On classification of finite-dimensional semisimple Hopf algebras
aa r X i v : . [ m a t h . R A ] M a r ON CLASSIFICATION OF FINITE-DIMENSIONALSEMISIMPLE HOPF ALGEBRAS.
LEONID KROP
Abstract.
We develop a mechanism for classication of isomor-phism types of non-trivial semisimple Hopf algebras whose groupof grouplikes G ( H ) is abelian of prime index p which is the small-est prime divisor of | G ( H ) | . We describe structure of the secondcohomology group of extensions of k C p by k G where C p is a cyclicgroup of order p and G a finite abelian group. We carry out anexplicit classification for Hopf algebras of this kind of dimension p for any odd prime p . The ground field is algebraically closed ofcharacteristic 0. Keywords
Hopf algebras, Abelian extensions, Crossed products,Cohomology Groups
Mathematics Subject Classification (2000)
Introduction
We work with Hopf algebras H over an algebraically closed field k of characteristic 0. We let G ( H ) denote the group of grouplikes of H .By the freeness theorem [26] dim H = m dim k G ( H ) for an integer m .We say that a prime p is small relative to a finite group G if p is theleast prime divisor of | G | . Unless stated otherwise, we assume that H is semisimple of dimension p | G ( H ) | for a prime number p , G ( H ) isabelian and p is small relative to G ( H ). For brevity, we name such Hopfalgebras almost abelian . As usual, a finite-dimensional Hopf algebra iscalled trivial if it or its dual is a group algebra. The goal of the paperis to classify semisimple, non-trivial almost abelian Hopf algebras.We introduce more notation. We denote by C p a cyclic group oforder p and by k G and k G the group algebra of G over k and its dual,respectively. We will write Ext( k C p , k G ) for the set of all equivalenceclasses of extensions of k C p by k G .The problem just stated reduces to that of classifying abelian exten-sions of a special kind. For by a result of [13], k G ( H ) is a normalsubHopf algebra, a fact that combined with the theorem of Kac-Zhu Date : 3/20/15. [9, 31] yields that H lies in Ext( k C p , k G ( H ) ). We will refer to elementsof Ext( k C p , k G ( H ) ) as Hopf algebras and extensions interchangeably.Our main concern becomes to understand the set of isomorphismtypes in Ext( k C p , k G ) where G is a finite abelian group and p is smallrelative to G . In general, that is for arbitrary finite groups F, G , there isno systematic procedure by which isomorphism classes of Hopf algebrasthat are extensions of k F by k G can be found. One purpose of thearticle is to fill this gap for the case in hand. In order to formulatethe statement we will require a few more notions. We write A p forthe group Aut( C p ) of automorphisms of C p . An action ⊳ of C p on G is a representation C p → Aut( G ). Let R = { ⊳ } denote the set of allrepresentations. The group Aut( G ) acts naturally on R by conjugationsplitting R into the union of sets eq( ⊳ ) of representations equivalent to ⊳ . In turn, the group A p also acts on R via ⊳ ⊳ α where, for every α ∈ A p , a ⊳ α x = a ⊳ α ( x ) , a ∈ G, x ∈ C p . This action is passed on thesets eq( ⊳ ) via eq( ⊳ ) α = eq( ⊳ α ) giving rise to classes of representations[ ⊳ ] = ∪ α eq( ⊳ ) α . We denote the stabilizer of eq( ⊳ ) by C ( ⊳ ).The splitting of R into the union of [ ⊳ ] induces a splitting ofExt( k C p , k G ). Namely, for every [ ⊳ ] we define Ext [ ⊳ ] ( k C p , k G ) as theset of all equivalence classes of extensions whose C p -action belongs to[ ⊳ ], and then we have Ext( k C p , k G ) = [ [ ⊳ ] Ext [ ⊳ ] ( k C p , k G ). It suffices toclassify isomorphism types in each Ext [ ⊳ ] ( k C p , k G ).To this end we bring in the second degree Hopf cohomology group[1, 23] denoted by H c ( k C p , k G , ⊳ ) after the work of M. Mastnak [16].We aim at constructing a subgroup G ( ⊳ ) of Aut( G ) and its action on H c ( k C p , k G , ⊳ ) compatible with isomorphism types of extensions in thesense that for any τ, τ ′ ∈ H c ( k C p , k G , ⊳ ) ( τ, ⊳ ) and ( τ ′ , ⊳ ) give rise toisomorphic Hopf algebras iff τ and τ ′ lie on the same orbit of G ( ⊳ ).To begin with, we introduce the group A ( ⊳ ) of all C p -automorphismsof ( G, ⊳ ). For every α ∈ C ( ⊳ ) we fix a C p -isomorphism λ α : ( G, ⊳ ) ∼ → ( G, ⊳ α ). We set G ( ⊳ ) to be the subgroup of Aut( G ) generated by A ( ⊳ )and the set { λ α | α ∈ C ( ⊳ ) } if ⊳ is nontrivial, and G ( ⊳ ) = Aut( G ) × A p ,otherwise. H c ( k C p , k G , ⊳ ) contains a distinguished subgroup H ( k C p , k G , ⊳ ) of(the images of) symmetric Hopf 2-cocycles parametrizing cocommu-tative extensions. Let us write H c ( k C p , k G , ⊳ ) / G ( ⊳ ) nc for the set of G ( ⊳ )-orbits not contained in H ( k C p , k G , ⊳ ). Reciprocally, we let nc Ext [ ⊳ ] ( k C p , k G ) / ∼ = stand for the set of isotypes of noncocommutativeextensions, and we put cl( H ) for the isomorphism class of H . LASSIFICATION OF HOPF ALGEBRAS 3
The principal result of the paper states: There is a bijection H c ( k C p , k G , ⊳ ) / G ( ⊳ ) nc ⇄ nc Ext [ ⊳ ] ( k C p , k G ) / ∼ =given by ( ⊳, τ ) G 7→ cl( H ( ⊳, τ )).Our next concern lies with structure of H c ( k C p , k G , ⊳ ). We want tofind a form of H c ( k C p , k G , ⊳ ) with good computational properties. Weneed several notions. Let H ( C p , b G, • ) be the second degree cohomol-ogy group of extensions of C p over b G , where b G is the dual group withthe C p -action • dual to ⊳ , and H ( G, k • ) be the Schur multiplier of G .There is a mapping N [15] acting on a Z C p -module M by N ( m ) = φ p .m where φ p is the p th cyclotomic polynomial. We denote the kernel of N in M by M N . The main result in the strong form states that for any odd p there is a C p -isomorphism(0.1) H c ( k C p , k G , ⊳ ) ≃ H ( C p , b G, • ) × H N ( G, k • )We remark that this isomorphism can be seen as the Hopf cohomologyversion of the Baer’s formula for cohomology of central extensions ofa group G by C p [2, p.34]. On the other hand its main utility liesin the fact that both factors in the right-hand side of it are nicelycomputable in any set of generators for G thanks to the classical iso-morphisms H ( C p , b G, • ) ≃ b G C p /N ( b G ) and H ( G, k • ) ≃ Alt( G ) [15]and [2], where Alt( G ) is the group of all bilinear, alternate mapping G × G → k • . We let X ( G, ⊳ ) denote the right-hand side of (0.1) andcall it the classifying group of Ext [ ⊳ ] ( k C p , k G ). For every odd p X ( G, ⊳ )acquires component-wise G ( ⊳ )-module structure via transport of actionalong the isomorphism (0.1). The new, most useful, formulation of themain theorem asserts that there is a bijection X ( G, ⊳ ) / G ( ⊳ ) nc ⇄ nc Ext [ ⊳ ] ( k C p , k G ) / ∼ =where X ( G, ⊳ ) / G ( ⊳ ) nc denotes the set orbits of G ( ⊳ ) not containedin b G C p /N ( b G ).For p = 2 less is known. We show only that the isomorphism (0.1)holds for elementary 2-groups, though it is not, in general, an A ( ⊳ )-isomorphism. The previous related works consist of the fundamental result of D.Stefan [29] to which this article provides concrete examples, and vari-ous classification theorems. The papers [8, 9, 31, 18, 19, 20, 21] treata number of instances of almost abelian Hopf algebras. Namely, it isknown that semisimple Hopf algebras of dimension p and p are trivial,nontrivial Hopf algebras of dimension p are almost abelian and the See Appendix 2
LEONID KROP number of their isomorphism types equals p + 1 for every odd p , whilethere is a unique 8-dimensional nontrivial Hopf algebra, and for any odd p Ext( k C , k Z p × Z p ) contains a unique Hopf algebra up to isomorphism.Information on the p -dimensional semisimple Hopf algebras is limitedto p = 2 and consists of a complete classification of 16-dimensionalsemisimple Hopf algebras and almost abelian Hopf algebras H of di-mension 2 n +1 with G ( H ) = Z n − × Z both due to Y. Kashina [10, 11].The paper is organized in six sections. In Section 1 we review thenecessary facts of the theory of abelian extension. Sections 2 and 3are devoted to the main results. We prove the structure theorem forthe groups H c ( k C p , k G , ⊳ ) and the isomorphism and bijection theoremsin Sections 3 and 4, respectively. Section 5 contains applications toclassification of Hopf algebras of dimensions p , p and an example ofa non self-dual semisimple Hopf algebra of dimension p . Howeverthe bulk of this Section is devoted to finding the exact number ofnontrivial almost abelian Hopf algebras of dimension p ; we show thatthere are 5 p + 23 distinct almost abelian Hopf algebras, if p >
3, and33, otherwise. In the course of the proof we extend the contents of[11] from p = 2 to an arbitrary prime. In the last section we revisita theorem of Kac-Masuoka on 8-dimensional Hopf algebras and give ageneralization of a result of A. Masuoka [21].0.1. Notation and Convention.
We adhere to the notation of [24] onHopf algeras and to [1, 16, 23] for the theory of Hopf algebra extensions.In addition to notation in the Introduction we will use the following. A • the group of units of a commutative ring A .Γ n direct product of n copies of group Γ.Fun(Γ , A • ) the group of all functions from Γ to A • with pointwisemultiplication. Z (Γ , A • , • )) , B (Γ , A • , • ) and H (Γ , A • , • ) are the groups of 2-cocycles,2-coboundaries, and the second degree cohomology group of Γ over A • with respect to an action • of Γ on A by ring automorphisms. δ Γ the differential of the standard cochain complex for cohomology ofthe triple (Γ , A • , • ) [15, IV.5]. Z n cyclic group of order n additively written.In order to simplify notation we will often use the same symbol foran element of Z (Γ , A • , ⊳ ) and its image in H (Γ , A • , ⊳ ). The contextmakes the intended meaning clear.Throughout the paper we treat the terms Γ-module, Γ-linear, etcas synonymous to Z Γ-module, Z Γ-linear, etc. We use the abbreviatedterm isotypes for isomophism types.
