twist [15]. In the past decades, the study ofNichols algebras are mainly focus on the Yetter-Drinfeld categories of group alge-bras. It’s an interesting problem to find new finite-dimensional Nichols algebrasover non-trivial semisimple Hopf algebras. In [6, section 3.7], Andruskiewitschand Giraldi found two class of 4 m and m -dimensional Nichols algebras (we callthose Nichols algebras are of type V abe ) which generally can’t be realized in thecategory of Yetter-Drinfeld modules over group algebras. Together with the resultsof [31], we will see that those Nichols algebras can be realized in the category ofYetter-Drinfeld modules over A µλ N n .Our classification program over Suzuki algebra is based on the lifting method which was introduced by Andruskiewitsch and Schneider [8]. The lifting method isa general framework to classify finite dimensional Hopf algebras with a fixed sub-Hopf algebra as coradical. Here let us recall the procedure for the lifting methodbriefly. Let H be a Hopf algebra whose coradical H is a Hopf subalgebra. The as-sociated graded Hopf algebra of H is isomorphic to R H where R = ⊕ n ∈ N R ( n ) isa braided Hopf algebra in the category H H YD of Yetter-Drinfield modules over H , biproduct or bosonization of R with H . As explained in[9], to classify finite-dimensional Hopf algebras H whose coradical is isomorphicto H we have to deal with the following questions:(a) Determine all Yetter-Drinfield modules V over H such that the Nicholsalgebra B ( V ) has finite dimension; find an e ffi cient set of relations for B ( V ).(b) If R = ⊕ n ∈ N R ( n ) is a finite-dimensional Hopf algebra in H H YD with V = R (1), decide if R ≃ B ( V ). Here V = R (1) is a braided vector space calledthe infinitesimal braiding .(c) Given V as in (a), classify all H such that gr H ≃ B ( V ) H (lifting).In this paper, we mainly focus on the part (a) of the lifting method. More pre-cisely, we deal with the Nichols algebras over simple Yetter-Drinfeld modules of A µλ Nn with n even. And in the sequel [31], we will study the case with n odd.In the part (a) of the lifting method, it involves that how to construct all Yetter-Drinfeld modules over a finite dimensional Hopf algebra. Majid [26] identifiedthe Yetter-Drinfeld modules with the modules of the Drinfeld double via the cat-egory equivalence HH YD ≃ H cop YD H cop ≃ D ( H cop ) M . The simple modules ofDrinfeld double D ( k G ) for a finite group G are completely described in [14] and[17]. Burciu generalized the case of finite groups to a general semisimple Hopfalgebra H . He showed that any simple representation of a Drinfeld double D ( H )can be obtained as an induced representation from a certain subalgebra of D ( H )[12]. Zhimin Liu and Shenglin Zhu [25] also generalized the results of Dijkgraaf,Pasquier and Roche and of Gould on Yetter-Drinfeld modules over finite group. ICHOLS ALGEBRAS OVER SUZUKI ALGEBRA A µλ N n I 3
In [28], for any Hopf algebra H (not necessarily finite-dimensional), Radford in-troduced an induction functor from the category of left H -modules to the categoryof Yetter-Drinfeld H -modules as well as an induction functor from the categoryof right H-comodules to the category of Yetter-Drinfeld H -modules. He showedthat every simple Yetter-Drinfeld H -module can be realized as a submodule(resp.,quotient module) of some induced module. Jun Hu and Yinhuo Zhang studiedRadford’s induced module H β [24], where β is a group-like element in H ∗ andintroduced a notion of β -character algebra C β ( H ) [23]. As an example, they ex-plicitly constructed all simple modules for the Drinfeld double of the Kac-Paljutkinalgebra.According to Radford’s method [28, Proposition 2], we constructed a com-plete set of simple Yetter-Drinfeld modules over A µλ N n . There are exactly 8 N one-dimensional, 2 N (4 n −
1) two-dimensional and 8 N n -dimensional non-isomorphic Yetter-Drinfeld modules over A µλ N n , see the Theorem 3.1.The involved Nichols algebras in the paper are of diagonal type, rack type, type V abe , the Nichols algebras B (cid:16) I sp jk (cid:17) for n = B (cid:16) K sjk , p (cid:17) for n ≥
2. Nicholsalgebras of diagonal type with finite dimension were classified by Heckenberger[20] via the concepts of Weyl groupoid and generalized root system [19] [22].Angiono obtained a presentation by generators and relations of any Nichols algebraof diagonal type with finite root system [10] [11]. Andruskiewitsch and Gra˜na[7] builded connections between Nichols algebras of group-type and racks. It’se ffi cient to study Nichols algebras in the language of racks [18] [21]. Nicholsalgebras associated to racks of type C , D or F are infinite-dimensional [4] [5] [3].The finite dimensional Nichols algebras of diagonal type over simple Yetter-Drinfeld modules of A µλ N n can be classified by(1) Cartan type A :(a) B (cid:16) A siik , p (cid:17) , N ∤ ks , see Lemma 4.1;(b) B (cid:16) ¯ A si jk , p (cid:17) , λ = , N ∤ k (2 s + n ) ,λ = − , N ∤ k (2 s + n ) , n even ,λ = − , N ∤ k (2 s + n ) + N , n odd , see the Lemma 4.3.(2) Cartan type A × A :(a) B (cid:16) B si jk (cid:17) , N | ks , N ∤ ks , see the Lemma 4.5;(b) B (cid:16) C sti jk , p (cid:17) , ( − ( i + j )( t + ω kn ( s + t + = −
1, see the Lemma 4.6,(c) B (cid:16) D stjk , p (cid:17) , 8 nN ∤ α , 8 nN | β , α = nk ( s + t + − jN ( t + β = nk ( s + t + + jN ( t + B (cid:16) E stjk , p (cid:17) , 8 nN ∤ α , 8 nN | β , α = nk ( s + t ) + jNt , β = nk ( s + t ) − jNt , see the Lemma 4.11;(e) B (cid:16) P sti jk , p (cid:17) , ( − p + ( i + j )( t + ) + j ˜ µ s + t + = −
1, see the Lemma 4.21;
SHI (f) B (cid:16) I sp jk (cid:17) , n = λ q = , q , q = ( − p ω k (2 s + , see the Lemma4.25;(g) B (cid:16) K sjk , p (cid:17) , λ q = , q = ( − p ¯ µ s ω kns , see the Lemma 4.26;(h) B (cid:16) G stjk , p (cid:17) , ae = b , b = − ae = ¯ µ s + t + ω kn (4 s + t + + jN ( − − t ) , b = ( − p ¯ µ s + t + ω nk (2 s + t + + jN (2 t + , see the Lemma 4.16;(i) B (cid:16) H sjk , p (cid:17) , ae = b , b = − ae = ¯ µ s + t + ω kn (4 t + s + + jN (4 t + , b = ( − p ¯ µ s + t + ω kn (2 t + s + + jN ( − − t ) , see the Lemma 4.19.(3) Cartan type A :(a) B (cid:16) B si jk (cid:17) , N | ks , N ∤ ks , N ∤ ks , see the Lemma 4.5;(b) B (cid:16) C sti jk , p (cid:17) , q = q − , q = ( − ( i + j )( t + ω nk ( s + t + , see the Lemma4.6;(c) B (cid:16) D stjk , p (cid:17) , 8 nN ∤ α , 8 nN | ( α + β ), α = nk ( s + t + − jN ( t + β = nk ( s + t + + jN ( t + B (cid:16) E stjk , p (cid:17) , 8 nN ∤ α , 8 nN | ( α + β ), α = nk ( s + t ) + jNt , β = nk ( s + t ) − jNt , see the Lemma 4.11;(e) B (cid:16) P sti jk , p (cid:17) , q = , q = ( − p + ( i + j )( t + ) + j ˜ µ s + t + , see the Lemma4.21;(f) B (cid:16) I sp jk (cid:17) , n = λ q = , q = ( − p ω k (2 s + , see the Lemma 4.25;(g) B (cid:16) K sjk , p (cid:17) , λ q = , q = ( − p ¯ µ s ω kns , see the Lemma 4.26;(h) B (cid:16) G stjk , p (cid:17) , ae = b , b = , b , ae = ¯ µ s + t + ω kn (4 s + t + + jN ( − − t ) , b = ( − p ¯ µ s + t + ω nk (2 s + t + + jN (2 t + , see the Lemma 4.16;(i) B (cid:16) H sjk , p (cid:17) , ae = b , b = , b , ae = ¯ µ s + t + ω kn (4 t + s + + jN (4 t + , b = ( − p ¯ µ s + t + ω kn (2 t + s + + jN ( − − t ) , see the Lemma 4.19.(4) Super type A ( q ; I ):(a) B (cid:16) D stjk , p (cid:17) , α ≡ nN mod 8 nN , 2 β . nN mod 8 nN , α = nk ( s + t + − jN ( t + β = nk ( s + t + + jN ( t + B (cid:16) E stjk , p (cid:17) , α ≡ nN mod 8 nN , 2 β . nN mod 8 nN , α = nk ( s + t ) + jNt , β = nk ( s + t ) − jNt , see the Lemma 4.11.(5) The Nichols algebra ufo (8):(a) B (cid:16) D stjk , p (cid:17) , α − β ≡ β ≡ nN mod 8 nN , 8 β . nN , α = nk ( s + t + − jN ( t + β = nk ( s + t + + jN ( t + B (cid:16) E stjk , p (cid:17) , α − β ≡ β ≡ nN mod 8 nN , 8 β . nN , α = nk ( s + t ) + jNt , β = nk ( s + t ) − jNt , see the Lemma 4.11. ICHOLS ALGEBRAS OVER SUZUKI ALGEBRA A µλ N n I 5
