On non-connected pointed Hopf algebras of dimension 16 in characteristic 2
aa r X i v : . [ m a t h . QA ] J un ON NON-CONNECTED POINTED HOPF ALGEBRAS OFDIMENSION 16 IN CHARACTERISTIC RONGCHUAN XIONG
Abstract.
Let k be an algebraically closed field. We give a complete isomorphismclassification of non-connected pointed Hopf algebras of dimension 16 with char k = 2that are generated by group-like elements and skew-primitive elements. It turns outthat there are infinitely many classes (up to isomorphism) of pointed Hopf algebras ofdimension 16. In particular, we obtain infinitely many new examples of non-commutativenon-cocommutative finite-dimensional pointed Hopf algebras. Keywords:
Nichols algebra; Pointed Hopf algebra; Positive characteristic; Lifting method. Introduction
Let k be an algebraically closed field of positive characteristic. It is a difficult questionto classify Hopf algebras over k of a given dimension. Indeed, the complete classificationshave been done only for prime dimensions (see [19]). One may obtain partial classificationresults by determining Hopf algebras with some properties. To date, pointed ones are theclass best classified.Let p, q, r be distinct prime numbers and char k = p . G. Henderson classified cocom-mutative connected Hopf algebras of dimension less than or equal to p [13]; X. Wangclassified connected Hopf algebras of dimension p [34] and pointed ones with L. Wang [33];V. C. Nguyen, L. Wang and X. Wang determined connected Hopf algebras of dimension p [20, 21]; Nguyen-Wang [22] studied the classification of non-connected pointed Hopfalgebras of dimension p and classified coradically graded ones; motivated by [27, 22], theauthor gave a complete classification of pointed Hopf algebras of dimension pq , pqr , p q ,2 q , 4 p and pointed Hopf algebras of dimension pq whose diagrams are Nichols algebras.It should be mentioned that S. Scherotzke classified finite-dimensional pointed Hopf al-gebras whose infinitesimal braidings are one-dimensional and the diagrams are Nicholsalgebras [26]; N. Hu, X. Wang and Z. Tong constructed many examples of pointed Hopfalgebras of dimension p n for some n ∈ N via quantizations of the restricted universalenveloping algebras of the restricted modular simple Lie algebras of Cartan type, see[14, 15, 30, 29]; C. Cibils, A. Lauve and S. Witherspoon constructed several examples offinite-dimensional pointed Hopf algebras whose diagrams are Nichols algebras of Jordantype [10]; N. Andruskiewitsch, et al. constructed some examples of finite-dimensional coradically graded pointed Hopf algebras whose diagram are Nichols algebras of non-diagonal type [1], which extends the work in [10]. Until now, it is still an open questionto give a complete classification of non-connected pointed Hopf algebras of dimension p or pointed ones of dimension pq whose diagrams are not Nichols algebras for odd primenumbers p, q .In this paper, we study the classification of non-connected pointed Hopf algebras over k of dimension 16 that are generated by group-like elements and skew-primitive elements.Indeed, S. Caenepeel, S. D˘asc˘alescu and S. Raianu classified all pointed complex Hopfalgebras of dimension 16 [9]. We mention that the classification of pointed Hopf algebras H over k with (dim H, char k ) = 1 yields similar isomorphism classes as in the case ofcharacteristic zero. Therefore, we deal with pointed Hopf algebras of dimension 16 withchar k = 2.The strategy follows the ideas in [2], that is, the so-called lifting method. Let H be afinite dimensional Hopf algebra such that the coradical H is a Hopf subalgebra, then gr H ,the graded coalgebra of H associated to the coradical filtration, is a Hopf algebra withprojection onto the coradical H . By [24, Theorem 2], there exists a connected gradedbraided Hopf algebra R = ⊕ ∞ n =0 R ( n ) in H H Y D such that gr H ∼ = R♯H . We call R and R (1) the diagram and infinitesimal braiding of H , respectively. Furthermore, the diagram R is coradically graded and the subalgebra generated by V is the so-called Nichols algebra B ( V ) over V := R (1), which plays a key role in the classification of pointed complex Hopfalgebras. In particular, pointed Hopf algebras are generated by group-like elements andskew-primitive elements if and only if the diagrams are Nichols algebras. See [5] for details.By means of the lifting method [2], we classify all non-connected Hopf algebras of di-mension 16 with char k = 2 whose diagrams are Nichols algebras. See Theorem 4.2 forthe classification results. Contrary to the case of characteristic zero, there exist infinitelymany isomorphism classes, which provides a counterexample to Kaplansky’s 10-th con-jecture, and there are infinitely many classes of pointed Hopf algebras of dimension 16with non-abelian coradicals.Besides, we also classify pointed Hopf algebras of dimension p with some properties,see e.g. Theorem 3.7. In particular, we obtain infinitely many new examples of non-commutative non-cocommutative finite-dimensional pointed Hopf algebras.The paper is organized as below: In section 2, we introduce necessary notations andmaterials that we will need to study pointed Hopf algebras in positive characteristic.In section 3, we study pointed Hopf algebras of dimension p with some properties. Insection 4, we classify non-connected pointed Hopf algebras of dimension 16 whose diagramsare Nichols algebras. The classification of pointed ones whose diagrams are not Nicholsalgebras is much more difficult and requires different techniques, such as the Hochschild OINTED HOPF ALGEBRAS OF DIMENSION 16 IN CHARACTERISTIC 2 3 cohomology of coalgebras (see e.g. [27, 22, 36]). We shall treat them in a subsequentwork. 2.
Preliminaries
Conventions.
We work over an algebraically closed field k of positive characteristic.Denote by char k the characteristic of k , by N the set of natural numbers, and by C n thecyclic group of order n . k × = k −{ } . Given n ≥ k ≥ I k,n = { k, k +1 , . . . , n } . Let C be acoalgebra. Then the set G ( C ) := { c ∈ C | ∆( c ) = c ⊗ c, ǫ ( c ) = 1 } is called the set of group-like elements of C . For any g, h ∈ G ( C ), the set P g,h ( C ) := { c ∈ C | ∆( c ) = c ⊗ g + h ⊗ c } is called the space of ( g, h )- skew primitive elements of C . In particular, the linear space P ( C ) := P , ( C ) is called the set of primitive elements . Unless otherwise stated, “pointed”refers to “nontrivial pointed” in our context.Our references for Hopf algebra theory are [25].2.1. Yetter-Drinfeld modules and bonsonizations.
Let H be a Hopf algebra withbijective antipode. A left Yetter-Drinfeld module M over H is a left H -module ( M, · ) anda left H -comodule ( M, δ ) satisfying δ ( h · v ) = h (1) v ( − S ( h (3) ) ⊗ h (2) · v (0) , ∀ v ∈ V, h ∈ H. (1)Let HH Y D be the category of Yetter-Drinfeld modules over H . Then HH Y D is braidedmonoidal. For
V, W ∈ HH Y D , the braiding c V,W is given by c V,W : V ⊗ W W ⊗ V, v ⊗ w v ( − · w ⊗ v (0) , ∀ v ∈ V, w ∈ W. (2)In particular, c := c V,V is a linear isomorphism satisfying the braid equation ( c ⊗ id)(id ⊗ c )( c ⊗ id) = (id ⊗ c )( c ⊗ id)(id ⊗ c ), that is, ( V, c ) is a braided vector space.
Remark 2.1.
Let V ∈ HH Y D such that dim V = 1 . Let { v } be a basis of V . By definition,there is an algebra map χ : H → k and g ∈ G ( H ) satisfying h (1) χ ( h (2) ) g = gh (2) χ ( h (1) ) , such that δ ( v ) = g ⊗ v , h · v = χ ( h ) v . Moreover, g lies in the center of G ( H ) . Remark 2.2.
Suppose that H = k [ G ] , where G is a group. We write GG Y D for thecategory of Yetter-Drinfeld modules over k [ G ] . Let V ∈ GG Y D . Then V as a G -comoduleis just a G -graded vector space V := ⊕ g ∈ G V g , where V g := { v ∈ V | δ ( v ) = g ⊗ v } . In thiscase, the condition (1) is equivalent to the condition g · V h ⊂ V ghg − .Assume in addition that the action of G is diagonalizable, that is, V = ⊕ χ ∈ b G V χ , where V χ := { v ∈ V | g · v = χ ( g ) v, ∀ g ∈ G } . Then V = ⊕ g ∈ G,χ ∈ b G V χg , where V χg = V g ∩ V χ . RONGCHUAN XIONG
Let G be a finite group. For any g ∈ G , we denote by O g the conjugacy class of g , by C G ( g ) the isotropy subgroup of g and by O ( G ) be the set of conjugacy classes of G . Forany Ω ∈ O ( G ), fix g Ω ∈ Ω, then G = ⊔ Ω ∈O ( G ) O g Ω is a decomposition of conjugacy classesof G . Let ψ : k [ C G ( g Ω )] → End( V ) be a representation of k [ C G ( g Ω )], denoted by ( V, ψ ).Then the induced module M ( g Ω , ψ ) := k [ G ] ⊗ k [ C G ( g Ω )] V can be an object in GG Y D by h · ( g ⊗ v ) = hg ⊗ v, δ ( g ⊗ v ) = gg Ω g − ⊗ ( g ⊗ v ) , h, g ∈ G, v ∈ V. In particular, dim M ( g Ω , ψ ) = [ G, C G ( g Ω )] × dim V . Furthermore, indecomposable objectsin GG Y D are indexed by the pairs (
V, ψ ), see e.g. [17, 37].
Theorem 2.3. [17, 37] M ( g Ω , ψ ) is an indecomposable object in GG Y D if and only if ( V, ψ ) is an indecomposable k [ C G ( g Ω )] -module. Furthermore, any indecomposable object in GG Y D is isomorphic to M ( g Ω , ψ ) for some Ω ∈ O ( G ) and indecomposable k [ C G ( g Ω )] -module ( V, ψ ) . Let C p s := h g i and char k = p . Then the p s non-isomorphic indecomposable C p s -modules consist of r -dimensional modules V r = k { v , v , · · · , v r } for r ∈ I ,p s , whosemodule structure given by g · v = v , g · v m = v m + v m − , < m ≤ r. The following well-known result follows directly by Theorem 2.3. See e.g.[11] for details.
Proposition 2.4.
Let C p s := h g i and char k = p . The indecomposable objects in C ps C ps Y D consist of r -dimensional objects M i,r := M ( g i , V r ) = k { v , v , · · · , v r } for r ∈ I ,p s , i ∈ I ,p s − , whose Yetter-Drinfeld module structure given by g · v = v , g · v m = v m + v m − , < m ≤ r ; δ ( v n ) = g i ⊗ v n , n ∈ I ,r . Let R be a braided Hopf algebra in HH Y D . We write ∆ R ( r ) = r (1) ⊗ r (2) for thecomultiplication to avoid confusions. The bosonization or Radford biproduct R♯H of R by H is a Hopf algebra over k defined as follows: R♯H = R ⊗ H as a vector space, and themultiplication and comultiplication are given by the smash product and smash coproduct,respectively:( r♯g )( s♯h ) = r ( g (1) · s ) ♯g (2) h, ∆( r♯g ) = r (1) ♯ ( r (2) ) ( − g (1) ⊗ ( r (2) ) (0) ♯g (2) . Clearly, the map ι : H → R♯H, h ♯h, h ∈ H is injective and the map π : R♯H → H, r♯h ǫ R ( r ) h, r ∈ R, h ∈ H is surjective such that π ◦ ι = id H . Furthermore, R = ( R♯H ) coH = { x ∈ R♯H | (id ⊗ π )∆( x ) = x ⊗ } .Conversely, if A is a Hopf algebra and π : A → H is a bialgebra map admitting abialgebra section ι : H → A such that π ◦ ι = id H , then A ≃ R♯H , where R = A coH is abraided Hopf algebra in HH Y D . See [25] for details.
OINTED HOPF ALGEBRAS OF DIMENSION 16 IN CHARACTERISTIC 2 5
Braided vector spaces and Nichols algebras.
We follows [5] to introduce thedefinition of Nichols algebras.Let (
V, c ) be a braided vector space. Then the tensor algebra T ( V ) = ⊕ n ≥ T n ( V ) := ⊕ n ≥ V ⊗ n admits a connected braided Hopf algebra structure with the comultiplicationdetermined by ∆( v ) = v ⊗ ⊗ v for any v ∈ V . The braiding can be extended to c : T ( V ) ⊗ T ( V ) → T ( V ) ⊗ T ( V ) in the usual way. Then the braided commutator isdefined by [ x, y ] c = xy − m T ( V ) · c ( x ⊗ y ) , x, y ∈ T ( V ) . Let B n be the braid group presented by generators ( τ j ) j ∈ I ,n − with the defining relations τ i τ j = τ j τ i , τ i τ i +1 τ i = τ i +1 τ i τ i +1 , for i ∈ I ,n − , j = i + 1 . (3)Then there exists naturally the representation ̺ n of B n on T n ( V ) for n ≥ ̺ n : σ j c j := id V ⊗ ( j − ⊗ c ⊗ id V ⊗ ( n − j − . Let M n : S n → B n be the (set-theoretical) Matsumoto section, that preserves the lengthand satisfies M n ( s j ) = σ j . Then the quantum symmetrizer Ω n : V ⊗ n → V ⊗ n is defined byΩ n = X σ ∈ S n ̺ n ( M n ( σ )) . Definition 2.5.
Let ( V, c ) be a braided vector space. The Nichols algebra B ( V ) is definedby B ( V ) = T ( V ) / J ( V ) , where J ( V ) = ⊕ n ≥ J n ( V ) and J n ( V ) = ker Ω n . (4)Indeed, J ( V ) coincides with the largest homogeneous ideal of T ( V ) generated by ele-ments of degree bigger than 2 that is also a coideal. Moreover, B ( V ) = ⊕ n ≥ B n ( V ) is aconnected N -graded Hopf algebra. Example 2.1.
A braided vector space ( V, c ) of rank m is said to be of diagonal type, ifthere exists a basis { x i } i ∈ I ,m such that c ( x i ⊗ x j ) = q ij x j ⊗ x i for q i,j ∈ k × . Rank 2 and 3Nichols algebras of diagonal type with finite PBW-generators were classified in [31, 32] . Example 2.2.
A braided vector space ( V, c ) of rank m > is said to be of Jordan type,denoted by V ( s, m ) , if there exists a basis { x i } i ∈ I ,m such that c ( x i ⊗ x ) = sx ⊗ x i , and c ( x i ⊗ x j ) = ( sx j + x j − ) ⊗ x i , i ∈ I ,m , j ∈ I ,m . Let char k = p . Then it is easy to see that dim B ( V (1 , m )) ≥ p m . See e.g. [1] for details. Proposition 2.6. A N -graded Hopf algebra R = ⊕ n ≥ R ( n ) in HH Y D is a Nichols algebraif and only if (1) R (0) ∼ = k , (2) P ( R ) = R (1) , (3) R is generated as an algebra by R (1) . RONGCHUAN XIONG
Recall that an object in the category of Yetter-Drinfeld modules is a braided vectorspace.
Proposition 2.7. [28, Theorem 5.7]
Let ( V, c ) be a rigid braided vector space. Then B ( V ) can be realized as a braided Hopf algebra in HH Y D for some Hopf algebra H . Remark 2.8.
By Definition 2.5 and Proposition 2.7, B ( V ) depends only on ( V, c
V,V ) andthe same braided vector space can be realized in HH Y D in many ways and for many H ’s. Several lemmas and propositions.
We introduce some important skills in positivecharacteristic. For more details, we refer to [16, 20, 21, 22, 26] and references therein.Let (ad L x )( y ) := [ x, y ] and ( x )(ad R y ) = [ x, y ]. The following propositions are veryuseful in positive characteristic. Proposition 2.9. [16]
Let A be any associative algebra over a field. For any a, b ∈ A , ( ad L a ) p ( b ) = [ a p , b ] , ( ad L a ) p − ( b ) = p − X i =0 a i ba p − − i ;( a )( ad R b ) p = [ a, b p ] , ( a )( ad R b ) p − = p − X i =0 b p − − i ab i . Furthermore, ( a + b ) p = a p + b p + p − X i =1 s i ( a, b ) , where is i ( a, b ) is the coefficient of λ i − in ( a )( ad R λa + b ) p − , λ an indeterminate. Lemma 2.10.
Let A be an associative algebra over k with generators g , x , subject to therelations g n = 1 , gx − xg = g (1 − g ) . Assume that char k = p > and p | n . Then (1): g i x = xg i + ig i − ig i +1 . In particular, g p x = xg p . (2): [22, Lemma 5.1(1)] ( g )( ad R x ) p − = g − g p , ( g )( ad R x ) p = [ g, x ] . (3): ( ad L x ) p − ( g ) = g − g p , [ x p , g ] = ( ad L x ) p ( g ) = [ x, g ] . Lemma 2.11. [22]
Let char k = p > , k ∈ N − { } and µ ∈ I ,pk − . Let A be anassociative algebra generated by g , x , y . Assume that the relations g pk = 1 , gx − xg = λ ( g − g ) , gy − yg = λ ( g − g µ +1 ) ,x p − λ x = 0 , y p − λ y = 0 , xy − yx + µλ y − λ x = λ (1 − g µ +1 ) , hold in A for some λ , λ ∈ I , , λ ∈ k . Then (1): ( x )( ad R y ) n = λ n − ( x )( ad R y ) − λ P n − i =0 λ i ( g µ +1 )( ad R y ) n − − i . In particular, if k = 1 , then ( x )( ad R y ) p = λ p − ( x )( ad R y ) . OINTED HOPF ALGEBRAS OF DIMENSION 16 IN CHARACTERISTIC 2 7 (2): ( ad L x ) n ( y ) = ( − µλ ) n − ( ad L x )( y ) − λ P n − i =0 ( − µλ ) i ( ad L x ) n − − i ( g µ +1 ) . Inparticular, if k = 1 , then ( ad L x ) p ( y ) = ( − µλ ) p − ( ad L x )( y ) . The following lemma extends [22, Proposition 3.9]
Lemma 2.12.
Let char k = p and G be a group of order p m for m ∈ I , . Let V ∈ GG Y D such that dim
V > − m . Then dim B ( V ) > p − m .Proof. The proof follows the same lines of [22, Proposition 3.9]. (cid:3)
Now we introduce the following proposition, which is useful to determine when a coal-gebra map is one-one.