LASSIFICATION OF HOPF ALGEBRAS 5 Abelian extensions
In this paper we are concerned with finite-dimensional Hopf algebrasover k . Let F and G be finite groups. A Hopf algebra H is an extensionof k F by k G if there is a sequence of Hopf mappings(1.1) k G ι H π ։ k F with ι monomorphism, π epimorphism, ι ( k G ) normal in H and Ker π = ι ( k G ) + H . We give a synopsis of basic results on abelian extensionsrefering to [23] for details.An abelian extension is characterized by a quadruple D = { σ, τ, ⊳, ⊲ } called a datum for H and we write H = H ( D ). This comes aboutfrom a crossed product splitting of H and H ∗ . For by [25], or generaltheorems [28, 2.4], [17, 3.5] H is a crossed product of k F over k G .Since H ∗ is an extension of k G by k F , see [5, 4.1] or [1, 3.3.1], H ∗ isa crossed product of k G over k F . Thus there are two module algebraactions · : k F ⊗ k G → k G and · : k F ⊗ k G → k F and a pair of group2-cocycles ( σ, τ ) ∈ Z ( F, ( k G ) • ) × Z ( G, ( k F ) • ) giving H and H ∗ analgebra structure with the multiplication( f x )( f ′ y ) = f ( x.f ′ ) σ ( x, y )) xy, x, y ∈ F, f, f ′ ∈ k G (1.2) ( aφ )( bφ ′ ) = ( ab ) τ ( a, b )( φ.b ) φ ′ , a, b ∈ G, φ, φ ′ ∈ k F (1.3)The standard identification k G ∼ = ( k G ) ∗ via a ev( a ) : f f ( a )allows us to define a right action ⊳ of k F on k G by the transpose ofaction · , viz. h a ⊳ x, f i = h ev( a ) , x.f i . That is(1.4) ( a ⊳ x )( f ) := f ( a ⊳ x ) = ( x.f )( a ) , for all f ∈ k G , a ∈ G, x ∈ F. Likewise we obtain an action ⊲ of k G on k F . In fact both ⊳ and ⊲ are permutation actions on G and F , respectively. In the dual bases { p a | a ∈ G } and { p x | x ∈ F } for k G and k F the two pairs of actions arerelated by the formulas x.p a = p a⊳x − (1.5) p x .a = p a − ⊲x . (1.6)We fuse both actions into the definition of a product on F × G via(1.7) ( xa )( yb ) = x ( a ⊲ y )( a ⊳ y ) b We use the standard notation
F ⊲⊳ G for the set F × G endowed withmultiplication (1.7). A short independent proof is given in the Appendix 1
LEONID KROP
Dualizing multiplication (1.3) endowes H with a coalgebra structure∆ H , ǫ H given by [23, 4.5]∆ H ( f x ) = X a,b ∈ G τ ( x, a, b ) f p a b ⊲ x ⊗ f p b x, (1.8) ǫ H ( f x ) = f (1 G ) . We say that two structures (1.2) and (1.8) are coherent if they turn H into a bialgebra. The coherence conditions are(1) F ⊲⊳ G is a group and (2) δ G σ − = δ F τ .Bialgebras so defined are always Hopf algebras, see [23, 4.7] for aformula for the antipode.In consequence the second Hopf cohomology group of extensions (1.1)with fixed actions ⊳, ⊲ is defined as(1.9) H ( k F, k G , ⊳, ⊲ ) = Z ( k F, k G , ⊳, ⊲ ) /B ( k f, k G , ⊳, ⊲ )where Z ( k F, k G , ⊳, ⊲ ) = { ( σ, τ ) | δ G σ − = δ F τ } is the group of Hopf2-cocycles and B ( k f, k G , ⊳, ⊲ ) = { ( δ F ζ − , δ G ζ ) | ζ : F × G → k • } is thegroup of Hopf 2-coboundaries.An extension (1.1) is called cocentral [12] if k F is a central subalgebraof H ∗ . Some equivalent conditions are ⊲ is trivial or G is normal in F ⊲⊳ G . Another consequence of cocentrality is that F acts by Hopfautomorphisms of k G (see e.g. [19, 11]).Our main interest lies with cocentral extensions (1.1) satisfying thecondition(1.10) H ( F, ( k G ) • , ⊳ ) = { } for every action ⊳ . We will call them special cocentral . Below we will write H = H ( τ, ⊳ )for a special cocentral extension with a datum { τ, ⊳ } .In the case of special cocentral extensions the definition of cohomol-ogy groups (1.9) can be simplified. This has been done by M. Mastnak[16] and we adopt his formulation. First we define an action of F onFun( F n × G m , k • ) extending the action ⊳ of F on G via(1.11) y.φ ( x , ..., x n , a , ..., a m ) = φ ( x , ..., x n , a ⊳ y, . . . , a m ⊳ y ) . Now we let Z c ( k F, k G , ⊳ ) and B c ( k F, k G , ⊳ ) denote the subgroups of Z ( G, ( k F ) • , id) and B ( G, ( k F ) • , id) of 2-cocycles τ and 2-coboundaries δ G η , respectively satisfying δ F τ = 1 = δ F η . This leads us to define H c ( k F, k G , ⊳ ) = Z c ( k F, k G , ⊳ ) /B c ( k F, k G , ⊳ ) . One can see immediately that the mapping τ (1 , τ ) carries out anisomorphism between H c ( k F, k G , ⊳ ) and H ( k F, k G , ⊳, id). Explicitly LASSIFICATION OF HOPF ALGEBRAS 7 both conditions δ F τ = ǫ and δ F η = ǫ are expressed by: τ ( xy ) = τ ( x )( x.τ ( y ))(1.12) η ( xy ) = η ( x )( x.η ( y ))(1.13)for all x, y ∈ F where F acts by (1.11). The equations say that each τ and η is a crossed homomorphism F → k G × G and F → k G , respectively.We call elements of Z c ( k F, k G , ⊳ ) and B c ( k F, k G , ⊳ ) Hopf Z c ( ⊳ ) , B c ( ⊳ ), etc for Z c ( k F, k G , ⊳ ), B c ( k F, k G , ⊳ ), etc. when the groups G and F are clear from the context. We single out a subgroup B ( ⊳ ) of Z c ( ⊳ ) by the equation B ( ⊳ ) = B ( G, ( k F ) • ) ∩ Z c ( ⊳ ). Clearly B c ( ⊳ ) ⊂ B ( ⊳ ) so we can form the subgroup H ( ⊳ ) = B ( ⊳ ) /B c ( ⊳ ) of H c ( ⊳ ).We note in passing that elements of H ( ⊳ ) parametrize cocommutativeextensions in Ext( k F, k G ).We add a remark on F -invariance of subgroups just defined. Lemma 1.1. If F is abelian, then subgroups Z c ( ⊳ ) , B cc ( ⊳ ) , and B c ( ⊳ ) are F -invariant. Proof:
For Z c ( ⊳ ) one has readily by (1.11)( z.τ )( xy ) = ( z.τ )( x )( zx.τ ( y ) = ( z.τ )( x )( x. (( z.τ )( y ))as x commutes with z . This shows z.τ ∈ Z c ( ⊳ ). For the remaining twocases it suffices to note that the operator δ G is F -linear on account of G acting trivially on k F . (cid:3) Structure of H c ( k C p , k G , ⊳ )From this point on H is an almost abelian Hopf algebra, G = G ( H ), F = C p , and p is small relative to G . Plainly G is normal in C p ⊲⊳ G ,hence the action ⊲ is trivial. In addition, H ( C p , ( k G ) • , ⊳ ) vanishes as k • is a divisible group by e.g. [16, 4.4]. All in all we see that H is aspecial cocentral extension of C p by k G . We begin with a simple fact. Lemma 2.1.
Let τ ∈ Z ( G, ( k C p ) • ) . Then for every x ∈ C p τ ( x ) is a -cocycle for G with coefficients in k • with the trivial action of G on k • . Proof:
The 2-cocycle condition for the trivial action is(2.1) τ ( a, bc ) τ ( b, c ) = τ ( ab, c ) τ ( a, b ) . Expanding both sides of the above equality in the basis { p x } and equat-ing coefficients of p x proves the assertion. (cid:3) LEONID KROP
Consider group F acting on an abelian group A , written multiplica-tively, by group automorphisms. Let Z F be the group algebra of F over Z . Z F acts on A via( X c i x i ) .a = Y x i . ( a c i ) , c i ∈ Z , x i ∈ F. For F = C p pick a generator t of C p and set φ i = 1 + t + · · · + t i − , i = 1 , . . . , p . Choose τ ∈ Z ( G, ( k C p ) • ) and expand τ in terms ofthe standard basis p t i for k C p , τ = P τ ( t i ) p t i with τ ( t i ) ∈ Z ( G, k • ).An easy induction on i shows that condition (1.12) implies(2.2) τ ( t i ) = φ i .τ ( t ) , for all i = 1 , . . . , p For i = p we have(2.3) φ p .τ ( t ) = 1in view of t p = 1 and τ (1) = 1.Let M be a Z C p -module. Following [15] we define the mapping N : M → M by N ( m ) = φ p ( t ) .m . We denote by M N the kernel of N in M . For M = Z ( G, k • ) , B ( G, k • ) or H ( G, k • ) we write Z N ( G, k • )for Z ( G, k • ) N and similarly for the other groups. We abbreviate Z N ( G, k • ) to Z N ( ⊳ ) and likewise for B N ( G, k • ) and H N ( G, k • ). Thusby definition Z N ( ⊳ ) is the set of all 2-cocycles satisfying(2.4) φ p .s = 1 . We want to compare abelian groups Z c ( ⊳ ) and Z N ( ⊳ ). This is done viathe mapping Θ : Z ( G, ( k C p ) • ) → Z ( G, k • ) , Θ( τ ) = τ ( t ) . Lemma 2.2.
The mapping Θ induces a C p -isomorphism between Z c ( ⊳ ) and Z N ( ⊳ ) . Proof:
We begin with an obvious equality x. ( τ ( y )) = ( x.τ )( y ). Taking y = t we get Θ( x.τ ) = x. Θ( τ ), that is C p -linearity of Θ. The relations(2.2) show that Θ is monic. It remains to establish that Θ is epic.Pick s ∈ Z N ( ⊳ ). Define τ : G × G → ( k C p ) • by setting τ ( t i ) = φ i ( t ) .s, ≤ i ≤ p . The proof will be complete if we demonstrate that τ satisfies (1.12).For any i, j ≤ p we have τ ( t i )( t i .τ ( t j )) = ( φ i ( t ) .s )( t i φ j ( t ) .s ) = ( φ i ( t ) + t i φ j ( t )) .s One sees easily that φ i ( t ) + t i φ j ( t ) = i + j − X k =0 t k . Hence if i + j < p we have φ i ( t ) + t i φ j ( t ) = φ i + j ( t ) and so τ ( t i )( t i .τ ( t j )) = τ ( t i + j ). If i + j = p + m LASSIFICATION OF HOPF ALGEBRAS 9 with m ≥
0, then p + m − X k =0 t k = φ p ( t ) + t p (1 + · · · + t m − ) which implies( p + m − X k =0 t k ) .s = φ p ( t ) .s · t p φ m ( t ) .s = φ m ( t ) .s = τ ( t i + j ) by (2.4) and as t p = 1. (cid:3) The next step is to describe structure of H ( ⊳ ). We need somepreliminaries. First, we write x.f for the left action of C p on k G dualto ⊳ as in (1.4). Since b G is the group of grouplikes of k G , and C p acts by Hopf automorphisms b G is C p -stable . Further, we use δ forthe differential on the group of 1-cochains of G in k • . We also note B N ( ⊳ ) = B ( G, k • ) ∩ Z N ( ⊳ ). By (2.4) δf ∈ B N ( ⊳ ) iff φ p ( t ) .δf = 1which, in view of δ being C p -linear, is the same as δ ( φ p ( t ) .f ) = 1.Since ( δf )( a, b ) = f ( a ) f ( b ) f ( ab ) − , Ker δ consists of characters of G ,whence δf ∈ B N ( ⊳ ) iff φ p ( t ) .f is a character of G . Say χ = φ p ( t ) .f ∈ b G . Then as tφ p ( t ) = φ p ( t ), χ is a fixed point of the C p -module b G .Letting b G C p stand for the set of fixed points in b G we have by [15,IV.7.1] an isomorphism H ( C p , b G, • ) ≃ b G C p /N ( b G ). We connect B N ( ⊳ )to H ( C p , b G ) via the homomorphism(2.5) Φ : B N ( ⊳ ) → H ( C p , b G, • ) , δf ( φ p .f ) N ( b G ) Lemma 2.3.
The following properties holds (i) Θ( B cc ( ⊳ )) = B N ( ⊳ ) , (ii) Θ( B c ( ⊳ )) = ker Φ , (iii) B N ( ⊳ ) / ker Φ ≃ H ( C p , b G, • ) , (iv) H cc ( ⊳ ) ≃ H ( C p , b G, • ) . Proof:
First we show that Φ is well-defined. For, δf = δg iff f g − = χ ∈ b G , henceΦ( δf ) = ( φ p .f ) N ( b G ) = ( φ p .gχ ) N ( b G )= ( φ p .g · φ p .χ ) N ( b G ) = ( φ p .g ) N ( b G ) = Φ( δg )(i) Take some δ G η ∈ B ( ⊳ ). Evidently for every x ∈ C p (*) ( δ G η )( x ) = δ ( η ( x )), hence Θ( δ G η ) = δ ( η ( t )) is a coboundary, and φ p .δ ( η ( t )) = 1 by (2.3), whence Θ( δ G η ) ∈ B N ( ⊳ ). Conversely, pick δf ∈ B N ( ⊳ ) and define ω = P pi =1 ( φ i .δf ) p t i . The argument of Lemma2.2 shows ω lies in Z c ( ⊳ ). Set η = P pi =1 ( φ i .f ) p t i . Using (*) again wederive δ G η = p X i =1 ( φ i .δf ) p t i = ω, hence δ G η ∈ B ( ⊳ ). Clearly Θ( δ G η ) = δf .(ii) The argument of Lemma 2.2 is applicable to 1-cocycles satisfying(1.13). It shows that η satisfies (1.13) iff(2.6) η ( t i ) = φ i .η ( t )For i = p we get φ p .η ( t ) = ǫ , hence the calculationΦ(Θ( δ G η )) = Φ( δ ( η ( t ))) = ( φ p .η ( t )) N ( b G ) = N ( b G ) . gives one direction. Conversely, Φ( δf ) ∈ N ( b G ) means φ p .f = φ p .χ which implies φ p .f χ − = ǫ . Set g = f χ − and define 1-cocycle η g = P pi =1 ( φ i .g ) p t i . Since φ p .g = ǫ , η g satisfies (1.13), whence δ G η g ∈ B c ( ⊳ ).As ( δ G η g )( t ) = δg = δf by construction, Θ( δ G η g ) = δf .(iii) We must show that Φ is onto. For every character χ in b G C p wewant to construct an f : G → k • satisfying φ p .f = χ . To this end weconsider splitting of G into the orbits under the action of C p . Sinceevery orbit is either regular, or a fixed point we have G = ∪ ri =1 { g i , g i ⊳ t, . . . , g i ⊳ t p − } ∪ G C p For every s ∈ G C p we pick a ρ s ∈ k satisfying ρ ps = χ ( s ). We define f by the rule f ( g i ) = χ ( g i ) , f ( g i ⊳ t j ) = 1 for all j = 1 and all i = 1 , . . . , r, and f ( s ) = ρ s for every s ∈ G C p By definition ( φ p .f )( g ) = Q p − j =0 f ( g ⊳ t j ). Therefore ( φ p .f )( s ) = ρ ps = χ ( s ) for every s ∈ G C p . If g = g i ⊳ t j for some i, j , then a calculation( φ p .f )( g ) = f ( g i ) = χ ( g i ) = χ ( g i ⊳ t j ) = χ ( g ), which uses the fact that χ is a fixed point under the action by C p , completes the proof.(iv) follows immediately from H ( ⊳ ) = B /B c ( ⊳ ) and parts (i)-(iii). (cid:3) Corollary 2.4.
Isomorphism Θ induces a C p -isomorphism Θ ∗ : H c ( ⊳ ) ≃ Z N ( ⊳ ) / ker Φ . (cid:3) We proceed to the main result of the section.
Proposition 2.5.