The Nichols algebras B (cid:16) G stjk , p (cid:17) and B (cid:16) H sjk , p (cid:17) are of type V abe , see section 4.2.When ae = b , then B ( V abe ) is of diagonal type. When ae , b , according to [6,section 3.7] or [30], we havedim B ( V abe ) = m , b = − , ae is m -th primitive root of unity , m , ae = , b is m -th primitive root of unity for m ≥ , unknown , ae , , b is m -th primitive root of unity for m > , ∞ , otherwise . When n >
2, then B (cid:16) I sp jk (cid:17) is of rack type and the associated rack is of type D ,see the Lemma 4.24.Unsolved cases in the paper:(1) B (cid:16) I sp jk (cid:17) , n =
2, see the section 4.3;(2) B ( V abe ), b , ae , b is m -th primitive root of unity for m > B (cid:16) K sjk , p (cid:17) , n ≥
2, see the section 4.4.The paper is organized as follows. In section 1, we introduce the motivation andbackground of the paper and summarize our main results. In section 2, we makean introduction for the Suzuki algebra and construct all simple representation of A µλ N n . In section 3, we construct all simple Yetter-Drinfeld modules over A µλ N n byusing Radford’s method and we put those Yetter-Drinfeld modules in the appendix.In section 4, we calculate Nichols algebras over simple Yetter-Drinfeld modulesof A µλ N n in cases: diagonal type, type V abe , the Nichols algebras B (cid:16) I sp jk (cid:17) and B (cid:16) K sjk , p (cid:17) . 2. T he H opf algebra A µλ Nn and the representations of A µλ N n Suzuki introduced a family of cosemisimple Hopf algebras A µλ Nn parametrizedby integers N ≥ n ≥ µ , λ = ±
1, and investigated various proper-ties and structures of them [32]. Wakui studied the Suzuki algebra A µλ Nn in per-spectives of polynomial invariant[34], braided Morita invariant[36] and coribbonstructures[33]. The Hopf algebra A µλ Nn is generated by x , x , x , x subject tothe relations: x = x , x = x , χ n = λχ n , χ n = χ n , x N + µ x N = , x i j x kl = i + j + k + l is odd , where we use the following notation for m ≥ χ m : = m z }| { x x x . . . . . ., χ m : = m z }| { x x x . . . . . .,χ m : = m z }| { x x x . . . . . ., χ m : = m z }| { x x x . . . . . . . SHI
The Hopf algebra structure of A µλ Nn is given by(2.1) ∆ ( χ ki j ) = χ ki ⊗ χ k j + χ ki ⊗ χ k j , ε ( x i j ) = δ i j , S ( x i j ) = x N − ji , for k ≥ i , j = , i , i + j = { i , i + , i + , · · · , i + j } be an index set. Then the basis of A µλ Nn canbe represented by(2.2) n x s χ t , x s χ t | s ∈ , N , t ∈ , n − o . Thus for s , t ≥ s + t ≥ ∆ ( x s χ t ) = x s χ t ⊗ x s χ t + x s χ t ⊗ x s χ t , ∆ ( x s χ t ) = x s χ t ⊗ x s χ t + x s χ t ⊗ x s χ t . The cosemisimple Hopf algebra A µλ Nn is decomposed to the direct sum of simplesubcoalgebras such as A µλ Nn = L g ∈ G k g ⊕ L ≤ s ≤ N − ≤ t ≤ n − C st [32, Theorem 3.1][34,lemma 5.5], where G = n x s ± x s , x s + χ n − ± √ λ x s + χ n − | s ∈ , N o , C st = k x s χ t + k x s χ t + k x s χ t + k x s χ t , s ∈ , N , t ∈ , n − . The set { k g | g ∈ G } ∪ n k x s χ t + k x s χ t | s ∈ , N , t ∈ , n − o is a full set ofnon-isomorphic simple left A µλ Nn -comodules, where the coactions of the comoduleslisted above are given by the coproduct ∆ . Denote the comodule k x s χ t + k x s χ t by Λ st . That is to say the comodule Λ st = k w + k w is defined as ρ ( w ) = x s χ t ⊗ w + x s χ t ⊗ w , ρ ( w ) = x s χ t ⊗ w + x s χ t ⊗ w . Proposition 2.1.
Let ω be a primitive nN-th root of unity. Set ˜ µ = ( , µ = ω n , µ = − , ¯ µ = ( , µ = ,ω n , µ = − . Then a full set of non-isomorphic simple left A µλ N n -modules is given by(1) V i jk = k v, i , j ∈ Z , k ∈ , N − . The action of A µλ N n on V i jk is given byx , x , x ( − i ω nk , x ( − j ω nk ; (2) V ′ i jk = k v, λ = , i , j ∈ Z , k ∈ , N − . The action of A µ + N n on V ′ i jk isgiven byx , x , x ( − i ω nk ˜ µ, x ( − j ω nk ˜ µ ; ICHOLS ALGEBRAS OVER SUZUKI ALGEBRA A µλ N n I 7 (3) V jk = k v ⊕ k v , k ∈ , N − , j ∈ , n − . The action of A µλ N n on the rowvector ( v , v ) is given byx ω kn − jN ) ω jN ! , x , x , x ω kn ! ; (4) V ′ jk = k v ′ ⊕ k v ′ , k ∈ , N − , j ∈ , n − , λ = , j + ∈ , n , λ = − . The action ofA µλ N n on the row vector ( v ′ , v ′ ) is given byx µω kn − jN ) ω jN ! , x , x , x µω kn ! . Remark . We left the proof to the reader since it’s easy and tedious.3. Y etter -D rinfeld modules over A µλ N n Similarly according to Radford’s method [28, Proposition 2], any simple leftYetter-Drinfeld module over a Hopf algebra H could be constructed by the sub-module of tensor product of a left module V of H and H itself, where the moduleand comodule structures are given by : h · ( ℓ ⊠ g ) = ( h (2) · ℓ ) ⊠ h (1) gS ( h (3) ) , (3.1) ρ ( ℓ ⊠ h ) = h (1) ⊗ ( ℓ ⊠ h (2) ) , ∀ h , g ∈ H , ℓ ∈ V . (3.2)Here we use ⊠ instead of ⊗ to avoid confusion by using too many symbols of thetensor product. we construct all simple left Yetter-Drinfeld modules over A µλ N n + inthis way and put them in the appendix without proof since it’s tedious verificationwith the definition of Yetter-Drinfeld modules. Let V be a simple left A µλ N n + mod-ule. The strategy is to break V ⊠ A µλ N n + into small sub-Yetter-Drinfeld moduleswhich can’t break any more, and single out a complete set simple Yetter-Drinfeldmodules over A µλ N n + from those submodules. The dimension distribution of thosesubmodules is 1, 2 and 2 n . From the appendix and table 1, it’s not di ffi cult tosee that there are 8 N pairwise non-isomorphic Yetter-Drinfeld modules of onedimension and 2 N (4 n −
1) pairwise non-isomorphic Yetter-Drinfeld modules oftwo dimension, see Theorem 3.1. There are seven class Yetter-Drinfeld modulesof 2 n dimension in total and they have the following relations.(1) I sp jk ≃ I sp j ′ k in case of j ≡ j ′ mod 4.(2) When n is even, then M si jk ≃ I sp j ′ k in case of i = p , j ′ | n is odd, then M si jk ≃ I sp j ′ k in case of i = p and j ′ ≡ ( , if i + j is even , , if i + j is odd . SHI (4) When n is even, then N si jk ≃ I sp j ′ k in case of j = p and j ′ ≡ ( , if λ = , , if λ = − . (5) When n is odd, then N si jk ≃ I sp j ′ k in case of j = p and j ′ ≡ , if λ = , i + j is even , , if λ = , i + j is odd , , if λ = − , i + j is even , , if λ = − , i + j is odd . (6) I sp jk ≃ J sp j ′ k in case of j ≡ ( j ′ mod 4 , if λ = , j ′ + , if λ = − . (7) K sjk , p ≃ K sj ′ k , p in case of j ≡ j ′ mod 4.(8) K sjk , p ≃ L sj ′ k , p in case of j + j ′ ≡ Q si jk , p ≃ L sj ′ k , p in case of j ′ ≡ , if i + j is even and n is odd , , if i + j is odd and n is odd , , if n is even . Since 8 N · + N (4 n − · + N · (2 n ) = (8 Nn ) , all the simple Yetter-Drinfeld modules are given by the following theorem. Theorem 3.1.
A set of full non-isomorphic simple Yetter-Drinfeld modules overA µλ N n is given by the following list.(1) There are N non-isomorphic Yetter-Drinfeld modules of one dimenion:(a) A siik , p , i , p ∈ Z , s ∈ , N, k ∈ , N − , see the Lemma 5.1;(b) ¯ A si i + k , p , i , p ∈ Z , s ∈ , N, k ∈ , N − , see the Lemma 5.2.(2) There are N (4 n − non-isomorphic Yetter-Drinfeld modules of twodimension:(a) B s k , s ∈ , N, k ∈ , N − , see the Lemma 5.3;(b) C sti jk , p , i j = or , k ∈ , N − , p ∈ Z , s ∈ , N, t ∈ , n − , seethe Lemma 5.6;(c) C sti jk , p , i = , j = ( i + , if λ = i , if λ = − , k ∈ , N − , s ∈ , N, p = ,t = n − , see the Lemma 5.6;(d) D stjk , p , j ∈ , n − , k ∈ , N − , p ∈ Z , s ∈ , N, t ∈ , n − , see theLemma 5.12;(e) E stjk , p , j ∈ , n − , k ∈ , N − , p ∈ Z , s ∈ , N, t ∈ , n − , see theLemma 5.14;(f) G stjk , p , j ∈ , n − , if λ = , j + ∈ , n , if λ = − , k ∈ , N − , p ∈ Z , s ∈ , N,t ∈ , n − , see the Lemma 5.20; ICHOLS ALGEBRAS OVER SUZUKI ALGEBRA A µλ N n I 9 (g) H stjk , p , j ∈ , n − , if λ = , j + ∈ , n , if λ = − , k ∈ , N − , p ∈ Z , s ∈ , N,t ∈ , n − , see the Lemma 5.21.(h) P sti jk , p , i j = or , k ∈ , N − , p ∈ Z , s ∈ , N, t ∈ , n − seethe Lemma 5.26;(3) There are N non-isomorphic Yetter-Drinfeld modules of n dimension:(a) I sp jk , j = or , k ∈ , N − , p ∈ Z , s ∈ , N, see the Lemma 5.16;(b) K sjk , p , j = ( or , if λ = − , or , if λ = , k ∈ , N − , p ∈ Z , s ∈ , N, seethe Lemma 5.22.