Proposition 2.13. [25, Proposition 4.3.3]
Let
C, D be coalgebras over k and f : C → D is a coalgebra map. Assume that C is pointed. Then the following are equivalent: (a) f is one-one. (b) For any g, h ∈ G ( C ) , f | P g,h ( C ) is one-one. (c) f | C is one-one. On pointed Hopf algebras of dimension p Let p be a prime number and char k = p . We study pointed Hopf algebras of dimension p with some properties, which will be used to obtain our main results. In particular,we obtain some classification results of pointed Hopf algebras of dimension p with someproperties. We mention that N. Andruskiewitsch and H. J. Schneider classified pointedcomplex Hopf algebra of p for an odd prime p [4]; S. Caenepeel, S. D˘asc˘alescu and S.Raianu classified all pointed complex Hopf algebras of dimension 16 [9]; and the Hopfsubalgebra of dimension p have already appeared in [22]. Lemma 3.1.
Let char k = p , C p s := h g i and V be an object in C ps C ps Y D such that dim B ( V ) = p . Then dim V = 2 . Furthermore, • If B ( V ) is of diagonal type, then V ∼ = M i, ⊕ M j, for i, j ∈ I ,p s − or M k, for p | k ∈ I ,p s − and hence B ( V ) ∼ = k [ x, y ] / ( x p , y p ) . • If B ( V ) is not of diagonal type, then p > , V ∼ = M i, for p ∤ i ∈ I ,p s − and hence B ( V ) ∼ = k h x, y i / ( x p , y p , yx − xy + x ) .Proof. Observe that dim B ( V ) = p if dim V = 1. Then by [22, Proposition 3.9], dim V = 2.By Proposition 2.4, V ∼ = M i, ⊕ M j, for i, j ∈ I ,p s − or M k, for k ∈ I ,p s − .Assume that V ∼ = M i, ⊕ M j, for i, j ∈ I ,p s − . Then V is of diagonal type with trivialbraiding, which implies that B ( V ) ∼ = k [ x, y ] / ( x p , y p ). RONGCHUAN XIONG
Assume that V ∼ = M k, := k { v , v } for k ∈ I ,p s − . Then the braiding of V is c ( " xy ⊗ h x y i ) = " x ⊗ x ( y + kx ) ⊗ xx ⊗ y ( y + kx ) ⊗ y . If p | k , then V is of diagonal type with trivial braiding and hence B ( V ) ∼ = k [ x, y ] / ( x p , y p ).If p ∤ k , then V is of Jordan type and hence by [10, Theorem 3.1 and 3.5], p > B ( V ) ∼ = k h x, y i / ( x p , y p , yx − xy + x ). (cid:3) Remark 3.2.
Let G be a finite group and V ∈ GG Y D . If dim V = 2 , then by [22, Propo-sition 3.3] , V is either of diagonal type or of Jordan type. Lemma 3.3.
Let char k = p , C p := h g i and V be a decomposable object in C p C p Y D suchthat dim B ( V ) = p . Then dim V = 3 . Furthermore, • If B ( V ) is of diagonal type, then V ∼ = M i, ⊕ M j, ⊕ M k, for i, j, k ∈ I ,p − or M , ⊕ M , and hence B ( V ) ∼ = k [ x, y, z ] / ( x p , y p , z p ) . • If B ( V ) is not of diagonal type, then p > , V ∼ = M i, ⊕ M , for i ∈ I ,p − andhence B ( V ) ∼ = k h x, y, z i / ( x p , y p , z p , yx − xy + x , [ x, z ] , [ y, z ]) .Proof. By Lemma 2.12, dim
V <
4. If dim V = 1, then B ( V ) ∼ = k [ x ] / ( x p ) and hencedim B ( V ) = p . If dim V = 2, then V ∼ = M i, ⊕ M j, or M i, for i, j ∈ I ,p − and hence V is of diagonal type or of Jordan type. Then by [22, Proposition 3.7], dim B ( V ) = p or16. Consequently, dim V = 3. Observe that V is a decomposable object in C p C p Y D . Then V ∼ = M i, ⊕ M j, ⊕ M k, or M i, ⊕ M j, for i, j, k ∈ I ,p − .Assume that V ∼ = M i, ⊕ M j, ⊕ M k, for i, j, k ∈ I ,p − . Then V is of diagonal typewith trivial braiding and hence B ( V ) ∼ = k [ x, y, z ] / ( x p , y p , z p ).Assume that V ∼ = M i, ⊕ M j, := k { x, y } ⊕ k { z } for i, j ∈ I ,p − . Then the braiding of V is c ( xyz ⊗ h x y z i ) = x ⊗ x ( y + ix ) ⊗ x z ⊗ xx ⊗ y ( y + ix ) ⊗ y z ⊗ yx ⊗ z, ( y + jx ) ⊗ z z ⊗ z . If i = 0 = j , then V has trivial braiding and hence B ( V ) ∼ = k [ x, y, z ] / ( x p , y p , z p ).If i = 0 and j = 0, then V is not of diagonal type, which also appeared in [1, 7.1]. Weclaim that dim B ( V ) > p . Indeed, if p >
2, then by [1, Proposition 7.1], dim B ( V ) =2 p p ; if p = 2, then the proof following the same lines. Indeed, it is easy to show that { x i y j [ z, x ] k z k } i,j,k,l ∈ I , is linearly independent in B ( V ).If i = 0 and j = 0, then without loss of generality, we assume that i = 1. In this case, V is not of diagonal type, which also appeared in [1]. If p = 2, then by [10, Theorem 3.1],dim B ( V ) >
16, a contradiction. If p >
2, then by [1], dim B ( V ) > p , a contradiction. OINTED HOPF ALGEBRAS OF DIMENSION 16 IN CHARACTERISTIC 2 9
Consequently, if V is not of diagonal type, then p > V ∼ = M i, ⊕ M , for i ∈ I ,p − . Clearly, c = id if and only if j = 0. Hence by [12, Theorem 2.2], B ( V ) ∼ = B ( M i, ) ⊗ B ( M , ). (cid:3) Remark 3.4. If p = 2 , then by Proposition 2.4, the objects of dimension greater than 2in C C Y D must be decomposable in C C Y D . Lemma 3.5.
Let p be a prime number and char k = p . Let H be a pointed Hopf algebraover k of dimension p . Assume that gr H = k [ g, x, y, z ] / ( g p − , x p , y p , z p ) with g ∈ G ( H ) and x, y, z ∈ P ,g ( H ) . Then the defining relations of H are g p = 1 , gx − xg = λ g (1 − g ) , gy − yg = λ g (1 − g ) , gz − zg = λ g (1 − g ) ,x p − λ x = 0 , y p − λ y = 0 , z p − λ z = 0 , xy − yx − λ x + λ y = λ (1 − g ) ,xz − zx − λ x + λ z = λ (1 − g ) , yz − zy − λ y + λ z = λ (1 − g ) . for some λ , λ , λ ∈ I , , λ , λ , λ ∈ k with ambiguity conditions λ λ = λ λ + λ λ . Proof.
It follows by a direct computation that∆( gx − xg ) = ( gx − xg ) ⊗ g + g ⊗ ( gx − xg ) ⇒ gx − xg ∈ P g,g ( H ) ∩ H . Hence gx − xg = λ g (1 − g ) for some λ ∈ k . By rescaling x , we can take λ ∈ I , . Thenby Proposition 2.9 and Lemma 2.10,∆( x p ) = ( x ⊗ g ⊗ x ) p = x p ⊗ ⊗ x p + λ ( g − ⊗ x, which implies that x p − λ x ∈ P ( H ). Since P ( H ) = 0, it follows that x p − λ x = 0 in H .Similarly, we have gy − yg = λ g (1 − g ) , y p − λ y = 0 , λ ∈ I , ; gz − zg = λ g (1 − g ) , z p − λ z = 0 , λ ∈ I , . Then a direct computation shows that∆( xy − yx ) = ( xy − yx ) ⊗ λ g (1 − g ) ⊗ x − λ g (1 − g ) ⊗ y + g ⊗ ( xy − yx ) , which implies that xy − yx − λ x + λ y ∈ P ,g ( H ). Since P ,g ( H ) = k { − g } , it followsthat xy − yx − λ x + λ y = λ (1 − g ) for some λ ∈ k . Similarly, we have xz − zx − λ x + λ z = λ (1 − g ) , yz − zy − λ y + λ z = λ (1 − g ) , for some λ , λ ∈ k . Applying the Diamond Lemma [7] to show that dim H = p , it suffices to show thatthe following ambiguities a p b = a p − ( ab ) , a ( b p ) = ( ab ) b p − , b < a, and a, b ∈ { g, x, y, z } , ( ab ) c = a ( bc ) , c < b < a and a, b, c ∈ { g, x, y, z } , are resolvable with the order z < y < x < h < g .By Lemma 2.10, [ g, x p ] = ( g )(ad R x ) p = λ p − [ g, x ] and [ g p , x ] = pg p − [ g, x ] = 0. Then adirect computation shows that the ambiguities ( g p ) x = g p − ( gx ) and g ( x p ) = ( gx ) x p − areresolvable. Similarly, ( g p ) a = g p − ( ga ) and g ( a p ) = ( ga ) a p − are resolvable for a ∈ { y, z } .By Lemma 2.11, [ x, y p ] = ( x )(ad R y ) p = λ p − [ x, y ] and [ x p , y ] = (ad L x ) p ( y ) = ( − λ ) p − [ x, y ].Then a direct computation shows that the ambiguity ( x p ) y = x p − ( xy ) and g ( x p ) =( gx ) x p − are resolvable. Similarly, a p b = a p − ( ab ) and a ( b p ) = ( ab ) b p − , for b < a , a, b ∈ { x, y, z } .Now we claim that the ambiguity g ( xy ) = ( gx ) y is resolvable. Indeed g ( xy ) = g ( yx + λ x − λ y + λ (1 − g )) = ( gy ) x + λ gx − λ gy + λ g (1 − g )= y ( gx ) + λ gx + λ xg − λ g x + λ g (1 − g ) − λ yg = yxg − λ yg + 2 λ xg + λ λ ( g − g ) − λ xg − λ λ g (1 − g ) + λ g (1 − g )= xyg + λ x ( g − g ) + λ yg + λ λ ( g − g ) − λ yg − λ λ g (1 − g )= x ( gy ) + λ gy − λ g y = ( gx ) y. Similarly, g ( xz ) = ( gx ) z and g ( yz ) = ( gy ) z are resolvable.We claim that the ambiguity x ( yz ) = ( xy ) z imposes λ λ = λ λ + λ λ . Indeed,( xy ) z = [ yx + λ x − λ y + λ (1 − g )] z = y ( xz ) + λ xz − λ yz + λ z − λ g z = ( yz ) x + λ yx − λ yz + λ xz + λ y (1 − g ) + λ z (1 − g ) − λ λ g (1 − g )= zyx + 2 λ yx + λ [ x, z ] − λ yz + λ y (1 − g ) + λ x (1 − g ) + λ z (1 − g ) − λ λ g (1 − g ) − λ λ g (1 − g )= zyx + 2 λ yx + λ λ x + λ λ z − λ zy − λ λ y + λ y (1 − g )+ λ x (1 − g ) + λ z (1 − g ) + λ λ (1 − g ) − λ λ (1 − g ) − λ λ g (1 − g ) − λ λ g (1 − g ); OINTED HOPF ALGEBRAS OF DIMENSION 16 IN CHARACTERISTIC 2 11 x ( yz ) = x ( zy + λ y − λ z + λ (1 − g )) = ( xz ) y + λ xy − λ xz + λ x (1 − g )= z ( xy ) + 2 λ xy − λ zy + λ (1 − g ) y − λ xz + λ x (1 − g )= zyx − λ [ x, z ] − λ zy + 2 λ xy + λ z (1 − g ) + λ x (1 − g )+ λ y (1 − g ) − λ λ g (1 − g )= zyx + 2 λ yx + λ λ x + λ λ z − λ zy − λ λ y + λ y (1 − g )+ λ x (1 − g ) + λ z (1 − g ) + 2 λ λ (1 − g ) − λ λ g (1 − g ) − λ λ (1 − g ) . (cid:3) Remark 3.6.
The Hopf subalgebras of H in Lemma 3.5 generated by g, x, y appeared in [22] as examples of pointed Hopf algebras over k of dimension p . Theorem 3.7.
Let p be a prime number and char k = p . Let H be a pointed Hopf algebraover k of dimension p . Assume that gr H = k [ g, x, y, z ] / ( g p − , x p , y p , z p ) with g ∈ G ( H ) , x ∈ P ,g and y, z ∈ P ( H ) . Then H is isomorphic to one of the following Hopf algebras: (1): H ( λ ) := k h g, x, y, z i / ( g p − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] − λx, [ x, z ] , [ y, z ] − z, x p , y p − y, z p ) , (2): k h g, x, y, z i / ( g p − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] − x, [ x, z ] − (1 − g ) , [ y, z ] − z, x p , y p − y, z p ) , (3): k h g, x i / ( g p − , gx − xg − g (1 − g ) , x p − x ) ⊗ k h y, z i / ( y p − y, z p , [ y, z ] − z ) , (4): k [ g, x ] / ( g p − , x p ) ⊗ k [ y, z ] / ( y p − y, z p − z ) , (5): H ( λ ) := k [ g, x, y, z ] / ( g p − , x p − y − λz, y p − y, z p − z ) , (6): k h g, x, y, z i / ( g p − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] − x, [ x, z ] , [ y, z ] , x p , y p − y, z p − z ) , (7): k h g, x, y, z i / ( g p − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] − x, [ x, z ] , [ y, z ] , x p − z, y p − y, z p − z ) , (8): H ( λ, γ ) := k h g, x, y, z i / ( g p − , [ g, x ] − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] , [ x, z ] , [ y, z ] , x p − x − λy − γz, y p − y, z p − z ) , (9): k [ g, x ] / ( g p − , x p ) ⊗ k [ y, z ] / ( y p − y, z p ) , (10): k [ g, x, y, z ] / ( g p − , x p − z, y p − y, z p ) , (11): k [ g, x, y, z ] / ( g p − , x p − y, y p − y, z p ) , (12): k [ g, x, y, z ] / ( g p − , x p − y − z, y p − y, z p ) , (13): k h g, x, y, z i / ( g p − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] , [ x, z ] − (1 − g ) , [ y, z ] , x p , y p − y, z p ) , (14): k h g, x, y, z i / ( g p − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] , [ x, z ] − (1 − g ) , [ y, z ] , x p − y, y p − y, z p ) , (15): k h g, x, y, z i / ( g p − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] − x, [ x, z ] , [ y, z ] , x p , y p − y, z p ) , (16): k h g, x, y, z i / ( g p − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] − x, [ x, z ] , [ y, z ] , x p − z, y p − y, z p ) , (17): H ( λ, i ) := k h g, x, y, z i / ( g p − , [ g, x ] − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] , [ x, z ] , [ y, z ] , x p − x − λy − iz, y p − y, z p ) , for i ∈ I , , (18): H ( λ ) := k h g, x, y, z i / ( g p − , [ g, x ] − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] , [ x, z ] − (1 − g ) , [ y, z ] , x p − x − λy, y p − y, z p ) , (19): k [ g, x ] / ( g p − , x p ) ⊗ k [ y, z ] / ( y p − z, z p ) , (20): k [ g, x, y, z ] / ( g p − , x p − z, y p − z, z p ) , (21): k [ g, x, y, z ] / ( g p − , x p − y, y p − z, z p ) , (22): k h g, x, y, z i / ( g p − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] − (1 − g ) , [ y, z ] , [ x, z ] , x p , y p − z, z p ) , (23): k h g, x, y, z i / ( g p − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] − (1 − g ) , [ y, z ] , [ x, z ] , x p − z, y p − z, z p ) , (24): k h g, x i / ( g p − , gx − xg − g (1 − g ) , x p − x ) ⊗ k [ y, z ] / ( y p − z, z p ) , (25): k h g, x, y, z i / ( g p − , gx − xg − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] , [ x, z ] , [ y, z ] , x p − x − z, y p − z, z p ) , (26): k h g, x, y, z i / ( g p − , gx − xg − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] , [ x, z ] , [ y, z ] , x p − x − y, y p − z, z p ) , (27): H ( λ ) := k h g, x, y, z i / ( g p − , gx − xg − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] − (1 − g ) , [ x, z ] , [ y, z ] , x p − x − λz, y p − z, z p ) , (28): k [ g, x ] / ( g p − , x p ) ⊗ k [ y, z ] / ( y p , z p ) , (29): k h g, x i / ( g p − , gx − xg − g (1 − g ) , x p − x ) ⊗ k [ y, z ] / ( y p , z p ) , (30): k [ g, x, y, z ] / ( g p − , x p − y, y p , z p ) , (31): k h g, x, y, z i / ( g p − , gx − xg = g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] , [ x, z ] , [ y, z ] , x p − x − y, y p , z p ) , (32): k h g, x, y, z i / ( g p − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] − (1 − g ) , [ x, z ] , [ y, z ] , x p , y p , z p ) , (33): k h g, x, y, z i / ( g p − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] − (1 − g ) , [ x, z ] , [ y, z ] , x p − z, y p , z p ) , (34): k h g, x, y, z i / ( g p − , gx − xg = g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] − (1 − g ) , [ x, z ] , [ y, z ] , x p − x, y p , z p ) , (35): k h g, x, y, z i / ( g p − , gx − xg = g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] − (1 − g ) , [ x, z ] , [ y, z ] , x p − x − z, y p , z p ) ,Furthermore, for λ, γ ∈ k , • H ( λ ) ∼ = H ( γ ) , if and only if, λ = γ ; • H ( λ ) ∼ = H ( γ ) , if and only if, there exist α , α , β , β ∈ k satisfying α pi − α i =0 = β pi − β i for i ∈ I , such that ( α + β λ ) γ = ( α + β λ ) and α β − α β = 0 ; • H ( λ, γ ) ∼ = H ( µ, ν ) , if and only if, there exist α i , β i ∈ k satisfying α pi − α i = 0 = β pi − β i for i ∈ I , such that α β − α β = 0 and λα + γβ = µ , λα + γβ = ν ; • H ( λ, i ) ∼ = H ( γ, j ) , if and only if, there is α = 0 ∈ k satsifying α p = α such that λα = γ and i = j ; • H ( λ ) ∼ = H ( γ ) , if and only if, there is α = 0 ∈ k satsifying α p = α such that λα = γ ; • H ( λ ) = H ( γ ) , if and only if, λ = γ . OINTED HOPF ALGEBRAS OF DIMENSION 16 IN CHARACTERISTIC 2 13
Proof.
