On the structure and cohomology ring of connected Hopf algebras

CONNECTED HOPF ALGEBRAS

KARIN ERDMANN, ØYVIND SOLBERG, AND XINGTING WANG

Abstract. Letpbe a prime, andkbe a field of characteristicp. We inves- tigate the algebra structure and the structure of the cohomology ring for the connected Hopf algebras of dimensionp³, which appear in the classification obtained in [22]. The list consists of 23 algebras together with two infinite families. We identify the Morita type of the algebra, and in almost all cases this is sufficient to clarify the structure of the cohomology ring.

1. Introduction

The classification of finite-dimensional Hopf algebras in characteristic zero is well investigated by many people, see survey paper [1]. While this work is stimulating in its own right, many Hopf algebras of interest are, however, defined over a fieldkof positive characteristic, where the classification is much less known. Some work has been done in this direction. In [28], Scherotzke classified finite-dimensional pointed rank one Hopf algebras in positive characteristic which are generated by group-like and skew-primitive elements. Many Hopf algebras in positive characteristic also come from Nichols algebras which are of much interest, see [7, 15].

We recall that a Hopf algebra is called connected when it admits only one iso- morphism class of simple comodules, or equivalently, its coradical is k. Note that finite-dimensional connected Hopf algebras only appear in characteristicp, for in- stance, group algebras of finite p-groups, restricted universal enveloping algebras of restricted Lie algebras, finite connected group schemes and others. In [16], all graded cocommutative connected Hopf algebras of dimension less than or equal to p³ are classified. In the recent work of Nguyen, Wang and the third author [22], connected Hopf algebras of dimensionp³ are classified over an algebraically closed field k of characteristic punder some assumption on the primitive space of these Hopf algebras. The result is given below in Theorem 2.1.

We are interested in understanding the structure of the cohomology rings for these algebras. This only depends on the algebra structure, not on the Hopf struc- ture. This motivates our work, namely we analyse the algebras, for those which are not local we obtain a presentation by quiver and relations. This allows us to obtain the cohomology rings in all cases except one.

Section 2 contains the description of the Hopf algebras in question, and a discus- sion of antipode, and Nakayama automorphism. Furthermore, we state our results on the algebra structure. It turns out that the algebras fall into six different classes, and in sections 3 to 8 we deal with these. Section 9 contains a reduction result which

Date: January 30, 2019.

2010Mathematics Subject Classification. 16E05, 16E40, 16T05.

Key words and phrases. cohomology, positive characteristic, (connected) Hopf algebras.

1

relates finite generation of Ext-algebras related via adjoint functors. This is formu- lated for abelian categories with enough injective and projective objects. Section 10 shows that some of the algebras in our list can be viewed as twisted tensor products (in the sense of [8]), and using Section 9 it follows that their cohomology is Noetherian (with one exception). Section 11 gives the cohomology rings of the various algebras occuring in the classification.

For general background on quivers, path algebras and admissible quotients of path algebras we refer the reader to [2, 3]. For basic homological algebra we refer the reader to [4].

2. Connected Hopf algebras of dimension p³

In this section we give the classification of connected Hopf algebras of dimension p³, and explain the origin of the classification from the primitive space of the Hopf algebras. We recall some facts about the antipode and the Nakayama automorphism of finite dimensional Hopf algebras in general, and we apply this knowledge to the algebras in the classification to describe the antipode and the order of the Nakayama automorphism. Finally we explain how we classify them as algebras, which is the base of our further investigations. The classification theorem below assumes k is algebraically closed. In this paper, the field is usually arbitrary of characteristicp.

2.1. Classification of connected Hopf algebras of dimension p³. The Hopf algebras in the classification of finite dimensional connected Hopf algebras of di- mensionp³ are always presented in the form ofkhx, y, zi/I, whereI is an ideal of khx, y, zidefined by giving generators. The comultiplication is given by

∆(x) =x⊗1 + 1⊗x,

∆(y) =y⊗1 + 1⊗y+Y,

∆(z) =z⊗1 + 1⊗z+Z,

for some elementsY andZin (khx, y, zi/I)⊗(khx, y, zi/I). In Theorem 2.1, proved in [22], the generators for the relations and the idealI and the elements Y and Z are given explicitly. IfY and/orZ are not given, then they are zero. In order to expressY andZ, we use the notation

ω(t) =

p−1

X

i=1

(p−1)!

i!(p−i)!tⁱ⊗t^p−i.

Theorem 2.1. Let k be an algebraically closed field of characteristicp. The con- nected Hopf algebras over kof dimensionp³ are precisely

A1: k[x, y, z]/hx^p−x, y^p−y, z^p−ziwithY =ω(x)andZ=ω(x)[y⊗1 + 1⊗y+ ω(x)]^p−1+ω(y).

A2: k[x, y, z]/hx^p, y^p−x, z^p−yi.

A3: k[x, y, z]/hx^p, y^p, z^pi.

A4: k[x, y, z]/hx^p, y^p, z^p−xi.

A5, p= 2: khx, y, zi/hx², y²,[x, y],[x, z],[y, z]−x, z²+xyi.

A5, p >2: A(β) = khx, y, zi/hx^p, y^p,[x, y],[x, z],[y, z]−x, z^p +x^p−1y−βxi for some β ∈k with Y =ω(x)and Z =ω(x)(y⊗1 + 1⊗y)^p−1+ω(y). Any twoA(β)andA(β⁰)are isomorphic as Hopf algebras if and only ifβ⁰=γβ for some(p²+p−1)-th root of unity γ.

B1: khx, y, zi/h[x, y]−y,[x, z],[y, z], x^p−x, y^p, z^piwith Z=ω(y).

B2: khx, y, zi/h[x, y]−y,[x, z],[y, z]−yf(x), x^p −x, y^p, z^p −zi, where f(x) = Pp−1

i=1(−1)ⁱ⁻¹(p−i)⁻¹xⁱ andZ =ω(x).

B3: khx, y, zi/h[x, y]−y,[x, z]−z,[y, z]−y², x^p−x, y^p, z^piwithZ =−2x⊗y for p >2.

C1: k[x, y, z]/(x^p−x, y^p−y, z^p−z).

C2: k[x, y, z]/(x^p−y, y^p−z, z^p).

C3: k[x, y, z]/hx^p, y^p−z, z^pi.

C4: k[x, y, z]/hx^p, y^p, z^pi.

C5: khx, y, zi/h[x, y]−z,[x, z],[y, z], x^p, y^p, z^pi.

C6: khx, y, zi/h[x, y]−z,[x, z],[y, z], x^p−z, y^p, z^piforp >2.