4. N ichols algebras over simple Y etter -D rinfeld modules In this section, we investigate Nichols algebras over simple Yetter-Drinfeldmodules of A µλ N n . So the Yetter-Drinfeld modules discussed in the section arethose listed in Theorem 3.1. For the knowledge about Nichols algebras, pleaserefer to [9] [1] [2].4.1. Nichols algebras of diagonal type.
Let V = L i ∈ I k v i be a vector space witha braiding c ( v i ⊗ v j ) = q i j v j ⊗ v i , then the Nichols algebra B ( V ) is of diagonal type.Our results in this section heavily rely on Heckenberger’s classification work [20].To keep the article concise, we don’t repeat this in the following proofs. For moredetails about Nichols algebras of diagonal type, please consult [2]. Lemma 4.1. dim B (cid:16) A siik , p (cid:17) = ( ∞ , N | ks , N ( N , ks ) , N ∤ ks . Proof. c ( w ⊗ w ) = h ( − i ω kn i s w ⊗ w = ω nks w ⊗ w . (cid:3) Corollary 4.2. (1) When N = , then dim B (cid:16) A siik , p (cid:17) = ∞ . (2) When N is a prime, then dim B (cid:16) A siik , p (cid:17) = ( ∞ , k = or s = N , N , otherwise . Lemma 4.3. dim B (cid:16) ¯ A si jk , p (cid:17) = ∞ , λ = , N | k (2 s + n ) , N (2 N , k (2 s + n )) , λ = , N ∤ k (2 s + n ) , ∞ , λ = − , N | k (2 s + n ) , n even , N (2 N , k (2 s + n )) , λ = − , N ∤ k (2 s + n ) , n even , ∞ , λ = − , N | k (2 s + n ) + N , n odd , N (2 N , k (2 s + n ) + N ) , λ = − , N ∤ k (2 s + n ) + N , n odd . Proof. c ( w ⊗ w ) = ( − ( i + j ) n ω kn (2 s + n ) w ⊗ w . (cid:3) YD-mod dim parameters mod comod B si jk i , j , p ∈ Z , j = i + s ∈ , N , k ∈ , N − V i jk ⊕ V jik k g + s ⊕ k g − s C sti jk , p i , j , p ∈ Z , s ∈ , N , t ∈ , n − k ∈ , N − V i jk ⊕ V i + j + k Λ s t + C sti jk , p i , j , p ∈ Z , j = ( i + , λ = , i , λ = − , s ∈ , N , t = n − k ∈ , N − V i jk ⊕ V i + j + k k h + s ⊕ k h − s D stjk , p s ∈ , N , t ∈ , n − j ∈ , n − p ∈ Z , k ∈ , N − V jk t , n − Λ s t + t = n − k h + s ⊕ k h − s E stjk , p s ∈ , N , t ∈ , n − j ∈ , n − p ∈ Z , k ∈ , N − V jk t , Λ s t t = k g + s ⊕ k g − s G stjk , p j ∈ , n − , λ = , j + ∈ , n , λ = − , s ∈ , N , t ∈ , n − k ∈ , N − p ∈ Z V ′ jk Λ s t + H stk , p j ∈ , n − , λ = , j + ∈ , n , λ = − , s ∈ , N , t ∈ , n − k ∈ , N − p ∈ Z V ′ jk Λ s t + P sti jk , p λ = i , j , p ∈ Z , s ∈ , N , t ∈ , n − k ∈ , N − V ′ i jk ⊕ V ′ i + j + k Λ s t + T able
1. Two dimensional simple Yetter-Drinfeld modules over A µλ N n . Here g ± s = x s ± x s , h ± s = x s χ n ± √ λ x s χ n . Corollary 4.4.
When N = , then dim B (cid:16) ¯ A si jk , p (cid:17) = ( , λ = − , n odd , ∞ , otherwise . ICHOLS ALGEBRAS OVER SUZUKI ALGEBRA A µλ N n I 11
Lemma 4.5. dim B (cid:16) B si jk (cid:17) = , N | ks , N ∤ ks , (Cartan type A × A ) , , N | ks , N ∤ ks , N ∤ ks , (Cartan type A ) , ∞ , otherwise . Proof. c ( w α ⊗ w β ) = ω nks w β ⊗ w α for α, β ∈ , (cid:3) Lemma 4.6.
Denote d = k ( s + t + , d = nk ( s + t + + Nn ( t + .(1) When i = j, then dim B (cid:16) C sti jk , p (cid:17) = , N | d , N ∤ d , (Cartan type A × A ) , , N | d , N ∤ d , N ∤ d , (Cartan type A ) , ∞ , otherwise . (2) When i = j + , then dim B (cid:16) C sti jk , p (cid:17) = , N | d , N ∤ d , (Cartan type A × A ) , , N | d , N ∤ d , N ∤ d , (Cartan type A ) , ∞ , otherwise . Proof. c ( w α ⊗ w β ) = qw β ⊗ w α for α, β ∈ , q = ( − ( i + j )( t + ω nk ( s + t + .dim B (cid:16) C sti jk , p (cid:17) < ∞ ⇐⇒ B (cid:16) C sti jk , p (cid:17) is of Cartan type A × A or A .(1) When i = j , then q = ω nk ( s + t + .(2) When i = j +
1, then q = ω nk ( s + t + + Nn ( t + . (cid:3) Lemma 4.7.
Denote α = nk ( s + t + − jN ( t + , β = nk ( s + t + + jN ( t + . dim B (cid:16) D stjk , p (cid:17) < ∞ if and only if one of the following conditions holds.(1) nN ∤ α , nN | β , Cartan type A × A ;(2) nN ∤ α , nN | ( α + β ) , Cartan type A ;(3) α ≡ nN mod 8 nN, β . and nN mod 8 nN, Super type A ( q ; I ) ;(4) α − β ≡ β ≡ nN mod 8 nN, β . nN. The Nichols algebras ufo (8) , see [2, Page 209] .Remark . When n = N ≤
5, then B (cid:16) D stjk , p (cid:17) is of the Cartan type A × A in case that j = N , k , s , t ) is in the set (1 , , , , (2 , , , , (2 , , , , (2 , , , , (3 , , , , (3 , , , , (3 , , , , (3 , , , , (3 , , , , (4 , , , , (4 , , , , (4 , , , , (4 , , , , (4 , , , , (4 , , , , (4 , , , , (4 , , , , (4 , , , , (4 , , , , (5 , , , , (5 , , , , (5 , , , , (5 , , , , (5 , , , , (5 , , , , (5 , , , , (5 , , , , (5 , , , When n ≤ N ≤
6, then B (cid:16) D stjk , p (cid:17) is of the Cartan type A in case that( n , N , j , k , s , t ) is in the set (2 , , , , , , (2 , , , , , , (2 , , , , , , (2 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , When n ≤ N ≤
6, then B (cid:16) D stjk , p (cid:17) is of the Super type A ( q ; I ) in case that( n , N , j , k , s , t ) is in the set (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , Proof.
The braiding is of diagonal type with the braiding given by c ( w ⊗ w ) = ω nk ( s + t + − jN ( t + w ⊗ w , c ( w ⊗ w ) = ω nk ( s + t + + jN ( t + w ⊗ w , c ( w ⊗ w ) = ω nk ( s + t + + jN ( t + w ⊗ w , c ( w ⊗ w ) = ω nk ( s + t + − jN ( t + w ⊗ w . (cid:3) Corollary 4.9. If B (cid:16) D stjk , p (cid:17) is isomorphic to the Nichols algebras ufo (8) , ∤ Nand ∤ n, then | N, | n.Remark . When n = N = j = k ≤
11, then B (cid:16) D stjk , p (cid:17) is the Nicholsalgebras ufo (8) in case that ( k , s , t ) is in the set (1 , , , (1 , , , (1 , , , (1 , , , (1 , , , (1 , , , (1 , , , (1 , , , (5 , , , (5 , , , (5 , , , (5 , , , (5 , , , (5 , , , (5 , , , (5 , , , (7 , , , (7 , , , (7 , , , (7 , , , (7 , , , (7 , , , (7 , , , (7 , , , (11 , , , (11 , , , (11 , , , (11 , , , (11 , , , (11 , , , (11 , , , (11 , , Proof.
Since α − β ≡ β ≡ nN mod 8 nN , ( × nk ( s + t + + × jN ( t + ≡ nN , × nk ( s + t + + × jN ( t + ≡ nN mod 8 nN , ⇒ n | × jN ( t + , N | × nk ( s + t + , ⇒ × jN ( t + = nr , × nk ( s + t + = Nr , ⇒ ( × nk ( s + t + + × nr ≡ nN mod 8 nN , · Nr + jN ( t + ≡ nN mod 8 nN , ICHOLS ALGEBRAS OVER SUZUKI ALGEBRA A µλ N n I 13 ⇒ n | nN , N | nN , ⇒ | N , | n . (cid:3) Lemma 4.11.