Similar to the proof of Lemma 3.5, we have gx − xg = λ g (1 − g ) , gy − yg = 0 , gz − zg = 0 ,x p − λ x ∈ P ( H ) , y p ∈ P ( H ) , z p ∈ P ( H ) ,xy − yx ∈ P ,g ( H ) , xz − zx ∈ P ,g ( H ) , yz − zy ∈ P ( H ) . for some λ ∈ I , . Since P ( H ) = k { y, z } and P ,g ( H ) = k { x, − g } , it follows that x p − λ x = µ y + µ z, y p = µ y + µ z, z p = µ y + µ z,xy − yx = ν x + ν (1 − g ) , xz − zx = ν x + ν (1 − g ) , yz − zy = ν y + ν z, for some µ , · · · , µ , ν , · · · , ν ∈ k .It follows by Lemmas 2.10–2.11 that[ g, x p ] = ( g )(ad R x ) p = [ g, x ] , g p x = xg p , [ x p , y ] = (ad L x ) p ( y ) = − ν (ad L x ) p − ( g ) = ν λ (1 − g ) , [ x p , z ] = (ad L x ) p ( z ) = − ν (ad L x ) p − ( g ) = ν λ (1 − g ) , [ x, y p ] = ( x )(ad R y ) p = ν ( x )(ad R y ) p − = ν p − [ x, y ] = ν p x + ν p − ν (1 − g ) , [ x, z p ] = ( x )(ad R z ) p = ν ( x )(ad R z ) p − = ν p − [ x, z ] = ν p x + ν p − ν (1 − g ) , [ y p , z ] = (ad L y ) p ( z ) = ν (ad L y ) p − ( z ) = ν p − [ y, z ] = ν p − ν y + ν p z, [ y, z p ] = ( y )(ad R z ) p = ν ( y )(ad R z ) p − = ν p − [ y, z ] = ν p y + ν p − ν z. Then the verification of ( a p ) b = a p − ( ab ) for a, b ∈ { g, x, y, z } and ( gx ) y = g ( xy ) , g ( xz ) =( gx ) z amounts to the conditions λ ν = µ ν = µ ν = 0 , λ ν = µ ν = µ ν = 0 ,µ ν + µ ν = ν p , µ ν + µ ν = ν p − ν , µ ν + µ ν = ν p , µ ν + µ ν = ν p − ν ,µ ν = ν p − ν , µ ν = ν p , µ ν = ν p , µ ν = ν p − ν ,µ ν + µ ν = 0 = µ ν + µ ν , µ ν = µ ν = µ ν = µ ν = 0 . Finally, the verification of ( xy ) z = x ( yz ) amounts to the conditions ν ν = ν ν = 0 , ν ν + ν ν + ν ν = ν ν . By the Diamond lemma, dim H = p .Let L be the subalgebra of H generated by y, z . It is clear that L is a Hopf subalgebra of H . Indeed, L ∼ = U L ( P ( H )), where U L ( P ( H )) is a restricted universal enveloping algebraof P ( H ). Then by [34, Proposition A.3], L is isomorphic to one of the following Hopfalgebras(a) k h y, z i / ( y p − y, z p , [ y, z ] − z ), (b) k [ y, z ] / ( y p − y, z p − z ),(c) k [ y, z ] / ( y p − y, z p ),(d) k [ y, z ] / ( y p − z, z p ),(e) k [ y, z ] / ( y p , z p ).Moreover, H ∼ = L + k h g, x i . Case (a).
Assume that L is isomorphic to the Hopf algebra described in ( a ). Withoutloss of generality, we can assume that µ − µ = µ = µ and ν = 0 = ν − µ = 0 = µ , λ ν = 0 = ν , ν p = ν , ν = ν ν , ν = ν p − ν and we can take ν ∈ I , by rescaling z .If λ = 0 = ν , then we can take ν = 0. Indeed, if ν = 0, then ν = 0, otherwise wecan take ν = 0 via the linear translation x := x + a (1 − g ) satisfying ν a = ν . Hence H ∼ = H ( ν ) described in (1). If λ = 0 = ν −
1, then ν = 1 and we can take ν = 0 viathe linear translation x := x + ν (1 − g ), which gives one class of H described in (2).If λ = 1, then ν = 0 = ν = ν , which gives one class of H described in (3). Claim: H ( λ ) ∼ = H ( γ ) for λ, γ ∈ k , if and only if, λ = γ .Suppose that φ : H ( λ ) → H ( γ ) for λ, γ ∈ k is a Hopf algebra isomorphism. Write g ′ , x ′ , y ′ , z ′ to distinguish the generators of H ( γ ). Observe that P ,g ′ ( H ( γ )) = k { x ′ } ⊕ k { − g ′ } and P ( H ( γ )) = k { y ′ , z ′ } . Then φ ( g ) = g ′ , φ ( x ) = α x ′ + α (1 − g ′ ) , φ ( y ) = β y ′ + β z ′ , φ ( z ) = γ y ′ + γ z ′ , (5)for some α i , β i , γ i ∈ k and i ∈ I , . Applying φ to the relation [ y, z ] − z = 0, we have γ = 0 , ( β − γ = 0 ⇒ β = 1 . Then applying φ to the relation [ x, y ] − λx = 0, we have λ = γ. Conversely, it is easy to see that H ( λ ) ∼ = H ( γ ) if λ = γ .Similarly, we can also show that the Hopf algebras described in (1)–(3) are pairwiseisomorphic. Indeed, direct computations show that there are no elements α i , β i , γ i ∈ k for i ∈ I , such that the morphism (5) is an isomorphism. Case (b).
Assume that L is isomorphic to the Hopf algebra described in ( b ). Withoutloss of generality, we can assume that µ − µ = µ = µ − ν = 0 = ν . Then λ ν = 0 = λ ν , ν p = ν , ν p = ν , ν = ν p − ν , ν = ν p − ν , µ ν + µ ν = 0 = µ ν + µ ν and ν ν = ν ν . Hence we can take ν , ν ∈ { , } by rescaling y, z .If λ = 0 and ν = 0 = ν , then ν = 0 = ν and we can take µ ∈ I , or µ ∈ I , byrescaling x . If µ = 0 = µ , then H is isomorphic to the Hopf algebra described in (4). If µ = 1, then H ∼ = H ( µ ) described in (5). If µ = 0 and µ = 0, then by rescaling x , wehave µ = 1, and hence by swapping x and y , H ∼ = H (0). OINTED HOPF ALGEBRAS OF DIMENSION 16 IN CHARACTERISTIC 2 15 If λ = 0 and ν − ν , then ν = 0 = µ , and we can take ν = 0 via the lineartranslation x := x + ν (1 − g ). Moreover, we can take µ ∈ I , by rescaling x , which givestwo classes of H described in (6)–(7).If λ = 0 and ν = 0 = ν −
1, then it can be reduced to the case λ = 0 and ν − ν by swapping x and y .If λ = 0 and ν = 1 = ν , then µ + µ = 0 = ν − ν and hence we can take ν = 0 = ν via the linear translations x := x + ν (1 − g ). Therefore, it can be reduced to the case λ = 0 and ν − ν via the linear translation z := z − y .If λ = 1, then ν = 0 = ν and hence ν = 0 = ν . Therefore H ∼ = H ( µ , µ ) describedin (8).Similar to the proof of Case ( a ), H ( λ ) ∼ = H ( γ ) for λ, γ ∈ k , if and only if, there exist α , α , β , β ∈ k satisfying α pi − α i = 0 = β pi − β i for i ∈ I , such that ( α + β λ ) γ =( α + β λ ) and α β − α β = 0. H ( λ, γ ) ∼ = H ( µ, ν ) if and only if, there exist α i , β i ∈ k satisfying α pi − α i = 0 = β pi − β i for i ∈ I , such that α β − α β = 0 and λα + γβ = µ , λα + γβ = ν . The Hopf algebras from the different items are pairwise non-isomorphic. Case (c).
Assume that L is isomorphic to the Hopf algebra described in ( c ). Withoutloss of generality, we can assume that µ − µ = µ = µ = ν = ν . Then λ ν = 0 = ν , ν = ν p , ν = ν p − ν , µ ν + µ ν = 0 = µ ν = ν ν and we can take ν ∈ I , by rescaling y .If λ = 0 = ν , then ν = 0 = µ ν and we can take ν ∈ I , by rescaling z . If ν = 0,then we can take µ , µ ∈ I , by rescaling x, z , which gives four classes of H described in(9)–(12). If ν = 1, then µ = 0 and we can take µ ∈ I , by rescaling x, z . Indeed, if µ = 0, then we can take µ = 1 via x := ax, z := a − z satisfying a p = µ . Therefore H is isomorphic to one of the Hopf algebras in (13)–(14).If λ = 0 = ν −
1, then µ = 0 = ν and we can take ν = 0 via the linear translation x := x + ν (1 − g ). Hence we can take µ ∈ I , by rescaling x , which gives two classes of H described in (15)–(16).If λ = 1, then ν = 0 = ν = µ ν and we can take ν ∈ I , by rescaling z . If ν = 0,then we can take µ ∈ I , by rescaling z and hence H ∼ = H ( µ , µ ) described in (17). If ν = 1, then µ = 0 and hence H ∼ = H ( µ ) described in (18).Similar to the proof of Case ( a ), H ( λ, i ) ∼ = H ( γ, j ) if and only if there is α = 0 ∈ k satsifying α p = α such that λα = γ and i = j . H ( λ ) ∼ = H ( γ ) if and only if there is α = 0 ∈ k satsifying α p = α such that λα = γ . The Hopf algebras from different itemsare pairwise non-isomorphic. Case (d).
Assume that L is isomorphic to the Hopf algebra described in ( d ). Withoutloss of generality, we can assume that µ = 0 = µ − µ = µ = ν = ν . Then ν = ν = ν = µ ν = 0. If λ = 0, then ν ∈ I , by rescaling x . If ν = 0, then we can take µ ∈ I , by rescaling x . If µ = 0, then we can take µ ∈ I , . If µ = 1, then we can take µ = 0 via the lineartranslation y := y + µ z . Therefore, we obtain three classes of H described in (19)–(21).If ν = 1, then µ = 0 and we can take µ ∈ I , , which gives two classes of H describedin (22)–(23). Indeed, if µ = 0, then we can take µ = 1 via x := ax, y := a − y, z := a − p z satisfying a − p = µ .If λ − ν , then we can take µ ∈ I , by rescaling y, z . If µ = 0, then wecan take µ ∈ I , by rescaling y, z . If µ = 1, then we can take µ = 0 via the lineartranslation y := y + µ z . Therefore, we obtain three classes of H described in (24)–(26).If λ = 1 and ν = 0, then µ = 0 and we can take ν = 1 by rescaling y, z . Therefore, H ∼ = H ( µ ) described in (27).Similar to the proof of Case ( a ), H ( λ ) = H ( γ ), if and only if, λ = γ . The Hopfalgebras from different items are pairwise non-isomorphic. Case (e).
Assume that L is isomorphic to the Hopf algebra described in ( e ). Withoutloss of generality, we can assume that µ = µ = µ = µ = ν = ν = 0. Then ν = 0 = ν , µ ν + µ ν = 0 and we can take ν , ν ∈ I , by rescaling y, z .If ν = 0 = ν and µ = 0 = µ , then H is isomorphic to one of the Hopf algebrasdescribed in (28)–(29).If ν = 0 = ν and µ = 0 or µ = 0, then H is isomorphic to one of the Hopf algebrasdescribed in (30)–(31). Indeed, if µ = 0, then we can take µ = 1 and µ = 0 via thelinear translation y := µ y + µ z , z := z ; if µ = 0, then we can take µ = 1 and µ = 0via the linear translation y := µ y + µ z , z := y ;If ν − ν , then µ = 0 and µ ∈ I , by rescaling z , which gives four classes of H described in (32)–(35).If ν = 0 = ν −
1, then it can be reduced to the case ν − ν by swapping y and z .If ν = 1 = ν , then µ + µ = 0 and hence it can be reduced to the case ν − ν via the linear translation z := z − y .Similar to the proof of Case (a), the Hopf algebras from different items are pairwisenon-isomorphic. (cid:3) Remark 3.8.
In Theorem 3.7, there are six infinite families of Hopf algebras of dimension p , which constitute new examples of Hopf algebras. Moreover, the Hopf algebras describedin (1) – (2) , (6) – (8) , (13) – (18) , (22) – (23) , (25) – (27) , (31) – (35) are not tensor product Hopfalgebras and constitute new examples of non-commutative and non-cocommutative pointedHopf algebras. In particular, up to isomorphism, there are infinitely many Hopf algebrasof dimension p that are generated by group-like elements and skew-primitive elements. OINTED HOPF ALGEBRAS OF DIMENSION 16 IN CHARACTERISTIC 2 17
Lemma 3.9.
Let p be a prime number and char k = p . Let H be a pointed Hopf algebraover k of dimension p . Assume that gr H = k [ g, x, y, z ] / ( g p − , x p , y p , z p ) with g ∈ G ( H ) , x, y ∈ P ,g ( H ) and z ∈ P ( H ) . Then the defining relations of H have the following form g p = 1 , gx − xg = λ g (1 − g ) , gy − yg = λ g (1 − g ) , gz − zg = 0 ,x p − λ x = λ z, y p − λ y = λ z, z p = λ z,xz − zx = γ x + γ y + γ (1 − g ) , yz − zy = γ x + γ y + γ (1 − g ) ,xy − yx − λ x + λ y = ( λ z, p = 2 ,λ (1 − g ) , p > . for some λ , · · · , λ , γ , · · · , γ ∈ k .Suppose that p = 2 . Then the ambiguity conditions are given by λ γ = λ γ , λ γ = λ γ , λ γ = λ γ , (6) λ γ = λ γ , λ γ = 0 , λ γ = λ γ , λ γ = 0 , (7) λ γ = λ γ , λ γ = λ γ , λ γ = λ γ , (8) ( λ − γ ) γ + γ γ = ( λ − γ ) γ + γ γ = ( λ − γ ) γ + γ γ = 0 , (9) ( λ − γ ) γ + γ γ = ( λ − γ ) γ + γ γ = ( λ − γ ) γ + γ γ = 0 , (10) λ γ = λ γ = λ γ = 0 = λ γ = λ γ = λ γ , (11) λ γ = λ γ . (12) Proof.
Similar to the proof of Lemma 3.5, we have gx − xg = λ g (1 − g ), gy − yg = λ g (1 − g )and gz − zg = 0 in H for some λ , λ ∈ I , . Moreover, x p − λ x, y p − λ y, z p ∈ P ( H ), xy − yx − λ x + λ y ∈ P ,g ( H ) and xz − zx, yz − zy ∈ P ,g ( H ). Since P ( H ) = k { z } and P ,g ( H ) = k { − g, x, y } , it follows that x p − λ x = λ z, y p − λ y = λ z, z p = λ z,xz − zx = γ x + γ y + γ (1 − g ) , yz − zy = γ x + γ y + γ (1 − g ) , for λ , λ , λ , γ , · · · , γ ∈ k .If g = 1, then xy − yx − λ x + λ y ∈ P ( H ) and hence xy − yx − λ x + λ y = λ z forsome λ ∈ k ; otherwise, xy − yx − λ x + λ y = λ (1 − g ) for some λ ∈ k . Assume that p = 2. Then it follows by a direct computation that[ x, [ x, y ]] − [ x , y ] = λ [ x, z ] − λ [ z, y ] , [ x, [ x, z ]] − [ x , z ] = γ [ x, y ] + γ [ g, x ] − λ [ x, z ] , [[ x, y ] , y ] − [ x, y ] = λ [ y, z ] − λ [ x, z ] , [[ x, z ] , z ] − [ x, z ] = γ [ x, z ] + γ [ y, z ] − λ [ x, z ] , [ y, [ y, z ]] − [ y , z ] = γ [ x, y ] + γ [ g, y ] − λ [ y, z ] , [[ y, z ] , z ] − [ y, z ] = γ [ x, z ] + γ [ y, z ] − λ [ y, z ] . Then the verification of ( a ) b = a ( ab ) and a ( b ) = ( ab ) b for a, b ∈ { g, x, y, z } amountsto the conditions (6)–(11). Then it follows by a direct computation that the ambiguities( ab ) c = a ( bc ) for a, b, c ∈ { g, x, y, z } give the conditions (12). (cid:3) Lemma 3.10.
Let p be a prime number and char k = p . Let H be a pointed Hopfalgebra over k . Assume that gr H = k [ g, h, x, y ] / ( g p − , h p n − , x p , y p ) with g, h ∈ G ( H ) , x ∈ P ,g ( H ) and y ∈ P ,g µ ( H ) for µ ∈ I ,p − . If µ = 0 , then the defining relations of H are g p = 1 , h p n = 1 , gx − xg = λ g (1 − g ) , gy − yg = 0 ,hx − xh = λ h (1 − g ) , hy − yh = 0 ,x p − λ x = µ y, y p = µ y, xy − yx = µ x + µ (1 − g ) , for λ ∈ I , , λ , µ , · · · , µ ∈ k with ambiguity conditions µ µ = 0 = µ µ , µ µ = µ p , µ µ = µ p − µ , λ µ = 0 = µ λ . If µ = 0 , then the defining relations are g p = 1 , h p n = 1 , gx − xg = λ g (1 − g ) , gy − yg = λ g (1 − g µ ) ,hx − xh = λ h (1 − g ) , hy − yh = λ h (1 − g µ ) ,x p − λ x = 0 , y p − λ y = 0 , xy − yx + µλ y − λ x = λ (1 − g µ +1 ) . for λ , λ ∈ I , , λ , · · · , λ ∈ k with ambiguity conditions λ λ (1 − g µ +1 ) = 0 = λ λ (1 − g µ +1 ) . Proof.