Suppose G is a finite abelian group. If | G | is odd,or G is a -group and either C -action is trivial, or G is an elementary -group, there exists a C p -isomorphism (2.7) H c ( ⊳ ) ≃ H ( C p , b G, • ) × H N ( G, k • ) . LASSIFICATION OF HOPF ALGEBRAS 11
Proof: (1) First we take up the odd case. By the preceeding Corollarywe need to decompose Z N ( ⊳ ) / ker Φ. We note that for any p and G there is a group splitting Z ( G, k • ) = B ( G, k • ) × H ( G, k • ) due tothe fact that the group of 1-cocycles k • G is injective, and hence so is B ( G, k • ). We aim at finding a C p -invariant complement to B ( G, k • ).To this end we recall a well-known isomorphism a : H ( G, k • ) ˜ → Alt( G ),see e.g. [30, § G ) is the group of all bimultiplicativealternating functions β : G × G → k • , β ( ab, c ) = β ( a, c ) β ( b, c ) , and β ( a, a ) = 1 for all a ∈ G. For the future applications we outline the construction of a . Namely, a is the antisymmetrization mapping sending z ∈ Z ( G, k • ) to a ( z )defined by a ( z )( a, b ) = z ( a, b ) z − ( b, a ). One can check that a ( z ) isbimultiplicative (cf. [30, (10)]) and it is immediate that a is C p -linear.Another verification gives im a = Alt( G ) and, moreover,ker a = B ( G, k • ), see [30, Thm.2.2]. Thus we obtain a C p -isomorphism H ( G, k • ) ≃ Alt( G ).Since elements of Alt( G ) are bimultiplicative mappings Alt( G ) ⊂ Z ( G, k • ). For every β ∈ Alt( G ) a simple calculation gives a ( β ) = β .Thus a ( β ) = 1 as the order of β divides the exponent of G . It follows B ( G, k • ) ∩ Alt( G ) = { } which gives a splitting of abelian groups Z ( G, k • ) = B ( G, k • ) × Alt( G )But now both subgroups B ( G, k • ) and Alt( G ) are C p -invariant hencethere holds Z N ( G, k • ) = B N ( G, k • ) × Alt N ( G ) which, in view ofAlt( G ) = H ( G, k • ), is the same as(2.8) Z N ( ⊳ ) = B N ( ⊳ ) × H N ( G, k • ) . Now part (iii) of Lemma 2.3 completes the proof of (1).(2) Here we prove the second claim of the Proposition. We decom-pose G into a product of cyclic groups h x i i , ≤ i ≤ m . For every α ∈ Alt( G ) we define s α ∈ Z ( G, k • ) via s α ( x i , x j ) = ( α ( x i , x j ) , if i ≤ j , else . Since s α · s β = s αβ the set S = { s α | α ∈ Alt( G ) } is a subgroup of Z ( G, k • ). One can see easily that s α = s β ⇔ α = β and a ( s α ) = α ,hence S is isomorphic to Alt( G ) under a . For every z ∈ Z N (triv) , a ( z ) ∈ Alt N ( G ), and therefore a ( z ) = a ( s ) for some s ∈ S N . We have zs − ∈ B ( G, k • ), but as zs − has order 2, zs − ∈ B N (triv). Thus Z N (triv) = B N (triv) × S N which proves (2.7). (3) We prove the last claim of the Proposition. Below G is an ele-mentary 2-group, and action of C is nontrivial. First we establish anintermediate result, namely Lemma 2.6.
If action ⊳ is nontrivial, then Z N ( ⊳ ) is a nonsplit exten-sion of Alt N ( G ) by B N ( ⊳ ) . Proof:
This will be carried out in steps.(i) We aim at finding a basis for Alt N ( G ). We begin by noting that asAlt( G ) has exponent 2, Alt N ( G ) is the set of all fixed points in Alt( G ).Put R = Z C . One can see easily that R -module G decomposes as(2.9) G = R × · · · × R m × G where R i ≃ R as a right C -module, and G = G C . Denote by t thegenerator of C . For each i let { x i − , x i } be a basis of R i such that x i − ⊳ t = x i . We also fix a basis { x m +1 , . . . , x n } of G .We associate to every subset { i, j } the bilinear form α ij by setting α ij ( x i , x j ) = α ij ( x j , x i ) = − , and α ij ( x k , x l ) = 1 for any { k, l } 6 = { i, j } . The set { α ij } forms a basis of Alt( G ). One can check easily that t actson basic elements as follows(2.10) t.α ij = α kl if and only if { x i , x j } ⊳ t := { x i ⊳ t, x j ⊳ t } = { x k , x l } . Recall the element φ = 1 + t ∈ Z C . We define forms β ij via(2.11) β ij = φ .α ij if t.α ij = α ij , and β ij = α ij , otherwise . The label ij on β ij is not unique as β ij = β kl whenever { x i , x j } ⊳ t = { x k , x l } . Of the two sets { i, j } and { k, l } labeling β ij we agree to usethe one with the smallest element, and call such minimal. We claim:(2.12) The elements { β ij } form a basis of Alt N ( G ) . Proof:
First we note that for every group M of exponent 2 M N = M C . Suppose β ∈ Alt( G ) C . Say β = Q α e ij ij , e ij = 0 ,
1. From t.β = Q ( t.α ij ) e ij = β we see that if α ij occurs in β , i.e. e ij = 1, thenso does t.α ij , hence β is a product of β ij . (cid:3) (ii) We want to show a ( Z N ( ⊳ )) = Alt N ( G ). The restriction a ∗ of a to Z N ( ⊳ ) induces a C -homomorphism Z N ( ⊳ ) a ∗ → Alt N ( G ) whose kernelequals B ( G, k • ) ∩ Z N ( ⊳ ) =: B N ( ⊳ ).We begin by showing φ . Alt( G ) ⊂ im a ∗ . For, if β = φ .α , pick an s ∈ Z ( G, k • ) with a ( s ) = α . Then ( t − .s ∈ Z N ( ⊳ ), and a (( t − .s ) =( t − .a ( s ) = ( t − .α = φ .α , as α = 1, which gives the inclusion. LASSIFICATION OF HOPF ALGEBRAS 13
By step (i) and definition (2.11) it remains to show that all fixedpoints α ij lie in im a ∗ . By formula (2.10) α ij is a fixed point if and onlyif either( a ) { i, j } ⊂ { m + 1 , . . . , n } or ( b ) { i, j } = { k − , k } for some k, ≤ k ≤ m . Below we find it convenient to write s i,j for s α ij .Consider case (a). We claim s i,j is a fixed point. For, t.s i,j is bi-multiplicative, hence is determined by its values at ( x k , x l ). It is im-mediate that t.s i,j ( x k , x l ) = s i,j ( x k , x l ) for all ( x k , x l ), whence the as-sertion. Since s i,j = 1 for all i, j , φ .s i,j = 1, hence s i,j ∈ Z N ( ⊳ ). As a ( s i,j ) = α ij , this case is done.We take up (b). Say z = s i − , i for some i, ≤ i ≤ m . An easyverification gives φ .z = α i − i = 1. Thus z / ∈ Z N ( ⊳ ). To prove(ii) we need to find a coboundary δg i such that zδg i ∈ Z N ( ⊳ ). Since a ( α i − i ) = 1 , α i − i = δf i for some f i : G → k • . Put G i for thesubgroup of G generated by all x j , j = 2 i − , i . We assert that onechoice is the function f i defined by(2.13) f i ( x j i − x j i x ′ ) = ( − j + j + j j for all x ′ ∈ G i For, on the one hand it is immediate that for any x ′ , x ′′ ∈ G i α i − i ( x j i − x j i x ′ , x k i − x k i x ′′ ) = ( − j k + j k On the other hand the definitions of f i and differential δ give δf i ( x j i − x j i x ′ , x k i − x k i x ′′ )= ( − j + j + j j ( − k + k + k k ( − j + k + j + k +( j + k )( j + k ) = ( − j k + j k Define the function g i : G → k • by g i ( x j i − x j i x ′ ) = ι j + j + j j where ι = −
1. One can check easily the equalities f i = 1 and t.g i = g i , g i = f i . Hence we have f i ( φ .g i ) = f i g i = f i = 1, and then a calculation φ . ( zδg i ) = ( φ .z )( φ .δg i ) = δf i · δ ( φ .g i ) = δ ( f i ( φ .g i )) = 1completes the proof of (ii).(iii) Suppose Z N ( ⊳ ) = B N ( ⊳ ) × C where C is a C -invariant subgroup.Then C is mapped isomorphically on Alt N ( G ) under a and so there isa unique z ∈ C such that a ( z ) = α . Since a ( s , ) = α , z = s , δg for some g : G → k • . Further, as α is a fixed point a ( t.z ) = α as well, hence t.z = z . In addition, since Alt( G ) is an elementary2-group, 1 = z = ( s , δg ) = ( δg ) = δ ( g ). It follows that g is acharacter of G . Moreover, t.z = z is equivalent to t.s , ( t.δg ) = s , δg which in turn gives s , ( t.s , )( t.δg ) = δg . As φ .s , = α = δf we have δf ( t.δg ) = δg which implies δf = δg ( t.δg ) on the accountof ( δg ) = δ ( g ) = 1 as g is a character. Equivalently we have theequality(2.14) f = g · ( t.g ) · χ for some χ ∈ b G. Noting that f is defined up to a character of G we can assume that f ( x ) = 1 = f ( x ) and f ( x x ) = −
1. For, f is defined as anyfunction satisfying δf = α . As δ ( f χ ) = δf for any χ ∈ b G , f canbe modified by any χ . By (2.13) f ( x j ) = − f ( x x ) , j = 1 , χ such that χ ( x ) = χ ( x ) = −
1. The equality (2.14) impliesthat for some χ ∈ b G there holds1 = f ( x j ) = g ( x ) g ( x ) χ ( x j ) , j = 1 , , and(*) − f ( x x ) = g ( x x ) χ ( x x )(**)as t swaps x and x . Since g is a character, g ( a ) = ± a ∈ G . It follows that g ( x ) = ι m and g ( x ) = ι k for some 0 ≤ m, k ≤
3. Then equation (*) gives 1 = ι m + k χ ( x j ). This equalityshows that χ ( x ) = χ ( x ) and m + k is even, because χ ( a ) = ± a . Now (**), and the fact that g is a character, gives − g ( x ) g ( x ) χ ( x ) χ ( x ) = ι m + k ) ι − m + k ) = 1, a contradiction. Thiscompletes the proof of the Lemma. (cid:3) Finally we prove (3). Let G be a group with a decomposition (2.9).Set C to be the subgroup of Z N ( ⊳ ) generated by the set B = B ′ ∪ B ′′ ∪ B ′′′ where B ′ = { φ .s i,j | α ij is not a fixed point, and { i, j } is minimal } B ′′ = { s i,j | i < j and { i, j } ⊂ { m + 1 , . . . , n }} B ′′′ = { s i − , i δg i | i = 1 , . . . , m } . There g i is chosen as in the case (ii) of Lemma 2.6. Passing on to Z N ( ⊳ ) / ker Φ we denote by B N ( ⊳ ) and C the images of these subgroupsin Z N ( ⊳ ) / ker Φ. Pick a v ∈ B . If v ∈ B ′ ∪ B ′′ then v = 1 because thecorresponding s i,j has order 2. For v = s i − , i δg i , v = δg i = δf i . Weknow t.f i = f i and f i = 1 and therefore φ .f i = 1, whence δf i ∈ ker Φby definition (2.5). It follows that v = 1 for all v ∈ B . Furthermore,by Lemma 2.6 the mapping a sends B to the basis (2.12) of Alt N ( G ).Therefore C is isomorphic to Alt N ( G ) at least as an abelian groupand forms a complement to B N ( ⊳ ) in Z N ( ⊳ ) / ker Φ. Since Alt N ( G )consists of fixed points the proof will be completed if we show thesame for C . The fact that B ′ ∪ B ′′ consists of fixed points follows from tφ = φ and part (a) of Lemma 2.6(ii). For an s i − , i δg i , the equality LASSIFICATION OF HOPF ALGEBRAS 15 φ .s i − , i = δf i gives t.s i − , i = s i − , i δf i . Since δf i ∈ ker Φ and t.δg i = δg i we see that s i − , i δg i is a fixed point in Z N ( ⊳ ) / ker Φ whichcompletes the proof. (cid:3) The Isomorphism Theorems
We begin with a general observation. Let H be be an extension oftype (A). The mapping π induces a k F -comodule structure ρ π on H via(3.1) ρ π : H → H ⊗ k F, ρ π ( h ) = h ⊗ π ( h ) .H becomes an F -graded algebra with the graded components H f = { h ∈ H | ρ π ( h ) = h ⊗ f } . Let χ : k F → H be a section of k F in H . Bydefinition χ is a convolution invertible k F -comodule mapping, that is(3.2) ρ π ( χ ( f )) = χ ( f ) ⊗ f, for every f ∈ F Set f = χ ( f ). The next lemma is similar to [23, 3.4] or [24, 7.3.4]. Lemma 3.1.
For every f ∈ F there holds H f = k G f Proof:
By definition of components H = H co π which equals to k G bythe definition of extension. By (3.2) ρ π ( f ) = f ⊗ f , hence k G f ⊂ H f .Since the containment holds for all f , the equalities H = ⊕ f ∈ F H f = ⊕ f ∈ F k G f force H f = k G f for all f ∈ F . (cid:3) Definition 3.2.
Given two F -graded algebras H = ⊕ H f and H ′ = ⊕ H ′ f and an automorphism α : F → F we say that a linear mapping ψ : H → H ′ is an α -graded morphism if ψ ( H f ) = H ′ α ( f ) for all f ∈ F . Lemma 3.3.