Denote α = nk ( s + t ) + jNt, β = nk ( s + t ) − jNt. dim B (cid:16) E stjk , p (cid:17) < ∞ if and only if one of the following conditions holds.(1) nN ∤ α , nN | β , Cartan type A × A ;(2) nN ∤ α , nN | ( α + β ) , Cartan type A ;(3) α ≡ nN mod 8 nN, β . and nN mod 8 nN, Super type A ( q ; I ) ;(4) α − β ≡ β ≡ nN mod 8 nN, β . nN. The Nichols algebras ufo (8) , see [2, Page 209] .Remark . When n = N ≤
6, then B (cid:16) E stjk , p (cid:17) is of the Cartan type A × A incase that j = N , k , s , t ) is in the set ( (2 , , , , (4 , , , , (4 , , , , (4 , , , , (4 , , , , (4 , , , , (4 , , , , (6 , , , , (6 , , , , (6 , , , , (6 , , , , (6 , , , ) When n = N ≤
6, then B (cid:16) E stjk , p (cid:17) is of the Cartan type A in case that j = N , k , s , t ) is in the set (3 , , , , (3 , , , , (3 , , , , (3 , , , , (6 , , , , (6 , , , , (6 , , , , (6 , , , , (6 , , , , (6 , , , , (6 , , , , (6 , , , , (6 , , , , (6 , , , , (6 , , , , (6 , , , When n ≤ N ≤
6, then B (cid:16) E stjk , p (cid:17) is of the Super type A ( q ; I ) in case that( n , N , j , k , s , t ) is in the set (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , Proof.
The braiding is of diagonal type with the braiding given by c ( w ⊗ w ) = ω nk ( s + t ) + jNt w ⊗ w , c ( w ⊗ w ) = ω nk ( s + t ) − jNt w ⊗ w , c ( w ⊗ w ) = ω nk ( s + t ) − jNt w ⊗ w , c ( w ⊗ w ) = ω nk ( s + t ) + jNt w ⊗ w . (cid:3) Corollary 4.13. If B (cid:16) E stjk , p (cid:17) is isomorphic to the Nichols algebras ufo (8) , ∤ Nand ∤ n, then | N, | n. Remark . When n = N = j = k ≤
11, then B (cid:16) E stjk , p (cid:17) is the Nicholsalgebras ufo (8) in case that ( k , s , t ) is in the set (1 , , , (1 , , , (1 , , , (1 , , , (1 , , , (1 , , , (1 , , , (1 , , , (5 , , , (5 , , , (5 , , , (5 , , , (5 , , , (5 , , , (5 , , , (5 , , , (7 , , , (7 , , , (7 , , , (7 , , , (7 , , , (7 , , , (7 , , , (7 , , , (11 , , , (11 , , , (11 , , , (11 , , , (11 , , , (11 , , , (11 , , , (11 , , Proof. If B (cid:16) E stjk , p (cid:17) is the Nichols algebras ufo (8), then ( × nk ( s + t ) − × jNt ≡ nN , × nk ( s + t ) − × jNt ≡ nN mod 8 nN . ⇒ N | × nk ( s + t ) , n | × jNt , ⇒ × nk ( s + t ) = Nr , × jNt = nr , ⇒ ( × Nr − × jNt ≡ nN mod 8 nN , × nk ( s + t ) − × nr ≡ nN mod 8 nN . ⇒ N | nN , n | nN , ⇒ | n , | N . (cid:3) Nichols algebra of type V abe . Let V abe = k v ⊗ k v be a vector space with abraiding given by c ( v ⊗ v ) = av ⊗ v , c ( v ⊗ v ) = bv ⊗ v , c ( v ⊗ v ) = bv ⊗ v , c ( v ⊗ v ) = ev ⊗ v , then the braided vector space ( V abe , c ) is of type V abe . The braided vector space V abe is isomorphic to V ae b via v
7→ √ ev , v v . When ae = b , then V abe isof diagonal type anddim B ( V abe ) = , b = − , ( B ( V abe ) is of Cartan type A × A ) , , b = , b , ( B ( V abe ) is of Cartan type A ) , ∞ , otherwise . According to [6, section 3.7] or [30], we have
Lemma 4.15. (1) Suppose b , ae and b = − , then B ( V abe ) < ∞ if ae is am-th primitive root of unity for m ≥ . In particular, dim B ( V abe ) = m.(2) Suppose b , ae = , then dim B ( V abe ) < ∞ if and only if b is a m-thprimitive root of unity for m ≥ . In particular, dim B ( V abe ) = m . Lemma 4.16. B (cid:16) G stjk , p (cid:17) is of type V abe , whereae = ¯ µ s + t + ω kn (4 s + t + + jN ( − − t ) , b = ( − p ¯ µ s + t + ω nk (2 s + t + + jN (2 t + . ICHOLS ALGEBRAS OVER SUZUKI ALGEBRA A µλ N n I 15
Remark . Denote α = ( n , N , j , s , t , k , p ).(1) Suppose λ = − ae =
1, then b = ω jN (2 t + for some suitable p ∈ Z since aeb = ω − jN (2 t + . If we choose j = t =
0, then Nn (8 Nn , N ) = n . Thatis to say, dim B (cid:16) G stjk , p (cid:17) = (4 n ) for suitable choice of ( n , N , j , s , t , k , p ). Forexample, in case µ = B (cid:16) G stjk , p (cid:17) = , α = (1 , , , , , , , (1 , , , , , , , · · · ;dim B (cid:16) G stjk , p (cid:17) = , α = (2 , , , , , , , (2 , , , , , , , · · · ;dim B (cid:16) G stjk , p (cid:17) = , α = (3 , , , , , , , (3 , , , , , , , · · · ;dim B (cid:16) G stjk , p (cid:17) = , α = (4 , , , , , , , (4 , , , , , , , · · · ;dim B (cid:16) G stjk , p (cid:17) = , α = (5 , , , , , , , (5 , , , , , , , · · · ;dim B (cid:16) G stjk , p (cid:17) = , α = (6 , , , , , , , (6 , , , , , , , · · · ;dim B (cid:16) G stjk , p (cid:17) = , α = (7 , , , , , , , (7 , , , , , , , · · · ;dim B (cid:16) G stjk , p (cid:17) = , α = (8 , , , , , , , (8 , , , , , , , · · · ;dim B (cid:16) G stjk , p (cid:17) = , α = (9 , , , , , , , (9 , , , , , , , · · · . (2) Suppose λ = − b = −
1, then ae = aeb = ω − jN (2 t + . So dim B (cid:16) G stjk , p (cid:17) = n for suitable choice of ( n , N , j , s , t , k , p ). For example, in case µ = B (cid:16) G stjk , p (cid:17) = , α = (1 , , , , , , , (1 , , , , , , , · · · ;dim B (cid:16) G stjk , p (cid:17) = , α = (2 , , , , , , , (2 , , , , , , , · · · ;dim B (cid:16) G stjk , p (cid:17) = , α = (3 , , , , , , , (3 , , , , , , , · · · ;dim B (cid:16) G stjk , p (cid:17) = , α = (4 , , , , , , , (4 , , , , , , , · · · ;dim B (cid:16) G stjk , p (cid:17) = , α = (5 , , , , , , , (5 , , , , , , , · · · . (3) Suppose λ = ae =
1, then j is even and b = ω jN (2 t + for some suit-able p ∈ Z since aeb = ω − jN (2 t + . If we choose j = t =
0, then Nn (8 Nn , N ) = (2 n ) . That is to say, dim B (cid:16) G stjk , p (cid:17) = (2 n ) for suitable choiceof ( n , N , j , s , t , k , p ).(4) Suppose λ = − b = −
1, then j is even and ae = aeb = ω − jN (2 t + . Sodim B (cid:16) G stjk , p (cid:17) = n for suitable choice of ( n , N , j , s , t , k , p ). Remark . If b = ae ⇒ ω jN (2 t + = ⇒ n | j (2 t + λ = −
1, then both j and 2 t + n | j (2 t + (2) In case λ = n ≤
5, then the formula 2 n | j (2 t +
1) implies ( n , j , t ) ∈{ (3 , , , (3 , , , (5 , , , (5 , , , (5 , , , (5 , , } .(a) Suppose N < µ =
1, then dim B (cid:16) G stjk , p (cid:17) = n , N , j , t , s , k , p ) is in the set (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (3 , , , , , , , (3 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (3 , , , , , , , (3 , , , , , , , (3 , , , , , , , (3 , , , , , , , (3 , , , , , , , (3 , , , , , , , (3 , , , , , , , (3 , , , , , , , (5 , , , , , , , (5 , , , , , , (b) Suppose N < µ =
1, then dim B (cid:16) G stjk , p (cid:17) =
27 in case that( n , N , j , t , s , k , p ) is in the set (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (3 , , , , , , , (3 , , , , , , , (3 , , , , , , , (3 , , , , , , (c) Suppose N < µ = −
1, then dim B (cid:16) G stjk , p (cid:17) =
27 in case that( n , N , j , t , s , k , p ) is in the set (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (3 , , , , , , , (3 , , , , , , , (3 , , , , , , , (3 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , (d) Suppose N < µ = −
1, then dim B (cid:16) G stjk , p (cid:17) = n , N , j , t , s , k , p ) is in the set (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (3 , , , , , , , (3 , , , , , , , (3 , , , , , , , (3 , , , , , , , (3 , , , , , , , (3 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , , (5 , , , , , , Lemma 4.19. B (cid:16) H sjk , p (cid:17) is is of type V abe withae = ¯ µ s + t + ω kn (4 t + s + + jN (4 t + , b = ( − p ¯ µ s + t + ω kn (2 t + s + + jN ( − − t ) . Remark . From observation, we have aeb = ω jN (2 t + .(1) Suppose ae =
1, then dim B (cid:16) H sjk , p (cid:17) = ( (4 n ) , λ = − , (2 n ) , λ = , under suitablechoice of ( n , N , j , s , t , k , p ). ICHOLS ALGEBRAS OVER SUZUKI ALGEBRA A µλ N n I 17 (2) Suppose b = −
1, then dim B (cid:16) H sjk , p (cid:17) = ( n , λ = − , n , λ = , under suitablechoice of ( n , N , j , s , t , k , p ). Lemma 4.21.