By similar computations as before, we have gx − xg = λ g (1 − g ) , gy − yg = λ g (1 − g µ ) ,hx − xh = λ h (1 − g ) , hy − yh = λ h (1 − g µ ) ,x p − λ x ∈ P ( H ) , y p − µ p − λ y ∈ P ( H ) , xy − yx + µλ y − λ x ∈ P ,g µ +1 ( H ) . for some λ , λ ∈ I , , λ , λ ∈ k . OINTED HOPF ALGEBRAS OF DIMENSION 16 IN CHARACTERISTIC 2 19 If µ = 0, then P ( H ) = k { y } and P ,g ( H ) = k { − g, x } . Hence x p − λ x = µ y, y p = µ y, xy − yx = µ x + µ (1 − g ) . for some µ , · · · , µ ∈ k . The verification of ( x p ) x = x ( x p ) and ( y p ) y = y ( y p ) amounts tothe conditions µ µ = 0 = µ µ . By induction, for any n >
1, we have ( x )(ad R y ) n = µ ( x )(ad R y ) n − and (ad L x ) n ( y ) =( − µ )(ad L x ) n − ( g ). Then by Lemma 2.10,[ x, y p ] = µ [ x, y ] = µ µ x + µ µ (1 − g ) , ( x )(ad R y ) p = µ ( x )(ad R y ) p − = µ p − [ x, y ] = µ p x + µ p − µ (1 − g );[ x p , y ] = λ [ x, y ] = λ µ x + λ µ (1 − g ) , (ad L x ) p ( y ) = − µ (ad L x ) p − ( g ) = − µ λ p − ( g −
1) = µ λ p − (1 − g ) . Hence by Proposition 2.9, [ x, y p ] = ( x )(ad R y ) p and [ x p , y ] = (ad L x ) p ( y ), which impliesthat µ µ = µ p , µ µ = µ p − µ , λ µ = 0 . Finally, it follows by a direct computation that a ( xy ) = ( ax ) y and ( gh ) b = g ( hb ) for a ∈ { g, h } , b ∈ { x, y } amounts to the conditions µ λ = 0 = µ λ . If µ = 0, then P ( H ) = 0 and P ,g µ +1 ( H ) = k { − g µ +1 } . By Fermat’s little theorem, µ p − = 1. Hence x p − λ x = 0 , y p − λ y = 0 , xy − yx + µλ y − λ x = λ (1 − g µ +1 ) . The verification of ( hx ) y = h ( xy ) amounts to the conditions λ λ (1 − g µ +1 ) = 0 = λ λ (1 − g µ +1 ) . Then using Lemmas 2.10 and 2.11, it follows by a direct computation that the ambiguities a p − ( ab ) = ( a p ) b , ( ab ) b p − = a ( b p ) for a, b ∈ { g, x, y } and g ( xy ) = ( gx ) y are resolvable.By the Diamond lemma, dim H = p n . (cid:3) Lemma 3.11.
Let p be a prime number and char k = p . Let H be a pointed Hopf algebraover k of dimension p . Assume that gr H = k [ g, h, x, y ] / ( g p − , h p − , x p , y p ) with g, h ∈ G ( H ) , x ∈ P ,g ( H ) and y ∈ P ,h µ ( H ) for µ ∈ I ,p − . Then the defining relations of H have the following form gx − xg = λ g (1 − g ) , hx − xh = λ h (1 − g ) , x p − λ x = 0 gy − yg = λ g (1 − h µ ) , hy − yh = λ h (1 − h µ ) , y p − λ y = 0 ,xy − yx − λ x + µλ y = λ (1 − gh µ ) . for some λ , λ ∈ I , , λ , λ , λ ∈ k .Proof. Observe that µ = 0, then P ( H ) = 0 and P ,gh µ ( H ) = k { − gh µ } . By Fermat’slittle theorem µ p − = 1. By similar computations as before, we have gx − xg = λ g (1 − g ) , hx − xh = λ h (1 − g ) , x p − λ x = 0 ,gy − yg = λ g (1 − h µ ) , hy − yh = λ h (1 − h µ ) , y p − λ y = 0 . for some λ , λ ∈ I , , λ , λ ∈ k . Now we determine ∆( xy − yx ). Observe that h µ x = xh µ + λ µh µ (1 − g ). Then∆( xy − yx ) = ( x ⊗ g ⊗ x )( y ⊗ h µ ⊗ y ) − ( y ⊗ h µ ⊗ y )( x ⊗ g ⊗ x )= ( xy − yx ) ⊗ gy − yg ) ⊗ x − ( h µ x − xh µ ) ⊗ y + gh µ ⊗ ( xy − yx )= ( xy − yx ) ⊗ λ g (1 − h µ ) ⊗ x − λ µh µ (1 − g ) ⊗ y + gh µ ⊗ ( xy − yx ) . One can check that xy − yx − λ x + µλ y ∈ P ,gh µ ( H ), which implies that xy − yx − λ x + µλ y = λ (1 − gh µ ) . for some λ ∈ k . (cid:3) Non-connected pointed Hopf algebras of dimension whose diagramsare Nichols algebras We classify non-connected pointed Hopf algebras of dimension 16 whose diagrams areNichols algebras. It turns out that there exist infinitely many such Hopf algebras up toisomorphism.
Lemma 4.1.
Let H be a pointed non-connected Hopf algebras over k of dimension .Then G ( H ) is isomorphic to the Dihedral group D , the quaternions group Q , C , C × C , C × C × C , C , C × C or C .Proof. By Nichols Zoeller theorem, | G ( H ) | must divide 16. By the assumption, | G ( H ) | ∈{ , , } and hence the lemma follows. (cid:3) Recall that D := h g, h | g = 1 , h = 1 , hg = g h i , Q := h g, h | g = 1 , hg = g h, g = h i . Now we give a complete classification of non-connected pointed Hopf algebras ofdimension 16 whose diagrams are Nichols algebras. OINTED HOPF ALGEBRAS OF DIMENSION 16 IN CHARACTERISTIC 2 21
Theorem 4.2.
Let H be a non-trivial non-connected pointed Hopf algebras over k ofdimension whose diagram is a Nichols algebra. Then H is isomorphic to one of thefollowing Hopf algebras (1): k [ D ] ⊗ k [ x ] / ( x ) , (2): k [ D ] ⊗ k [ x ] / ( x − x ) , with x ∈ P ( H ) ; (3): k h g, h, x i / ( g − , h − , hg − g h, [ g, x ] , [ h, x ] , x ) , (4): k h g, h, x i / ( g − , h − , hg − g h, [ g, x ] , [ h, x ] − h (1 − g ) , x ) , (5): e H ( λ ) := k h g, h, x i / ( g − , h − , hg − g h, [ g, x ] − g (1 − g ) , [ h, x ] − λh (1 − g ) , x ) , for λ ∈ k , with g, h ∈ G ( H ) and x ∈ P ,g ( H ) ; moreover, • e H ( λ ) ∼ = e H ( γ ) for λ, γ ∈ k , if and only if, λ = γ + i for some i ∈ I , ; (6): k [ Q ] ⊗ k [ x ] / ( x ) , (7): k [ Q ] ⊗ k [ x ] / ( x − x ) , with x ∈ P ( H ) ; (8): k h g, h, x i / ( g − , hg − g h, g − h , [ g, x ] , [ h, x ] , x ) , (9): e H ( λ ) := k h g, h, x i / ( g − , hg − g h, g − h , [ g, x ] − g (1 − g ) , [ h, x ] − λh (1 − g ) , x ) , for λ ∈ k , with g, h ∈ G ( H ) and x ∈ P ,g ( H ) ; moreover, • e H ( λ ) ∼ = e H ( γ ) for λ, γ ∈ k , if and only if, λ = γ + i or ( λ − j )( γ − i ) = 1 for some i, j ∈ I , ; (10): k [ C ] ⊗ k [ x ] / ( x ) , (11): k [ C ] ⊗ k [ x ] / ( x − x ) , with x ∈ P ( H ) ; (12): k [ g, x ] / ( g − , x ) , (13): k h g, x i / ( g − , [ g, x ] − g (1 − g µ ) , x − µx ) for µ ∈ { , } ,with g ∈ G ( H ) and x ∈ P ,g µ ( H ) for µ ∈ { , , } ; (14): k [ C × C ] ⊗ k [ x ] / ( x ) , (15): k [ C × C ] ⊗ k [ x ] / ( x − x ) , with x ∈ P ( H ) ; (16): k [ g, h, x ] / ( g − , h − , x ) , (17): k h g, h, x i / ( g − , h − , [ g, h ] , [ g, x ] , [ h, x ] − h (1 − g µ ) , x ) , (18): e H ,µ ( λ ) := k h g, h, x i / ( g − , h − , [ g, h ] , [ g, x ] − g (1 − g µ ) , [ h, x ] − λh (1 − g µ ) , x − µx ) for λ ∈ k ,with g, h ∈ G ( H ) and x ∈ P ,g µ ( H ) for µ ∈ { , } ; (19): k [ g, h, x ] / ( g − , h − , x ) , (20): k h g, h, x i / ( g − , h − , [ g, h ] , [ g, x ] − g (1 − h ) , [ h, x ] , x ) , (21): e H ( λ ) := k h g, h, x i / ( g − , h − , [ g, h ] , [ g, x ] − λg (1 − h ) , [ h, x ] − h (1 − h ) , x − x ) for λ ∈ k , with g, h ∈ G ( H ) and x ∈ P ,h ( H ) ; moreover, • e H , ( λ ) ∼ = e H , ( γ ) , if and only if, λ = γ ; • e H , ( λ ) ∼ = e H , ( γ ) , if and only if, λ = γ or λγ = λ + γ ; • e H ( λ ) ∼ = e H ( γ ) , if and only if, λ = γ + i for i ∈ I , ; (22): k [ C × C × C ] ⊗ k [ x ] / ( x ) , (23): k [ C × C × C ] ⊗ k [ x ] / ( x − x ) , with x ∈ P ( H ) ; (24): k [ g, h, k, x ] / ( g − , h − , k − , x ) , (25): e H ( λ ) := k h g, h, k, x i / ( g − , h − , k − , [ g, h ] , [ g, k ] , [ h, k ] , [ g, x ] , [ h, x ] − h (1 − g ) , [ k, x ] − λk (1 − g ) , x ) for λ ∈ k , (26): e H ( λ, γ ) := k h g, h, k, x i / ( g − , h − , k − , [ g, h ] , [ g, k ] , [ h, k ] , [ g, x ] − g (1 − g ) , [ h, x ] − λh (1 − g ) , [ k, x ] − γk (1 − g ) , x − x ) for λ, γ ∈ k , with g, h, k ∈ G ( H ) and x ∈ P ,g ( H ) ; moreover, • e H ( λ ) ∼ = e H ( γ ) , if and only if, λγ = λ + γ, or (1 + λ ) γ = 1 , or λ = γ + i, or iγ = λγ, i ∈ I , ; • e H ( λ , λ ) ∼ = e H ( γ , γ ) , if and only if, there exist q, r, ν, ι ∈ I , such that qι + rν = 1 , qγ + rγ = λ , νγ + ιγ = λ ; (27): k [ C ] ⊗ k [ x, y ] / ( x , y ) , (28): k [ C ] ⊗ k [ x, y ] / ( x − x, y ) , (29): k [ C ] ⊗ k [ x, y ] / ( x − y, y ) , (30): k [ C ] ⊗ k [ x, y ] / ( x − x, y − y ) , (31): k [ C ] ⊗ k h x, y i / ([ x, y ] − y, x − x, y ) , with x, y ∈ P ( H ) ; (32): k [ g, x, y ] / ( g − , x , y ) , (33): k h g, x, y i / ( g − , [ g, x ] , [ g, y ] , x , y , [ x, y ] − (1 − g )) , (34): k [ g, x, y ] / ( g − , x − x, y ) , (35): k h g, x, y i / ( g − , [ g, x ] , [ g, y ] , x − x, y , [ x, y ] − y ) , (36): k h g, y i / ( g − , [ g, y ] − g (1 − g ) , y − y ) ⊗ k [ x ] / ( x ) , (37): k h g, y i / ( g − , [ g, y ] − g (1 − g ) , y − y ) ⊗ k [ x ] / ( x − x ) , with g ∈ G ( H ) , x ∈ P ( H ) and y ∈ P ,g ( H ) ; (38): k [ g, y ] / ( g − , y ) ⊗ k [ x ] / ( x ) , (39): k h g, y i / ( g − , [ g, y ] − g (1 − g ) , y ) ⊗ k [ x ] / ( x ) , (40): k [ g, x, y ] / ( g − , x , y − x ) , (41): k h g, x, y i / ( g − , [ g, x ] , [ g, y ] − g (1 − g ) , x , y − x, [ x, y ]) , (42): k h g, x, y i / ( g − , [ g, x ] , [ g, y ] , x , y , [ x, y ] − (1 − g )) , (43): k h g, x, y i / ( g − , [ g, x ] , [ g, y ] − g (1 − g ) , x , y , [ x, y ] − (1 − g )) , (44): k [ g, y ] / ( g − , y ) ⊗ k [ x ] / ( x − x ) , (45): k [ g, x, y ] / ( g − , x − x, y − x ) , (46): e H ( λ ) := k h g, x, y i / ( g − , [ g, x ] , [ g, y ] − g (1 − g ) , x − x, y − λx, [ x, y ]) , (47): k h g, x, y i / ( g − , [ g, x ] , [ g, y ] , x − x, y , [ x, y ] − y ) , (48): k h g, x, y i / ( g − , [ g, x ] , [ g, y ] − g (1 − g ) , x − x, y , [ x, y ] − y ) , with g ∈ G ( H ) , x ∈ P ( H ) and y ∈ P ,g ( H ) ; • e H ( λ ) ∼ = e H ( γ ) , if and only if, λ = γ ; OINTED HOPF ALGEBRAS OF DIMENSION 16 IN CHARACTERISTIC 2 23 (49): k [ g, x, y ] / ( g − , x , y ) , (50): k h g, x, y i / ( g − , [ g, x ] , [ g, y ] , x , y , [ x, y ] − (1 − g )) , (51): k h g, x, y i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , x − x, y , [ x, y ] + y ) , (52): k h g, x, y i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , x − x, y , [ x, y ] + y − (1 − g )) , with g ∈ G ( H ) , x, y ∈ P ,g ( H ) ; (53): k [ g, x, y ] / ( g − , x , y ) , (54): k [ g, x, y ] / ( g − , x − y, y ) , (55): k h g, x, y i / ( g − , [ g, x ] , [ g, y ] , x , y , [ x, y ] − (1 − g )) , (56): k h g, x, y i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , x − x, y , [ x, y ]) , (57): k h g, x, y i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , x − x − y, y , [ x, y ]) , (58): k h g, x, y i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , x − x, y , [ x, y ] − (1 − g )) , with g ∈ G ( H ) , x ∈ P ,g ( H ) and y ∈ P ,g ( H ) ; (59): k [ g, x, y ] / ( g − , x , y ) , (60): k h g, x, y i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , x − x, y , [ x, y ] + y ) , (61): k h g, x, y i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] − g (1 − g ) , x − x, y − y, [ x, y ] + y − x ) ,with g ∈ G ( H ) , x ∈ P ,g ( H ) and y ∈ P ,g ( H ) ; (62): k [ g, x, y ] / ( g − , x , y ) , (63): k h g, x, y i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , x , y , [ x, y ]) , with g ∈ G ( H ) , x, y ∈P ,g ( H ) ; (64): k h g, x, y i / ( g − , [ g, x ] , gy − ( y + x ) g, [ x, y ] , x , y ) (65): k h g, x, y i / ( g − , [ g, x ] , gy − ( y + x ) g, [ x, y ] , x − x, y ) (66): k h g, x, y i / ( g − , [ g, x ] , gy − ( y + x ) g, [ x, y ] , x − y, y ) (67): k h g, x, y i / ( g − , [ g, x ] , gy − ( y + x ) g, [ x, y ] , x − x, y − y ) (68): k h g, x, y i / ( g − , [ g, x ] , gy − ( y + x ) g, [ x, y ] − y, x − x, y ) with g ∈ G ( H ) , x, y ∈ P ( H ) ; (69): k h g, x, y i / ( g − , [ g, x ] , gy − ( y + x ) g, [ x, y ] , x , y ) , (70): k h g, x, y i / ( g − , [ g, x ] − g (1 − g ) , gy − ( y + x ) g, [ x, y ] , x , y ) , with g ∈ G ( H ) , x, y ∈ P ,g ( H ) ; (71): k [ C × C ] ⊗ k [ x, y ] / ( x , y ) , (72): k [ C × C ] ⊗ k [ x, y ] / ( x − x, y ) , (73): k [ C × C ] ⊗ k [ x, y ] / ( x − y, y ) , (74): k [ C × C ] ⊗ k [ x, y ] / ( x − x, y − y ) , (75): k [ C × C ] ⊗ k h x, y i / ([ x, y ] − y, x − x, y ) ,with x, y ∈ P ( H ) ; (76): k [ g, h, x, y ] / ( g − , h − , x , y ) , (77): k [ g, h, x, y ] / ( g − , h − , x − y, y ) , (78): k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] , [ h, x ] − h (1 − g ) , [ g, y ] , [ h, y ] , x , y , [ x, y ]) , (79): k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] , [ h, x ] − h (1 − g ) , [ g, y ] , [ h, y ] , x − y, y , [ x, y ]) , (80): k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] , [ h, x ] , [ g, y ] , [ h, y ] , x , y , [ x, y ] − (1 − g )) , (81): k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] , [ h, x ] − h (1 − g ) , [ g, y ] , [ h, y ] , x , y , [ x, y ] − (1 − g )) , (82): k [ g, h, x, y ] / ( g − , h − , x , y − y ) (83): k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] , [ h, x ] − h (1 − g ) , [ g, y ] , [ h, y ] , x , y − y, [ x, y ]) , (84): k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] , [ h, x ] , [ g, y ] , [ h, y ] , x − y, y − y, [ x, y ]) , (85): k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] , [ h, x ] − h (1 − g ) , [ g, y ] , [ h, y ] , x − y, y − y, [ x, y ]) , (86): k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] , [ h, x ] , [ g, y ] , [ h, y ] , x , y − y, [ x, y ] − x ) , (87): k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] , [ h, x ] , [ g, y ] , [ h, y ] , x − y, y − y, [ x, y ] − x ) , (88): e H ( λ ) := k h g, h, x i / ( g − , h − , [ g, h ] , [ g, x ] − g (1 − g ) , [ h, x ] − λh (1 − g ) , x − x ) ⊗ k [ y ] / ( y ) , (89): e H ( λ ) := k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] − g (1 − g ) , [ h, x ] − λh (1 − g ) , [ g, y ] , [ h, y ] , x − x − y, y , [ x, y ]) , (90): e H ( λ ) := k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] − g (1 − g ) , [ h, x ] − λh (1 − g ) , [ g, y ] , [ h, y ] , x − x, y , [ x, y ] − (1 − g )) , (91): e H ( λ, γ ) := k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] − g (1 − g ) , [ h, x ] − λh (1 − g ) , [ g, y ] , [ h, y ] , x − x − γy, y − y, [ x, y ]) , with g, h ∈ G ( H ) , x ∈ P ,g ( H ) and y ∈ P ( H ) ; • e H n ( λ ) ∼ = e H n ( γ ) for n ∈ I , , if and only if, λ = γ + i for i ∈ I , ; • e H ( λ, µ ) ∼ = e H ( γ, ν ) if and only if λ = γ + i for i ∈ I , and µ = ν ; (92): k [ g, h, x, y ] / ( g − , h − , x , y ) , (93): k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] , [ g, y ] , [ h, x ] − h (1 − g ) , [ h, y ] , x , y , [ x, y ]) , (94): e H ( λ ) := k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] − g (1 − g ) , [ g, y ] , [ h, x ] − λh (1 − g ) , [ h, y ] , x − x, y , [ x, y ] + y ) , (95): e H ( λ ) := k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] − g (1 − g ) , [ g, y ] , [ h, x ] − λh (1 − g ) , [ h, y ] − h (1 − g ) , x − x, y , [ x, y ] + y ) ,with g, h ∈ G ( H ) and x, y ∈ P ,g ( H ) ; moreover, • e H n ( λ ) ∼ = e H n ( γ ) for n ∈ I , , if and only if, λ = γ + i for i ∈ I , ; (96): k [ g, h, x, y ] / ( g − , h − , x , y ) , (97): k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] , [ g, y ] , [ h, x ] , [ h, y ] , x , y , [ x, y ] − (1 − gh )) , (98): k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] − g (1 − g ) , [ g, y ] , [ h, x ] , [ h, y ] , x − x, y , [ x, y ]) , (99): k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] − g (1 − g ) , [ g, y ] , [ h, x ] − h (1 − g ) , [ h, y ] , x − x, y , [ x, y ] + y ) , OINTED HOPF ALGEBRAS OF DIMENSION 16 IN CHARACTERISTIC 2 25 (100): k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] − g (1 − g ) , [ g, y ] , [ h, x ] , [ h, y ] − h (1 − h ) , x − x, y − y, [ x, y ]) , (101): k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] − g (1 − g ) , [ g, y ] − g (1 − h ) , [ h, x ] − h (1 − g ) , [ h, y ] − h (1 − h ) , x − x, y − y, [ x, y ] − x + y ) , with g, h ∈ G ( H ) , x ∈ P ,g ( H ) and y ∈ P ,h ( H ) ; (102): k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] , gy − ( y + x ) g, [ h, x ] , hy − ( y + λx ) h, [ x, y ] , x , y ) , (103): k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] , gy − ( y + x ) g, [ h, x ] , hy − ( y + λx ) h, [ x, y ] , x − x, y ) , (104): k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] , gy − ( y + x ) g, [ h, x ] , hy − ( y + λx ) h, [ x, y ] , x − y, y ) , (105): k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] , gy − ( y + x ) g, [ h, x ] , hy − ( y + λx ) h, [ x, y ] , x − x, y − y ) , (106): k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] , gy − ( y + x ) g, [ h, x ] , hy − ( y + λx ) h, [ x, y ] − y, x − x, y ) , λ ∈ k , with g ∈ G ( H ) , x, y ∈ P ( H ) ; (107): k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] , [ h, x ] , gy − ( y + x ) g, [ h, y ] , [ x, y ] , x , y ) , (108): k h g, h, x, y i / ( g − , h − , [ g, h ] , [ g, x ] , [ h, x ] , gy − ( y + x ) g, [ h, y ] − h (1 − h ) , [ x, y ] − x, x , y − y ) , with g ∈ G ( H ) , x, y ∈ P ,h ( H ) ; (109): k [ C ] ⊗ k [ x, y, z ] / ( x , y , z ) , (100): k [ C ] ⊗ k [ x, y, z ] / ( x − x, y − y, z − z ) , (111): k [ C ] ⊗ k [ x, y, z ] / ( x − y, y − z, z ) , (112): k [ C ] ⊗ k [ x, y, z ] / ( x , y − z, z ) , (113): k [ C ] ⊗ k [ x, y, z ] / ( x , y , z − z ) , (114): k [ C ] ⊗ k [ x, y, z ] / ( x , y − y, z − z ) , (115): k [ C ] ⊗ k [ x, y, z ] / ( x − y, y , z − z ) , (116): k [ C ] ⊗ k h x, y, z i / ([ x, y ] − z, [ x, z ] , [ y, z ] , x , y , z ) , (117): k [ C ] ⊗ k h x, y, z i / ([ x, y ] − z, [ x, z ] , [ y, z ] , x , y , z − z ) , (118): k [ C ] ⊗ k h x, y, z i / ([ x, y ] − y, [ x, z ] , [ y, z ] , x − x, y , z ) , (119): k [ C ] ⊗ k h x, y, z i / ([ x, y ] − y, [ x, z ] , [ y, z ] , x − x, y − z, z ) , (120): k [ C ] ⊗ k h x, y, z i / ([ x, y ] − y, [ x, z ] , [ y, z ] , x − x, y , z − z ) , (121): k [ C ] ⊗ k h x, y, z i / ([ x, y ] − y, [ x, z ] , [ y, z ] , x − x, y − z, z − z ) , (122): k [ C ] ⊗ k h x, y, z i / ([ x, y ] , [ x, z ] = x, [ y, z ] = y, x , y , z − z ) , with x, y ∈ P ( H ) ; (123): k [ g, x, y, z ] / ( g − , x , y , z ) , (124): k h g, x, y, z i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] = y, [ x, z ] = z, [ y, z ] , x − x, y , z ) , with g ∈ G ( H ) and x, y, z ∈ P ,g ( H ) ; (125): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] − z, [ x, z ] , [ y, z ] , x , y , z ) , (126): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] − z, [ x, z ] , [ y, z ] , x , y , z − z ) , (127): k [ g, x, y ] / ( g − , x , y ) ⊗ k [ z ] / ( z ) , (128): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] , [ x, z ] − (1 − g ) , [ y, z ] , x , y , z ) , (129): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] , [ x, z ] − y, [ y, z ] , x , y , z ) , (130): k [ g, x, y ] / ( g − , x , y ) ⊗ k [ z ] / ( z − z ) , (131): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] , [ x, z ] − x, [ y, z ] , x , y , z − z ) , (132): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] , [ x, z ] − x, [ y, z ] − y, x , y , z − z ) , (133): k [ g, x, y, z ] / ( g − , x − z, y , z ) , (134): k [ g, x, y, z ] / ( g − , x − z, y , z − z ) , (135): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] − z, [ y, z ] , [ x, z ] , x − z, y , z ) (136): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] − z, [ y, z ] , [ x, z ] , x − z, y , z − z ) (137): k h g, x, y, z i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] − y − z, [ x, z ] , [ y, z ] , x − x, y , z ) , (138): k h g, x, y, z i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] − y − z, [ x, z ] , [ y, z ] , x − x, y , z − z ) , (139): k h g, x, y i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , [ x, y ] − y, x − x, y ) ⊗ k [ z ] / ( z ) , (140): k h g, x, y, z i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] − y, [ x, z ] , [ y, z ] − (1 − g ) , x − x, y , z ) , (141): k h g, x, y, z i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] − y, [ x, z ] − (1 − g ) , [ y, z ] , x − x, y , z ) , (142): k h g, x, y, z i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] − y, [ x, z ] − y, [ y, z ] , x − x, y , z ) , (143): k h g, x, y, z i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] − y, [ x, z ] , [ y, z ] , x − x − z, y , z ) , (144): k h g, x, y, z i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] − y, [ x, z ] , [ y, z ] − y, x − x, y , z − z ) , (145): e H ( λ ) := k h g, x, y, z i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] − y, [ x, z ] , [ y, z ] , x − x − λz, y , z − z ) , (146): k h g, x, y, z i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] − y, [ x, z ] , [ y, z ] , x − x, y − z, z ) , (147): k h g, x, y, z i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] − y − z, [ x, z ] , [ y, z ] , x − x, y − z, z ) , (148): e H ( λ ) := k h g, x, y, z i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] − y − λz, [ x, z ] , [ y, z ] , x − x, y − z, z − z ) , with g ∈ G ( H ) , x, y ∈ P ,g ( H ) and z ∈ P ( H ) ; Moreover, • e H ( λ ) ∼ = e H ( γ ) or e H ( λ ) ∼ = e H ( γ ) , if and only if, λ = γ ; (149): e H ( λ ) := k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] − λx, [ x, z ] , [ y, z ] − z, x p , y − y, z ) , (150): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] − x, [ x, z ] − (1 − g ) , [ y, z ] − z, x , y − y, z ) , OINTED HOPF ALGEBRAS OF DIMENSION 16 IN CHARACTERISTIC 2 27 (151): k h g, x i / ( g − , [ g, x ] − g (1 − g ) , x − x ) ⊗ k h y, z i / ( y − y, z , [ y, z ] − z ) , (152): k [ g, x ] / ( g − , x ) ⊗ k [ y, z ] / ( y − y, z − z ) , (153): e H ( λ ) := k [ g, x, y, z ] / ( g − , x − y − λz, y − y, z − z ) , (154): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] − x, [ x, z ] , [ y, z ] , x , y − y, z − z ) , (155): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] − x, [ x, z ] , [ y, z ] , x − z, y − y, z − z ) , (156): e H ( λ, γ ) := k h g, x, y, z i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] , [ x, z ] , [ y, z ] , x − x − λy − γz, y − y, z − z ) , (157): k [ g, x ] / ( g − , x ) ⊗ k [ y, z ] / ( y − y, z ) , (158: k [ g, x, y, z ] / ( g − , x − z, y − y, z ) , (159): k [ g, x, y, z ] / ( g − , x − y, y − y, z ) , (160): k [ g, x, y, z ] / ( g − , x − y − z, y − y, z ) , (161): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] , [ x, z ] − (1 − g ) , [ y, z ] , x , y − y, z ) , (162): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] , [ x, z ] − (1 − g ) , [ y, z ] , x − y, y − y, z ) , (163): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] − x, [ x, z ] , [ y, z ] , x , y − y, z ) , (164: k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] − x, [ x, z ] , [ y, z ] , x − z, y − y, z ) , (165): e H ( λ, i ) := k h g, x, y, z i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] , [ x, z ] , [ y, z ] , x − x − λy − iz, y − y, z ) , for i ∈ I , , (166): e H ( λ ) := k h g, x, y, z i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] , [ x, z ] − (1 − g ) , [ y, z ] , x − x − λy, y − y, z ) , (167): k [ g, x ] / ( g − , x ) ⊗ k [ y, z ] / ( y − z, z ) , (168): k [ g, x, y, z ] / ( g − , x − z, y − z, z ) , (169): k [ g, x, y, z ] / ( g − , x − y, y − z, z ) , (170): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] − (1 − g ) , [ y, z ] , [ x, z ] , x , y − z, z ) , (171): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] − (1 − g ) , [ y, z ] , [ x, z ] , x − z, y − z, z ) , (172): k h g, x i / ( g − , [ g, x ] − g (1 − g ) , x − x ) ⊗ k [ y, z ] / ( y − z, z ) , (173): k h g, x, y, z i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] , [ x, z ] , [ y, z ] , x − x − z, y − z, z ) , (174): k h g, x, y, z i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] , [ x, z ] , [ y, z ] , x − x − y, y − z, z ) , (175): e H ( λ ) := k h g, x, y, z i / ( g − , gx − xg − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] − (1 − g ) , [ x, z ] , [ y, z ] , x − x − λz, y − z, z ) , (176): k [ g, x ] / ( g − , x ) ⊗ k [ y, z ] / ( y , z ) , (177): k h g, x i / ( g − , [ g, x ] − g (1 − g ) , x − x ) ⊗ k [ y, z ] / ( y , z ) , (178): k [ g, x, y, z ] / ( g − , x − y, y , z ) , (179): k h g, x, y, z i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] , [ x, z ] , [ y, z ] , x − x − y, y , z ) , (180): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] − (1 − g ) , [ x, z ] , [ y, z ] , x , y , z ) , (181): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , [ g, z ] , [ x, y ] − (1 − g ) , [ x, z ] , [ y, z ] , x − z, y , z ) , (182): k h g, x, y, z i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] − (1 − g ) , [ x, z ] , [ y, z ] , x − x, y , z ) , (183): k h g, x, y, z i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] − (1 − g ) , [ x, z ] , [ y, z ] , x − x − z, y , z ) ,with g ∈ G ( H ) x ∈ P ,g ( H ) and y, z ∈ P ( H ) ; Moreover, • e H ( λ ) ∼ = e H ( γ ) for λ, γ ∈ k , if and only if, λ = γ ; • e H ( λ ) ∼ = e H ( γ ) for λ, γ ∈ k , if and only if, there exist α , α , β , β ∈ k satisfying α i − α i = 0 = β i − β i for i ∈ I , such that ( α + β λ ) γ = ( α + β λ ) and α β − α β = 0 ; • e H ( λ, γ ) ∼ = e H ( µ, ν ) if and only if, there exist α i , β i ∈ k satisfying α i − α i =0 = β i − β i for i ∈ I , such that α β − α β = 0 and λα + γβ = µ , λα + γβ = ν ; • e H ( λ, i ) ∼ = e H ( γ, j ) if and only if λ = γ and i = j ; • e H ( λ ) ∼ = e H ( γ ) or e H ( λ ) = e H ( γ ) , if and only if, λ = γ ; (184): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , gz − ( z + y ) g, [ x, y ] , [ x, z ] , [ y, z ] , x , y , z ) , (185): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , gz − ( z + y ) g, [ x, y ] , [ x, z ] , [ y, z ] , x − x, y − y, z − z ) , (186): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , gz − ( z + y ) g, [ x, y ] , [ x, z ] , [ y, z ] , x − y, y − z, z ) , (187): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , gz − ( z + y ) g, [ x, y ] , [ x, z ] , [ y, z ] , x , y − z, z ) , (188): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , gz − ( z + y ) g, [ x, y ] , [ x, z ] , [ y, z ] , x , y , z − z ) , (189): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , gz − ( z + y ) g, [ x, y ] , [ x, z ] , [ y, z ] , x , y − y, z − z ) , (190): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , gz − ( z + y ) g, [ x, y ] , [ x, z ] , [ y, z ] , x − y, y , z − z ) , (191): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , gz − ( z + y ) g, [ x, y ] − z, [ x, z ] , [ y, z ] , x , y , z ) , (192): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , gz − ( z + y ) g, [ x, y ] − z, [ x, z ] , [ y, z ] , x , y , z − z ) , (193): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , gz − ( z + y ) g, [ x, y ] − y, [ x, z ] , [ y, z ] , x − x, y , z ) , (194): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , gz − ( z + y ) g, [ x, y ] − y, [ x, z ] , [ y, z ] , x − x, y − z, z ) , (195): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , gz − ( z + y ) g, [ x, y ] − y, [ x, z ] , [ y, z ] , x − x, y , z − z ) , OINTED HOPF ALGEBRAS OF DIMENSION 16 IN CHARACTERISTIC 2 29 (196): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , gz − ( z + y ) g, [ x, y ] − y, [ x, z ] , [ y, z ] , x − x, y − z, z − z ) , (197): k h g, x, y, z i / ( g − , [ g, x ] , [ g, y ] , gz − ( z + y ) g, [ x, y ] , [ x, z ] = x, [ y, z ] = y, x , y , z − z ) , with g ∈ G ( H ) , x, y ∈ P ( H ) . Remark 4.3.