Suppose H and H ′ are two extensions of k F by k G and ψ : H → H ′ a Hopf isomorphism sending k G to k G . Then ψ is an α -graded mapping for some α . Proof:
Suppose H and H ′ are given by sequences k G ι H π ։ k F, and k G ι ′ H ′ π ′ ։ k F By definition of extension Ker π = H ( k G ) + and likewise Ker π ′ = H ′ ( k G ) + . By assumption ψ ( k G ) = k G , hence ψ induces a Hopf iso-morphism α : H/H ( k G ) + → H ′ /H ′ ( k G ) + . Replacing H/H ( k G ) + and H ′ /H ′ ( k G ) + by k F we can treat α as a Hopf isomorphism α : k F → k F . α is in fact an automorphism of F . We arrive at a commutativediagram k G ι −−−→ H π −−−→ k F ψ y ψ y α y k G ι ′ −−−→ H ′ π ′ −−−→ k F Since ψ is a coalgebra mapping for every f ∈ F we have∆ H ′ ( ψ ( f )) = ( ψ ⊗ ψ )∆ H ( f ) = ψ (( f ) ) ⊗ ψ (( f ) ) , hence ρ π ′ ( ψ ( f )) = ψ (( f ) ) ⊗ π ′ ψ (( f ) ) = ψ (( f ) ) ⊗ απ (( f ) )On the other hand, applying ψ ⊗ α to the equality ρ π ( f ) = ( f ) ⊗ π (( f ) ) = f ⊗ f gives ψ (( f ) ) ⊗ απ (( f ) ) = ψ ( f ) ⊗ α ( f )whence we deduce ρ π ′ ( ψ ( f )) = ψ ( f ) ⊗ α ( f ). Thus ψ ( f ) ∈ H ′ α ( f ) whichshows the inclusion ψ ( H f ) = ψ ( k G f ) = k G ψ ( f ) ⊆ H ′ α ( f ) = k G α ( f )Since both sides of the above inclusion have equal dimensions, the proofis complete. (cid:3) In what follows H is an almost abelian Hopf algebra, G = G ( H ), F = C p , and p is small relative to G . Let ⊳ and ⊳ ′ be two actions of C p on G . We denote ( G, ⊳ ) and (
G, ⊳ ′ ) the corresponding C p -modulesand we use the notation ‘ • ’ and ‘ ◦ ’ for the actions of C p on k G cor-responding by (1.4) to ⊳ and ⊳ ′ , respectively. We let I ( ⊳, ⊳ ′ ) denotethe set of all automorphisms of G intertwining actions ⊳ and ⊳ ′ , thatis automorphisms λ : G → G satisfying(3.3) ( a ⊳ x ) λ = aλ ⊳ ′ x, a ∈ G, x ∈ C p We make every λ act on functions τ : C p × G → k • by( τ.λ )( x, a, b ) = τ ( x, aλ − , bλ − ) . Lemma 3.4. (i)
The group Z ( G, ( k C p ) • ) is invariant under the actioninduced by any automorphism of G , (ii) A C p -isomorphism λ : ( G, ⊳ ) → ( G, ⊳ ′ ) induces C p -isomorphismsbetween the groups Z c ( ⊳ ) , B c ( ⊳ ) , H c ( ⊳ ) and Z c ( ⊳ ′ ) , B c ( ⊳ ′ ) , H c ( ⊳ ′ ) , re-spectively. Proof: (i) is immediate.(ii) We must check condition (1.12) for τ.λ and Z C p -linearity of theinduced map. First we note λ − is a C p - isomorphism between ( G, ⊳ ′ ) LASSIFICATION OF HOPF ALGEBRAS 17 and (
G, ⊳ ), as one can check readily. Next we verify (1.12) and C p -linearity in a single calculation( τ.λ )( xy )( a, b ) = τ ( xy, aλ − , bλ − )= τ ( x, aλ − , bλ − )( x • τ ( y, aλ − , bλ − ))= τ ( x, aλ − , bλ − ) τ ( y, aλ − ⊳ x, bλ − ⊳ x )= τ ( x, aλ − , bλ − ) τ ( y, ( a ⊳ ′ x ) λ − , ( b ⊳ ′ x ) λ − )= ( τ.λ )( x )( x ◦ ( τ.λ )( y ))( a, b ) . In the case of B c ( ⊳ ), first one checks the equality( δ G η ) .λ = δ G ( η.λ ) for any η : C p × G → k • . It remains to verify the condition (1.13) for η.λ . That is done similarlyto the calculation in (ii). (cid:3)
Let (
G, ⊳ ) be a C p -module. Recall that A ( ⊳ ) denotes the group of C p -automorphisms of ( G, ⊳ ). By the above Lemma Z c ( ⊳ ) is an A ( ⊳ )-module. Symmetrically, the group A p = Aut( C p ) of automorphisms of C p acts on Map( C p × G , k • ) via τ.α ( x, a, b ) = τ ( α ( x ) , a, b )We want to know the effect of this action on Z c ( ⊳ ). Let ( G, ⊳ ) be a C p -module. For α ∈ A p we define a C p -module ( G, ⊳ α ) via a ⊳ α x = a ⊳ α ( x ) , a ∈ G, x ∈ C p Similarly, an action ‘ • ’ of C p on k G can be twisted by α into ‘ • α ’ via x • α r = α ( x ) • r, r ∈ k G One can see easily that if • and ⊳ correspond to each other by (1.4),then so do • α and ⊳ α . Lemma 3.5. (i) If λ ∈ I ( ⊳, ⊳ ′ ) , then λ ∈ I ( ⊳ α , ⊳ ′ α ) for every α ∈ A p , (ii) The mapping τ τ.α induces an A ( ⊳ )-isomorphism between Z c ( ⊳ ) , B c ( ⊳ ) , H c ( ⊳ ) and Z c ( ⊳ α ) , B c ( ⊳ α ) , H c ( ⊳ α ), respectively for every α ∈ A p . Proof: (i) For every a ∈ G, x ∈ C p we have( a⊳ α ) λ = ( a ⊳ α ( x )) λ = aλ ⊳ ′ α ( x ) = aλ ⊳ ′ α x (ii) First we note that A ( ⊳ ) can be identified with A ( ⊳ α ) for any α by the folllowing calculation( g ⊳ α x ) φ = ( g ⊳ α ( x )) φ = ( gφ ) ⊳ α ( x ) = gφ ⊳ α x for every φ ∈ A ( ⊳ ) . Thus we will treat every Z c ( ⊳ α ) as an A ( ⊳ )- module. Our next step isto show that for every τ ∈ Z c ( ⊳ ), τ.α lies in Z c ( ⊳ α ). This boils downto checking (1.12) for τ.α with the ⊳ α -action:( τ.α )( xy ) = τ ( α ( x ) α ( y )) = τ ( α ( x ))( α ( x ) • τ ( α ( y ))= τ ( α ( x ))( x • α τ ( α ( y )) = ( τ.α )( x )( x • α ( τ.α )( y )) . As for A ( ⊳ )-linearity, for every φ ∈ A ( ⊳ ), we have(( τ.α ) .φ )( x, a, b ) = ( τ.α )( x, aφ − , bφ − ) = τ ( α ( x ) , aφ − , bφ − )= ( τ.φ )( α ( x ) , a, b ) = (( τ.φ ) .α )( x, a, b ) . (cid:3) We need several short remarks.
Lemma 3.6.
Suppose τ is a -cocycle. Assume r ∈ ( k G ) • is such that φ p .r = ǫ . Set r i = φ i .r, ≤ i ≤ p . Define a -cocycle ζ : C p → ( k G ) • by ζ ( t i ) = r i and a -cocycle τ ′ = τ ( δ G ζ ) . Then the mapping ι : H ( τ, ⊳ ) → H ( τ ′ , ⊳ ) , ι ( p a t i ) = p a r i t i , a ∈ G, ≤ i ≤ p is an equivalence of extensions. Proof:
It suffices to show show δ G ζ ∈ B c for then [23, 5.2] yields theconclusion of the lemma. Now δ G ζ ∈ B c means that ζ satisfies (1.13).The argument of Lemma 2.2 used to derive (1.12) from the condition(2.4) works verbatim for ζ . (cid:3) Lemma 3.7. H ( τ, ⊳ ) is cocommutative iff τ lies in H cc ( ⊳ ) . Proof: H ∗ ( τ, ⊳ ) is commutative iff ab = ba which is equivalent to τ ( a, b ) = τ ( b, a ). This condition is equivalent to τ ( t ) : G × G → k • being a symmetric 2-cocycle. Indeed, one implication is trivial, whileif τ ( t ) is symmetric, then as pointed out in the odd case of Proposition2.5 τ ( t ) is a coboundary, that is an element of B N / ker Φ. A referenceto Lemma 2.3(i) completes the proof. (cid:3) Unless stated otherwise, H ( τ, ⊳ ) is a noncocommutative Hopf al-gebra. We pick another algebra H ( τ ′ , ⊳ ′ ) isomorphic to H ( τ, ⊳ ) via ψ : H ( τ, ⊳ ) → H ( τ ′ , ⊳ ′ ). The next observation is noted in [18, p. 802]. Lemma 3.8.
Mapping ψ induces an Hopf automorphism of k G . (cid:3) Let G be a finite group and Aut Hf ( k G ) be the group of Hopf au-tomorphisms of k G . Identifying ( k G ) ∗ with k G as in §
1, for every φ ∈ Aut Hf ( k G ) the transpose mapping φ ∗ is a Hopf automorphism of k G , hence an automorphism of G . This leads up to LASSIFICATION OF HOPF ALGEBRAS 19
Lemma 3.9.
Let G be a finite group. The mapping φ φ ∗ is anisomorphism between Aut Hf ( k G ) and Aut( G ) . φ is a C p -isomorphism ( k G , • ) → ( k G , ◦ ) if and only if φ ∗ is a C p -isomorphism ( G, ⊳ ′ ) → ( G, ⊳ ) . Proof:
The first assertion is clear by the opening remark. Next werecall that φ ∗ acts on G via(3.4) ( aφ ∗ )( f ) := f ( aφ ∗ ) = φ ( f )( a ) , f ∈ k G . Let ⊳ and ⊳ ′ be actions related to • and ◦ by (1.4). The last conclusionfollows from the calculation(( a ⊳ ′ x ) φ ∗ )( f ) = φ ( f )( a ⊳ ′ x ) = ( x ◦ φ ( f ))( a ) = φ ( x • f )( a )= ( aφ ∗ )( x • f ) = ( aφ ∗ ⊳ x )( f ) , for all f ∈ k G . (cid:3) We proceed to the formulation of isomorphism theorems. First werephrase definitions of [ ⊳ ] and C ( ⊳ ). Let ‘ ≃ ’ denote equivalence ofactions of C p on G . With R defined in the Introduction we have [ ⊳ ] = { ⊳ ′ ∈ R| ⊳ ′ ≃ ⊳ α for some α ∈ A p } and C ( ⊳ ) = { α ∈ A p | ⊳ α ≃ ⊳ } .Furthermore we denote by G ( ⊳ ) the subgroup of Aut( G ) generated by A ( ⊳ ) and a set of automorphisms λ α ∈ I ( ⊳, ⊳ α ) one for every α ∈ C ( ⊳ )if ⊳ is nontrivial, and A ( ⊳ ) × A p , otherwise. Proposition 3.10. G ( ⊳ ) is a crossed product of A ( ⊳ ) with C ( ⊳ ) . Proof:
The claim holds by definition for the trivial action. Else, wenote that λ A ( ⊳ ) λ − = A ( ⊳ ) for every λ ∈ I ( ⊳, ⊳ α ) by Lemmas 3.4(ii),3.5(i). In addition, for every λ, µ ∈ I ( ⊳, ⊳ α ) , λ − µ ∈ A ( ⊳ ). Thus wehave I ( ⊳, ⊳ α ) = A ( ⊳ ) λ α . It follows that λ α · λ β = φ ( α, β ) λ αβ for some φ ( α, β ) ∈ A ( ⊳ ). It remains to show that the kernel of π : G ( ⊳ ) → C ( ⊳ ) , π ( φλ α ) = α equals A ( ⊳ ). Pick α : x → x k , k = 1. Clearly λ ∈ I ( ⊳, ⊳ α ) iff tλ = λt k where we treat t ∈ C p as automorphism of G .Since elements of A ( ⊳ ) commute with t , I ( ⊳, ⊳ α ) ∩ A ( ⊳ ) = ∅ . (cid:3) Our next goal is to define a G ( ⊳ )-module structure on H c ( ⊳ ). As wementioned above H c ( ⊳ ) is A ( ⊳ )-module. Further, for every λ ∈ I ( ⊳, ⊳ α )Lemmas 3.4(ii), 3.5(ii) show that the mapping(3.5) ω λ,α : τ τ.λα − , τ ∈ H c ( ⊳ )is an automorphism of H c ( ⊳ ). We denote by φ the automorphism of H c ( ⊳ ) induced by φ ∈ A ( ⊳ ) and we abbreviate ω λ α ,α to ω α . Lemma 3.11.