C7: k[x, y, z]/hx^p, y^p, z^p−zi.

C8: k[x, y, z]/hx^p−y, y^p, z^p−zi.

C9: k[x, y, z]/hx^p, y^p−y, z^p−zi.

C10: khx, y, zi/h[x, y]−z,[x, z],[y, z], x^p, y^p, z^p−zi.

C11: khx, y, zi/h[x, y]−y,[x, z],[y, z], x^p−x, y^p, z^pi.

C12: khx, y, zi/h[x, y]−y,[x, z],[y, z], x^p−x, y^p−z, z^pi.

C13: khx, y, zi/h[x, y]−y,[x, z],[y, z], x^p−x, y^p, z^p−zi.

C14: khx, y, zi/h[x, y]−y,[x, z],[y, z], x^p−x, y^p−z, z^p−zi.

C15: khx, y, zi/h[x, y]−z,[x, z]−x,[y, z] +y, x^p, y^p, z^p−ziforp >2.

C16: C(λ, δ) =khx, y, zi/h[x, y],[x, z]−λx,[y, z]−λ⁻¹y, x^p, y^p, z^p−δzifor some λ ∈ k^× such that δ = λ^p−1 = ±1. Any two C(λ1, δ1) and C(λ2, δ2) are isomorphic as Hopf algebras if and only ifδ1=δ2andλ1=λ2orλ1λ2= 1.

Remark. 1) The antipode of the above Hopf algebras always exists, because the coalgebra structure is connected.

2) LetH be a connected Hopf algebra of dimensionp³ and denote by P(H) ={h∈H |∆(h) =h⊗1 +h⊗1},

the primitive space of H. The isomorphism classes of H given above satisfy the following conditions:

(A) dim_kP(H) = 1.

(B) dim_kP(H) = 2 with non-commuting elements.

(C) dim_kP(H) = 3.

3) Hopf algebras of typeCare exactly the restricted universal enveloping algebras of restricted Lie algebras of dimension 3. It includes all thep-nilpotent restricted Lie algebras of dimension 3 classified in [29, Theorem 2.1 (3/1),(3/2)], i.e., (up to isomorphisms) (3/1) (a) is C4, (3/1) (b) is C3, (3/1) (c) is C2; (3/2) (a) is C5 (p≥3) and C10(p= 2), and (3/2) (b) isC6.

2.2. Antipode and Nakayama automorphism. In this subsection we review some known results on the antipode and the Nakayama automorphism of finite dimensional Hopf algebras in general, and then apply them to the algebras in The- orem 2.1.

LetH = (H, m, u,∆, , S) be any finite-dimensional Hopf algebra over an arbi- trary base fieldk. Aleft integralΛ inH is an element ofH such thathΛ =(h)Λ for allh∈H; and aright integral inH is an element Λ⁰∈H such that Λ⁰h=(h)Λ⁰ for allh∈H. The space of left integrals and the space of right integrals are denoted byRl

HandRr

H, respectively. A well-known result of Larson and Sweedler shows that dimRl

H = dimRr

H = 1. We sayH isunimodular ifRl H=Rr

H.

For any algebra mapα: H →k, we can define the left winding automorphism of H associated to α as Ξ^l[α](h) = P

α(h1)h2 for any h ∈ H. Similarly, the right winding automorphism of H associated to α can be defined as Ξ^r[α](h) = Ph₁α(h₂) for anyh∈H. Note thatRl

H ⊂H is a rightH-submodule ofH. Since it is one-dimensional, the rightH-module structure gives an algebra mapα:H →k such that Λh=α(h)Λ for 06= Λ∈Rl

Handh∈H. We will use the term left winding automorphism ofRl

H instead ofα.

The following results are due to Larson-Sweedler [20], Pareigis [25], and Brown- Zhang [6].

Theorem 2.2. Let H be any finite-dimensional Hopf algebra. Then

(a) H is Frobenius with a nondegenerate associated bilinear formh−,−i:H⊗ H →kgiven by ha, bi=λ(ab), where06=λ∈Rl

H^∗ anda, b∈H.

(b) The Nakayama automorphism is given byS²ξ, where ξ is the left winding automorphism of Rl

H.

(c) H is symmetric if and only ifH is unimodular andS² is inner.

This has the following consequence for finite dimensional involutory Hopf alge- bras. Recall that a Hopf algebraH with antipodeS isinvolutory ifS²= 1.

Corollary 2.3. The following finite-dimensional involutory Hopf algebras are sym- metric:

(a) commutative algebras, (b) group algebras, (c) local algebras, (d) semisimple algebras.

Proof. Parts (a) and (b) are clear. Parts (c) and (d) hold because any local or

semisimple Hopf algebra is unimodular.

The next result for restricted Lie algebras is from [30, 18].

Theorem 2.4. Let gbe a finite dimensional restricted Lie algebra overk of char- acteristic p > 0. Then the restricted enveloping algebra u(g) is Frobenius with Nakayama automrophism σ given by σ(x) = x+ Tr(adx) for all x ∈ g. As a consequence, σ^p = 1 and u(g) is symmetric if and only if Tr(adx) = 0 for all x∈g.

A direct calculation by using the antipode axiom m(S⊗1)∆ =m(1⊗S)∆ =u

yields that the antipode of Hopf algebras for allA, B,Ctypes is given by S(x) =−x, S(y) =−y, S(z) =−z

except forB3whose antipode is given by

S(x) =−x, S(y) =−y, S(z) =−z−2xy.

Using this we have the following.

Proposition 2.5. For the Hopf algebras listed in Theorem 2.1, the following are true.

(a) All Hopf algebras are involutory except for B3whereS^2p= 1.

(b) The Hopf algebras of type Aare unimodular and symmetric.

(c) The Hopf algebras of type B1and B3are not symmetric.

(d) The Hopf algebras C1–C10 and C15 are symmetric, C11–C14 are not symmetric, and C16is symmetric if and only ifλ²=−1.

Proof. (a) is clear since forB3, we haveS²ⁿ(z) =z+ 2ny for alln≥1.

(b) The algebras A1–A4are commutative. One checks easily thatA5is local.

Then it follows from Corollary 2.3(c).

(c) B1 and C11 are isomorphic as augmented algebras so it follows from (d) later. ForB3, one sees thatS²(z) =z+xyis not inner.

(d) This follows from Theorem 2.4.

Remark. From a Hopf algebra view point it is not known whetherB2is symmetric or not because it is hard to compute its left or right integral and to check whether it is unimodular. However we shall show that through finding another presentation of algebras of typeB2none of them are symmetric (See Proposition 5.1 (a)).