By Theorem 4.2, there are 197 types of non-connected pointed Hopf algebrasof dimension with char k = 2 whose diagrams are Nichols algebras. Up to isomorphism,there are infinitely many classes of such Hopf algebras. In particular, we obtain infinitelymany new examples of non-commutative non-cocommutative pointed Hopf algebras. Let H be a non-trivial non-connected pointed Hopf algebra of dimension 16. By Lemma4.1, G ( H ) is isomorphic to D , Q , C , C × C , C × C × C , C , C × C or C . Wewill subsequently prove Theorem 4.2 by a case by case discussion. In what follows, R isthe diagram of H and V := R (1).4.1. Coradical of dimension . Observe that dim H = 8. Then dim R = 2. By Lemma2.12, dim V = 1 with a basis { x } satisfying c ( x ⊗ x ) = x ⊗ x . Therefore, R ∼ = k [ x ] / ( x ).4.1.1. G ( H ) ∼ = D . Observe that \ G ( H ) = { ǫ } and Z ( D ) = { , g } . Then by Remark2.1, x ∈ V ǫg µ for µ ∈ I , . Therefore,gr H = k h g, h, x | g = h = 1 , hg = g h, gx = xg, hx = xh, x = 0 i , with g, h ∈ G ( H ) and x ∈ P ,g µ ( H ) for µ ∈ I , . Now we determine the liftings of gr H .By similar computations as before, we have gx − xg = λ g (1 − g µ ) , hx − xh = λ h (1 − g µ ) , x − µλ x = x ∈ P ( H ) , for some λ ∈ I , , λ ∈ k .If µ = 0, then gx − xg = 0 = hx − xh in H and P ( H ) = k { x } , which implies that x = λ x for some λ ∈ k . Observe that H is the tensor product Hopf algebra between k [ D ] and k [ x ] / ( x − λ x ). Then dim H = 16. By rescaling x , we can take λ ∈ I , ,which gives two classes of H described in (1)–(2). Clearly, they are non-isomorphic.If µ = 1, then P ( H ) = 0 and hence x = 0 in H . Applying the Diamond Lemma [7] toshow that dim H = 16, it suffices to show that the following ambiguities( g ) x = g ( gx ) , ( h ) x = h ( hx ) , ( gh ) x = g ( hx ) , are resolvable with the order x < h < g . By Lemma 2.10, we have [ g , x ] = 0 = [ h , x ]and hence the first two ambiguities are resolvable. Now we show that the ambiguity( gh ) x = g ( hx ) is resolvable: g ( hx ) = g ( xh + λ h (1 − g )) = ( gx ) h + λ gh (1 − g ) = xhg + ( λ + λ ) hg (1 − g ) , = ( xh + λ h (1 − g )) g + λ hg (1 − g ) = ( hx ) g + λ hg (1 − g ) = ( hg ) x = ( gh ) x. If λ = 0, then by rescaling x , we can take λ ∈ I , , which gives two classes of H described in (3)–(4). If λ = 1, then H ∼ = e H ( λ ) described in (5).Now we prove that e H ( λ ) ∼ = e H ( γ ) for λ, γ ∈ k , if and only if, λ = γ + i for some i ∈ I , .Observe that Aut( D ) ∼ = D with generators ψ , ψ , where ψ ( g ) = g, ψ ( h ) = gh ; ψ ( g ) = g − , ψ ( h ) = h. Suppose that φ : e H ( λ ) → e H ( γ ) for λ, γ ∈ k is a Hopf algebra isomorphism. Write g ′ , h ′ , x ′ to distinguish the generators of e H ( γ ). Therefore, φ ( g ) ∈ { g ′ , ( g ′ ) } , φ ( h ) = ( g ′ ) i h ′ for i ∈ I , and φ ( x ) ∈ P ,φ ( g ) ( e H ( γ )). Note that spaces of the skew-primitive elementsof e H ( γ ) are trivial except P , ( g ′ ) ( e H ( γ )) = k { x ′ } ⊕ k { − ( g ′ ) } . Then φ ( x ) = a (1 − ( g ′ ) ) + bx ′ for some a, b = 0 ∈ k . Applying φ to relation gx − xg = g (1 − g ), then φ ( gx − xg − g (1 − g )) = φ ( g ) φ ( x ) − φ ( x ) φ ( g ) − φ ( g )(1 − ( g ′ ) )= bφ ( g ) x ′ − bx ′ φ ( g ) − φ ( g )(1 − ( g ′ ) ) = ( b − φ ( g )(1 − ( g ′ ) ) = 0 . Therefore, b = 1. Then applying φ to the relations hx − xh = λh (1 − g ), then we have φ ( h ) x ′ − x ′ φ ( h ) − λφ ( h )(1 − ( g ′ ) ) = 0 . If φ ( h ) = ( g ′ ) µ h ′ for µ ∈ I , , then φ ( h ) x ′ − x ′ φ ( h ) = γφ ( h )(1 − ( g ′ ) ) and hence γ = λ .If φ ( h ) = ( g ′ ) i h ′ for i ∈ { , } , then φ ( h ) x ′ − x ′ φ ( h ) = ( γ + 1) φ ( h )(1 − ( g ′ ) ) and hence γ + 1 = λ . Consequently, we have γ = λ + i, for i ∈ I , . Conversely, for any λ ∈ k , i ∈ I , , let ψ : e H ( λ ) → e H ( λ + i ) be the algebra map givenby ψ ( g ) = g ′ , ψ ( h ) = ( g ′ ) i h ′ , ψ ( x ) = x ′ + b (1 − ( g ′ ) ) , b ∈ k . Then it is easy to see that it is an epimorphism of Hopf algebras and ψ | ( e H ( λ )) is injective.Hence ψ is a Hopf algebra isomorphism.4.1.2. G ( H ) ∼ = Q . Observe that c Q = { ǫ } and Z ( Q ) = { , g } . Then by Remark 2.1, x ∈ V ǫg µ for µ ∈ I , . Therefore,gr H = k h g, h, x | g = 1 , hg = g h, g = h , gx = xg, hx = xh, x = 0 i , with g, h ∈ G ( H ) and x ∈ P ,g µ ( H ). Similar to the case G ( H ) ∼ = D , the definingrelations of H are given by g = 1 , hg = g h, g = h ,gx − xg = λ g (1 − g µ ) , hx − xh = λ h (1 − g µ ) , x − λ x = 0 , OINTED HOPF ALGEBRAS OF DIMENSION 16 IN CHARACTERISTIC 2 31 for some λ ∈ I , , λ ∈ k with ambiguity conditions λ = 0 if µ = 1.If µ = 0, then gx − xg = 0 = hx − xh in H . Observe that H is the tensor product Hopfalgebra between k [ Q ] and k [ x ] / ( x − λ x ). Then dim H = 16. By rescaling x , we cantake λ ∈ I , , which gives two classes of H described in (6)–(7).If µ = 1, then it follows by a direct computation that the ambiguities ( g ) x = g ( gx ),( h ) x = h ( hx ), ( gh ) x = g ( hx ), are resolvable with the order x < h < g and hencedim H = 16. If λ = 0, then by rescaling x , we can take λ ∈ I , , which gives two classesof H described in (8)–(9). Indeed, if λ = 1, then H ∼ = e H (0) by swapping g and h . If λ = 1, then H ∼ = e H ( λ ) described in (9).Now we prove that e H ( λ ) ∼ = e H ( γ ) for λ, γ ∈ k , if and only if, λ = γ + i or ( λ − j )( γ − i ) =1 for i, j ∈ I , .Observe that Aut( Q ) ∼ = S with generators ψ , ψ , ψ where ψ ( g ) = g − , ψ ( h ) = gh ; ψ ( g ) = h, ψ ( h ) = g ; ψ ( g ) = gh, ψ ( h ) = g h. Suppose that φ : e H ( λ ) → e H ( γ ) for λ, γ ∈ k is a Hopf algebra isomorphism. Then φ | Q : Q → Q is an automorphism. Hence φ ( g ) ∈ { g, g , h, g h, gh, g h } and φ ( h ) ∈{ g, g , h, g h, gh, g h } − { φ ( g ) , g φ ( g ) } . Write g ′ , h ′ , x ′ to distinguish the generators of e H ( γ ). Since spaces of skew-primitive elements of e H ( γ ) are trivial except P , ( g ′ ) ( e H ( γ )) = k { x ′ } ⊕ k { − ( g ′ ) } , φ ( x ) = a (1 − ( g ′ ) ) + bx ′ for some a, b = 0 ∈ k .If φ ( g ) = ( g ′ ) µ g ′ for µ ∈ I , , then φ ( h ) = ( g ′ ) ν ( g ′ ) i h ′ for i, ν ∈ I , . Applying φ to therelations gx − xg = g (1 − g ) , hx − xh = λh (1 − g ), we have a = 1 , λ = γ + i. If φ ( g ) = ( g ′ ) µ h ′ for µ ∈ I , , then φ ( h ) = ( g ′ ) ν g ′ ( h ′ ) i for i, ν ∈ I , . Applying φ to therelations gx − xg = g (1 − g ) , hx − xh = λh (1 − g ), we have aγ = 1 , a (1 + iγ ) = λ ⇒ ( λ − i ) γ = 1 . If φ ( g ) = ( g ′ ) µ g ′ h ′ for µ ∈ I , , then φ ( h ) = ( g ′ ) ν ( g ′ ) i ( h ′ ) j for i, j, ν ∈ I , satisfying i + j = 1. Applying φ to the relations gx − xg = g (1 − g ) , hx − xh = λh (1 − g ), we have a (1 + γ ) = 1 , a ( i + jγ ) = λ ⇒ ( λ − j )( γ + 1) = 1 . Conversely, if λ = γ + i or ( λ − j )( γ − i ) = 1 for i, j ∈ I , , then we can build analgebra map ψ : e H ( λ ) → e H ( γ ) in the form of φ , it is easy to see that ψ is a Hopf algebraepimorphism and Ψ | ( e H ( λ )) is injective. Hence ψ is a Hopf algebra isomorphism.Assume that G ( H ) ∼ = C . Then c C = { ǫ } and Z ( C ) = C := h g i . Then by Remark2.1, x ∈ V ǫg µ for µ ∈ I , . Therefore,gr H = k h g, x | g = 1 , gx = xg, x = 0 i , with g ∈ G ( H ) and x ∈ P ,g µ ( H ). Up to isomorphism, we can take µ ∈ { , , , } . Thenby a similar computation as before, we have gx − xg = λ g (1 − g µ ) , x − µλ x ∈ P ,g µ ( H ) , λ ∈ I , . By [3, Proposition 6.3] and [8, Theorem 2.2], up to isomorphism, we can take µ ∈{ , , , } .If µ = 0, then gx − xg = 0 in H and P ( H ) = k { x } . Hence x = λ x for λ ∈ k .Observe that H ∼ = k [ C ] ⊗ k [ x ] / ( x − λ x ). Then dim H = 16. By rescaling x , we cantake λ ∈ I , , which gives two classes of H described in (10)–(11). Clearly, they arenon-isomorphic.If µ = 0, then P ,g µ = k { − g µ } and hence x − µλ x = λ (1 − g µ ) for λ ∈ k . Thenwe take λ = 0 via the linear translation x := x − a (1 − g µ ) satisfying a − µλ a = λ .Indeed, it is easy to see that the linear translation is a Hopf algebra isomorphism. ByLemma 2.10, we have [ g, x ] = 0, which implies that the ambiguity ( g ) x = g ( gx ) isresolvable. By Proposition 2.9,[ g, x ] = [[ g, x ] , x ] = λ g (1 − g µ ) − λ ( µ + 1) g µ +1 (1 − g µ ) . Hence the ambiguity g ( x ) = ( gx ) x imposes the condition λ = 0 if µ = 2. Then byDiamond Lemma, dim H = 16 with ambiguity condition: λ = 0 if µ = 2.If λ = 0, then H is the Hopf algebra described in (12). If λ = 1, then µ ∈ { , } and H is the Hopf algebra described in (13). Obviously, the two Hopf algebras with µ = 1and µ = 4 are non-isomorphic since they are not isomorphic as coalgebras.4.1.3. G ( H ) ∼ = C × C = h g i × h h i . Then \ C × C = { ǫ } and Z ( C × C ) = C × C .Then by Remark 2.1, x ∈ V ǫg µ h ν for µ ∈ I , , ν ∈ I , . Therefore,gr H = k h g, h, x | g = 1 , h = 1 , gh = gh, gx = xg, x = 0 i , with g ∈ G ( H ) and x ∈ P ,g µ h ν ( H ).Observe that Aut( C × C ) ∼ = D with generators ψ , ψ , where ψ ( g ) = gh, ψ ( h ) = g h ; ψ ( g ) = gh, ψ ( h ) = h. Then up to isomorphism, we can take ( µ, ν ) ∈ { (0 , , (1 , , (2 , , (0 , } . By similarcomputations as before, we have gx − xg = λ g (1 − g µ h ν ) , hx − xh = λ h (1 − g µ h ν ) , for some λ , λ ∈ k . Then∆( x ) = ( x ⊗ g µ h ν ⊗ x ) = x ⊗ g µ h ν , x ] ⊗ x + g µ ⊗ x = x ⊗ µλ + νλ )( g µ h ν − g µ ) ⊗ x + g µ ⊗ x . OINTED HOPF ALGEBRAS OF DIMENSION 16 IN CHARACTERISTIC 2 33
It is easy to see that x − ( µλ + νλ ) x ∈ P ,g µ ( H ).If ( µ, ν ) = (0 , gx = xg, hx = xh in H and P ( H ) = k { x } , which implies that x = λ x for λ ∈ I , . In this case, H ∼ = k [ C × C ] ⊗ k [ x ] / ( x − λ x ), which are describedin (14)–(15).If ( µ, ν ) ∈ { (1 , , (2 , } , then P ,g µ ( H ) = k { − g µ } and hence x − µλ x = λ (1 − g µ )for some λ ∈ k . We can take λ = 0 via the linear translation x := x − a (1 − g µ ) satisfying a − µλ µ = λ . Similar to the case G ( H ) ∼ = C , it follows by a direct computation thatthe ambiguities ( g ) x = g ( gx ), ( h ) x = h ( hx ), g ( x ) = ( gx ) x , h ( x ) = ( hx ) x and( gh ) x = g ( hx ) are resolvable. Then by Diamond lemma, dim H = 16. By rescaling x ,we can take λ ∈ I , . If λ = 0, then by rescaling x , λ ∈ I , , which gives two classesof H described in (16)–(17). If λ = 1, then H ∼ = e H ,µ ( λ ) described in (18). Obviously, e H , ( λ ) and e H , ( γ ) for any λ, γ ∈ k are non-isomorphic since their coalgebra structureare not isomorphic.We claim that e H , ( λ ) ∼ = e H , ( γ ), if and only if, λ = γ ; e H , ( λ ) ∼ = e H , ( γ ), if and onlyif, λ = γ or λγ = λ + γ .Suppose that φ : e H , ( λ ) → e H , ( γ ) for λ, γ ∈ k is a Hopf algebra isomorphism. Then φ | C × C : C × C → C × C is an automorphism. Then φ ( g ) ∈ { g, g , gh, g h } and φ ( h ) ∈ { h, g h } . Write g ′ , h ′ , x ′ to distinguish the generators of e H , ( γ ). Since spaces ofskew-primitive elements of e H , ( γ ) are trivial except P ,g ′ ( e H , ( γ )) = k { x ′ } ⊕ k { − g ′ } ,it follows that φ ( g ) = g ′ , φ ( x ) = a (1 − g ′ ) + bx ′ for some a, b = 0 ∈ k . Applying φ to the relations gx − xg = g (1 − g ) and x − x = 0,then we have b = 1. Observe that φ ( h ) ∈ { h ′ , ( g ′ ) h ′ } . Applying φ to the relations hx − xh = λh (1 − g ), then we have γ = λ . Similarly, we have e H , ( λ ) ∼ = e H , ( γ ), if andonly if, λ = γ or λγ = λ + γ .If ( µ, ν ) = (0 , P ( H ) = 0 and hence x − λ x = 0. Then by rescaling x , λ ∈ I , . Similar to the last case, it follows by a direct computation that the ambiguities( g ) x = g ( gx ), ( h ) x = h ( hx ), g ( x ) = ( gx ) x , h ( x ) = ( hx ) x and ( gh ) x = g ( hx ) areresolvable. Then by Diamond lemma, dim H = 16. If λ = 0, then we can take λ ∈ I , ,which gives two classes of H described in (19)–(20). If λ = 1, then H ∼ = e H ( λ ) describedin (21). Similar to the last case, e H ( λ ) ∼ = e H ( γ ), if and only if, λ = γ + i for i ∈ I , .4.1.4. Case G ( H ) ∼ = C × C × C . Then \ C × C × C = { ǫ } and Z ( C × C × C ) = C × C × C := h g i × h h i × h k i . Then by Remark 2.1, x ∈ V ǫg µ h ν k ι for µ, ν, ι ∈ I , .Therefore,gr H = k h g, h, k, x | g = 1 , h = 1 , k = 1 , gx = xg, hx = xh, kx = xk, x = 0 i , with g, h, k ∈ G ( H ) and x ∈ P ,g µ h ν k ι ( H ). Then by a similar computation as before, wehave gx − xg = λ g (1 − g µ h ν k ι ) , hx − xh = λ h (1 − g µ h ν k ι ) , kx − xk = λ k (1 − g µ h ν k ι ) ,x − ( µλ + νλ + ιλ ) x ∈ P ( H ) . for some λ , λ , λ ∈ k . Observe that C × C × C is 2-torsion. Then we can take( µ, ν, ι ) = (0 , , , (1 , , µ, ν, ι ) = (0 , , gx − xg = hx − xh = kx − xk = 0 in H and P ( H ) = k { x } ,which implies that x = λ x . By rescaling x , λ ∈ I , . Then H ∼ = k [ C × C × C ] ⊗ k [ x ] / ( x − λ x ), which gives two classes of H described in (22)–(23).If ( µ, ν, ι ) = (1 , , P ( H ) = 0 and hence x − λ x = 0 in H . It follows by a directcomputation that the ambiguities ( a ) b = a ( ab ) and ( ab ) c = a ( bc ) for a, b, c ∈ { g, h, k, x } are resolvable. By Diamond lemma, dim H = 16. By rescaling x , we can take λ ∈ I , .If λ = 0, then we can take λ ∈ I , by rescaling x . If λ = 0, then we can also take λ ∈ I , , which gives two classes of H described in (24) and (25). In fact, if λ = 1, then H ∼ = e H (0). If λ = 1, then H ∼ = e H ( λ ). If λ = 1, then H ∼ = e H ( λ , λ ) described in(26).We claim that e H ( λ ) ∼ = e H ( γ ), if and only if, λγ = λ + γ, or (1 + λ ) γ = 1 , or λ = γ + i, or 1 + iγ = λγ, i ∈ I , . Suppose that φ : e H ( λ ) → e H ( γ ) for λ, γ ∈ k is a Hopf algebra isomorphism. Then φ | C × C × C : C × C × C → C × C × C is an automorphism. Write g ′ , h ′ , x ′ todistinguish the generators of e H ( γ ). Since spaces of skew-primitive elements of e H ( γ ) aretrivial except P ,g ′ ( e H ( γ )) = k { x ′ } ⊕ k { − g ′ } , it follows that φ ( g ) = g ′ , φ ( x ) = a (1 − g ′ ) + bx ′ for some a, b = 0 ∈ k . Let φ ( h ) = ( g ′ ) p ( h ′ ) q ( k ′ ) r for p, q, r ∈ I , . Then applying φ to therelation hx − xh = h (1 − g ), we have ( q + rγ ) b = 1 . Let φ ( k ) = ( g ′ ) µ ( h ′ ) ν ( k ′ ) ι for µ, ν, ι ∈ I , . Then applying φ to the relation kx − xk = λk (1 − g ), we have ( ν + γι ) b = λ. Observe that φ | G ( e H ( λ )) is an isomorphism if and only if qι + rν = 1. Hence by a case bycase discussion, we have λγ = λ + γ, or (1 + λ ) γ = 1 , or λ = γ + i, or 1 + iγ = λγ, i ∈ I , . OINTED HOPF ALGEBRAS OF DIMENSION 16 IN CHARACTERISTIC 2 35