The mapping φλ α φω α , φ ∈ A ( ⊳ ) , α ∈ C ( ⊳ ) defines G ( ⊳ ) -module structure on H c ( ⊳ ) . Proof: H c ( ⊳ ) is a subquotient of Z ( G, ( k C p ) • ), and the action of A ( ⊳ ) and ω α on H c ( ⊳ ) are induced from their action on Z ( G, ( k C p ) • ).Furthermore it is elementary to check that every λ ∈ Aut( G ) com-mutes with every β ∈ A p as mappings of Z ( G, ( k C p ) • ). It followsthat the equalities ω α ω β = φ ( α, β ) ω αβ and ω α φω − α = λ α φλ − α hold inAut( Z ( G, ( k C p ) • ). This shows that the mapping of the Lemma is ahomomorphism, as needed. (cid:3) Theorem 3.12. (I) . Noncocommutative extensions H ( τ, ⊳ ) and H ( τ ′ , ⊳ ′ ) are isomorphic if and only if (i) There exist an α ∈ A p and a C p -isomorphism λ : ( G, ⊳ ) → ( G, ⊳ ′ α ) such that (ii) τ ′ = τ. ( λα − ) in H c ( ⊳ ′ ) . (II) . There is a bijection between the orbits of G ( ⊳ ) in H c ( ⊳ ) not con-tained in H cc ( ⊳ ) and the isomorphism types of noncocommutative ex-tensions in Ext [ ⊳ ] ( k C p , k G ) . Proof: (I). In one direction, suppose ψ : H ( τ, ⊳ ) → H ( τ ′ , ⊳ ′ ) is anisomorphism. By Lemma 3.8 ψ induces an automorphism φ : k G → k G ,and from Lemma 3.3 we have the equality ψ ( t ) = rt k for some k and r ∈ k G . The equality ψ ( t p ) = 1 implies ( rt k ) p = φ p ( t k ) ◦ r = 1 and,as φ p ( t k ) = φ p ( t ), we have φ p ◦ r = 1 which shows r ∈ ( k G ) • . Let α : x x k , x ∈ C p be this automorphism of C p , and set φ = ψ | k G .Then the calculation φ ( t • f ) = ψ ( tf t − ) = rα ( t ) φ ( f ) α ( t ) − r − = α ( t ) ◦ φ ( f ) , f ∈ k G shows φ : ( k G , • ) → ( k G , ◦ α ) is a C p -isomorphism. It follows by Lemma3.9 that ( G, ⊳ ′ α ) is isomorphic to ( G, ⊳ ) under φ ∗ , hence λ = ( φ ∗ ) − :( G, ⊳ ) → ( G, ⊳ ′ α ) is a required isomorhism.It remains to establish the second condition of the theorem. To thisend we first modify ψ . Namely, set s = φ − ( r ) and observe that, as φ − is a C p -mapping and φ p ◦ α r = 1, we get φ p • s = φ − ( φ p ◦ α r ) = 1.Therefore by Lemma 3.6 there is an equivalence ι : H ( τ, ⊳ ) → H ( e τ , ⊳ )with ι ( t ) = st . Notice that ι is an algebra map with ι ( s ) = s for all s ∈ k G , hence ι − ( t ) = s − t . Thus we have ( ψι − )( t ) = t k by the choiceof s . It follows we can assume ψ ( t ) = t k hence ψ ( x ) = x k for all x ∈ C p .Abbreviating H ( τ, ⊳ ) , H ( τ ′ , ⊳ ′ ) to H, H ′ , respectively, we take up theidentity. ∆ H ′ ( ψ ( x )) = ( ψ ⊗ ψ )∆ H ( x ) , x ∈ C p , LASSIFICATION OF HOPF ALGEBRAS 21 expressing comultiplicativity of ψ on elements of C p . By (1.8) thistranslates into(3.6) X a,b τ ′ ( x k , a, b ) p a x k ⊗ p b x k = X c,d τ ( x, c, d ) φ ( p c ) x k ⊗ φ ( p d ) x k . Next we connect φ ( p b ) to the action of φ ∗ . This is given by the formula(3.7) φ ( p b ) = p b ( φ ∗ ) − . For, since φ is an algebra map, φ ( p b ) = p c where c is such that φ ( p b )( c ) = 1. By definition of action φ ∗ , φ ( p b )( c ) = ( cφ ∗ )( p b ) = p b ( cφ ∗ ),hence cφ ∗ = b , whence c = b ( φ ∗ ) − .Switching summation symbols c, d to l = c ( φ ∗ ) − and m = d ( φ ∗ ) − ,the right-hand side of (3.6) takes on the form X l,m τ ( x, lφ ∗ , mφ ∗ ) p l x k ⊗ p m x k Thus ψ is comultiplicative on C p iff(3.8) τ ′ ( α ( x ) , a, b ) = τ ( x, aφ ∗ , bφ ∗ ) = τ ( φ ∗ ) − ( x, a, b ) = τ.λ ( x, a, b ) . Applying α − to the last displayed equation we arrive at(3.9) τ ′ ( x, a, b ) = τ.λα − ( x, a, b ) . as needed.Conversely, let us assume hypotheses of part (I). Using Lemma 3.9we infer that λ − induces a Hopf C p -isomorphism φ = ( λ − ) ∗ : ( k G , • ) → ( k G , ◦ α ). We define ψ : H ( τ, ⊳ ) → H ( τ ′ , ⊳ ′ ) via ψ ( f x ) = φ ( f ) α ( x ) , f ∈ k G , x ∈ C p . First we verify that ψ is an algebra map utilizing φ ( x • f ) = α ( x ) ◦ φ ( f ),namely ψ (( f x )( f ′ x ′ )) = ψ ( f ( x • f ′ ) xx ′ ) = φ ( f ) φ ( x • f ′ ) α ( x ) α ( x ′ )= φ ( f )( α ( x ) ◦ φ ( f ′ )) α ( x ) α ( x ′ ) = φ ( f ) α ( x ) φ ( f ′ ) α − ( x ) α ( x ) α ( x ′ )= ( φ ( f ) α ( x ))( φ ( f ′ ) α ( x ′ ) = ψ ( f x ) ψ ( f ′ x ′ ) . To see comultiplicativity of ψ we need to verify(3.10) ∆ H ′ ( ψ ( f x )) = ( ψ ⊗ ψ )∆ H ( f x ) . By the multiplicativity of ∆ H ′ , ψ, ∆ H it suffices to check (3.10) sep-arately for any f and for every x . Now the first case holds as φ isa coalgebra mappping, and the second follows from τ ′ = τ.λα − bycalculations (3.6) and (3.9). (II). Let H denote the set of all pairs ( τ ′ , ⊳ ′ ) with ⊳ ′ and τ ′ runningover [ ⊳ ] and H c ( ⊳ ′ ) \ H ( ⊳ ′ ), respectively. We define an equivalencerelation on H by( τ ′ , ⊳ ′ ) ∼ ( τ ′′ , ⊳ ′′ ) iff H ( τ ′ , ⊳ ′ ) ≃ H ( τ ′′ , ⊳ ′′ ) . Let H / ∼ stand for the set of equivalence classes. By construction H / ∼ is just a copy of nc Ext [ ⊳ ] ( k C p , k G ) / ∼ =. We select the subset H ( ⊳ ) = { ( τ, ⊳ ) | τ ∈ H c ( ⊳ ) } of H and define the orbit ( τ, ⊳ ) G ( ⊳ ) as theset { ( τ ′ , ⊳ ) | τ ′ ∈ τ G ( ⊳ ) } . The proof will be complete if we show that theset { C ∩ H ( ⊳ ) | C ∈ H / ∼} coincides with the set of orbits of G ( ⊳ ) in H ( ⊳ ). Now pick ( τ ′ , ⊳ ′ ) ∈ C . Since ⊳ ′ ∈ [ ⊳ ], there exists an isomorphism µ ∈ I ( ⊳ ′ , ⊳ α ) hence setting τ = τ ′ .µα − we have ( τ, ⊳ ) ∈ C by part (I).Moreover C ∩ H ( ⊳ ) ∋ ( σ, ⊳ ) if and only if H ( τ, ⊳ ) ≃ H ( σ, ⊳ ), hence bypart (I) again we have σ = τ.ω λ,α , that is σ ∈ τ G ( ⊳ ). Same argumentshows that the equivalence class generated by ( τ, ⊳ ) intersect H ( ⊳ ) inthe orbit of ( τ, ⊳ ). (cid:3) Corollary 3.13.
For every τ ∈ H c ( ⊳ ) the cardinality of the orbit τ G ( ⊳ ) satisfies | τ A ( ⊳ ) | ≤ | τ G ( ⊳ ) | ≤ | C ( ⊳ ) || τ A ( ⊳ ) | . Proof:
Since A ( ⊳ ) ⊆ G ( ⊳ ) the lower bound is clear. By Propositon3.10 G ( ⊳ ) = S α ∈ C ( ⊳ ) ω α A ( ⊳ ) hence τ G ( ⊳ ) = S α ∈ C ( ⊳ ) τ ω α A ( ⊳ ). It re-mains to note that for every α the cardinality of τ ω α A ( ⊳ ) coincideswith that of τ A ( ⊳ ). (cid:3) With some extra effort we can extend the bijection theorem to theentire set Ext [ ⊳ ] ( k C p , k G ) provided G is an elementary p -group for any p . Since our prime interest lies with nontrivial Hopf algebras we statethe result without proof. Theorem 3.14.
Let G be a finite elementary p -group. The numberof isotypes of cocommutative Hopf algebras in Ext [ ⊳ ] ( k C p , k G ) equals tothe number of orbits of A ( ⊳ ) in H cc ( ⊳ ) . (cid:3) We comment briefly on the dual case of commutative Hopf algebras.First, Ext [ ⊳ ] ( k C p , k G ) contains a commutative Hopf algebra iff ⊳ = triv.Second, we introduce the group Cext( G, C p ) of central extensions of G by C p [2]. We outline properties of Ext [triv] ( k C p , k G ) again withoutproof. Theorem 3.15. (1)
The group
Ext [triv] ( k C p , k G ) is isomorphic to thegroup Cext(
G, C p ) under the map H ( τ, triv) ⇆ k L ( τ ) where L ( τ ) is thecentral extension defined by the -cocycle τ . LASSIFICATION OF HOPF ALGEBRAS 23 (2)
For G elementary p -group of rank n with an odd p the number ofisotypes in Ext [triv] ( k C p , k G ) equals ⌊ n +22 ⌋ . For calculation of orbits of G ( ⊳ ) in H c ( ⊳ ) we prefer to use its isomor-phic copy Z N ( ⊳ ) / ker Φ which we denote by X ( ⊳ ) and refer to it as theclassifying group for Ext [ ⊳ ] ( k C p , k G ).We turn X ( ⊳ ) into a G ( ⊳ )-module by transfering its action from Z c ( ⊳ )to Z N ( ⊳ ) along Θ. Pick some ω λ,α and suppose α − : x x l , x ∈ C p .For s ∈ Z N ( ⊳ ) we put(3.11) s.ω λ,α = ( φ l • s ) .λ. Lemma 3.16. (i)
For every prime and any action ‘ ⊳ ’ the isomorphism Θ ∗ : H c ( ⊳ ) ≃ X ( ⊳ ) of Corollary 2.4 is G ( ⊳ ) -linear. (ii) For every prime and any action X ( ⊳ ) fits into the exact sequence (3.12) b G C p /N ( b G ) X ( ⊳ ) ։ a ( Z N ( ⊳ )) . (iii) For every odd p there is a G ( ⊳ ) splitting (3.13) X ( ⊳ ) ≃ b G C p /N ( b G ) × Alt N ( G ) . Proof: (i) We begin by noting that for every λ ∈ I ( ⊳, ⊳ α ) there holds(*) x • α ( s.λ ) = ( x • s ) .λ, x ∈ C p . Still assuming α − : x → x l ,the conclusion (i) follows from (2.2) and the opening remark by thecalculationΘ( τ.ω λ,α ) = ( τ.ω λ,α )( t ) = ( τ.λ )( t l ) = (by (2.2)) φ l • α ( τ.λ )( t )= φ l • α ( τ ( t ) .λ ) = (by (*)) ( φ l • τ ( t )) .λ = Θ( τ ) .ω λ,α This equation demonstrates that definition (3.11) turnes Z N ( ⊳ ) into a G ( ⊳ )-module. It is immediate that B c ( ⊳ ) is a G ( ⊳ )-subgroup of Z c ( ⊳ ).By Lemma 2.3(ii) ker Φ is a G ( ⊳ )-subgroup, which proves part (i).(ii) The mapping a : Z ( G, k • ) → Alt( G ) of Proposition 2.5 re-stricted to Z N ( ⊳ ) gives rise to an exact sequence B N ( ⊳ ) → Z N ( ⊳ ) → a ( Z N ( ⊳ ). Thanks to the G ( ⊳ )-isomorphism b G C p /N ( b G ) ≃ B N ( ⊳ ) / ker Φinduced by Φ (see Lemma 2.3) we arrive at the exact sequence (3.12)of G ( ⊳ )-modules.(iii) For an odd p splitting (2.8) is carried out by the mapping s sa ( s − ) × a ( s ) which is clearly a G ( ⊳ )-map. It remains to note thathomomorphism Φ is also a G ( ⊳ )-map. (cid:3) We point out that part (ii) fails in general for 2-groups. See Appendix 2 Almost Abelian Hopf Algebras of Dimension ≤ p Hopf algebras of dimension ≤ p . We begin by revisiting clas-sification of semisimple Hopf algebras of dimension p , p due to [20, 18].If dim H = p , then by a Kac-Masuoka theorem [9, 20] H contains acentral subHopf algebra k C p hence H ∈ Ext [triv] ( k C p , k C p ). Thus H iscommutative, and as Alt( C p ) = 1, H is cocommutative. It follows that H = k L where L is a group of order p , that is L = C p or C p × C p .Suppose dim H = p . By the Kac-Masuoka theorem, loc.cit, appliedto H ∗ we have that H ∗ is a central extension of the form k C p H ∗ ։ Q where dim Q = p . By the foregoing Q = k G with G = C p or C p × C p . By duality H is a cocentral extension of k C p by k G . If G = C p , then Alt( G ) = 1, hence H is cocommutative. It follows thata nontrivial H belongs to Ext( k C p , k C p × C p ) with a nontrivial action of C p on C p × C p .Before moving on we introduce algebras R i = Z p C p / h ( t − i i , ≤ i ≤ p − α k for themapping x x k , x ∈ C p , ⊳ k for ⊳ α k and ω k for ω α k . The arguments inthe next proposition will be used throughout § Proposition 4.1. ([18])
There are up to isomorphism p + 7 Hopf al-gebras in
Ext( k C p , k C p × C p ) , p + 1 of which are nontrivial. Proof:
We run the procedure for computing the number of isoclassesfor G = C p × C p . Let ⊳ r denote the right regular action of C p on R .Every nontrivial C p -module ( C p × C p , ⊳ ) is isomorphic to ( R , ⊳ r ). Inconsequence Ext( k C p , k G ) = Ext [ ⊳ r ] ( k C p , k G ) ∪ Ext [triv] ( k C p , k G ). ByTheorem 3.15(2) Ext [triv] ( k C p , k G ) contributes four nonisomorphic al-gebras. It remains to show that Ext [ ⊳ r ] ( k C p , k G ) contains p +3 isotypes.To simplify notation we put ⊳ = ⊳ r .(i) The classifying group X ( ⊳ ). Set G = R and let e = 1 , f = t − r is the image of r ∈ R in R . The matrix of t in the basis { e, f } is T = (cid:18) (cid:19) . Let { e ∗ , f ∗ } be the dual basis for b G . Themapping induced by t in b G has the matrix T tr relative to the dualbasis. Hence e ∗ is fixed by t and N ( b G ) = ( t − p − . b G = 0, as p > b G C p /N ( b G ) = h e ∗ i . FurtherAlt( G ) = b G ∧ b G , where ∧ denote the multiplication in the Grassmanalgebra over b G , is generated by e ∗ ∧ f ∗ , and the latter is a fixed by t .Therefore φ p ( t ) .e ∗ ∧ f ∗ = p ( e ∗ ∧ f ∗ ) = 0, hence Alt N ( G ) = Alt( G ). Allin all we arrive at the equality X ( ⊳ ) = h e ∗ , e ∗ ∧ f ∗ i LASSIFICATION OF HOPF ALGEBRAS 25 (ii) Groups A ( ⊳ ) , C ( ⊳ ) and G ( ⊳ ). By definition φ ∈ A ( ⊳ ) iff thematrix Φ of φ satisfies Φ T = T Φ and det Φ = 0. This condition isequivalent to Φ = (cid:18) c d c (cid:19) , c ∈ Z • p . By the opening remark C ( ⊳ ) = Z • p as ( G, ⊳ k ) ≃ ( G, ⊳ ) for every k ∈ Z • p .The group G ( ⊳ ) is generated by A ( ⊳ ) and a set { λ k | k ∈ Z • p } with λ k ∈ I ( ⊳, ⊳ k ). An easy verification gives that λ k defined via e.λ k = e, f.λ k = kf lies in I ( ⊳, ⊳ k ).(iii) Orbits of G ( ⊳ ) in X ( ⊳ ). First we determine the orbits of A ( ⊳ ).Pick φ ∈ A ( ⊳ ) and suppose it has the matrix Φ relative to { e, f } . It isan elementary fact that the mapping induced by φ in b G has the matrixΦ tr in the dual basis. If Φ is written as in (ii) then we have e ∗ .φ − = ce ∗ , and e ∗ ∧ f ∗ .φ − = c e ∗ ∧ f ∗ . Let us identify ae ∗ + be ∗ ∧ f ∗ ∈ X ( ⊳ ) with the vector ( a, b ) ∈ Z p . Bythe above φ ∈ A ( ⊳ ) acts in Z p via ( a, b ) .φ − = ( ca, c b ).By Corollary 3.13 the G ( ⊳ )-orbit of ( a, b ) is the union of A ( ⊳ )-orbitsof elements ( a, b ) .ω k , k ∈ C ( ⊳ ). There ω k = λ k α − k , and for every x ∈ X ( ⊳ ) there holds by (3.11) x.ω k = ( φ l .x ) .λ k where l = k − . Since e ∗ and e ∗ ∧ f ∗ are fixed by t we have φ l .x = lx for x = e ∗ , e ∗ ∧ f ∗ .Moreover it is immediate that e.λ k = e ∗ and e ∗ ∧ f ∗ .λ k = le ∗ ∧ f ∗ .We conclude that ( a, b ) .ω k = ( la, l b ) ∈ ( a, b ) A ( ⊳ ). It follows that G ( ⊳ )-orbits coincide with A ( ⊳ )-orbits. We compute the latter.The subset Z • p of Z p is stable under action of A ( ⊳ ). For every m ∈ Z • p the set (1 , m ) A ( ⊳ ) has p − , m ) A ( ⊳ ) ∩ (1 , n ) A ( ⊳ ) = ∅ if m = n . Since | Z • p | = ( p − the family { (1 , m ) A ( ⊳ ) | m ∈ Z • p } accounts for all orbits in Z • p . Thus we ob-tained p − Z p \ Z • p is the unionof { (0 , b ) | b ∈ Z • p } and { ( a, | a ∈ Z p } . Let ζ be a generator of Z • p .It follows readily that { (0 , b ) | b ∈ Z • p } is the union of (0 , A ( ⊳ ) and(0 , ζ ) A ( ⊳ ) which supplies two more nontrivial orbits. The second set isthe union of two trivial orbits, viz. { (0 , } and its complement. (cid:3) To recover the full strength of [18] we would need to show that every H ( τ, ⊳ ) is self-dual. However, such a theorem is unattainable due tothe next Remark 4.2.