2.3. Algebra structure. The cohomology ring of a finite dimensional Hopf alge- bra only depends on the algebra structure, not the Hopf structure. So in order to determine the structure of the cohomology ring it is enough to study the algebra structure. We will do this in the following six sections, where we show that the algebras in this classification fall into 6 different classes, (0) semisimple algebras, (1) group algebras (tensored or direct sum with a semisimple algebra), (2) (direct sums of) selfinjective Nakayama algebras, (3) enveloping algebra of restricted Lie algebras, (4) coverings of local algebras, (5) other local algebras. We will show that

Semisimple algebras A1=C1

Group algebras A2 =C2, A3= C4, A4 = C3, C7,

C8,C9,C10 Selfinjective Nakayama algebras B2,C12,C13, C14 Enveloping algebra of re-

stricted Lie algebras C5,C6,C15

Coverings of local algebras B1=C11,B3,C16

Other local algebras A5

Note that here we focus on the algebra structure (and we do not consider the comultiplication).

3. Semisimple algebras

In this section we classify the semisimple algebras occurring among the algebras in the classification of the finite dimensional connected Hopf algebras of dimension p³.

We show the following.

Proposition 3.1. Letkbe a field of characteristicp. The algebras of type A1and C1are equal, and they are isomorphic to k^p3.

Proof. The algebra of typeA1(orC1) are given as Λ =k[x, y, z]/hx^p−x, y^p−y, z^p−zi

for a fieldkof characteristicp. The polynomialu^p−uinuhaspdifferent roots in Zp, so thatk[u]/hu^p−ui 'k^p. Furthermore we have that

Λ'k[x]/(x^p−x)⊗kk[y]/(y^p−y)⊗kk[z]/(z^p−z)'k^p3.

Remark 3.2. As we already said, thep-dimensional algebraB =k[u]/hu^p−uiis isomorphic tok^p forkof characteristicp. There is an elementary explicit formula for the orthogonal primitive idempotents of this algebra, and we will use this later.

Namely forr∈Z^p let

e_r:=

p−1

Y

s6=rs=0

(u−s) (r−s).

Then ue_r =re_r, and the e_r for r ∈ Zp are pairwise orthogonal idempotents and their sum is the identity of the algebra. We mention two consequences.

(1) Suppose 06=ζ∈B and uζ =rζ, thenζ is a scalar multiple ofer. Namely, we haveζ=P

iλiei forλi∈k, and then 0 = (u−r)ζ=P

iλi(u−r)ei. From the formula, (u−r)ei = 0 fori=rand otherwise it is a non-zero scalar multiple ofei, and the claim follows.

(2) SupposeBis contained in some algebra Λ, andy∈Λ satisfies [u, y] =y. Then yei=ei+1yei. Namely we haveuy=y(u+ 1) and thereforeu(yei) =y(u+ 1)ei= (i+ 1)yei. Variations of this argument will be used later.

4. Group algebras

In this section we find group algebras occurring in the classification of the finite dimensional connected Hopf algebras of dimension p³. Here we denote by C_n the cyclic group of ordern, and for a finite groupGwe denote bykGthe group algebra ofGover a fieldk.

Proposition 4.1. Let k be a field of characteristicp.

(a) The algebras A2and C2are equal, and they are isomorphic to kC_p3. (b) The algebras A3and C4are equal, and they are isomorphic to

k(C_p×C_p×C_p).

(c) The algebras A4and C3are isomorphic, and they are isomorphic to k(Cp×Cp²).

(d) The algebra C7is isomorphic to the direct sum of pcopies ofk(Cp×Cp).

(e) The algebra C8 is isomorphic to the direct sum ofpcopies ofkC_p2. (f) The algebra C9 is isomorphic to the direct sum ofp² copies ofkCp. (g) The algebra C10is isomorphic to the direct sum ofk(Cp×Cp)and p−1

copies ofMp(k).

Proof. (a) The algebras of typeA2andC2are given as (with a possible change of variables)

Λ =k[x, y, z]/hx^p−y, y^p−z, z^pi.

The algebra homomorphism k[u] → Λ sending u 7→ x+hx^p−y, y^p −z, z^pi is surjective with kernelhu^p3i. Hence Λ'k[u]/hu^p3i, which is isomorphic tokCp³.

(b) The algebras of type A3 and C4 are given as Λ = k[x, y, z]/hx^p, y^p, z^pi.

Then

Λ'k[x]/hx^pi ⊗kk[y]/hy^pi ⊗kk[z]/hz^pi.

This is in turn isomorphic tok(C_p×C_p×C_p).

(c) The algebra of type A4 and C3 are given as (with a possible change of variables) Λ =k[x, y, z]/hx^p, y^p−z, z^pi. Then

Λ'k[x]/hx^pi ⊗kk[y, z]/hy^p−z, z^pi 'k[x]/hx^pi ⊗kk[u]/hu^p2i,

since the algebra homomorphismk[u]→k[y, z]/hy^p−z, z^pisendingu7→y+hy^p− z, z^piis surjective with kernelhu^p2i. This algebra is isomorphic tok(Cp×C_p2).

(d) The algebra of typeC7is given as Λ =k[x, y, z]/hx^p, y^p, z^p−zi. Then Λ'k[x]/hx^pi ⊗kk[y]/hy^pi ⊗kk[z]/hz^p−zi

'k[x]/hx^pi ⊗kk[y]/hy^pi ⊗kk^p

The last algebra is isomorphic tok(Cp×Cp)⊗kk^p, which is isomorphic topcopies ofk(Cp×Cp).

(e) The algebra of typeC8is given as Λ =k[x, y, z]/hx^p−y, y^p, z^p−zi. Then Λ'k[x, y]/hx^p−y, y^pi ⊗kk[z]/hz^p−zi

'k[x, y]/hx^p−y, y^pi ⊗kk^p 'k[u]/hu^p2i ⊗kk^p,

since the algebra homomorphismk[u]→k[x, y]/hx^p−y, y^pisendingu7→x+hx^p− y, y^pi is surjective with kernel hu^p2i. This last algebra occurring above is in turn isomorphic tokC_p2⊗kk^p, which is isomorphic topcopies ofkC_p2.

(f) The algebra of typeC9is given as Λ =k[x, y, z]/hx^p, y^p−y, z^p−zi. Then Λ'k[x]/hx^pi ⊗_kk[y]/hy^p−yi ⊗_kk[z]/hz^p−zi

'k[x]/hx^pi ⊗kk^p2

This last algebra is isomorphic to kC_p⊗_kk^p2, which in turn is isomorphic to p² copies ofkCp.