Conversely, if λγ = λ + γ , then let ψ : e H ( λ ) → e H ( γ ) be the algebra given by ψ ( g ) = g ′ , ψ ( h ) = h ′ k ′ , ψ ( k ) = k ′ , ψ ( x ) = (1 − λ ) x ′ ;if (1 + γ ) λ = 1, then let ψ : e H ( λ ) → e H ( γ ) be the algebra given by ψ ( g ) = g ′ , ψ ( h ) = h ′ k ′ , ψ ( k ) = h ′ , ψ ( x ) = λx ′ ;if i + γ = λ for i ∈ I , , then let ψ : e H ( λ ) → e H ( γ ) be the algebra given by ψ ( g ) = g ′ , ψ ( h ) = h ′ , ψ ( k ) = ( h ′ ) i k ′ , ψ ( x ) = x ′ ;if 1 + iγ = λγ for i ∈ I , , then let ψ : e H ( λ ) → e H ( γ ) be the algebra given by ψ ( g ) = g ′ , ψ ( h ) = k ′ , ψ ( k ) = h ′ ( k ′ ) i , ψ ( x ) = γ − x ′ . It follows by a direct computation that ψ is a well-defined Hopf algebra epimorphism.Observe that ψ | P ,g ( H ( λ )) is injective. Then ψ is a Hopf algebra isomorphism.We claim that e H ( λ , λ ) ∼ = e H ( γ , γ ), if and only if, there exists q, r, ν, ι ∈ I , suchthat qι + rν = 1 , qγ + rγ = λ , νγ + ιγ = λ . (13)Suppose that φ : e H ( λ , λ ) → e H ( γ , γ ) for λ , λ , γ , γ ∈ k is a Hopf algebra isomor-phism. Similar to the last case, we have φ ( g ) = g ′ , φ ( x ) = a (1 − g ′ ) + bx ′ for some a, b = 0 ∈ k . Applying φ to the relations gx − xg = g (1 − g ) , x − x = 0, wehave b = 1.Let φ ( h ) = ( g ′ ) p ( h ′ ) q ( k ′ ) r and φ ( k ) = ( g ′ ) µ ( h ′ ) ν ( k ′ ) ι for µ, ν, ι ∈ I , , p, q, r ∈ I , .Observe that qι + rν = 1 since φ is an isomorphism. Then applying φ to the relations hx − xh = λ h (1 − g ) and kx − xk = λ k (1 − g ), we have qγ + rγ = λ , νγ + ιγ = λ . Conversely, if there exist q, r, ν, ι satisfying conditions (13), then let ψ : e H ( λ , λ ) → e H ( γ , γ ) be the algebra defined by ψ ( g ) = g ′ , ψ ( h ) = ( h ′ ) q ( k ′ ) r , φ ( k ) = ( h ′ ) ν ( k ′ ) ι , ψ ( x ) = x ′ . It follows by a direct computation that ψ is a well-defined Hopf algebra epimorphism.Observe that ψ | P ,g ( H ( λ ,λ )) is injective. Then ψ is a Hopf algebra isomorphism. Coradical of dimension . In this case, G ( H ) ∼ = C or C × C . Then dim R = 4.Observe that \ G ( H ) = { ǫ } . Then there is an element x ∈ V such that c ( x ⊗ x ) = x ⊗ x .Hence dim B ( k { x } ) = 2. By assumption, R ∼ = B ( V ) and hence dim V >
1. If dim
V > B ( V ) >
4, a contradiction. Therefore, dim V = 2. Observe that V is eitherof diagonal type or of Jordan type. If V is of Jordan type, then by [10, Theorem 3.1],dim B ( V ) = 16. Hence V is of diagonal type. Moreover R ∼ = k [ x, y ] / ( x , y ).4.2.1. G ( H ) ∼ = C := h g i . Then by Lemma 3.1, V ∼ = M i, ⊕ M j, for i, j ∈ I , or M k, for k ∈ { , } .Assume that V ∼ = M i, ⊕ M j, for i, j ∈ I , , that is, x ∈ V ǫg i , y ∈ V ǫg j . Thengr H := k h g, x, y | g = 1 , gx = xg, gy = yg, x = 0 , y = 0 , xy − yx i , with g ∈ G ( H ), x ∈ P ,g i ( H ) and y ∈ P ,g j ( H ). Observe that Aut( C ) ∼ = C . Up toisomorphism, we can take( i, j ) ∈ { (0 , , (0 , , (0 , , (1 , , (1 , , (1 , , (2 , } . By similar computations as before, we have gx − xg = λ g (1 − g i ) , gy − yg = λ g (1 − g j ) ,x − iλ x ∈ P ,g i ( H ) , y − jλ y ∈ P ,g j ( H ) ,xy − yx + λ jy − λ ix ∈ P ,g i + j ( H ) . for λ , λ ∈ I , .Assume that ( i, j ) = (0 , gx = xg , gy = yg in H and P ( H ) = { x, y } . Then x = µ x + µ y, y = µ x + µ y, xy − yx = µ x + µ y, for some µ , µ , · · · , µ ∈ k . Observe that P ( H ) is a two-dimensional restricted Liealgebra and H ∼ = k [ C ] ⊗ U L ( P ( H ), where U L ( P ( H )) is a restricted universal envelopingalgebra. Then by [34, Theorem 7.4], we obtain five classes of H described in (27)–(31).Assume that ( i, j ) = (0 , P ( H ) = k { x } , P ,g ( H ) = k { − g, y } and P ,g ( H ) = k { − g } . Hence x = µ x, y − λ y = µ (1 − g ) , xy − yx = µ y + µ (1 − g ) , for some µ , µ , µ , µ ∈ k . We can take µ ∈ I , and µ = 0 by rescaling x, y and via thelinear translation y := y − a (1 − g ) satisfying a − λ a = µ . Then it follows by a directcomputation that[ x, [ x, y ]] = µ [ x, y ] = µ y + µ µ (1 − g ) , [ x , y ] = [ µ x, y ] = µ µ y + µ µ (1 − g ) , [[ x, y ] , y ] = − µ [ g, y ] = − µ λ g (1 − g ) , [ x, y ] = λ [ x, y ] = λ µ y + λ µ (1 − g ) . OINTED HOPF ALGEBRAS OF DIMENSION 16 IN CHARACTERISTIC 2 37
By Proposition 2.9, [ x, [ x, y ]] = [ x , y ] and [[ x, y ] , y ] = [ x, y ], which implies that( µ − µ ) µ = 0 , ( µ − µ ) µ = 0 , λ µ = 0 , λ µ = 0 . Then it is easy to verify that the ambiguities ( g ) x = g ( gx ), ( g ) y = g ( gx ), ( x ) y = x ( xy ), ( xy ) y = x ( y ), ( gx ) y = g ( xy ), ( x ) x = x ( x ) and ( y ) y = y ( y ) are resolvable. ByDiamond lemma, dim H = 16.If λ = 0 = µ , then µ = 0 and we can take µ ∈ I , by rescaling x , which gives twoclasses of H described in (32) and (33).If λ = 0 = µ −
1, then µ = µ and µ = µ µ and hence we can take µ ∈ I , by rescaling x . If µ = 0, then µ = 0, which gives one class of H described in (34). If µ = 1, then we can take µ = 0 via the linear translation y := y − µ (1 − g ), which givesone class of H described in (35).If λ = 1, then µ = 0 = µ , which gives two classes of H described in (36)–(37).Assume that ( i, j ) = (0 , P ( H ) = k { x } , P ,g ( H ) = k { − g , y } . Hence x = µ x, y = µ x, xy − yx = µ y + µ (1 − g ) . From [ x, [ x, y ]] = [ x , y ], [[ x, y ] , y ] = [ x, y ], ( x ) x = x ( x ) and ( y ) y = y ( y ), we have( µ − µ ) µ = 0 , ( µ − µ ) µ = 0 , µ µ = 0 = µ µ . Then it is easy to verify that the ambiguities ( g ) x = g ( gx ), ( g ) y = g ( gy ), ( x ) y = x ( xy ), ( xy ) y = x ( y ), ( gx ) y = g ( xy ) are resolvable. By Diamond lemma, dim H = 16.By rescaling x, y , λ , µ ∈ I , .If µ = 0, then µ = 0 and µ µ = 0. If µ = 0, then we can take µ ∈ I , by rescaling x . If µ = 0, then we can take µ ∈ I , . Therefore, ( µ , µ ) admits three possibilities and λ ∈ I , , which gives six classes of H described in (38)–(43).If µ = 1, then µ = µ and µ = µ µ , which implies that µ ∈ I , by rescaling x . • If µ = 0, then µ = 0, which impies that xy − yx = 0 in H . If λ = 0, thenby rescaling y , we can take µ ∈ I , , which gives two classes of H described in(44)–(45). If λ = 1, then H ∼ = e H ( µ ) described in (46). • If µ = 1, then µ = 0, that is, y = 0 in H . Hence we can take µ = 0 via thelinear translation y := y − µ (1 − g ). Indeed, it is easy to see that the translationis a well-defined Hopf algebra isomorphism. Therefore, we obtain two classes of H described in (47)–(48).Now we claim that e H ( λ ) ∼ = e H ( γ ), if and only if, λ = γ .Suppose that φ : e H ( λ ) → e H ( γ ) for λ, γ ∈ k is a Hopf algebra isomorphism. Write g ′ , x ′ , y ′ to distinguish the generators of e H ( γ ). Observe that spaces of skew-primitiveelements of e H ( γ ) are trivial except P , ( g ′ ) ( e H ( γ )) = k { y ′ }⊕ k { − ( g ′ ) } and P ( e H ( γ )) = k { x ′ } . Then φ ( g ) = g ′ ± , φ ( x ) = αx ′ , φ ( y ) = a (1 − ( g ′ ) ) + by ′ for some α = 0 , a, b = 0 ∈ k . Applying φ to the relation x − x = 0, we have α = 1.Applying φ to the relation gy − yg = g (1 − g ), we have b = 1. Then applying φ to therelation y − λx = 0, we have φ ( y − λx ) = ( y ′ ) − λx ′ = ( γ − λ ) x ′ = 0 ⇒ γ = λ. Assume that ( i, j ) = (1 , P ,g ( H ) = k { − g, x, y } and P ,g = k { − g } .Hence x − λ x = µ (1 − g ) , y − λ y = µ (1 − g ) , xy − yx + λ y − λ x = µ (1 − g ) , for µ , µ , µ ∈ k . It follows by a direct computation that all ambiguities are resolvableand hence by the Diamond lemma, dim H = 16. We can take µ = 0 = µ via the lineartranslation x := x − a (1 − g ), y := y − b (1 − g ) satisfying a − λ a = µ and b − λ b = µ .If λ = 0 or λ = 0, then we can take µ ∈ I , by rescaling x or y .If λ = 0 = λ , then µ ∈ I , , which gives two classes of H described in (49)–(50).If λ − λ , then µ ∈ I , , which gives two classes of H described in (51)–(52).If λ = 0 = λ −
1, then µ ∈ I , , which gives two classes of H described in (51)–(52)by swapping x and y . If λ = λ = 1, then H is isomorphic to one of the Hopf algebrasdescribed in (51)–(52). Indeed, in this case, consider the translation y := y + x + a (1 − g )satisfying a = µ , it is easy to see that H is isomorphic to the Hopf algebras defined by k h g, x, y | g = 1 , [ g, x ] = g (1 − g ) , [ g, y ] = 0 , x = x, y = 0 , [ x, y ] = y + ( a + µ )(1 − g ) i . If a + µ = 0, then H is isomorphic to the Hopf algebra described in (51). If a + µ = 0,then by rescaling y , H is isomorphic to the Hopf algebra described in (52).Assume that ( i, j ) = (1 , P ( H ) = 0, P ,g ( H ) = { − g, x } , P ,g ( H ) = { − g , y } and P ,g ( H ) = k { − g } . Hence, x − λ x = µ y + µ (1 − g ) , y = 0 , xy − yx − λ x = µ (1 − g ) , for some µ , µ , µ ∈ k . The verification of the ambiguities ( a ) b = a ( ab ) and ( ab ) b = a ( b )for all a, b ∈ { g, x, y } and ( gx ) y = g ( xy ) amount to the conditions µ λ = 0 = µ µ , λ = 0 . Then by Diamond lemma, dim H = 16. We can take µ = 0 via the linear translation x := x − a (1 − g ) satisfying a − λ a = µ and take µ ∈ I , by rescaling y .If λ = 0, then we can take µ ∈ I , by rescaling x , which gives three classes of H described in (53)–(55). If λ − µ , then by rescaling y , we can take µ ∈ I , , OINTED HOPF ALGEBRAS OF DIMENSION 16 IN CHARACTERISTIC 2 39 which gives two classes of H described in (56)–(57). If λ − µ −
1, then µ = 0,which gives one class of H described in (58).Assume that ( i, j ) = (1 , P ( H ) = 0, P ,g ( H ) = k { − g, x } , P ,g ( H ) = k { − g } and P ,g ( H ) = k { − g , y } . Hence x − λ x = µ (1 − g ) , y − λ y = µ (1 − g ) , [ x, y ] + λ y − λ x = 0 , for some µ , µ ∈ k . It follows by a direct computation that all ambiguities are resolvableand hence by Diamond lemma, dim H = 16. Then we can take µ = 0 = µ via the lineartranslation x := x − a (1 − g ) , y := y − b (1 − g ) satisfying a − aλ = µ , b − bλ = µ .Therefore, the structure of H depends on λ , λ ∈ I , , denoted by H ( λ , λ ).We claim that H (0 , ∼ = H (1 , φ : H (0 , → H (1 ,
0) given by φ ( g ) = g , φ ( x ) = y and φ ( y ) = x . It follows by a direct computationthat φ is a Hopf algebra morphism. Obviously, φ is an epimorphism and φ | ( H (0 , isinjective. Therefore, φ is an isomorphism. It is easy to see that H (0 , H (1 ,
0) and H (1 ,
1) are pairwise non-isomorphic. Therefore, we obtain three classes of H describedin (59)–(61).Assume that ( i, j ) = (2 , P ( H ) = 0. Hence x = 0 , y = 0 , xy − yx = 0 . Then it is easy to see that all ambiguities are resolvable and hence by the Diamond lemma,dim H = 16. Similar to the last case, we obtain two classes of H described in (62)–(63).Assume that V ∼ = M k, for k ∈ { , } . Thengr H := k h g, x, y | g − , gx = xg, gy = ( y + x ) g, xy − yx, x , y i ;with g ∈ G (gr H ) , x, y ∈ P ,g k (gr H ) for k ∈ I , . By similar computations as before, wehave gx − xg = λ ( g − g k +1 ) , gy − ( y + x ) g = λ ( g − g k +1 ) , xy − yx, x , y ∈ P ( H ) , for some λ , λ ∈ k .If k = 0, then P ( H ) = k { x, y } , which implies that x = α x + α y, y = α x + α y, xy − yx = α x + α y ;for some α , · · · , α ∈ k . Observe that P ( H ) is a two-dimensional restricted Lie algebraand H ∼ = k [ C ] ♯U L ( P ( H ), where U L ( P ( H )) is a restricted universal enveloping algebra.Then by [34, Theorem 7.4], we obtain five classes of H described in (64)–(68).If k = 1, then P ( H ) = 0 and hence the defining relations of H are gx − xg = λ ( g − g ) , gy − ( y + x ) g = λ ( g − g ) , xy − yx = x = y = 0 . The verification of the ambiguities ( a ) b = a ( ab ) and ( ab ) b = a ( b ) for all a, b ∈ { g, x, y } and ( gx ) y = g ( xy ) gives no ambiguity conditions. Then by Diamond lemma, dim H = 16.We write H ( λ , λ ) := H for convenience. Cliam: H ( λ , λ ) ∼ = H ( γ , γ ), if and only if, there exist α , α = 0 , β ∈ k such that α γ = λ and β γ − α + α γ − λ = 0.Suppose that φ : H ( λ , λ ) → H ( γ , γ ) for λ , λ , γ , γ ∈ k is a Hopf algebra isomor-phism. Write g ′ , x ′ , y ′ to distinguish the generators of H ( γ , γ ). Then φ ( g ) = g ′ ± , φ ( x ) = α (1 − ( g ′ ) ) + α x ′ + α y ′ , φ ( y ) = β (1 − ( g ′ ) ) + β x ′ + β y ′ for some α , α , α , β , β , β ∈ k . Applying φ to the relation gx − xg = λ g (1 − g ), wehave α = 0 = α γ − γ . Then applying φ to the relation gy − ( y + x ) g = λ g (1 − g ),we have β = α , β γ − α + γ β − λ = 0 . Then it is easy to check that φ is a well-defined bialgebra map. Since φ is an isomorphism,it follows that α = 0. Consequently, the claim follows.By rescaling x , we can take λ ∈ I , . Then from the last claim, we have H ( λ , ∼ = H ( λ , λ ) for λ ∈ I , and H (0 , = H (1 , H described in (69)–(70).4.2.2. G ( H ) ∼ = C × C := h g i × h h i . If V is a decomposable object in C × C C × C Y D , then V := k { x, y } must be the sum of two one-dimensional objects in C × C C × C Y D such that x ∈ V ǫg i h j , y ∈ V ǫg µ h µ for i, j, µ, ν ∈ I , . If V is an indecomposable object in C × C C × C Y D ,then by [6] and Theorem 2.3, V := k { x, y } ∈ C × C C × C Y D by g · x = x, g · y = y + x, h · x = x, h · y = y + λx, λ ∈ k ; δ ( x ) = g k h l ⊗ x, δ ( y ) = g k h l ⊗ y, for some k, l ∈ I , . We claim that ( k, l, λ ) ∈ { (0 , , λ ) , (0 , , , (1 , , } ; otherwise, V is of Jordan type, acontradication.Assume that V is a decomposable object in C × C C × C Y D . Then x ∈ V ǫg i h j , y ∈ V ǫg µ h µ for i, j, µ, ν ∈ I , . Without loss of generality, we may assume that x, y ∈ V , x ∈ V g , y ∈ V g i for i ∈ I , or x ∈ V g , y ∈ V h .Assume that x, y ∈ V ǫ . Then H ∼ = k [ C × C ] ⊗ U L ( P ( H )), where U L ( P ( H )) is arestricted universal enveloping algebra of P ( H ). Then by [34, Theorem 7.4], we obtainfive classes of H described in (71)–(75). OINTED HOPF ALGEBRAS OF DIMENSION 16 IN CHARACTERISTIC 2 41
Assume that x ∈ V ǫg , y ∈ V ǫ . Then by Lemma 3.10, the defining relations of H are g = 1 , h = 1 , gx − xg = λ g (1 − g ) , gy − yg = 0 ,hx − xh = λ h (1 − g ) , hy − yh = 0 ,x − λ x = µ y, y = µ y, xy − yx = µ x + µ (1 − g ) , for λ ∈ I , , λ , µ , · · · , µ ∈ k with ambiguity conditions µ µ = 0 = µ µ , µ µ = µ , µ µ = µ µ , λ µ = 0 = λ µ . By rescaling y , we can take µ ∈ I , .If λ = 0 = µ , then µ = 0 = µ µ and we can take λ , µ ∈ I , by rescaling x, y . If µ = 0, then by rescaling y , µ ∈ I , . If µ = 0, then µ = 0 and we can take µ = 1 byrescaling y . Therefore, we obtain six classes of H described in (76)–(81).If λ = 0 = µ −