Let τ ( t ) = e ∗ ∧ f ∗ and H ( τ, ⊳ ) be the correspondingHopf algebra. H ( τ, ⊳ ) ∗ ≃ H ( τ, ⊳ ) if and only if p − is a square in Z • p . Proof:
By general theory H ( τ, ⊳ ) ∗ ≃ H ( τ ′ , ⊳ ) for some τ ′ ∈ H c ( ⊳ ). Anisomorphism H ( τ, ⊳ ) ∗ ≃ H ( τ, ⊳ ) exists if and only if τ ′ ( t ) and τ ( t ) lie onthe same orbit. The 2-cocycle τ ′ is the multiplication cocycle for H ( τ, ⊳ ) written as an element of Ext( k G, k C p ). Since H ( τ, ⊳ ) = k ( b G ⋊ C p ) and e ∗ is a fixed point under the action of C p , k h e ∗ i is a normal subHopfalgebra of H ( τ, ⊳ ) giving rise to an exact sequence(4.1) k h e ∗ i ֒ → H ( τ, ⊳ ) Π ։ k G where G = h x, y i with x = Π( f ∗ ) , y = Π( t ). Clearly xy = yx sothat G = G . Let ρ Π = (id ⊗ Π)∆ H : H ( τ, ⊳ ) → H ( τ, ⊳ ) ⊗ k G be thecoaction induced by Π. We want to find a section γ : k G → H ( τ, ⊳ )splitting (4.1). This is the matter of finding T satisfying ρ Π ( T ) = T ⊗ y .It is not hard to see that T must be of the form ut for some unit u ∈ k G in fact a tedious but straightforward verification shows that for u = P i,j ζ − ij p e i f j T = ut is a desired element. Since f ∗ is a group-likeelement of H ( τ, ⊳ ), ρ Π ( f ∗ i T j ) = f ∗ i T j ⊗ x i y j , and therefore γ : x i y j f ∗ i T j defines a section of k G in H ( τ, ⊳ ). Let τ ′ : G × G → k h e ∗ i be the2-cocycle associated to γ . By definition τ ′ ( a, b ) = γ ( a ) γ ( b ) γ ( ab ) − . Wewill write below a = x i y j , b = x k y l . A simple calculation using T f ∗ = f ∗ e ∗ T gives τ ′ ( a, b ) = e ∗ jl . Viewing e ∗ as the functional e ∗ ( t k ) = ζ k on C p ( t ) we conclude that τ ′ ( t, a, b ) = ζ jl .We need to find a decomposition of τ ′ ( t ) according to (2.8), that is τ ′ ( t ) = b · λ with b ∈ B N ( ⊳ ) and λ ∈ Alt N ( G ). Set β = a ( τ ′ ( t )) andnote that by the definition of a , β ( a, b ) = τ ′ ( t, a, b ) τ ′ ( t, b, a ) − whichgives β ( a, b ) = ζ jk − il . Observe that β = a ( λ ) = λ , hence λ = β andtherefore b = τ ′ β − . It follows that b ( a, b ) = ( ζ ) jk + il . Let us select f : G → k • , f ( x i y j ) = ( ζ − ) ij . A straightforward calculation produces theequality δf ( a, b ) = b . We want to find the image of b in X ( ⊳ ) under Φ,that is Φ( b ) = φ p ( t ) .t . Since φ p ( t ) .f ( a ) = Q p − i =0 f ( a i ⊳t i ), φ p ( t ) .f ( x ) = 1as x ⊳ t = x , and φ p .f ( y ) = Q p − i =0 f ( a i b ) = Q p − i =0 ( ζ − ) i = 1. Since φ p ( t ) .f is a character of G , φ p ( t ) .f = 1. Thus b ∈ ker Φ which means τ ′ ( t ) = β in X ( ⊳ ). But β = τ − as τ ( a, b ) = ( e ∗ ∧ f ∗ )( a, b ) = ζ il − jk .Thus τ ′ ( t ) = τ ( t ) p − or ( p − ) e ∗ ∧ f ∗ in the additive notation. ByProposition 4.1(iii) τ ′ ( t ) lies on the orbit of τ ( t ) iff p − is a square. (cid:3) Hopf algebras of dimension p . . From now on we assume that H is of dimension p with an abelian group G of grouplikes of order p . Theorem 4.3.
There are p + 23 distinct nontrivial almost abelianHopf algebras of dimension p if p > , if p = 3 , e ≥ and ,otherwise. Proof:
This will carried out in steps. In the additive notation G = Z p or G = Z p ⊕ Z p , and the theory splits into two parts. LASSIFICATION OF HOPF ALGEBRAS 27 G = Z p . .There are up to isomorphism two nontrivial Z p C p -module structureson G . Namely, if C p -module G is decomposable, then G ≃ R ⊕ R ,and G ≃ R , otherwise.(I) Suppose G ≃ R ⊕ R , and let ⊳ d be the action of C p on G composed of regular actions of C p on R and R . We aim to prove Theorem 4.4.
Ext [ ⊳ d ] ( k C p , k C p ) contains p +11 isotypes of extensions p + 8 of which are nontrivial. Proof:
We carry out the procedure for computing the number of iso-types for C p -module ( G, ⊳ d ). To simplify notation we put ⊳ = ⊳ d .(1) The classifying group X ( ⊳ ). Select a basis { e, g, f } for G where { e, f } is the basis for R as in Proposition 4.1, and R = Z p g . Clearlythe matrix T of t in that basis is T = . Let { e ∗ , g ∗ , f ∗ } bethe dual basis for b G . We fix a basis { e ∗ ∧ g ∗ , e ∗ ∧ f ∗ , g ∗ ∧ f ∗ } for b G ∧ b G .We refer to the above bases as standard. Proposition 4.5. X ( ⊳ ) = h e ∗ , g ∗ i ⊕ b G ∧ b G . Proof:
Recall X ( ⊳ ) = b G C p /N ( b G ) ⊕ Alt N ( G ). We use the well knownidentification Alt( G ) = b G ∧ b G . One can see easily that the matrix of t in the standard basis of b G is T tr . By general principles [4, III,8.5] thematrix of t in the standard basis of b G ∧ b G is T tr ∧ T tr = − .It follows that ( t − p − • b G = 0 and ( t − p − • b G ∧ b G = 0, thatis N ( b G ) = 0 and ( b G ∧ b G ) N = b G ∧ b G . Further, one can see easily b G C p = h e ∗ , g ∗ i . (cid:3) (2) Groups A ( ⊳ ) , C ( ⊳ ) and G ( ⊳ ). By definition φ ∈ A ( ⊳ ) iff its matrixΦ satisfies Φ T = T Φ and det Φ = 0. By a straighforward calculationone can see that φ ∈ A ( ⊳ ) iff(4.2) Φ = a a a a a a , a ij , ∈ Z p , a a = 0It is easy to see that ( G, ⊳ k ) ≃ ( G, ⊳ ) for every k ∈ Z • p which gives C ( ⊳ ) = Z p • . Likewise one can check directly that λ k : e e, g g, f kf lies in I ( ⊳, ⊳ k ) for every k ∈ Z p • . This determines G ( ⊳ ) asthe latter is generated by A ( ⊳ ) and the λ k . (3) Orbits of A ( ⊳ ) in X ( ⊳ ). In order to simplify notation we changecoordinates of matrices (4.2) by setting u = a , v = a , a = u − q,a = u − r, a = s . We treat the tuple ( u, v, q, r, s ) as the coordinate ofeither φ or its matrix Φ. On general principles [4, III,8.5] the matricesof φ − in the standard bases for b G and b G ∧ b G are Φ tr and Φ tr ∧ Φ tr ,respectively. For Φ = Φ( u, v, q, r, s ) a routine calculation gives(4.3) Φ tr = u u − q v s u r u , and(4.4) Φ tr ∧ Φ tr = uv r u z q uv , where z = det (cid:18) u − q vs u − r (cid:19) . Next we identify X ( ⊳ ) with Z p via theassignment x = a e ∗ + a g ∗ + b e ∗ ∧ g ∗ + b e ∗ ∧ f ∗ + b g ∗ ∧ f ∗ v ( x ) = ( a , a , b , b , b ). We use the notation e ′ i , e ′′ j , i = 1 , , j = 1 , , Z p , Z p , respectively. We begin with A ( ⊳ )-orbits in b G C p and b G ∧ b G . We define Z ′ i , Z ′′ j , ≤ i ≤ , ≤ j ≤ Z ′ i = { ( a , a ) | a i = 0 and a k = 0 for k > i > } ,Z ′′ j = { ( b , b , b ) | b j = 0 and b k = 0 for k > j > } , and Z ′ = { (0 , } , Z ′′ = { (0 , , } . Furthermore we split Z ′′ into theunion of Z ′′ ,k , k = 0 , Z ′′ ,k = { ( b , ζ k b , | b ∈ Z p • } . We let κ denote an element of { , (2 , , (2 , , } . Lemma 4.6.
The sets Z ′ i , Z ′′ κ are all the orbits of A ( ⊳ ) in b G C p and b G ∧ b G , respectively. Proof:
First note Z p = ∪ Z ′ i and Z p = ∪ Z ′′ κ . The equalities e ′ i A ( ⊳ ) = Z ′ i , i = 1 , e ′′ κ A ( ⊳ ) = Z ′′ κ for κ = 1 ,
3, and ζ k e ′′ A ( ⊳ ) = Z ′′ ,k , k = 0 , (cid:3) Let us write Z ′ i × Z ′′ κ for the set of vectors ( v , v ) with v ∈ Z ′ i , v ∈ Z ′′ κ . These sets are A ( ⊳ )-stable and some of them are orbits itself. Welist those that are in Lemma 4.7.
For all ( i, κ ) = (1 , (2 , k )) , (2 , , k = 0 , Z ′ i × Z ′′ κ is anorbit. LASSIFICATION OF HOPF ALGEBRAS 29
Proof:
The claim is that for generators e ′ , e ′′ of Z ′ i , Z ′′ κ in the nonex-ceptional cases, ( e ′ , e ′′ ) generates Z ′ i × Z ′′ κ . We give details for Z ′ × Z ′′ ,other cases are treated similarly. Combining (4.3) with (4.4) we obtain(1 , , , , . A ( ⊳ ) = { ( u, , z, q, uv ) } Now for every element ( a , , b , b , b ) ∈ Z ′ × Z ′′ the equations u = a , uv = b , uvr = b , q = b , are obviously solvable. A solution to theequation z = b is provided by r = 0 and s = − v − b . (cid:3) We pick up p − Lemma 4.8.
Each set Z ′ × Z ′′ ,k , k = 0 , is a union of ( p − / orbits. Proof:
Say k = 0. By definition Z ′ × Z ′′ , = { ( a , b , b , | a ∈ Z p • , b ∈ Z p • , b arbitrary } , hence | Z ′ × Z ′′ , | = ( p − ( p − p . Forevery m ∈ Z p • we let z m = (1 ,
0; 0 , m, φ = φ ( u, v, q, r, s ) we have z m .φ − = ( u, , mr, mu , | z m A ( ⊳ ) | = ( p − p , and one can verify directly that z m . A ( ⊳ ) ∩ z n . A ( ⊳ ) = ∅ for m = n . Since there are p − orbits of this size, this caseis done. For i = 1 one should take z ′ m = (1 ,
0; 0 , ζ m, (cid:3)
We summarize
Lemma 4.9.
There are p + 8 nontrivial orbits of A ( ⊳ ) in X ( ⊳ ) . Proof:
The previous two lemmas give p + 8 nontrivial orbits. The restwill come from splitting of the remaining set Z ′ × Z ′′ . The latter isdefined as { ( a , a , b , b , b ) | a , b ∈ Z p • , a , b , b arbitrary } . For every k ∈ Z p we define w k = ( k, , , , w k .φ − = ( uk − u − q, v, z, q, uv ) . where ( u, v, q, r, s ) are the parameters of φ . This formula shows that w k .φ − does not depend on r , Setting r = 0 we have z = − sv . Itfollows easily that w k .φ − is uniquely determined by ( u, v, q, s ), hence | w k . A ( ⊳ ) | = ( p − p . Furthermore, we claim that w k . A ( ⊳ ) ∩ w l . A ( ⊳ ) = ∅ for k = l . For, suppose( uk − u − q, v, − sv, q, uv ) = ( u ′ l − u ′− q ′ , v ′ , − s ′ v ′ , q ′ , u ′ v ′ )for some ( u, v, q, s ) and ( u ′ , v ′ , q ′ , s ′ ). Then v = v ′ , q = q ′ give u = u ′ ,hence uk = ul and therefore k = l , a contradiction. We conclude that | ∪ ≤ k ≤ p − w k . A ( ⊳ ) | = p ( p − . As this is the number of elements in Z ′ × Z ′′ , the proof is complete. (cid:3) (4) Orbits of G ( ⊳ ). We need to know the action of ω k = λ k α − k where λ k are defined in part (2). Set l = k − (mod p ). Lemma 4.10.