Denote q = ( − p + ( i + j )( t + ) + j ˜ µ s + t + , then dim B (cid:16) P sti jk , p (cid:17) = , q = − , ( B (cid:16) P sti jk , p (cid:17) is of Cartan type A × A ) , , q = , q , ( B (cid:16) P sti jk , p (cid:17) is of Cartan type A ) , ∞ , otherwise . Remark . When n = µ = N ≤ q = −
1, then ( N , j , s , t , k , p ) is in theset (1 , , , , , , (2 , , , , , , (2 , , , , , , (2 , , , , , , (2 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (4 , , , , , , (4 , , , , , , (4 , , , , , , (4 , , , , , , (4 , , , , , , (4 , , , , , , (4 , , , , , , (4 , , , , , , (5 , , , , , , (5 , , , , , , (5 , , , , , , (5 , , , , , , (5 , , , , , , (5 , , , , , , (5 , , , , , , (5 , , , , , , (5 , , , , , When n = µ = N ≤ q = , q , then ( N , j , s , t , k , p ) is in the set (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (6 , , , , , , (6 , , , , , , (6 , , , , , , (6 , , , , , , (6 , , , , , , (6 , , , , , , (6 , , , , , , (6 , , , , , , (6 , , , , , , (6 , , , , , , (6 , , , , , , (6 , , , , , , (6 , , , , , , (6 , , , , , , (6 , , , , , , (6 , , , , , When n = µ = − N ≤ q = −
1, then ( N , j , s , t , k , p ) is in the set (1 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (5 , , , , , , (5 , , , , , , (5 , , , , , , (5 , , , , , , (5 , , , , , , (5 , , , , , , (5 , , , , , , (5 , , , , , , (5 , , , , , When n = µ = − N ≤
10 and q = , q , then ( N , j , s , t , k , p ) is in the set (3 , , , , , , (3 , , , , , , (3 , , , , , , (3 , , , , , , (9 , , , , , , (9 , , , , , , (9 , , , , , , (9 , , , , , , (9 , , , , , , (9 , , , , , , (9 , , , , , , (9 , , , , , , (9 , , , , , , (9 , , , , , , (9 , , , , , , (9 , , , , , , (9 , , , , , , (9 , , , , , , (9 , , , , , , (9 , , , , , , (9 , , , , , , (9 , , , , , , (9 , , , , , , (9 , , , , , , (9 , , , , , , (9 , , , , , , (9 , , , , , , (9 , , , , , Proof.
The braiding of P sti jk , p is given by c ( w α ⊗ w β ) = qw ⊗ w qw ⊗ w qw ⊗ w qw ⊗ w ! , α, β ∈ , , where q = ( − p + ( i + j )( t + ) + j ˜ µ s + t + ω nk (2 s + t + . So B (cid:16) P sti jk , p (cid:17) is finite dimen-sional i ff q = − q = , q . q = ( ω nN [2 p + ( i + j )(2 t + + j ] + nk (2 s + t + , µ = ,ω nN [2 p + ( i + j )(2 t + + j ] + n (2 s + t + k + , µ = − . (cid:3) Nichols algebras over I sp jk .Lemma 4.23. Let ( V , c ) be a braided vector space such that c ( x ⊗ y ) ∈ k f x ( y ) ⊗ x,where the map f x : V → V is bijective for any x ∈ V under a fixed basis. Then ( V , ⊲ ) is a rack with x ⊲ y = f x ( y ) .Proof. For any x , y , z ∈ V , c c c ( x ⊗ y ⊗ z ) ∈ k [( x ⊲ y ) ⊲ ( x ⊲ z )] ⊗ ( x ⊲ y ) ⊗ x , c c c ( x ⊗ y ⊗ z ) ∈ k [ x ⊲ ( y ⊲ z )] ⊗ ( x ⊲ y ) ⊗ x . So x ⊲ ( y ⊲ z ) = ( x ⊲ y ) ⊲ ( x ⊲ z ). (cid:3) Lemma 4.24.
When n > , then dim B (cid:16) I sp jk (cid:17) = ∞ .Proof. When n >
2, let X = { w r , m r | r ∈ , n } , then ( X , ⊲ ) is a rack as defined inLemma 4.23. It’s easy to see that { w r | r ∈ , n } and { m r | r ∈ , n } are two subracksof X . X is of type D , since w ⊲ ( m ⊲ ( w ⊲ m )) = ( m , if n > , m , if n = . According to [5, Theorem 3.6], dim B (cid:16) I sp jk (cid:17) = ∞ . (cid:3) Lemma 4.25.
Let n = and q = ( − p ω k (2 s + , then dim B (cid:16) I sp jk (cid:17) < ∞ ⇐⇒ ( λ q = , q , Cartan type A × A ,λ q = , q , Cartan type A . Proof.
When n =
1, then the braiding is given by c ( w ⊗ w ) = ( − p ω kn (2 s + w ⊗ w , c ( w ⊗ m ) = λ ( − p ω n (2 k + jN )(2 s + m ⊗ w , c ( m ⊗ w ) = ( − p ω n (2 k + jN ) + kns w ⊗ m , c ( m ⊗ m ) = ( − p ω kn (2 s + m ⊗ m . So B (cid:16) I sp jk (cid:17) is of diagonal type and its Dynkin diagram is q ◦ λ q q ◦ ,where q = ( − p ω k (2 s + = ω N p + k (2 s + . (cid:3) ICHOLS ALGEBRAS OVER SUZUKI ALGEBRA A µλ N n I 19
When n =
2, then the braiding is given by c ( w ⊗ w ) = ω kn (2 s + w ⊗ w , c ( w ⊗ w ) = ω n (2 k + jN )(2 s + w ⊗ w , c ( w ⊗ w ) = ( − p ω n (2 k + jN ) + kns w ⊗ w , c ( w ⊗ w ) = ( − p ω n (2 k + jN )( s − + kn w ⊗ w , c ( w ⊗ m ) = ω kns m ⊗ w , c ( w ⊗ m ) = ω kn ( s + m ⊗ w , c ( w ⊗ m ) = λω n (2 k + jN ) + kn (2 s − m ⊗ w , c ( w ⊗ m ) = λω n (2 k + jN ) + kn (2 s + m ⊗ w , c ( m ⊗ m ) = ( − p ω kn (2 s + m ⊗ m , c ( m ⊗ m ) = λ ( − p ω n (2 k + jN ) + kns m ⊗ m , c ( m ⊗ m ) = λ ( − p ω n (2 k + jN ) + kns m ⊗ m , c ( m ⊗ m ) = ( − p ω n (2 k + jN ) + kn (2 s + m ⊗ m , c ( m ⊗ w ) = ω n (2 k + jN ) s w ⊗ m , c ( m ⊗ w ) = ω kn (2 s + w ⊗ m , c ( m ⊗ w ) = ( − p ω n (2 k + jN )(2 s − + kn w ⊗ m , c ( m ⊗ w ) = ω n (2 k + jN ) + kn (2 s + w ⊗ m . Nichols algebras over K sjk , p . Let b + a − = nr + d , r ∈ N , ≤ d ≤ n − , n + − b + a − = ne + f , e ∈ N , ≤ f ≤ n − , then the braiding of K sjk , p is given by c ( w a ⊗ w b ) = ( − p (cid:16) ¯ µω nk (cid:17) s w b ⊗ w , a = , ( − p λ r ¯ µ s + n ( r − ω n ( r − nk + jN ) + nks w n ⊗ w n − a + , a > , d = , | ( a + b ) , ( − p λ r + ¯ µ s + n ( r − ω n ( r − nk + jN ) + nks w d ⊗ w n − a + , a > , d > , | ( a + b ) , ( − p λ e (cid:16) ¯ µω nk (cid:17) s − ne − + a ω − jNne w n ⊗ w n − a + , a > , f = , ∤ ( a + b ) , ( − p λ e + ( ¯ µω nk ) s − n ( e + − + a ω jNn ( e + w n + − f ⊗ w n − a + , a > , f > , ∤ ( a + b ) . Lemma 4.26.
Let q = ( − p ¯ µ s ω kns , then (1) When n = , then dim B (cid:16) K sjk , p (cid:17) < ∞ ⇐⇒ ( λ q = , q , Cartan type A × A ,λ q = , q , Cartan type A . (2) dim B (cid:16) K sjk , p (cid:17) < ∞ = ⇒ q = − or q = , q.Proof. When n =
1, then the braiding of B (cid:16) K sjk , p (cid:17) is given by c ( w ⊗ w ) = ( − p ¯ µ s ω kns , c ( w ⊗ w ) = ( − p ¯ µ s ω kns , c ( w ⊗ w ) = ( − p ¯ µ s ω kns − jN , c ( w ⊗ w ) = ( − p ¯ µ s ω kns . B (cid:16) K sjk , p (cid:17) is of diagonal type and its Dynkin diagram is q ◦ λ q q ◦ , where q = ( − p ¯ µ s ω kns .When n =
2, then the braiding of B (cid:16) K sjk , p (cid:17) is given by c ( w ⊗ w ) = ( − p ¯ µ s ω kns w ⊗ w , c ( w ⊗ w ) = ( − p ¯ µ s ω kns w ⊗ w , c ( w ⊗ w ) = ( − p ¯ µ s ω kns w ⊗ w , c ( w ⊗ w ) = ( − p ¯ µ s ω kns w ⊗ w , c ( w ⊗ w ) = ( − p ¯ µ s − ω kn ( s − − jN w ⊗ w , c ( w ⊗ w ) = ( − p λ ¯ µ s − ω kn ( s − − jN w ⊗ w , c ( w ⊗ w ) = ( − p λ ¯ µ s ω kns − jN w ⊗ w , c ( w ⊗ w ) = ( − p ¯ µ s ω kns − jN w ⊗ w , c ( w ⊗ w ) = ( − p ¯ µ s ω kns − jN w ⊗ w , c ( w ⊗ w ) = ( − p ¯ µ s ω kns w ⊗ w , c ( w ⊗ w ) = ( − p ¯ µ s ω kns w ⊗ w , c ( w ⊗ w ) = ( − p ¯ µ s ω kns w ⊗ w , c ( w ⊗ w ) = ( − p ¯ µ s ω kns w ⊗ w , c ( w ⊗ w ) = ( − p λ ¯ µ s ω kns − jN w ⊗ w , c ( w ⊗ w ) = ( − p ¯ µ s + ω kn ( s + − jN w ⊗ w , c ( w ⊗ w ) = ( − p λ ¯ µ s + ω kn ( s + + jN w ⊗ w . ICHOLS ALGEBRAS OVER SUZUKI ALGEBRA A µλ N n I 21 W = k w ⊕ k w and W = k w ⊕ k w are braided vector spaces. B ( W ) is of diag-onal type and its Dynkin diagram is q ◦ q q ◦ , where q = ( − p ¯ µ s ω kns . B ( W ) is of type V abe with b = q and ae = q . So B ( W )(or B ( W )) is finitedimensional i ff q = − q = , q . (cid:3)
5. A ppendix
Lemma 5.1.