1, then µ = µ , ( µ − µ = 0 and µ µ = 0 = λ µ . We can take µ ∈ I , by rescaling y . If µ = 0, then µ = 0 and we can take λ ∈ I , by rescaling x ,which gives four classes of H described in (82)–(85). If µ = 1, then λ = 0 and we cantake µ ∈ I , by rescaling x . If µ = 0, then we can take µ = 0 via the linear translation x := x + µ (1 − g ), which gives one class of H described in (86). If µ = 1, then µ = 0,which gives one class of H described in (87).If λ − µ , then µ = 0 and µ µ = 0. If µ = 0 = µ , then H ∼ = e H ( λ )described in (88). If µ = 0, then µ = 0 and we can take µ = 1 by rescaling y , whichimplies that H ∼ = e H ( λ ) described in (89). If µ = 0 and µ = 0, then by rescaling y , µ = 1, which implies that H ∼ = e H ( λ ) described in (90).If λ = µ = 1, then µ = 0 = µ and hence H ∼ = e H ( λ , µ ) described in (91). Claim: e H n ( λ ) ∼ = e H n ( γ ) for n ∈ I , , if and only if, λ = γ + i for i ∈ I , ; e H ( λ, µ ) ∼ = e H ( γ, ν ) if and only if λ = γ + i for i ∈ I , and µ = ν .Suppose that φ : e H ( λ ) → e H ( γ ) for λ, γ ∈ k is a Hopf algebra isomorphism. Then φ | C × C : C × C → C × C is an automorphism. Write g ′ , h ′ , x ′ to distinguish thegenerators of e H ( γ ). Since spaces of skew-primitive elements of e H ( γ ) are trivial except P ,g ′ ( e H ( γ )) = k { x ′ } ⊕ k { − g ′ } and P ( e H ( γ )) = k { y ′ } , it follows that φ ( g ) = g ′ , φ ( h ) = ( g ′ ) i h ′ , φ ( x ) = a (1 − g ′ ) + bx ′ , φ ( y ) = cy ′ for some a, b = 0 , c = 0 ∈ k and i ∈ I , . Then applying φ to the relations gx − xg = g (1 − g )and hx − xh = λh (1 − g ), we have b = 1 b ( i + γ ) = λ ⇒ i + γ = λ. Conversely, if λ = γ + i for i ∈ I , , then consider the algebra map ψ : e H ( λ ) → e H ( γ ) , g → g, h → g i h, x → x, y → y . It is easy to see that ψ is a Hopf algebraepimorphism and ψ | ( e H ( λ ) ) is injective. Therefore, e H ( λ ) ∼ = e H ( γ ). Similarly, e H n ( λ ) ∼ = e H n ( γ ) for n ∈ I , , if and only if, λ = γ + i for i ∈ I , ; e H ( λ, µ ) ∼ = e H ( γ, ν ) if and only if λ = γ + i for i ∈ I , and µ = ν .Assume that x, y ∈ V ǫg . Then by Lemma 3.10, the defining relations of H are g = 1 , h = 1 , gx − xg = λ g (1 − g ) , gy − yg = λ g (1 − g ) ,hx − xh = λ h (1 − g ) , hy − yh = λ h (1 − g ) ,x − λ x = 0 , y − λ y = 0 , xy − yx + λ y − λ x = 0 . for λ , λ ∈ I , , λ , · · · , λ ∈ k .If λ = 0 = λ , then we can take λ , λ ∈ I , by rescaling x, y , which gives two classes of H described in (92)–(93). Let H := H ( λ , λ ) for convenience. Indeed, H (1 , ∼ = H (0 , x and y ; H (1 , ∼ = H (1 ,
1) via the Hopf algebra isomorphism φ : H (1 , → H (1 ,
1) defined by φ ( g ) = g, φ ( h ) = h, φ ( x ) = x, φ ( y ) = x + y. Moreover, H (0 ,
0) and H (1 ,
0) are not isomorphic since H (0 ,
0) is commutative while H (1 ,
0) is not commutative.If λ − λ , then we can take λ ∈ I , by rescaling y . If λ = 0, then H ∼ = e H ( λ )described in (94). If λ = 1, then H ∼ = e H ( λ ) described in (95).If λ = 0 = λ −
1, then H is isomorphic to one of the Hopf algebras described in(94)–(95) by swapping x and y .If λ = λ = 1, then H is isomorphic to one of the Hopf algebras described in (94)–(95).Indeed, consider the translation y := x + y , it is easy to see that H is isomorphic to theHopf algebra defined by g = 1 , h = 1 , gx − xg = g (1 − g ) , gy − yg = 0 , hx − xh = λ h (1 − g ) ,hy − yh = ( λ + λ ) h (1 − g ) , x − x = 0 , y = 0 , xy − yx + y = 0 . If λ + λ = 0, then H is isomorphic to the Hopf algebra described in (94). If λ + λ = 0,then by rescaling y , H is isomorphic to the Hopf algebra described in (95). Claim: e H n ( λ ) ∼ = e H n ( γ ) for n ∈ I , , if and only if, λ = γ + i for i ∈ I , .Assume that x ∈ V ǫg , y ∈ V ǫh . Then by Lemma 3.11, the defining relations of H are gx − xg = λ g (1 − g ) , hx − xh = λ h (1 − g ) , x − λ x = 0 gy − yg = λ g (1 − h ) , hy − yh = λ h (1 − h ) , y − λ y = 0 ,xy − yx − λ x + λ y = λ (1 − gh ) . OINTED HOPF ALGEBRAS OF DIMENSION 16 IN CHARACTERISTIC 2 43 for some λ , λ ∈ I , , λ , λ , λ ∈ k . The verifications of ( a ) b = a ( ab ) , a ( b ) = ( ab ) b for a, b ∈ { g, h, x, y } and a ( xy ) = ( ax ) y for a ∈ { g, h } amounts to the conditions( λ + λ ) λ = ( λ + λ ) λ = ( λ + λ ) λ = 0 , ( λ − λ ) λ = ( λ − λ ) λ = ( λ − λ ) λ = 0 . Then by the Diamond lemma, dim H = 16.If λ = 0 = λ , then λ = 0 = λ and hence we can take λ ∈ I , by rescaling x , whichgives two classes of H described in (96)–(97).If λ − λ , then λ = λ , λ = 0 and ( λ − λ = 0. Hence we can take λ , λ ∈ I , by rescaling x, y . If λ = 0, then λ = 0, which gives one class of H describedin (98). If λ = 1, then we can take λ = 0 via the linear translation y := y − λ (1 − h ),which gives one class of H described in (99).If λ = 0 = λ −
1, then we obtain two classes of H described in in (98)–(99) via thelinear translation g := h, h := g, x := y, y := x .If λ = 1 = λ , then λ = λ ∈ I , and (1 + λ ) λ = 0. If λ = 0 = λ , then λ = 0,which gives one class of H described in (100). If λ = λ = 1, then we can take λ = 0 viathe linear translation y := y − λ (1 − h ), which gives one class of H described in (101).Assume that V is an indecomposable object in C × C C × C Y D . Then gr H = k h g, h, x, y i ,subject to the relations g = h = x = y = 1 , [ g, x ] = [ h, x ] = [ g, h ] = 0 , gy = ( y + x ) g, hy = ( y + λx ) h, with g, h ∈ G (gr H ) , x, y ∈ P g k h l (gr H ), where ( k, l, λ ) ∈ { (0 , , λ ) , (0 , , , (1 , , } . Itis easy to see that gr H with ( k, l, λ ) ∈ { (0 , , , (1 , , } are isomorphic. Hence we cantake ( k, l, λ ) ∈ { (0 , , λ ) , (0 , , } . By similar computations as before, we have gx − xg = λ g (1 − g k h l ) , gy − ( y + x ) g = λ g (1 − g k h l ); hx − xh = λ h (1 − g k h l ) , hy − ( y + λx ) h = λ h (1 − g k h l ) . If ( k, l, λ ) = (0 , , λ ), then P ( H ) = k { x, y } and x , y , [ x, y ] ∈ P ( H ). Hence H ∼ = k [ C ] ♯U L ( P ( H ), where U L ( P ( H )) is a restricted universal enveloping algebra of P ( H ).Then by [34, Theorem 7.4], we obtain five classes of H described in (102)–(106).If ( k, l, λ ) = (0 , , x − λ x, y − λ y, xy − yx − λ x + λ y ∈ P ( H ). Therefore, the defining relations of H are g = h = 1 , gh = hg, gx − xg = λ g (1 − h ) , gy − ( y + x ) g = λ g (1 − h ) ,hx − xh = λ h (1 − h ) , hy − yh = λ h (1 − h ) ,xy − yx − λ x + λ y = 0 , x − λ x = 0 , y − λ y = 0 . The verifications of ( a ) b = a ( ab ) , ( ab ) b = a ( b ) for a, b ∈ { g, h, x, y } and a ( xy ) = ( ax ) y for a ∈ { g, h } amounts to the conditions λ = 0 = λ . By Diamond Lemma, dim H = 16. We can take λ = 0 via the linear translation x := x + λ (1 − h ) and take λ ∈ I , by rescaling x, y , which gives two classes of H describedin (107)–(108).4.3. Coradical k [ C ] . Then by Lemma 3.3, V ∼ = M i, ⊕ M j, ⊕ M k, for i, j, k ∈ I ,p − or M , ⊕ M , and hence B ( V ) ∼ = k [ x, y, z ] / ( x p , y p , z p ).Assume that V ∼ = M i, ⊕ M j, ⊕ M k, for i, j, k ∈ I , . Thengr H = k h g, x, y, z | g = 1 , [ g, x ] = [ g, y ] = [ g, z ] = x = y = z = [ x, y ] = [ x, z ] = [ y, z ] = 0 i , with g ∈ G ( H ), x ∈ P ,g i ( H ), y ∈ P ,g j ( H ) and z ∈ P ,g k ( H ). Up to isomorphism, wemay assume that ( i, j, k ) = (0 , , , , , ,
0) and (1 , , i, j, k ) = (0 , , H ∼ = k [ C ] ⊗ U L ( P ( H )), where U L ( P ( H )) is arestricted universal enveloping algebra of P ( H ). Then by [20, Theorem 1.4], we obtainfourteen classes of H described in (109)–(122).Assume that ( i, j, k ) = (1 , , H are g = 1 , gx − xg = λ g (1 − g ) , gy − yg = λ g (1 − g ) , gz − zg = λ g (1 − g ) ,x − λ x = 0 , y − λ y = 0 , z − λ z = 0 , xy − yx − λ x + λ y = 0 ,xz − zx − λ x + λ z = 0 , yz − zy − λ y + λ z = 0 . for λ , λ , λ ∈ I , . Let H ( λ , λ , λ ) := H for convenience. We claim that H (1 , , ∼ = H (1 , , φ : H (1 , , → H (1 , , , g → g, x → x, y → x + y, z → z . Then it is easy to see φ is a Hopf algebra epimorphism and φ | ( H (1 , , is injective, which implies that the claim follows. Similarly, H (1 , , ∼ = H (1 , , ∼ = H (1 , , H (0 , ,
0) is commutative and H (1 , ,
0) is not commutative.Hence H ∼ = H (0 , ,
0) or H (1 , ,
0) described in (123) or (124).Assume that ( i, j, k ) = (1 , , H are g = 1 , gx − xg = λ g (1 − g ) , gy − yg = λ g (1 − g ) , gz − zg = 0 ,x − λ x = λ z, y − λ y = λ z, z = λ z,xz − zx = γ x + γ y + γ (1 − g ) , yz − zy = γ x + γ y + γ (1 − g ) ,xy − yx − λ x + λ y = λ z. for λ , λ , λ ∈ I , and λ , λ , λ , γ , · · · , γ ∈ k with the ambiguity conditions given by(6)–(12).Suppose that λ = 0 = λ . Then by rescaling x, y , we can take λ , λ ∈ I , . OINTED HOPF ALGEBRAS OF DIMENSION 16 IN CHARACTERISTIC 2 45 If λ = 0 = λ , then λ γ i = 0 for all i ∈ I , and by rescaling x , we can take λ ∈ I , .If λ = 1, then γ i = 0 for all i ∈ I , , that is, [ x, z ] = 0 = [ y, z ] in H . Then H dependson λ ∈ I , , that is, H is isomorphic to one of the Hopf algebras described in (125)–(126).If λ = 0 = λ , then γ = γ γ = γ , γ γ = γ γ , γ γ = γ γ , ( γ − γ ) γ =0 = ( γ − γ ) γ and by rescaling x, y , we can take γ , γ ∈ I , . If γ = 0 = γ , then γ = 0 = γ and we can take γ , γ ∈ I , . Let H ( γ , γ ) := H for convenience. It iseasy to see that H (0 , ∼ = H (1 ,
0) by swapping x and y and H (1 , ∼ = H (1 ,
0) via thelinear translation y := y + x . Observe that H (0 ,
0) is commutative while H (1 ,
0) is notcommutative. Therefore, H is isomorphic to one of the Hopf algebras described in (127)–(128). If γ − γ , then γ = γ = γ = 0 and hence we can take γ = 0 via thelinear translation y := y + γ (1 + g ), which gives one class of H described in (129). If γ = 0 = γ −
1, then H is isomorphic to the Hopf algebra described in (129) by swapping x and y . If γ = 1 = γ , then H is isomorphic to the Hopf algebra described in (129) viathe linear translation y := y + x .If λ = 0 = λ −
1, then (1 − γ ) γ = γ γ = (1 − γ ) γ , (1 + γ + γ ) γ = 0 =(1 + γ + γ ) γ , (1 − γ ) γ = γ γ , (1 − γ ) γ = γ γ . If γ = 0 = γ , then γ , γ ∈ I , ,(1 − γ ) γ = 0 = (1 − γ ) γ . Moreover, we can take γ = 0 = γ . Indeed, if γ = 0 or γ = 0, then γ = 0 or γ = 0; if γ = 1 or γ = 1, then we can take γ = 0 or γ = 0via the linear translation x := x + γ (1 − g ) or y := y + γ (1 − g ). Observe that theHopf algebras with γ − γ and γ = 0 = γ − x and y . Then H is isomorphic to one of the Hopf algebras described in (130)–(132). If γ − γ , then γ , γ ∈ I , , γ + γ = 1, (1 − γ ) γ = γ , (1 − γ ) γ = 0. If γ = 1, then γ = 0 = γ and hence H is isomorphic to the Hopf algebra described in(131) via the linear translation x := x + y + γ (1 − g ). If γ = 0, then γ = 1, γ = γ andhence H is isomorphic to the Hopf algebra described in (131) via the linear translation x := y + γ (1 − g ) , y := x + y + γ (1 − g ). Similarly, if γ = γ − γ = 1 = γ , H isisomorphic to the Hopf algebra described in (131).If λ − λ , then γ i = 0 for all i ∈ I , and hence H is isomorphic to one of theHopf algebras described in (133)–(136). If λ = 0 = λ − λ = 1 = λ , then similarto the last case, H is isomorphic to one of the Hopf algebra described in (133)–(136).Suppose that λ − λ . Then γ = 0 = γ and by rescaling y , we can take λ ∈ I , .If λ = 0, then λ γ i = 0 for all i ∈ I , − { } and by rescaling y , we can take λ ∈ I , .Observe that γ = λ γ and λ γ = 0. If λ = 1, then γ i = 0 for all i ∈ I , and we cantake λ = 0 via the linear translation x := x − λ y . Therefore, we obatin two classes of H described in (137)–(138). If λ = 0 = λ , then λ γ i = 0 for all i ∈ I , . If λ = 0, then γ = 0, γ γ = 0 andby rescaling y, z , we can take γ , γ ∈ I , . If γ = 0, then we can take γ ∈ I , . Let H ( γ , γ ) := H for convenience. Then it is easy to see that H (1 , ∼ = H (0 ,
1) via thelinear translation x := x + y . Therefore, H is isomorphic to one of the Hopf algebrasdescribed in (139)–(141). If γ = 1, then γ = 0 and hence we can take γ = 0 via thelinear translation y := y + γ (1 − g ), which gives one class of H described in (142). If λ = 0, then γ i = 0 for all i ∈ I , and we can take λ = 1 by rescaling z , which gives twoclasses of H described in (143).If λ = 0 = λ −
1, then λ γ = 0, (1 − γ ) γ = 0, (1 + γ ) γ = 0, γ = γ γ ,(1 − γ ) γ = 0. If γ = 1, then λ = 0, γ = γ γ and we can take γ = 0 = γ via thelinear translation y := y + γ (1 − g ). Indeed, if γ = 0, then γ = 0; if γ = 0, then γ = γ γ and hence the translation is well-defined. Then H is isomorphic to the Hopfalgebra described as follows: • k h g, x, y, z i / ( g − , [ g, x ] − g (1 − g ) , [ g, y ] , [ g, z ] , [ x, y ] − y, [ x, z ] − γ y, [ y, z ] − y, x − x, y , z − z ).We can take γ = 0 via the linear translation x := x + γ y . Indeed, it follows by a directcomputation that the translation is a well-defined Hopf algebra isomorphism. Therefore, H is isomorphic to the Hopf algebra described in (144). If γ = 0, then γ = 0 = γ = γ ,and hence H ∼ = e H ( λ ) described in (145).If λ = 1, then γ i = 0 for i ∈ I , . If λ = 0 = λ , then we can take λ = 0 via thelinear translation x := x + αy satisfying α = λ , which gives one class of H described in(146). If λ = 0 and λ = 0, then by rescaling y, z , we can take λ = 1. Moreover, we cantake λ = 0 via the linear translation x := x + αy satisfying α + α = λ , which gives oneclass of H described in (147). If λ = 1, then we can take λ = 0 via the linear translation x := x + αy satisfying α + λ α = λ and hence H ∼ = e H ( λ ) described in (148).Suppose that λ = 0 = λ − λ = 1 = λ . Then it can be reduced to the case λ − λ by swapping x and y or via the linear translation y := x + y , respectively. Claim: e H ( λ ) ∼ = e H ( γ ) or e H ( λ ) ∼ = e H ( γ ), if and only if, λ = γ .Suppose that φ : e H ( λ ) → e H ( γ ) for λ, γ ∈ k is a Hopf algebra isomorphism. Then φ | C : C → C is an automorphism. Write g ′ , x ′ , y ′ , z ′ to distinguish the generators of e H ( γ ). Since spaces of skew-primitive elements of e H ( γ ) are trivial except P ,g ′ ( e H ( γ )) = k { x ′ , y ′ } ⊕ k { − g ′ } and P ( e H ( γ )) = k { z ′ } , it follows that φ ( g ) = g ′ , φ ( x ) = α x ′ + α y ′ + α (1 − g ′ ) , φ ( y ) = β x ′ + β y ′ + β (1 − g ′ ) , φ ( z ) = kz ′ for some α i , β i , k ∈ k and i ∈ I , . Then applying φ to the relations gx − xg = g (1 − g ), z = z , x = x and [ g, y ] = 0, we have α = 0 , k = 1 , α + α γ = 0 , β = 0 . OINTED HOPF ALGEBRAS OF DIMENSION 16 IN CHARACTERISTIC 2 47
Then applying φ to the relation [ x, y ] − y − λz , we have λ = γ. Conversely, it is easy to see that e H ( λ ) ∼ = e H ( γ ) if λ = γ . Similarly, e H ( λ ) ∼ = e H ( γ ) ifand only if λ = γ .Assume that ( i, j, k ) = (1 , , H is isomorphic to one of theHopf algebras described in (149)–(183).Assume that V ∼ = M , ⊕ M , . Then gr H = k h g, x, y, z i , subject to the relations g = x = y = z = 1 , [ g, x ] = [ g, y ] = [ x, y ] = [ x, z ] = [ y, z ] = 0 , gz − ( z + y ) g = 0 , with g ∈ G (gr H ) , x, y, z ∈ P (gr H ). It follows by a direct computation that gx − xg = 0 , gy − yg = 0 , gz − ( z + y ) g = 0; x , y , z , [ x, y ] , [ x, z ] , [ y, z ] ∈ P ( H ) . Then H ∼ = k [ C ] ♯U L ( P ( H )), where U L ( P ( H )) is a restricted universal enveloping algebraof P ( H ). Then by [20, Theorem 1.4], we obtain fourteen classes of H described in (184)–(197). ACKNOWLEDGMENT
The essential part of this article was written during the visit of the author to Universityof Padova supported by China Scholarship Council (Grant No. 201706140160) and theNSFC (Grant No. 11771142). The author would like to thank his supervisors Profs. G.Carnovale, N. Hu and Prof. G. A. Garcia so much for the help and encouragement.
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