(g) The algebra of typeC10is given as

Λ =khx, y, zi/h[x, y]−z,[x, z],[y, z], x^p, y^p, z^p−zi.

It is clear from the relations that the subalgebra generated byzis in the center of Λ and is giving rise toporthogonal central idempotents{e_i}^p−1_i=0 withze_i=ie_i as in Remark 3.2. Then Λ =⊕^p−1_i=0Λei, and we have thatiei =zei= [x, y]ei. Further- more Λei is generated by {xei, yei}, hence we have a surjective homomorphism of ringsϕi:khx, yi →Λeisendingx7→xeiandy7→yei. The kernel ofϕiis generated by the elementsx^p, y^p,[x, y]−igiving that Λei'khx, yi/hx^p, y^p,[x, y]−ii.

Fori = 0 we get that Λe0 'k[x, y]/hx^p, y^pi 'k[x]/hx^pi ⊗kk[y]/hy^pi, which is isomorphic tok(C_p×C_p).

Now consider the case wheniis in [1, . . . , p−1]. We have that Λ'k⊗_ZpZphx, y, zi/h[x, y]−z,[x, z],[y, z], x^p, y^p, z^p−zi,

and all the arguments we used above are valid over Zp. So we first analyze the algebra over Zp and then induce up tok. So to reduce the amount of notation we also call the algebra Λ when considering it given over the fieldZ^p.

Any element in Λei can be written as a linear combination of{x^ry^sei}^p−1,p−1_r,s=0 . Given any non-zero ideal I in Λe_i, we can assume that x^p−1y^p−1e_i is in I by multiplying with a power of x from the left and a power of y from the right.

Using the identityxyei−iei =yxei, we obtain thaty^seix=xy^sei−siy^s−1ei for s > 1. Starting with multiplying the element x^p−1y^p−1ei with xfrom the right, we infer thatx^p−1y^p−2ei is inI. Inductively it follows thatx^p−1ei(and by similar arguments thaty^p−1e_i) is inI. Consequently,x^p−1y^se_iis inIfor all 16s6p−1.

By multiplyingx^p−1e_i from the left withy, we infer thatx^p−2e_i is inI. Continuing this way we obtain that e_i is in I and I = Λe_i. It follows that Λe_i is a simple ring for iin [1, . . . , p−1]. By Wedderburn-Artin Λe_i is isomorphic to M_n(D) for n ≥1 and a division ringD. Using that dim_ZpΛe_i =p², one can show that the only possibility for Λei is Mp(Z^p). Inducing up to k, we infer that the algebra of typeC10is isomorphic to k(Cp×Cp)⊕Mp(k)^p−1. In fact, a simple Λei-module S with basis B ={b0, b1, . . . , b_p−1} can be constructed in the following way. Let xbj=

(bj−1, j6= 0

0, j= 0 andybj=

(λj+1bj+1, j6=p−1

0, j=p−1 forλj+1=−(j+ 1)i. Then one can show that the vector space spanned byBis a simple Λe_i-module.

5. Selfinjective Nakayama algebras

In this section we find selfinjective Nakayama algebras occurring in the classifi- cation of the finite dimensional connected Hopf algebras of dimension p³. Here we denote byAen the quiver consisting ofn+ 1 vertices with one arrow starting in each vertex forming an oriented cycle. For a fieldkand an integern≥1 we denote by J the ideal inkeAn generated by the arrows.

Proposition 5.1. Let k be a field of characteristicp.

(a) The algebra of type B2 is isomorphic tokAep²−1/J^p, which is not a sym- metric algebra.

(b) The algebra of type C12is isomorphic to keAp−1/J^p2.

(c) The algebra of type C13 is isomorphic to a direct sum of p copies of the selfinjective Nakayama algebra keAp−1/J^p of dimension p².

(d) The algebra of type C14is isomorphic to the direct sum of keAp−1/J^p and p−1 copies ofM_p(k).

Proof. (a) The algebra of typeB2is given as

Λ =khx, y, zi/h[x, y]−y,[x, z],[y, z]−yf(x), x^p−x, y^p, z^p−zi, where f(x) =Pp−1

i=1(−1)ⁱ⁻¹(p−i)⁻¹xⁱ. We have that yx =xy−y,zx =xz and zy=yz−yf(x). This implies that any monomial in the variables{x, y, z}can be written as a linear combination of monomials of the form x^ry^sz^tfor r, sand t in {0,1, . . . , p−1}. Hence the set B={x^ry^sz^t}^p−1_r,s,t=1 spans Λ, which consists of p³ elements. Since the dimension of Λ isp³, the setBis a basis for Λ.

All the relations for the algebra are homogeneous iny. It follows from this that hyiⁱ =hyⁱi. Sincey^p= 0, it follows thathyi^p= (0). Furthermore,

Λ/hyi 'k[x]/hx^p−xi ⊗kk[z]/hz^p−zi 'k^p2. It follows that rad Λ =hyi.

The subalgebra of Λ generated byxandz is

khx, zi/hx^p−x,[x, z], z^p−zi 'k[x]/hx^p−xi ⊗_kk[z]/hz^p−zi,

which is commutative. Lete₀, . . . , e_p−1 be the orthogonal primitive idempotents of the algebra generated byxso thatxe_i=ie_i. Let alsog₀, . . . , g_p−1be the orthogonal

primitive idempotents of the algebra generated byz so thatzgj=jgj. Then these span a commutative semisimple subalgebra of dimensionp² of Λ, with orthogonal primitive idempotentseigj. These correspond to the vertices in the quiver we shall construct (as they are liftings of the primitive idempotents in Λ/rad Λ).

These orthogonal idempotents decompose Λ =⊕i,jΛe_ig_j into a direct sum ofp² left modules. The summand Λeigj has basis{y^reigj}^p−1_r=0} and the radical is gener- ated byye_ig_j (and Λe_ig_j is indecomposable as a left module). Hence rad Λ/rad²Λ is generated by{yeigj}^p−1,p−1_i,j=0 , and they correspond to the arrows in the quiver we shall construct.

We claim thatyeigj= (ei+1g_j−f(i))yeigj(eigj). Clearlyyeigj=yeigj(eigj).

We have the relationxy =y(x+ 1) which implies thatx·yei = (i+ 1)yei and hencex·(yeigj) = (i+ 1)yeigj. Using Remark 3.2 we obtain that

e_i+1ye_ig_j =ye_ig_j. We havezy=yz−y·f(x) andz, xcommute, so

z(yg_j) =yzg_j−yf(x)g_j =j·yg_j−yg_j·f(x) and therefore

z(yeigj) =z(ygjei) = (jygj)ei−(ygjf(x))ei

=jyg_je_i−yg_jf(i)e_i

= (j−f(i))·(yeigj)

and again by Remark 3.2, we have that ygjei = g_j−f(i)·ygjei. It follows that yeigj= (ei+1g_j−f(i))yeigj(eigj).