Let s ∈ , N, p ∈ Z , V iik = k v, denote w = v ⊠ h x s + ( − p x s i ,then A siik , p = k w is a one-dimensional Yetter-Drinfeld module over A µλ N n with A siik , p ≃ V iik as A µλ N n -module and ρ ( w ) = h x s + ( − p x s i ⊗ w. Lemma 5.2.
Let s ∈ , N , p ∈ Z , V i jk = k v where ( i = j , λ = , i ≡ j + , λ = − , and i , j ∈ Z . Denotew = v ⊠ h x s + χ n − + ( − p √ λ x s + χ n − i , then ¯ A si jk , p = k w is a one-dimensional Yetter-Drinfeld module over A µλ N n with ¯ A si jk , p ≃ V i jk as A µλ N n -module and ρ ( w ) = h x s + χ n − + ( − p √ λ x s + χ n − i ⊗ w. Lemma 5.3.
Let s ∈ , N, i , j ∈ Z , i ≡ j + , V i jk = k v, denotew = v ⊠ h x s + x s i , w = v ⊠ h x s − x s i , then B si jk = k w ⊕ k w is a two-dimensional Yetter-Drinfeld module over A µλ N n with B si jk ≃ k ( w + w ) ⊕ k ( w − w ) ≃ V i jk ⊕ V jik as A µλ N n -module and ρ ( w p ) = h x s + ( − p + x s i ⊗ w p for p = or .Remark . The actions on the row vector ( w , w ) given by x − i ω nk ( − i ω nk ! , x , x , x − j ω nk ( − j ω nk ! . Remark . B si jk ≃ B sjik via w w ′ , w
7→ − w ′ . Lemma 5.6.
Let s ∈ , N, t ∈ , n − , V i jk = k v, i , j , p ∈ Z . Denotew = v ⊠ h x s + χ t + + ( − p p ( − i + j x s + χ t + i , w = v ⊠ x s χ t + + ( − p p ( − i + j x s χ t + , then C sti jk , p = k w ⊕ k w is a two-dimensional Yetter-Drinfeld module over A µλ N n with actions on the row vector ( w , w ) given byx − i ω nk ( − i ω nk ! , x , x , x − j ω nk ( − j ω nk ! , and the comodule structure given by ρ ( w ) = x s + χ t + ⊗ w + ( − p p ( − i + j x s + χ t + ⊗ w ,ρ ( w ) = x s χ t + ⊗ w + ( − p p ( − i + j x s χ t + ⊗ w . Remark . (1) C sti jk , p ≃ C sti + j + k , p + via w w ′ , w
7→ − w ′ .(2) When t = n − j = ( i , λ = i + , λ = − , C sti jk , p is not a simple Yetter-Drinfeld module.(3) When t = n − j = ( i + , λ = i , λ = − , then w + w + w − w · √ λ √− λ = v ⊠ h x s χ n + ( − p √ λ x s χ n i , w + w − w − w · √ λ √− λ = v ⊠ h x s χ n + ( − p + √ λ x s χ n i . Lemma 5.8.
Let s ∈ , N, r ∈ , n and V i jk = k v. Denotew r = v ⊠ x s + , r = ,χ r − · w , r even ,χ r − · w , r odd , m r = v ⊠ x s χ , r = ,χ r − · m , r even ,χ r − · m , r odd , then M si jk = L nr = ( k w r ⊕ k m r ) is a n-dimensional Yetter-Drinfeld module overA µλ N n with the module structure given byx · w r = ( − i ω nk w , r = , w r + , r even , r < n ,ω nk w r − , r odd , ( − i ω nk w n , r = n even , x · w r = w r + , r odd , r < n ,ω nk w r − , r even , ( − j ω nk w n , r = n odd , x pq · w r = , pq = or , ≤ r ≤ n , ICHOLS ALGEBRAS OVER SUZUKI ALGEBRA A µλ N n I 23 x · m r = m r + , r odd , r < n ,ω nk m r − , r even ,λ ( − j ω nk m n , r = n odd , x pq · m r = , pq = or , ≤ r ≤ n , x · m r = ( − i ω nk m , r = , m r + , r even , r < n ,ω nk m r − , r odd ,λ ( − i ω nk m n , r = n even , and the comodule structure given by ρ ( w r ) = x s − r + χ r − ⊗ w r + x s − r + χ r − ⊗ m r , r even , x s − r + χ r − ⊗ w r + x s − r + χ r − ⊗ m r , r odd ,ρ ( m r ) = x s − r + χ r − ⊗ m r + x s − r + χ r − ⊗ w r , r even , x s − r + χ r − ⊗ m r + x s − r + χ r − ⊗ w r , r odd . Remark . M si jk ≃ M si j + k when n is even. Lemma 5.10.
Let s ∈ , N, V i jk = k v. Denotew r = v ⊠ x s + , r = ,χ r − · w , r even ,χ r − · w , r odd , m r = v ⊠ x s χ , r = ,χ r − · m , r even ,χ r − · m , r odd , then N si jk = L nr = ( k w r ⊕ k m r ) is a n-dimensional Yetter-Drinfeld module overA µλ N n with the module structure given byx · w r = ( − j ω nk w , r = , w r + , r even , r < n ,ω nk w r − , r odd ,λ ( − j ω nk w n , r = n even , x · w r = w r + , r odd , r < n ,ω nk w r − , r even ,λ ( − i ω nk w n , r = n odd , x pq · w r = , pq = or , ≤ r ≤ n , x · m r = m r + , r odd , r < n ,ω nk m r − , r even , ( − i ω nk m n , r = n odd , x pq · m r = , pq = or , ≤ r ≤ n , x · m r = ( − j ω nk m , r = , m r + , r even , r < n ,ω nk m r − , r odd , ( − j ω nk m n , r = n even , and the comodule structure given by ρ ( w r ) = x s − r + χ r − ⊗ w r + x s − r + χ r − ⊗ m r , r even , x s − r + χ r − ⊗ w r + x s − r + χ r − ⊗ m r , r odd ,ρ ( m r ) = x s − r + χ r − ⊗ m r + x s − r + χ r − ⊗ w r , r even , x s − r + χ r − ⊗ m r + x s − r + χ r − ⊗ w r , r odd . Remark . (1) When n is even, then M si jk ≃ N si ′ j ′ k in case of i = j ′ , λ = n is odd, then M si jk ≃ N si ′ j ′ k in case i = j ′ and j = ( i ′ , λ = , i ′ + , λ = − . Lemma 5.12.
Let V jk = k v ⊕ k v , s ∈ , N, t ∈ , n − , p ∈ Z , denotew = v ⊠ x s + χ t + + ( − p ω jN − kn v ⊠ x s + χ t + , w = v ⊠ x s χ t + + ( − p ω kn − jN v ⊠ x s χ t + , then D stjk , p = k w ⊕ k w is a two-dimensional Yetter-Drinfeld module over A µλ N n with D stjk , p ≃ V jk as A µλ N n -module and the comodule structure given by ρ ( w ) = x s + χ t + ⊗ w + ( − p ω jN − kn x s + χ t + ⊗ w ,ρ ( w ) = x s χ t + ⊗ w + ( − p ω kn − jN x s χ t + ⊗ w . Remark . Let w ′ = ( − p ω jN − kn w . When t = n −
1, then ρ ( ± √ λ w + w ′ ) = (cid:16) x s + χ n − ± √ λ x s + χ n − (cid:17) ⊗ ( ± √ λ w + w ′ ) Lemma 5.14.
Let V jk = k v ⊕ k v , s ∈ , N, t ∈ , n − , p ∈ Z , denotew = v ⊠ x s χ t + ( − p ω jN − kn v ⊠ x s χ t , w = v ⊠ x s + χ t − + ( − p ω kn − jN v ⊠ x s + χ t − , = v ⊠ x s χ t + ( − p ω kn − jN v ⊠ x s χ t , then E stjk , p = k w ⊕ k w is a two-dimensional Yetter-Drinfeld module over A µλ N n with E stjk , p ≃ V jk as A µλ N n -module and the comodule structure given by ρ ( w ) = x s χ t ⊗ w + ( − p ω jN − kn x s χ t ⊗ w ,ρ ( w ) = x s + χ t − ⊗ w + ( − p ω kn − jN x s + χ t − ⊗ w , = x s χ t ⊗ w + ( − p ω kn − jN x s χ t ⊗ w . ICHOLS ALGEBRAS OVER SUZUKI ALGEBRA A µλ N n I 25
Lemma 5.15.