Action of ω k is described by e ∗ .ω k = le ∗ , g ∗ .ω k = lg ∗ e ∗ ∧ g ∗ .ω k = le ∗ ∧ g ∗ e ∗ ∧ f ∗ .ω k = l e ∗ ∧ f ∗ g ∗ ∧ f ∗ .ω k = − (cid:18) l (cid:19) e ∗ ∧ g ∗ + l g ∗ ∧ f ∗ Proof:
By (3.11) for x ∈ X ( ⊳ ), x.ω k = ( φ l • x ) .λ k . For x = e ∗ , g ∗ , e ∗ ∧ g ∗ , e ∗ ∧ f ∗ φ l • x = lx as these elements are fixed by C p . Because( t − • b G ∧ b G = 0 we expand φ l in powers of t −
1, namely φ l = l + (cid:0) l (cid:1) ( t −
1) + higher terms. One can check ( t − • g ∗ ∧ f ∗ = − e ∗ ∧ g ∗ which gives φ l • g ∗ ∧ f ∗ = lg ∗ ∧ f ∗ − (cid:18) l (cid:19) e ∗ ∧ g ∗ , By definition of λ k its matrix is Λ k = diag(1 , , k ) (that is the diagonalmatrix with entries 1 , , k ). It follows (see part (3)) that the matrix of λ k in the standard basis of b G is (Λ − k ) tr = diag(1 , , l ). Applying λ k to φ l • x as x runs over the standard bases of b G and X ( ⊳ ) we complete theproof of the Lemma. (cid:3) The next Proposition completes the proof of Theorem 4.4.
Proposition 4.11.
The sets of G ( ⊳ ) and A ( ⊳ ) -orbits coincide. Proof:
By Corollary 3.13 for every x ∈ X ( ⊳ ), x G ( ⊳ ) is a union of orbits x.ω k A ( ⊳ ) for 1 ≤ k ≤ p −
1. Thus it suffices to show x.ω k ∈ x A ( ⊳ )for every k and generators x of every orbit of A ( ⊳ ). We give a samplecalculation for x = w m of Lemma 4.9. By Lemma 4.10 w m .ω k = ( lm, l, − (cid:18) l (cid:19) , , l ) . Now take φ with coordinates u = l, v = l, q = 0 , r = 0 , s = l − (cid:0) l (cid:1) .Then by (4.5) w m .φ − = w m .ω k as needed. (cid:3) .We move on to the next case(II) G ≃ R . We denote by ⊳ r the right multiplication in R . Thiscase is sensitive to the prime p . Let us agree to write X p for X ( ⊳ r ) if G is a p -group. For r ∈ Z p C p we denote by r the image of r in R . Theelements e = 1 , f = ( t − , g = ( t − form a basis for R in whichaction of t is defined by T = . Let { e ∗ , f ∗ , g ∗ } be the dual LASSIFICATION OF HOPF ALGEBRAS 31 basis for b G , and { e ∗ ∧ f ∗ , e ∗ ∧ g ∗ , f ∗ ∧ g ∗ } the induced basis for b G ∧ b G .We call all these bases standard. We aim to prove Theorem 4.12.
For p > [ ⊳ r ] ( k C p , k C p ) contains p + 9 isoclasses, p + 7 of which are nontrivial, and three nontrivial isoclasses if p = 3 . Proof:
Proof will be carried out in steps following the procedure forcomputing the number of isoclasses.(1) Classifying groups X p . Lemma 4.13. If p = 3 , then X = h e ∗ ∧ f ∗ , e ∗ ∧ g ∗ i For every p > X p = Z p e ∗ ⊕ b G ∧ b G Proof:
The matrices of t in the standard bases of b G and b G ∧ b G are T tr and T tr ∧ T tr , respectively, with T tr ∧ T tr = . Fromthis one computes directly ( t − • b G = ( t − • b G ∧ b G = 0. Since φ p ( t ) = ( t − p − , it follows that N ( G ) = 0 and ( b G ∧ b G ) N = b G ∧ b G for any p >
3. Furhermore b G C p = Z p e ∗ for every p . Thus as X p = b G C p /N ( b G ) ⊕ ( b G ∧ b G ) N the second statement of the Lemma follows.Say p = 3. Then N ( b G ) = ( t − • b G = Z p e ∗ , hence b G C p /N ( b G ) = 0.Another verification gives ( b G ∧ b G ) N = h e ∗ ∧ f ∗ , e ∗ ∧ g ∗ i . (cid:3) (2) Groups A ( ⊳ r ) and C ( ⊳ r ). For any ring R with unity viewedas a right regular R -module and any right R -module M the mapping λ M : M → Hom R ( R, M ) defined by x.λ M ( m ) = mx, x ∈ R is an R -isomorphism. Setting M = R = R we have A ( ⊳ r ) = { λ R ( m ) | m ∈ R } . Expand m in the standard basis of R , m = ue + qf + rg . Then thematrix of φ = λ R ( m ) is Φ = u q r u q u . The matrices of mappingsinduced by φ − in b G and b G ∧ b G are Φ tr and Φ tr ∧ Φ tr . Explicitly(4.6) Φ tr = u q u r q u and Φ tr ∧ Φ tr = u uq u q − ur uq u We will show that C ( ⊳ r ) = Z p • by constructing a family of isomor-phisms λ k : ( G, ⊳ r ) → ( G, ⊳ kr ) for every k ∈ Z p • . To this end, let us take M = ( R , ⊳ kr ) and set λ k = λ M ( e ). By definition of λ k we have e.λ k = e, f.λ k = e ( t k − , g.λ k = e ( t k − Using the expansion t k − k ( t −
1) + (cid:0) k (cid:1) ( t − (mod ( t − ) weconclude that Λ k = k (cid:0) k (cid:1) k is the matrix of λ k in the standardbasis. We shall need an explicit form of the associated matrices de-scribing the action of λ k in b G and b G ∧ b G , respectively. Put l = k − (mod p ) as usual. Then an easy calculation gives(4.7) (Λ − k ) tr = l (cid:0) l (cid:1) l , (4.8) (Λ − k ) tr ∧ (Λ − k ) tr = l (cid:0) l (cid:1) l
00 0 l . Unless stated otherwise we assume below that p >
3. The degeneratecase p = 3 follows easily from the general one.(3) Orbits of A ( ⊳ r ) in X p . We identify X p with Z p via x = ae ∗ + b e ∗ ∧ f ∗ + b e ∗ ∧ g ∗ + b g ∗ ∧ f ∗ ( a, b , b , b ). We begin by listing allorbits in b G C p and b G ∧ b G , respectively: Z ′ = { (0) } , Z ′ = { ( a ) | a = 0 } , Z ′′ = { (0 , , } ,Z ′′ ij = { ( ∗ , . . . , ∗ , ζ j b i , , . . . , | b i ∈ Z p • } , i = 1 , , j = 0 , ∗ denotes an arbitrary element of Z p . For more complexorbits we need vectors v k ( m ) = (1 , , . . . , m, . . . , ∈ Z with the m filling the ( k + 1)th slot, k = 1 , , Z p • . Lemma 4.14.
There are p + 5 orbits of A ( ⊳ r ) in X p , namely Z ′ × Z ′′ , Z ′ × Z ′′ , Z ′ × Z ′′ ij , and v k ( m ) A ( ⊳ r ) , k = 1 , , Proof:
The first two sets are clearly orbits. By (4.6) and every i, j (0 , . . . , ζ ji +1 , , . . . , .φ = (0 , ∗ , . . . , ∗ , ζ j u , . . . ,
0) with the ∗ standingfor an arbitrary element of Z p . This shows Z ′ × Z ′′ ij is the orbit of(0 , . . . , ζ ji +1 , , . . . , v k ( m ) .φ = ( u, ∗ , . . . , ∗ , u m, , . . . , v k ( m ) A ( ⊳ r ) has ( p − p k − elements.Another verification gives v k ( m ) A ( ⊳ r ) ∩ v k ( n ) A ( ⊳ r ) = ∅ for m = n . Let LASSIFICATION OF HOPF ALGEBRAS 33 us define Z ′′ i = Z ′′ i ∪ Z ′′ i and observe that | Z ′′ i | = ( p − p i − whichgives | Z ′ × Z ′′ i | = ( p − p i − . Evidently v i ( m ) ∈ Z ′ × Z ′′ i for all m andtherefore comparing cardinalities we arrive at the equality Z ′ × Z ′′ i = S m v i ( m ) A ( ⊳ r ). But clearly X p = S Z ′ l × Z ′′ i , l = 0 ,
1; 0 ≤ i ≤ (cid:3) (4) End of the proof. Proposition 4.15.
The nonzero orbits of G ( ⊳ r ) in X p are as follows: Z ′ × Z ′′ ij , Z ′ × Z ′′ , Z ′ × Z ′′ , Z ′ × Z ′′ , Z ′ × Z ′′ j , and v ( m ) A ( ⊳ r ) , where i = 1 , , j = 0 , and m runs over Z p • . Proof:
By Corollary 3.13 we need to determine the A ( ⊳ r )-orbit con-taining vω k where v runs over a set of generators of A ( ⊳ r )-orbits ofLemma 4.14, and ω k = λ k α − k , ≤ k ≤ p − A ( ⊳ r )-orbits Z ′ × Z ′′ and Z ′ × Z ′′ ij generators are e ∗ and v ij = (0 , , . . . , ζ i +1 j , . . . , e ∗ and e ∗ ∧ f ∗ beingfixed points for the action of t , and by (4.7), (4.8) it is immediate that(4.10) e ∗ ω k = le ∗ and v j ω k = l v j , hence Z ′ × Z ′′ j and Z ′ × Z ′′ are G ( ⊳ r )-orbits.(ii) Next we take the generator v = e ∗ ∧ g ∗ . Noting that ( t − • e ∗ ∧ g ∗ = 0, we use the expansion φ l = l + (cid:0) l (cid:1) ( t −
1) (mod ( t − ) toderive φ l • e ∗ ∧ g ∗ = ce ∗ ∧ f ∗ + le ∗ ∧ g ∗ , c ∈ Z p . Applying λ k to the last equation we find with the help from (4.8)(4.11) e ∗ ∧ g ∗ ω k = c ′ e ∗ ∧ f ∗ + l e ∗ ∧ g ∗ , for some c ′ ∈ Z p . The last equation shows that v .ω k ∈ v A ( ⊳ r ) if l , hence k , is nota square, and v .ω k ∈ v A ( ⊳ r ), otherwise. This means v G ( ⊳ r ) = Z ′ × ( Z ′′ ∪ Z ′′ ) = Z ′ × Z ′′ as needed.The argument for the generator v j = (0 , , , ζ j ) = ζ j f ∗ ∧ g ∗ of Z ′ × Z ′′ j is almost identical. Using the expansion φ l = l + c ( t − c ( t − (mod ( t − ) we derive φ l • f ∗ ∧ g ∗ = ( c + c ) e ∗ ∧ f ∗ + c e ∗ ∧ g ∗ + lf ∗ ∧ g ∗ .Applying λ k we have by (4.8)(4.12) f ∗ ∧ g ∗ .ω k = c ′ e ∗ ∧ f ∗ + c l e ∗ ∧ g ∗ + l f ∗ ∧ g ∗ , c ′ , c ∈ Z p . which shows ζ j f ∗ ∧ g ∗ .ω k ∈ Z j for every k , hence Z ′ × Z ′′ j is a G ( ⊳ r )-orbit.(iii) We pause to mention that the above arguments settle the p = 3-case. For, since X = h e ∗ ∧ f ∗ , e ∗ ∧ g ∗ i , by parts (i) and (ii) it has threenonzero orbits, namely Z ′′ j , Z ′′ , j = 0 , (iv) Here we take v ( m ) = (1 , m, , v ( m ) .ω k = ( l, l m, , ∈ v ( m ) A ( ⊳ r ) by (4.9). That is, v ( m ) A ( ⊳ r ) isa G ( ⊳ r )-orbit for every m ∈ Z p • .It remains to show that the last three sets of the Proposition are G ( ⊳ r )-orbits.(v) Z ′ × Z ′′ is an orbit. By Lemma 4.14 Z ′ × Z ′′ = S m v ( m ) A ( ⊳ r )where v ( m ) = e ∗ + me ∗ ∧ g ∗ . Note that by (4.10) and (4.11) there holds v ( m ) .ω k = ( l, c ′ , l m, v ( n ) .φ =( u, uq, u n,
0) where u, q run over Z • p and Z p , respectively. For every l choosing φ = φ ( l, u − c ′ ,
0) and n = lm we obtain v ( m ) .ω k = v ( n ) φ − .Letting k hence l run over Z • p we see that v ( m ) G ( ⊳ r ) = S n v ( n ) A ( ⊳ r )which completes the proof.(vi) Here we show that each Z ′ × Z ′′ j is an orbit. By (4.10) and(4.12) v ( m ) .ω k = (1 , , , m ) .ω k = ( l, c ′ , c ′′ , ml ) for some c ′ , c ′′ ∈ Z p . We seek an n such that(4.13) v ( m ) .ω k = v ( n ) .φ for some φ ∈ A ( ⊳ r ) . By (4.6) v ( n ) .φ = ( u, q − ur, uq, u n ) where u, q, r take arbitraryvalues in Z • p and Z p , respectively. Choosing u, q, r such that u = l, q − ur = c ′ , uq = c ′′ and n = ml fullfils (4.13). This yields theequality (*) v ( m ) G ( ⊳ r ) = [ n ∈ m Z • p (1 , , , n ) A ( ⊳ r ). Therefore dependingon m ∈ Z p • , or m / ∈ Z p • the right hand side of (*) equals to Z ′ × Z ′′ or Z ′ × Z ′′ , respectively. (cid:3) G = Z p e ⊕ Z p . .Our immediate goal is to classify nontrivial Hopf algebras inExt( k C p , k Z p ⊕ Z p ). We find it convenient to enlarge the scope of theproblem by taking G = Z p e ⊕ Z p for any e ≥ e = 2. As before our prime is odd, the even case isdone in [11]. The end result is- Theorem 4.16.