It follows from the above that ifQis a quiver withp²verticesvi,j =yeigj andp² arrowsai,j:vi,j→v_i+1,j−f(i), then Λ is a quotient ofkQ. To show thatQisAep²−1, we need to show that the orbit of (0,0) under the map (i, j)7→(i+ 1, j−f(i)) is all ofZp×Zp. Afterkpsteps one gets

(1) (kp,−f(1)−f(2)− · · · −f(kp−1)) To compute this, note that for 1 ≤ m ≤ p−1 one has Pp−1

j=1j^m ≡ −1 modp if m=p−1 and is zero otherwise (To see this, one can for example use [12, Lemma 4.3]). Then one gets that (1) is equal to (0,−k) and it follows that the orbit of (0,0) must have sizep² as required. Hence the quiverQisAep²−1 with the orientation as an oriented cycle. Since the Loewy length of the indecomposable projectives Λeigj

are all equal, the relation ideal isJ^p.

A Nakayama algebrakAen/J^tis symmetric if and only ifn+ 1 dividest−1 by [31, Corollary IV.6.16]. Sincep²-p−1 for every primep, all the algebras of type B2are not symmetric.

Remark. Note that the fieldk can be arbitrary of characteristicp.

(b) The algebra of typeC12is given as

Λ =khx, y, zi/h[x, y]−y,[x, z],[y, z], x^p−x, y^p−z, z^pi.

Any element in Λ can be written as a linear combination of elements in the set {x^ry^s}^p−1,p_r,s=0²⁻¹. Let a = hyi in Λ. Since the relations are homogeneous in y, it follows from the above thatais a nilpotent ideal in Λ. Furthermore it is easy to see that Λ/a 'k[x]/hx^p−xi 'k^p. We infer that rad Λ =a. By the above comments

we have that rad Λ has basis {x^ry^s}^p−1,p_r=0,s>²⁻¹₁ (dimension p³−p) and rad²Λ has basis{x^ry^s}^p−1,p_r=0,s>²⁻¹₂ (dimension p³−2p). The rad Λ/rad²Λ has a basis given by the residue classes of the elements {x^ry}^p−1_r=0. Hence Λ is isomorphic to a quotient of a path algebrakQoverk, whereQhaspvertices

( vα=

Q

β∈Zp\{α}(x−β) Q

β∈Zp\{α}(α−β) )

α∈Zp

andparrows given by{vαy}α∈Zp. By part (2) of Remark 3.2 we have that vαy=yv_α−1,

so thatv_α+1y is an arrow from vertexv_αto vertexv_α+1 for allαin Zp. It follows that Λ'keAp−1/J^p2.

(c) The algebra of typeC13is given as

Λ =khx, y, zi/h[x, y]−y,[x, z],[y, z], x^p−x, y^p, z^p−zi.

This is isomorphic to the tensor product

Λ∼=khx, yi/h[x, y]−y, x^p−x, y^pi ⊗kk[z]/hz^p−zi.

The second tensor factor is semisimple and isomorphic tok^p, hence Λ is the direct sum of p copies of B = khx, yi/h[x, y]−y, x^p−x, y^pi. Let a = hyi in B. Since the relations are homogeneous iny, it follows from the above thatais a nilpotent ideal inB. Furthermore it is easy to see that B/a'k[x]/hx^p−xi 'k^p. We infer that radB = a. The subalgebra generated by x gives rise to a complete set of orthogonal idempotentsei withxei=iei (see Remark 3.2), so thatB=⊕^p−1_i=0Bei. ThenBeihas basis{y^sei}^p−1_s=0 and its radical is generated byyei. We havex(yei) = y(x+ 1)e_i = (i+ 1)ye_i, hence by Remark 3.2 ye_i =e_i+1ye_i andye_i gives an arrow e_i →e_i+1. It follows from this that B is isomorphic to the algebra keAp−1/J^p of dimensionp². By the above, this completes the proof of (c).

(d) The algebra of typeC14is given as

Λ =khx, y, zi/h[x, y]−y,[x, z],[y, z], x^p−x, y^p−z, z^p−zi.

It is clear from the relations that the subalgebra generated byzis in the center of Λ and is giving rise to porthogonal central idempotents {ei}^p−1_i=0 with zei = iei. Then Λ = ⊕^p−1_i=0Λei. We claim that Λe0 is isomorphic tokAep−1/J^p, while Λei is isomorphic to Mp(k) fori ≥1. To do this, we first consider the algebra Λ as an algebra over the prime fieldZp. The idempotents constructed above are given over this field. By abuse of notation we still denote the algebra by Λ when considered over the fieldk⁰ =Zp.

The subalgebra generated byxgives rise to a complete set of orthogonal idem- potents {fi}^p−1_i=1 with xfi = ifi. This implies that {y^reifs}^p−1,p−1_r,s=0 is basis for Λei. Let i≥1, then we want to show that Λei is a simple ring. Let I be a non- zero ideal in Λei with a non-zero elementm=Pp−1,p−1

r,s=0 αr,sy^reifs. Assume that αr₀,s₀ 6= 0. Thenmfs₀ =Pp−1

r=0αr,s₀y^reifs₀ is in I. Sincefjy =yf_j−1, it follows that fr₀+s₀mfs₀ =αr₀,s₀y^r0eifs₀ is inI and consequentlyy^r0eifs₀ is in I. Multi- plying this last element from the right with powers ofy, we obtain thaty^rseifs is in I for some rs for s = 0,1, . . . , p−1. Multiplying from the left with y^p−rs we obtain that e_if_s is inI for s = 0,1, . . . , p−1, hence e_i is inI and I = Λe_i and Λe_i is a simple ring. Furthermore, Λe_i ' M_n(D) for some n ≥ 1 and a division

ring D. A simple module over Λei then has dimension ndimk⁰D over k⁰. Since Λei is non-commutative and D is commutative, n > 1. Using similar arguments as above one can show that St = k⁰{y^reift}^p−1_r=0 is a simple Λei-module. Hence ndimk⁰D =p. It follows thatn=pandD =k⁰, and therefore when inducing up to the fieldk we get that Λei'Mp(k) fori≥1.

For Λe0, let ϕ:khx, yi →Λe0 be induced by the inclusionkhx, yi,→khx, y, zi.