Let V jk = k v ⊕ k v , s ∈ , N, p ≡ or , denotew = v ⊠ x s + ( − p ω jN − kn v ⊠ x s , w = v ⊠ x s + ( − p ω kn − jN v ⊠ x s , then F sjk , p = k w ⊕ k w is a two-dimensional Yetter-Drinfeld module over A µλ N n with F sjk , p ≃ V jk as A µλ N n -module and the comodule structure given by ρ ( w ) = x s ⊗ w + ( − p ω jN − kn x s ⊗ w ρ ( w ) = x s ⊗ w + ( − p ω kn − jN x s ⊗ w . Lemma 5.16.
Let V jk = k v ⊕ k v , s ∈ , N, denotew r = h v + ( − p ω jN − kn ) v i ⊠ x s + , r = ,χ r − · w , r even ,χ r − · w , r odd , m r = h v + ( − p ω jN − kn ) v i ⊠ x s x , r = ,χ r − · w , r even ,χ r − · w , r odd , then I sp jk = L nr = ( k w r ⊕ k m r ) is a n-dimensional Yetter-Drinfeld module overA µλ N n with the module structure given byx · w r = ( − p ω kn w , r = , w r + , r even , < r < n ,ω kn w r − , r odd , < r ≤ n , ( − p ω n (2 k + jN ) w n , r = n even , x · w r = ω kn w r − , r even , < r ≤ n , w r + , r odd , ≤ r < n , ( − p ω n (2 k + jN ) w n , r = n odd , x pq · w r = , pq = or , ≤ r ≤ n , x · m r = ω kn m r − , r even , < r ≤ n , m r + , r odd , ≤ r < n ,λ ( − p ω n (2 k + jN ) m n , r = n odd , x pq · m r = , pq = or , ≤ r ≤ n , x · m r = ( − p ω kn m , r = , m r + , r even , < r < n ,ω kn m r − , r odd , < r ≤ n ,λ ( − p ω n (2 k + jN ) m n , r = n even , and the comodule structure given by ρ ( w r ) = x s − r + χ r − ⊗ w r + x s − r + χ r − ⊗ m r , r even , x s − r + χ r − ⊗ w r + x s − r + χ r − ⊗ m r , r odd ,ρ ( m r ) = x s − r + χ r − ⊗ m r + x s − r + χ r − ⊗ w r , r even , x s − r + χ r − ⊗ m r + x s − r + χ r − ⊗ w r , r odd . Remark . I sjkp ≃ I sj ′ kp in case of j ≡ j ′ mod 4.(1) When n is even, then M si jk ≃ I sj ′ kp in case of i = p , j ′ | n is odd, then M si jk ≃ I sj ′ kp in case of i = p and j ′ ≡ ( , i + j is even , , i + j is odd . Lemma 5.18.
Let V jk = k v ⊕ k v , s ∈ , N, denotew = h v + ( − p ω − kn v i ⊠ x s + , w r = ( χ r − · w , r even ,χ r − · w , r odd , m = h v + ( − p ω − kn v i ⊠ x s x , m r = ( χ r − · w , r even ,χ r − · w , r odd , then J sp jk = L nr = ( k w r ⊕ k m r ) is a n-dimensional Yetter-Drinfeld module overA µλ N n with the module structure given byx · w r = ( − p ω kn w , r = , w r + , r even , < r < n ,ω kn w r − , r odd , < r ≤ n ,λ ( − p ω n (2 k + jN ) w n , r = n even , x · w r = ω kn w r − , r even , < r ≤ n , w r + , r odd , ≤ r < n ,λ ( − p ω n (2 k + jN ) w n , r = n odd , x pq · w r = , pq = or , ≤ r ≤ n , x · m r = ω kn m r − , r even , < r ≤ n , m r + , r odd , ≤ r < n , ( − p ω n (2 k + jN ) m n , r = n odd , x pq · m r = , pq = or , ≤ r ≤ n , ICHOLS ALGEBRAS OVER SUZUKI ALGEBRA A µλ N n I 27 x · m r = ( − p ω kn m , r = , m r + , r even , < r < n ,ω kn m r − , r odd , < r ≤ n , ( − p ω n (2 k + jN ) m n , r = n even , and the comodule structure given by ρ ( w r ) = x s − r + χ r − ⊗ w r + x s − r + χ r − ⊗ m r , r even , x s − r + χ r − ⊗ w r + x s − r + χ r − ⊗ m r , r odd ,ρ ( m r ) = x s − r + χ r − ⊗ m r + x s − r + χ r − ⊗ w r , r even , x s − r + χ r − ⊗ m r + x s − r + χ r − ⊗ w r , r odd . Remark . I sp jk ≃ J sp j ′ k in case of j ≡ ( j ′ mod 4 , λ = , j ′ + , λ = − . Lemma 5.20.
Let V ′ jk = k v ′ ⊕ k v ′ , s ∈ , N, t ∈ , n − , denotew = v ′ ⊠ x s + χ t + ( − p ω jN − kn √ ¯ µ v ′ ⊠ x s + χ t , w = v ′ ⊠ x s χ t + + ( − p p ¯ µω kn − jN v ′ ⊠ x s χ t + , then G stjk , p = k w ⊕ k w is a two-dimensional Yetter-Drinfeld module over A µλ N n with G stjk , p ≃ V ′ jk as A µλ N n -module and the comodule structure given by ρ ( w ) = x s + χ t ⊗ w + ( − p ω jN − kn √ ¯ µ x s + χ t ⊗ w ,ρ ( w ) = x s χ t + ⊗ w + ( − p p ¯ µω kn − jN x s χ t + ⊗ w . Lemma 5.21.
Let V ′ jk = k v ′ ⊕ k v ′ , s ∈ , N, t ∈ , n − , denotew = v ′ ⊠ x s χ t + + ( − p ω jN − kn √ ¯ µ v ′ ⊠ x s χ t + , w = v ′ ⊠ x s + χ t + ( − p p ¯ µω kn − jN v ′ ⊠ x s + χ t , then H stjk , p = k w ⊕ k w is a two-dimensional Yetter-Drinfeld module over A µλ N n with H stjk , p ≃ V ′ jk as A µλ N n -module and the comodule structure given by ρ ( w ) = x s χ t + ⊗ w + ( − p ω jN − kn √ ¯ µ x s χ t + ⊗ w ,ρ ( w ) = x s + χ t ⊗ w + ( − p p ¯ µω kn − jN x s + χ t ⊗ w . Lemma 5.22.
Let V ′ jk = k v ′ ⊕ k v ′ , s ∈ , N, denotew r = v ′ ⊠ h x s + ( − p x s i , r = ,χ r − · w , r even and ≤ r ≤ n ,χ r − · w , r odd and ≤ r ≤ n , then K sjk , p = L nr = w r is a n-dimensional Yetter-Drinfeld module over A µλ N n withthe module structure given byx · w r = ( w r + , r odd and ≤ r < n , ¯ µω kn w r − , r even and ≤ r ≤ n , x · w r = λ ¯ µ − n ω kn − n (4 kn + jN ) w n , r = , ¯ µω kn w r − , r odd and ≤ r < n , w r + , r even and ≤ r < n ,λ ¯ µ n ω n (4 kn + jN ) w , r = n , x pq · w r = , pq = or , ≤ r ≤ n , and the comodule structure given by ρ ( w r ) = x s − r + χ r − ⊗ w r + Ω x s − r + χ r − ⊗ w n − r + , ≤ r even , x s − r + χ r − ⊗ w r + Ω x s − r + χ r − ⊗ w n − r + , ≤ r odd , h x s + ( − p x s i ⊗ w , r = , where Ω = ( − p λ ¯ µ r − − n ω kn ( r − − n ) − jNn .Remark . K sjk , p ≃ K sj ′ k , p in case of j ≡ j ′ mod 4. Lemma 5.24.
Let V ′ jk = k v ′ ⊕ k v ′ , s ∈ , N, p ∈ Z , denotew r = v ′ ⊠ h x s + ( − p x s i , r = ,χ r − · w , r even and ≤ r ≤ n ,χ r − · w , r odd and ≤ r ≤ n , then L sjk , p = L nr = k w r is a n-dimensional Yetter-Drinfeld module over A µλ N n with the module structure given byx · w r = ( w r + , r odd and ≤ r < n , ¯ µω kn w r − , r even and ≤ r ≤ n , x · w r = λ ¯ µ − n ω kn − n (4 kn − jN ) w n , r = , ¯ µω kn w r − , r odd and ≤ r < n , w r + , r even and ≤ r < n ,λ ¯ µ n ω n (4 kn − jN ) w , r = n , x pq · w r = , pq = or , ≤ r ≤ n , ICHOLS ALGEBRAS OVER SUZUKI ALGEBRA A µλ N n I 29 and the comodule structure given by ρ ( w r ) = x s − r + χ r − ⊗ w r + Ω x s − r + χ r − ⊗ w n − r + , ≤ r even , x s − r + χ r − ⊗ w r + Ω x s − r + χ r − ⊗ w n − r + , ≤ r odd , h x s + ( − p x s i ⊗ w , r = , where Ω = ( − p λ ¯ µ r − − n ω kn ( r − − n ) + jNn .Remark . L sjk , p ≃ L sj ′ k , p in case of j ≡ j ′ mod 4 and K sjk , p ≃ L sj ′ k , p in caseof j ≡ − j ′ mod 4. Lemma 5.26.