Then{[x, y]−y, x^p−x, y^p} are contained in Kerϕ. Since

dim_kΛe₀= dim_kkhx, yi/h[x, y]−y, x^p−x, y^pi=p²,

it follows that Λe0'khx, yi/h[x, y]−y, x^p−x, y^pi, which we consider as an iden- tification. The ideal generated by y is the radical of Λe0. View the primitive orthogonal idempotents {f_i}^p−1_i=0 as elements in Λe₀. Then {y^rf_s}^p−1,p−1_r,s=0 is a k- basis for Λe0. Therefore {fs}^p−1_s=0 is a k-basis for Λe0/rad Λe0 and {yfs}^p−1_s=0 is a k-basis for rad Λe0/rad²Λe0. We let these sets define the vertices and the arrows in a quiver Q, which is Aep−1. All the indecomposable projective modules have the same Loewy length and Λe₀ has dimension p², so that Λe₀ is isomorphic to kAep−1/J^p.

It follows from the above that Λ is isomorphic to direct sum of keAp−1/J^p and

p−1 copies ofMp(k).

6. Enveloping algebra of restricted Lie algebras

In this section we find enveloping algebras of restricted Lie algebras occurring in the classification of the finite dimensional connected Hopf algebras of dimension p³.

First we give a quick review of the definition of the enveloping algebra of a re- stricted Lie algebraL. A restricted Lie algebraLis a (finite dimensional) Lie algebra over a fieldkof characteristicpwith Lie bracket [−,−] :L×L→Land ap-operation (−)^[p]:L→L(for details see [19]). Suppose Lhas ak-basis {x1, x2, . . . , xn}. By abuse of notation we letkhx1, x2, . . . , xnibe the free algebra of the indeterminants {x1, x₂, . . . , x_n}. Then the universial enveloping algebraU^[p](L) ofLis given by

U^[p](L) =khx1, x2, . . . , xni/h{xixj−xjxi−[xi, xj]}^n,n_i,j=1,{x^p_i −x^[p]_i }ⁿ_i=1i.

Next we give the description of the algebras C5, C6 and C15 as enveloping algebras of restricted Lie algebras all of which are 3-dimensional.

Proposition 6.1. (a) LetLbe the3-dimensional restrictedp-nilpotent Lie al- gebra with basis {x1, x₂, x₃}, the only non-zero bracket being [x₁, x₂] =x₃ and the p-operation given by x^[p]_i = 0 for i = 1,2,3 for p > 2. Then the algebra of type C5is isomorphic to the enveloping algebra of the restricted Lie algebra ofL.

(b) LetLbe the3-dimensional restricted Lie algebra with basis{x1, x2, x3}with the only non-zero bracket being [x1, x2] =x3 and the p-operation given by x^[p]₁ =x3 and x^[p]_i = 0 for i= 2,3, given in [29, Theorem 2.1 (3/2) (b)].

Then the algebra of type C6is isomorphic to the enveloping algebra of the restricted Lie algebra of L.

(c) Let Lbe the restricted Lie algebra of type sl2(k). The algebra of type C15 is isomorphic to the enveloping algebra of the restricted Lie algebra ofL.

Proof. (a) The algebra of typeC5is given by

Λ =khx, y, zi/h[x, y]−z,[x, z],[y, z], x^p, y^p, z^pi.

LetL be the 3-dimensional Lie algebra with basis{x1, x₂, x₃} with the only non- zero bracket being [x₁, x₂] = x₃. We want to show that we can choose a zero p-operation to obtain a restricted Lie algebra.

We have that

adx₁= [−, x1] =_{0 0 0}

0 0 0 0−1 0

and (adx₁)²= 0, adx2= [−, x2] =_{0 0 0}

0 0 0 1 0 0

and (adx2)²= 0, adx₃= [−, x3] =_{0 0 0}

0 0 0 0 0 0

and adx₃= 0.

By [19, Chapter V, Theorem 11] there is a p-operation (−)^[p]:L → L such that (xi)^[p] = 0 for i = 1,2,3. This shows that L is the 3-dimensional restricted Lie algebra, and therefore Λ is the enveloping algebra of the 3-dimensional restricted Lie algebraL.

(b) The algebra of typeC6is given by

Λ =khx, y, zi/h[x, y]−z,[x, z],[y, z], x^p−z, y^p, z^pi

forp >2. Recall the 3-dimensional restricted Lie algebraLwith basis{x1, x₂, x₃} with the only non-zero bracket being [x₁, x₂] = x₃ and the p-operation given by x^[p]₁ =x3 andx^[p]_i = 0 fori= 2,3, given in [29, Theorem 2.1 (3/2) (b)]. Then the algebra Λ is isomorphic to the restricted enveloping algebra of the 3-dimensional restricted Lie algebraL.

(c) The algebra of typeC15is given by

Λ =khx, y, zi/h[x, y]−z,[x, z]−x,[y, z] +y, x^p, y^p, z^p−zi

forp >2. We shall prove that this is the enveloping algebra of the 3-dimensional restricted Lie of typesl2(k).

Let L= sl2(k) = ke+⊕kh⊕ke−, where the standard presentation usually is give as

[h, e±] =±2e±, [e+, e−] =h.

By letting

h⁰=−1

2h, e⁰₊=αe₊, e⁰₋ =βe₋, we obtain the equations

[h⁰, e⁰₊] =−e₊, [h⁰, e⁰₋] =e⁰₋, [e⁰₊, e⁰₋] =h⁰, wheneverαβ=−¹₂ ink.

Now we investigate if there exists ap-operation onL. We have that ade⁰₊= [−, e⁰₊] =0 0 −1

0 0 0 0−1 0

and (ade⁰₊)³= 0, ade⁰₋= [−, e⁰₋] =_{0 0 0}

0 0 1 1 0 0

and (ade⁰₋)³= 0, adh⁰= [−, h⁰] =_{1 0 0}

0−1 0 0 0 0

and (adh⁰)^p=₁^p _{0 0}

0 (−1)^p0 0 0 0

= adh⁰,

sincepis odd. By [19, Chapter V, Theorem 11] there is ap-operation (−)^[p]:L→L such that

(e⁰_±)^[p]= 0 and (h⁰)^[p] =h⁰.

This shows that L is the 3-dimensional restricted Lie algebra of type sl2(k), and therefore Λ is the enveloping algebra of the 3-dimensional restricted Lie algebra of

typesl2(k).

7. Coverings of local algebras

In this section we find algebras in the classification of the finite dimensional connected Hopf algebras of dimension p³ which are coverings of local algebras by a cyclic group. We start off by recalling the notion of a covering of a path algebra (for details see [13]).

Let Λ =kQ/I be an admissible quotient of the path algebrakQ over a fieldk.

LetGbe a (finite) group, and letw:Q1→Gbe a weight function, that is, just a functionw:Q1→G. This is extended to a weight function on all non-trivial paths by defining theweight of a path p=a_na_n−1· · ·a₂a₁in Qto be

n

Y

i=1

w(ai) =w(a1)w(a2)· · ·w(a_n−1)w(an),

and for a trivial path p the weight is w(p) = e, the identity in G. Assume that the relations inIare homogeneous with respect to the weight. With these data we define a coveringQ(w) ofe Qwith the vertex set

Q(w)e ₀={(v, g)|v∈Q₀, g∈G}

and the arrow set

Q(w)e 1={(a, g) : (o(a), g)→(t(a), gw(a))|a∈Q1, g∈G},

whereo(a) andt(a) denote the origin and the terminus of the arrowa, respectively.

Define a mapπ:Q(w)e →Qinduced from letting π(v, g) =v and π(a, g) =a

for all vertices v in Q₀, all arrows a in Q₁ and all g in G. It is straightforward to see that given an element g in G and a path p in Q, there is a unique path pe_g= (p, g) : (o(p), g)→(t(p), gw(p)) inQ(w) such thate π(pe_g) =p. This we extend to a lifting from kQ to kQ(w). Hence, givene g in G and a weight homogeneous relation σ in I, there is a unique lifting eσ_g of σ to kQ(w). This is a uniforme element, that is, all paths occuring in eσ_g start in the same vertex and end in the same vertex. Furthermore, ifI⁰ is the ideal generated by the liftings of a minimal generating set of I andI(w) is the set of liftings of elements ine I, thenI⁰ =I(w)e and I(w) is an ideal ine kQ(w). In particular, there is a well-defined algebra mape π:kQ(w)/ee I(w)→kQ/I.

We give the following example which will occur in the main result of this section.

Example 7.1. Let

Λ =k(^y ::1dd ^z)/hy^p, yz−zy, z^pi,

and define w: {y, z} → Cp = hgi by letting w(y) = g and w(z) = e. Let Q: ^y ::1dd ^z and I = hy^p, yz −zy, z^pi. Then the covering kQ(w)/e I(w) ise

given by the quiver

(1, e)

(y,e)

//

(z,e)

(1, g)

(y,g) ##

(z,g)

(1, g^p−1)

(y,g^p−1)

::

(z,g^p−1)

(1, g²)

(y,g²)

(z,g²)

(1, g^p−2)

(y,g^p−2)

OO

(z,g^p−2)

GG (1, g³)

yy ^(z,g3)GG

ff

and relations

(y, g^i+p−1)· · ·(y, gⁱ⁺²)(y, gⁱ⁺¹)(y, gⁱ), (z, gⁱ)^p,

(y, gⁱ)(z, gⁱ)−(z, gⁱ⁺¹)(y, gⁱ), for alliin{0,1, . . . , p−1}.

Now we are ready to give the algebras in the classification of finite dimensional connected Hopf algebras of dimensionp³ which are coverings of a local algebra by a cyclic group.

Proposition 7.2. Let k be a field of characteristicp. Let

Λ₁=khy, zi/hy^p, yz−zy, z^pi and Λ₂=khy, zi/hy^p, yz−zy−y², z^pi.

Consider the groupG=hσi=C_p.

(a) The algebras of type B1and C11are isomorphic, and they are isomorphic to a covering of the local algebraΛ1with the weight functionw:{y, z} →G given byw(y) =σandw(z) =e.

(b) The algebra of type B3is a covering of the local algebraΛ2 with the weight function w(y) =w(z) =σ.

(c) Assume that k contains the field of order p² if p > 2. The algebra of type C16is a covering of the local algebraΛ1 with a weight functionw of the form w(y) = σ⁻¹ and w(z) =σ^−a for some a (precisely, a such that λ⁻¹ =aλwhen λ is the defining parameter occuring in an algebra of type C16).

Proof. (a) The algebra of typeB1andC11are given as

Λ =khx, y, zi/h[x, y]−y,[x, z],[y, z], x^p−x, y^p, z^pi.

Any element in Λ can be written as a linear combination of elements in the set {x^ry^sz^t}p−1,p−1,p−1

r,s,t=0 . Let a =hy, zi in Λ. Since the relations are homogeneous in y andz, it follows from the above that ais a nilpotent ideal in Λ. Furthermore it is easy to see that Λ/a ' k[x]/hx^p−xi ' k^p. We infer that rad Λ = a. By the above comments we have that rad²Λ has basis {x^ry^sz^t}p−1,p−1,p−1

r=0,s+t>2 . The factor rad Λ/rad²Λ has a basis given by the residue classes of the elements{x^ry, x^rz}^p−1_r=0.

Hence Λ is isomorphic to a quotient of a path algebra kQ overk, where Qhasp vertices

( vα=

Q

β∈Zp\{α}(x−β) Q

β∈Zp\{α}(α−β) )

α∈Zp

and 2parrows spanned by {x^ry, x^rz}^p−1_r=0. The linear span of {xⁱ}_i∈Zp is equal to the linear span of {v_α}_α∈Zp. Hence, a basis for rad Λ/rad²Λ is {v_αy, v_αz}_α∈Zp. Since z is in Z(Λ), we have thatv_αz =zv_α. By Remark 3.2 (2) the elementv_αy gives rise to an arrow from vertexv_αto vertexv_α+1, and the elementv_αz to a loop at vertexvα. Hence the quiver Qof Λ is

v0 y₀ //

z₀

v1

y1

z₁

v_p−1

yp−1

<<

zp−1

// v2

y₂

z₂

zz

vp−2 yp−2

OO

zp−2

// v3

}}

z3

zz

cc

The relations are{yi+p−1· · ·yi+2yi+1yi, z_i^p, yizi−zi+1yi}^p−1_i=0. Hence Λ'kQ/h{y_i+p−1· · ·yi+2yi+1yi, z^p_i, yizi−zi+1yi}^p−1_i=0i.

Therefore Λ is isomorphic to the covering of the local algebra Λ1 with the weight functionw:{y, z} →Cp =hσigiven byw(y) =σand w(z) =e(see Example 7.1 takingσ=g).

(b) The algebra of typeB3is given by

Λ =khx, y, zi/h[x, y]−y,[x, z]−z,[y, z]−y², x^p−x, y^p, z^pi.