Let V ′ i jk = k v, s ∈ , N, t ∈ , n − , denotew = v ⊠ x s + χ t + ( − p p ( − i + j x s + χ t , w = v ⊠ h x s χ t + + p ( − i + j ( − p x s χ t + i , then P sti jk , p = k w ⊕ k w is a two-dimensional Yetter-Drinfeld module over A µ + N n with P sti jk , p = k ( w + w ) ⊕ k ( w − w ) ≃ V ′ i jk ⊕ V ′ i + j + k as A µ + N n -modules and thecomodule structure given by ρ ( w ) = x s + χ t ⊗ w + ( − p p ( − i + j x s + χ t ⊗ w ,ρ ( w ) = x s χ t + ⊗ w + p ( − i + j ( − p x s χ t + ⊗ w . Remark . The action of A µ + N n on the row vector ( w , w ) is given by x µ ( − j ω kn ˜ µ ( − j ω kn ! , x , x , x µ ( − i ω kn ˜ µ ( − i ω kn ! . P sti jk , p = k w ⊕ k ( − p √ ( − i + j w ! ≃ Λ s t + as comodules.If i ′ = i + j ′ = j + p ′ = p +
1, then P sti jk , p ≃ P sti ′ j ′ k , p ′ via w w ′ , w
7→ − w ′ . Lemma 5.28.
Let V ′ i jk = k v, s ∈ , N, i , j , p ∈ Z , k ∈ , N − , and denotew r = v ⊠ h x s + ( − p x s i , r = ,χ r − · w , r even and ≤ r ≤ n ,χ r − · w , r odd and ≤ r ≤ n , then Q si jk , p = L nr = w r is a n-dimensional Yetter-Drinfeld module over A µ + N n withthe module structure given byx · w r = ( w r + , r odd and ≤ r < n , ˜ µ ω kn w r − , r even and ≤ r ≤ n , x · w r = λ ˜ µ − n ) ω kn − kn ( − ( i + j ) n w n , r = , ˜ µ ω kn w r − , r odd and ≤ r < n , w r + , r even and ≤ r < n ,λ ˜ µ n ω kn ( − ( i + j ) n w , r = n , x pq · w r = , pq = or , ≤ r ≤ n , and the comodule structure given by ρ ( w r ) = x s − r + χ r − ⊗ w r + Ω x s − r + χ r − ⊗ w n − r + , ≤ r even , x s − r + χ r − ⊗ w r + Ω x s − r + χ r − ⊗ w n − r + , ≤ r odd , h x s + ( − p x s i ⊗ w , r = , where Ω = ( − p λ ˜ µ r − − n ) ω kn ( r − − n ) ( − ( i + j ) n .Remark . When ( − ( i + j ) n = ω n j ′ N , then Q si jk , p ≃ L sj ′ k , p .(1) When i = j , then ( − ( i + j ) n = ω n j ′ N ⇒ | j ′ .(2) When i = j +
1, then ( − ( i + j ) n = ω n j ′ N ⇒ j ′ ≡ . Q si jk , p ≃ L sj ′ k , p in case that n is odd and j ′ ≡ ( , if i + j is even , , if i + j is odd . Q si jk , p ≃ L sj ′ k , p in case that n is even and j ′ ≡ eferences [1] N. Andruskiewitsch. An introduction to Nichols algebras. In Quantization, geometry andnoncommutative structures in mathematics and physics , Math. Phys. Stud., pages 135–195.Springer, Cham, 2017.[2] N. Andruskiewitsch and I. Angiono. On finite dimensional Nichols algebras of diagonal type.
Bull. Math. Sci. , 7(3):353–573, 2017.[3] N. Andruskiewitsch, G. Carnovale, and G. Garc´ıa. Finite-dimensional pointed Hopf algebrasover finite simple groups of Lie type I. Non-semisimple classes in
PSL n ( q ). J. Algebra , 442:36–65, 2015.[4] N. Andruskiewitsch, G. Carnovale, and G. Garc´ıa. Finite-dimensional pointed Hopf algebrasover finite simple groups of Lie type III. Semisimple classes in
PSL n ( q ). Rev. Mat. Iberoam. ,33(3):995–1024, 2017.[5] N. Andruskiewitsch, F. Fantino, M. Gra˜na, and L. Vendramin. Finite-dimensional pointed Hopfalgebras with alternating groups are trivial.
Ann. Mat. Pura Appl. (4) , 190(2):225–245, 2011.[6] N. Andruskiewitsch and J. Giraldi. Nichols algebras that are quantum planes.
Linear and Mul-tilinear Algebra , 66(5):961–991, 2018.[7] N. Andruskiewitsch and M. Gra˜na. From racks to pointed Hopf algebras.
Adv. Math. ,178(2):177–243, 2003.[8] N. Andruskiewitsch and H.-J. Schneider. Lifting of quantum linear spaces and pointed Hopfalgebras of order p . J. Algebra , 209(2):658–691, 1998.[9] N. Andruskiewitsch and H-J. Schneider. Pointed Hopf algebras. In
New directions in Hopf alge-bras , volume 43 of
Math. Sci. Res. Inst. Publ. , pages 1–68. Cambridge Univ. Press, Cambridge,2002.[10] I. Angiono. On Nichols algebras of diagonal type.
J. Reine Angew. Math. , 683:189–251, 2013.
ICHOLS ALGEBRAS OVER SUZUKI ALGEBRA A µλ N n I 31 [11] I. Angiono. A presentation by generators and relations of Nichols algebras of diagonal typeand convex orders on root systems.
J. Eur. Math. Soc. (JEMS) , 17(10):2643–2671, 2015.[12] S. Burciu. On the Grothendieck rings of generalized Drinfeld doubles.
J. Algebra , 486:14–35,2017.[13] C. C˘alinescu, S. D˘asc˘alescu, A. Masuoka, and C. Menini. Quantum lines over non-cocommutative cosemisimple Hopf algebras.
J. Algebra , 273(2):753–779, 2004.[14] R. Dijkgraaf, V. Pasquier, and P. Roche. Quasi Hopf algebras, group cohomology and orb-ifold models.
Nuclear Phys. B Proc. Suppl. , 18B:60–72 (1991), 1990. Recent advances in fieldtheory (Annecy-le-Vieux, 1990).[15] P. Etingof and S. Gelaki. The classification of triangular semisimple and cosemisimple Hopfalgebras over an algebraically closed field.
Internat. Math. Res. Notices , (5):223–234, 2000.[16] F. Fantino, G. Garc´ıa, and M. Mastnak. On finite-dimensional copointed Hopf algebras overdihedral groups.
J. Pure Appl. Algebra , 223(8):3611–3634, 2019.[17] M. D. Gould. Quantum double finite group algebras and their representations.
Bull. Austral.Math. Soc. , 48(02):275–301, 1993.[18] M. Gra˜na, I. Heckenberger, and L. Vendramin. Nichols algebras of group type with manyquadratic relations.
Adv. Math. , 227(5):1956–1989, 2011.[19] I. Heckenberger. The Weyl groupoid of a Nichols algebra of diagonal type.
Invent. Math. ,164(1):175–188, 2006.[20] I. Heckenberger. Classification of arithmetic root systems.
Adv. Math. , 220(1):59–124, 2009.[21] I. Heckenberger, A. Lochmann, and L. Vendramin. Braided racks, Hurwitz actions and Nicholsalgebras with many cubic relations.
Transform. Groups , 17(1):157–194, 2012.[22] I. Heckenberger and H. Yamane. A generalization of Coxeter groups, root systems, and Mat-sumoto’s theorem.
Math. Z. , 259(2):255–276, 2008.[23] Jun Hu and Yinhuo Zhang. The β -character algebra and a commuting pair in Hopf algebras. Algebr. Represent. Theory , 10(5):497–516, 2007.[24] Jun Hu and Yinhuo Zhang. Induced modules of semisimple Hopf algebras.
Algebra Colloq. ,14(4):571–584, 2007.[25] Zhimin Liu and Shenglin Zhu. On the structure of irreducible Yetter-Drinfeld modules overquasi-triangular Hopf algebras.
J. Algebra , 539:339–365, 2019.[26] S. Majid. Doubles of quasitriangular Hopf algebras.
Comm. Algebra , 19(11):3061–3073, 1991.[27] A. Masuoka. Cocycle deformations and Galois objects for some cosemisimple Hopf algebrasof finite dimension. In
New trends in Hopf algebra theory (La Falda, 1999) , volume 267 of
Contemp. Math. , pages 195–214. Amer. Math. Soc., Providence, RI, 2000.[28] D. E. Radford. On oriented quantum algebras derived from representations of the quantumdouble of a finite-dimensional Hopf algebra.
J. Algebra , 270(2):670–695, 2003.[29] Yuxing Shi. Finite-dimensional Hopf algebras over the Kac-Paljutkin algebra H . Rev. Un. Mat.Argentina , 60(1):265–298, 2019.[30] Yuxing Shi. A class Nichols algebras of undiagonal type and enumerative combinatorics. toappear in arXiv , 2020.[31] Yuxing Shi. Finite dimensional Nichols algebras over Suzuki algebra A µλ N n + I: over simpleYetter-Drinfeld modules. to appear in arXiv , 2020.[32] S. Suzuki. A family of braided cosemisimple Hopf algebras of finite dimension.
Tsukuba J.Math. , 22(1):1–29, 1998.[33] M. Wakui. The coribbon structures of some finite dimensional braided Hopf algebras generatedby 2 × Noncommutative geometry and quantum groups (Warsaw, 2001) ,volume 61 of
Banach Center Publ. , pages 333–344. Polish Acad. Sci. Inst. Math., Warsaw,2003. [34] M. Wakui. Polynomial invariants for a semisimple and cosemisimple Hopf algebra of finitedimension.
J. Pure Appl. Algebra , 214(6):701–728, 2010.[35] M. Wakui. Triangular structures of Hopf algebras and tensor Morita equivalences.
Rev. Un.Mat. Argentina , 51(1):193–210, 2010.[36] M. Wakui. Braided Morita equivalence for finite-dimensional semisimple and cosemisimpleHopf algebras. In
Proceedings of the Meeting for Study of Number Theory, Hopf Algebras andRelated Topics , pages 157–183. Yokohama Publ., Yokohama, 2019.S chool of M athematics and S tatistics , J iangxi N ormal U niversity , N anchang hina Email address ::