The relations [x, y]−y, [x, z]−z and [y, z]−y² imply that x’s can be moved across y’s, x’s can be moved across z’s and that z’s can be moved across y’s.

From this we obtain that hy, zi is a nilpotent ideal in Λ. It is easy to see that Λ/hy, zi ' k[x]/hx^p−xi ' k^p. Hence hy, zi is the radical in Λ, and Λ ' kQ/I, whereQis a quiver withpvertices given by

( v_α=

Q

β∈Zp\{α}(x−β) Q

β∈Zp\{α}(α−β) )

α∈Zp

.

The arrows are given byv_αyv_β andv_αzv_β which are non-zero when αandβ runs throughZp. Using Remark 3.2 we get thatv_αy=yv_α−1andv_αz=zv_α−1for allα

inZp. Hence the quiver is

v0 y₀ //

z₀ //v1 y₁

z1

v_p−1

yp−1

<<

zp−1

<<

v₂

y2

z2

vp−2

yp−2

OO

zp−2

OO

v3

y3

}}

z3

}}

cccc

The relations are

{yi+1zi−zi+1yi−yi+1yi, y_i+p−1y_i+p−2· · ·yi+1yi, z_i+p−1z_i+p−2· · ·zi+1zi}_i∈Zp. Therefore Λ is a covering of the local algebra Λ2 with the weight function w(y) = w(z) =σ.

(c) The algebra of typeC16is given as

Λ =C(λ, δ) =khx, y, zi/h[x, y],[x, z]−λx,[y, z]−λ⁻¹y, x^p, y^p, z^p−δzi for someλ∈k^× such that δ=λ^p−1 =±1. Any element in Λ can be written as a linear combination of elements in the set{x^ry^sz^t}p−1,p−1,p−1

r,s,t=0 ofp³ elements, hence it is ak-basis for Λ. Leta =hx, yiin Λ. Since the relations are homogeneous in x and y, it follows from the above that a is a nilpotent ideal in Λ. Furthermore it is easy to see that Λ/a ' k[z]/hz^p−δzi ' k^p (note that λ lies in the field of order p² if δ = −1 6= 1). We infer that rad Λ = a. By the above comments we have that rad Λ has basis {x^ry^sz^t}p−1,p−1,p−1

r+s>1,t=0 (dimension p³−p) and rad²Λ has basis{x^ry^sz^t}p−1,p−1,p−1

r+s>2,t=0 (dimensionp³−3p). The vector space rad Λ/rad²Λ has a basis given by the residue classes of the elements{xz^t, yz^t}^p−1_t=0.

Consider the mapϕ:k[u]→Λ given byϕ(u) =z. Then Kerϕ=hu^p−δui, so thatk[z]/hz^p−δziis a subalgebra of Λ. This is a commutative semisimple algebra, since the polynomialz^p−δz is separable.

Note that the set of roots ofz^p−δzis closed under addition and that 0 is a root, so it is an additive subgroup of orderpof the field. We takeλas in the definition of the algebra. This is a non-zero root of the polynomial z^p−δz, and with this, the set of roots is precisely

{0, λ,2λ, . . . ,(p−1)λ}.

Sinceλ⁻¹ also is a root, there is a unique integerawith 1≤a≤p−1 such that λ⁻¹=aλ.

Next we find a set of complete orthogonal idempotents in Λ. We have fixed the roots ofz^p−δzabove. With this, we have a set of complete orthogonal idempotents {e0, e₁, . . . , e_p−1}ofk[z]/hz^p−δzisuch that

zei =iλ·ei.

(for a formula take a variation of Remark 3.2, but we don’t need it). So Λ =

⊕^p−1_i=0Λei as a left module.

The set{x^uy^ve_i}^p−1,p−1_u,v=0 a k-basis for Λe_i (recall thatx, ycommute). In partic- ularxe_i andye_i are both non-zero and they generate the radical of Λe_i.

We claim thatxei andyeiare eigenvectors for left multiplication withz: To see this, we use the relations

zx=xz−λx and zy=yz−λ⁻¹y.

Therefore

zxei = (xz−λx)ei=x(iλei)−λ(xei) = (i−1)λ(xei)

and the eigenvalue is (i−1)λ. This implies thatxei=e_i−1xei, by the Remark 3.2.

Similarly

zyei= (yz−λ⁻¹y)ei= (iλ−aλ)yei

and the eigenvalue is (i−a)λ. Soyei=e_i−ayei.

As we have found a basis for the algebra modulo its radical, and the radical mod- ulo the radical square, we can find the quiverQof Λ. The vertices ofQcorrespond to{e_i}^p−1_i=1, and we take as arrowsα_i=xe_iandβ_i=ye_istarting ati. By the above, αiends at vertexi−1 andβi ends at vertexi−a, whereλ⁻¹=aλ. It follows from the above thatα_i−(p−1)· · ·α_i−1αi,β_i−(p−1)a· · ·β_i−aβiandβ_i−1αi−α_i−aβiare rela- tions. It is easy to see thatkQ/h{α_i−(p−1)· · ·α_i−1αi, β_i−(p−1)a· · ·β_i−aβi, β_i−1αi− α_i−aβi}^p−1_i=0ihas dimension p³, hence

Λ'kQ/h{α_i−(p−1)· · ·αi−1αi, β_i−(p−1)a· · ·βi−aβi, βi−1αi−αi−aβi}^p−1_i=0i.

Therefore Λ is isomorphic to the covering of the local algebra Λ1 with weight func- tionwof the formw(y) =σ⁻¹ andw(z) =σ^−a forCp=hσi.

8. Other local algebras

In this section we discuss the algebras which we cannot classify as we have done with all the other algebras. The only class of algebras left are the algebras of type A5. Here it is only in characteristic 2 that we can identify this algebra.

Proposition 8.1. Forp= 2 the algebra of type A5is isomorphic to the semidi- hedral algebra of dimension 8 labelled as Alg III.1 (d)in [9, page 298].

Proof. Forp= 2 the algebra of typeA5is given as

Λ =khx, y, zi/hx², y²,[x, y],[x, z],[y, z]−x, z²+xyi.

We observe thatxis central and not a generator. We substitutex= [y, z], this is central, and the other relations translate to

[y, z]², y², z²+ [y, z]y.

Noting that char(k) = 2, the last two relations mean that in the algebra y² = 0 and z² =yzy. With these, the first relation is equivalent to (yz)² = (zy)² in the algebra. Moreover ify²= 0 andz²=yzy, then it follows that [y, z] is central. We get that

Λ'khy, zi/h(yz)²−(zy)², y², z²−yzyi, and these relations imply that (zy)²z= 0.