There are p +8 distinct Hopf algebras in Ext( k C p , k G ) if either p > or e ≥ , and if p = 3 and e = 2 . Proof:
We break up the proof in steps.(1) Our first task is to describe the set of classes [ ⊳ ] and their as-sociated groups A ( ⊳ ) , C ( ⊳ ). We need several preliminary observations.Every representation ⊳ : C p → Aut( G ) is determined by ⊳ ( t ). Letus write Γ e = Aut( Z p e ⊕ Z p ) and Γ e ( p ) for the set of all elements of LASSIFICATION OF HOPF ALGEBRAS 35 order p in Γ e . It is clear that the mapping ⊳ ⊳ ( t ) sets up a bi-jection between the set { ⊳ } and Γ e ( p ), and we will identify both sets.Furthermore the class eq( ⊳ ) of representations equivalent to ⊳ corre-sponds to the Γ e -conjugacy class of ⊳ ( t ) denoted ⊳ ( t ) Γ . It follows that[ ⊳ ] = S ≤ k ≤ p − ⊳ ( t k ) Γ . G has a natural basis e , e comprised of generators of Z p e , Z p , re-spectively. Let ǫ be an endomorphism of G . We use the standardmatrix representation of endomorphisms of direct sums to associateto ǫ a matrix M ( ǫ ) = (cid:18) a bcp e − d (cid:19) relative to the basis { e , e } with a, b, c, d ∈ Z p e and the bar over an n ∈ Z p e denoting the image of n in Z p . The correspondence ǫ M ( ǫ ) extends to an isomorphism underthe multiplication rule (cid:18) a bcp e − d (cid:19) (cid:18) a ′ b ′ c ′ p e − d ′ (cid:19) = (cid:18) aa ′ + c ′ bp e − ab ′ + bd ′ ( ca ′ + dc ′ ) p e − dd ′ (cid:19) Lemma 4.17. Γ e is the set of all matrices (cid:18) a bcp e − d (cid:19) satisfying ad = 0 Proof:
The natural epimorphism G → Z p ⊕ Z p induces a homomor-phism π : Γ e → Aut( Z p ) via (cid:18) a bcp e − d (cid:19) (cid:18) a b d (cid:19) . If γ is invertiblethen so is π ( γ ), and the latter is equivalent to ad = 0. Conversely, if ad = 0, then a, d are units in Z p e . One can check easily a factorization(4.14) (cid:18) a bcp e − d (cid:19) = (cid:18) a − cp e − (cid:19) (cid:18) a d (cid:19) (cid:18) a a − b d (cid:19) which completes the proof. (cid:3) Lemma 4.18. (i) Γ e ( p ) is the set of all matrices (cid:18) ip e − jkp e − (cid:19) ; (ii) | Γ e ( p ) | = p regardless of e ; (iii) Γ e ( p ) is a normal subgroup of Γ e . Proof: (i) Assume M = (cid:18) a bcp e − d (cid:19) has order p . Then π ( M ) has alsoorder p which implies a p = 1 = d p , hence d = 1 and a ≡ p ). Astraightforward induction on r gives(4.15) M r = (cid:18) a r + bc (cid:0) r (cid:1) p e − rbrcp e − (cid:19) whence M p = I iff a p = 1. But this condition on a is equivalent to a = 1 + ip e − . (ii) and (iii) are easy consequences of (i). (cid:3) By the above Lemma Γ e ( p ) does not depend on e . We will omit e from its notation below. Remark 4.19.
All parts of this Lemma fail for p = 2. Proposition 4.20.
The set { [ ⊳ ] } consists of five nontrivial elements. Proof: (1) The first class of action is the one generated by ⊳ with ⊳ ( t ) = diag(1 + p e − , p e − , kp e − ,
1) form the center of Γ( p ). Since ⊳ ( t k ) = diag(1 + kp e − ,
1) it follows that [ ⊳ ] = { ⊳ k | ≤ k ≤ p − } .As ⊳ ( t ) is in the center A ( ⊳ ) = Γ e , C ( ⊳ ) = { } hence G ( ⊳ ) = A ( ⊳ ).(2) Let T ℓ be the subset of lower triangular matrices in Γ( p ), Z the center of Γ( p ) and T ′ ℓ = T ℓ \ Z . Fix one action ⊳ ℓ defined by ⊳ ℓ ( t ) = (cid:18) p e − (cid:19) . Lemma 4.21. (i) T ′ ℓ = ⊳ Γ ℓ ; (ii) I ( ⊳ ℓ , ⊳ kℓ ) = ∅ for every k . In particular, diag (1 , k − ) ∈ I ( ⊳ ℓ , ⊳ kℓ ) ; (iii) A ( ⊳ ℓ ) = (cid:26)(cid:18) a cp e − a (cid:19)(cid:27) and C ( ⊳ ℓ ) = A p . Proof: (i) Pick another action ⊳ with ⊳ ( t ) = (cid:18) ip e − jp e − (cid:19) , j = 0.Matrices (cid:18) kp e − (cid:19) lie in the center of T ℓ . By (4.14) ⊳ Γ ℓ equals to { ⊳ γℓ } where γ runs over all upper triangular matrices in Γ e . Choose a γ = (cid:18) a b d (cid:19) and observe that γ ∈ I ( ⊳ ℓ , ⊳ ) iff (*) ⊳ ℓ ( t ) γ = γ ⊳ ( t ). Onecan see by a direct calculation that (*) holds iff ai + bj ≡ p ) a ≡ jd (mod p ) . These congruences are equivalent to the conditions b ≡ − aij − (mod p ), d ≡ aj − (mod p ) which gives (**) I ( ⊳ ℓ , ⊳ ) = { (cid:18) a − aij − cp e − aj − (cid:19) } .(ii) Take ⊳ = ⊳ kℓ and observe that i = 0 , j = k for this action.Specifying a = 1 , c = 0 in (**) yields (2).(iii) Set ⊳ = ⊳ ℓ and note that i = 0 , j = 1 in this case. Then (**)gives the assertion. (cid:3) LASSIFICATION OF HOPF ALGEBRAS 37 (3) It remains to describe conjugacy classes in Γ( p ) \ T ℓ . Elementsof this set are distinguished by the property- Lemma 4.22. ⊳ ( t ) ∈ Γ( p ) \ T ℓ iff the C p -module ( G, ⊳ ) is cyclic. Proof:
In one direction take ⊳ ( t ) = (cid:18) ip e − jkp e − (cid:19) ∈ Γ( p ) \ T ℓ . Then j = 0 and therefore from e ⊳ t = (1 + ip e − ) e + je we have e = j − e ⊳ ( t − (1 + ip e − )) showing that G is generated by e .Conversely, assume j = 0. The subgroup h pe i is a C p -submoduleof G . Further, G/ h pe i is a trivial C p -module isomorphic to Z p ⊕ Z p which proves ( G, ⊳ ) is not cyclic. (cid:3)
We associate to an action ⊳ ∈ Γ( p ) \ T ℓ with ⊳ ( t ) = (cid:18) ip e − jkp e − (cid:19) the element m ( ⊳ ) = jk of Z p e . For an n ∈ Z p e we define I ( n ) to be theideal of R generated by p ( t − , ( t − − np e − and ( t − . m ( ⊳ ) isan invariant of ⊳ ( t ) Γ according to Lemma 4.23. (i)
In the foregoing notation ( G, ⊳ ) ≃ R/I ( m ) . (ii) Two actions ⊳, ⊳ ′ in Γ( p ) \ T ℓ are equivalent iff m ( ⊳ ) = m ( ⊳ ′ ) . Proof:
Let R = Z p e C p . Since G = e R by the preceeding Lemma,both the assertions follow from the equality I ( m ) = ann R e for m = m ( ⊳ ). In one direction, a simple calculation gives that pe is a fixedpoint and e ⊳ ( t − = jkp e − e . It follows that e ⊳ g ( t ) = 0 for everygenerator g ( t ) of I ( m ) from the above list, whence I ( m ) ⊂ ann R e . Inthe opposite direction we note every element of R is congruent to some n + m ( t − , n, m ∈ Z p e modulo I ( m ). Were ann R e = I ( m ), therewould be an n + m ( t −
1) with e ⊳ ( n + m ( t − n = 0 or m p ). But e ⊳ ( n + m ( t − n + mip e − ) e + mje = 0 holdsiff m ≡ p ) and n = 0 proving the equality in question. (cid:3) We single out three actions in Γ( p ) \ T ℓ ,(4.16) ⊳ = (cid:18) p e − (cid:19) , ⊳ = (cid:18) p e − (cid:19) , ⊳ ζ = (cid:18) ζp e − (cid:19) . The next lemma completes the proof of the Proposition
Lemma 4.24. Γ( p ) \ T ℓ is the union of [ ⊳ ] , [ ⊳ ] and [ ⊳ ζ ] . Proof:
By the formula (4.15) we have m ( ⊳ r ) = r m ( ⊳ ). The preceed-ing Lemma makes it clear that sets [ ⊳ q ] , q = 0 , , ζ correspond to theorbits of Z p • in Z p , namely { } , Z p • , ζ Z p • . (cid:3) (4) We complete the proof of the main theorem of this section bycomputing the classifying groups and orbits for each of the five classesof actions. To begin with we select a basis for b G dual to { e i } denoted by { e ∗ i } . C p and Γ e act in b G by (1.4) and ( f.γ )( g ) = f ( gγ − ) , f ∈ b G, g ∈ G ,respectively. These actions extend to Alt( G ) = b G ∧ b G in the usual way.We note that Alt( G ) is generated by β = e ∗ ∧ e ∗ and the latter formhas order p . For the future references we record Lemma 4.25. (i)
Let (cid:18) a bcp e − d (cid:19) be the matrix of either γ ∈ Γ or t relative to { e i } . The matrix of γ − or t relative to { e ∗ i } is (cid:18) a cbp e − d (cid:19) (ii) There holds β.γ − = adβ, t.β = β , and Alt N ( G ) = Alt( G ) . Proof: (i) is seen by a simple calculation. For (ii) we use part (i)to calculate e ∗ ∧ e ∗ .γ − = ( ae ∗ + ce ∗ ) ∧ ( bp e − e ∗ + de ∗ ) = ade ∗ ∧ e ∗ .Similartly t.e ∗ ∧ e ∗ = ade ∗ ∧ e ∗ . However in the case of t , a = 1 + ip e − and d = 1 by Lemma 4.18, which gives the second formula. Therefore φ p ( t ) .β = pβ = 0 which proves the last assertion. (cid:3) (i) We take up the action ⊳ of Proposition 4.20(1). Lemma 4.26.
Ext [ ⊳ ] ( k C p , k G ) contains two distinct nontrivial Hopfalgebras. Proof:
A simple calculation gives b G C p = h pe ∗ , e ∗ i . As for N ( b G ) wehave φ p ( t ) .e ∗ = pe ∗ = 0 and φ p ( t ) .e ∗ = ( P p − i =0 (1 + p e − ) i ) e ∗ = pe ∗ . Itfollows that b G C p /N ( b G ) = h e ∗ i where e ∗ = e ∗ + N ( b G ). As noted inProposition 4.20(1) G ( ⊳ ) = A ( ⊳ ) = Γ e . By Lemma 4.25 e ∗ .γ − = de ∗ and β.γ − = adβ . We conclude that X ( ⊳ ) ≃ Z p ⊕ Z p with theaction ( c , c ) .γ = ( dc , adc ). Now it is immediate that there are twonontrivial (i.e. c = 0) orbits, viz. { (0 , c ) } and { c , c | c c = 0 } . (cid:3) (ii) Next we consider ⊳ ℓ from Proposition 4.20(2). Lemma 4.27.
There are p + 1 distinct nontrivial Hopf algebras in Ext [ ⊳ ℓ ] ( k C p , k G ) . Proof:
One can see easily with the help from Lemma 4.25 b G C p = h pe ∗ , e ∗ i . Further N ( e ∗ ) = pe ∗ = 0 and N ( e ∗ ) = pe ∗ . All in all wehave b G C p /N ( b G ) = h e ∗ i and X ( ⊳ ℓ ) = h e ∗ , β i . Using definition (3.11)we have e ∗ .ω k = ( φ k − ( t ) .e ∗ ) .λ k where λ k = diag(1 , k − ) by Lemma4.21. Since e ∗ is a fixed point, φ k − ( t ) .e ∗ = k − e ∗ and by Lemma4.25 e ∗ .λ k = ke ∗ , hence e ∗ is fixed by ω k . A similar calculation gives β.ω k = β . Thus G ( ⊳ ℓ )-orbits coincide with A ( ⊳ ℓ )-orbits. For the latter LASSIFICATION OF HOPF ALGEBRAS 39 we take φ ∈ A ( ⊳ ℓ ) as in Lemma 4.21(iii) and apply Lemma 4.25 to get e ∗ .φ − = ae ∗ and β.φ − = a β . It transpires that X ( ⊳ ℓ ) ≃ Z p with theaction on the right by ( c , c ) .φ − = ( ac , a c ). Now the argument inProposition 4.1 completes the proof. (cid:3) (iii) Finally we tackle actions (4.16). We determine the groups A ( ⊳ q ) , C ( ⊳ q ) , q = 0 , , ζ and sets of intertwiners { λ k | k ∈ C ( ⊳ q ) } . Lemma 4.28. (i) A ( ⊳ q ) = (cid:26)(cid:18) a bbqp e − a (cid:19)(cid:27) ; (ii) C ( ⊳ ) = A p and for every ≤ k ≤ p − I ( ⊳ , ( ⊳ ) k ) ∋ (cid:18) k (cid:19) ; (iii) If q = 0 , then C ( ⊳ q ) = { , p − } and I ( ⊳ q , ( ⊳ q ) p − ) ∋ (cid:18) qp e − − (cid:19) Proof: (i) A ( ⊳ q ) is the group of units of End R ( R/I ( q )). We pointedout in Theorem 4.12(2) that End R ( R/I ( q )) consists of mappings λ ( u ) : x ux, u, x ∈ R/I ( q ). By Lemma 4.23(i) u = a b ( t − r = r + I ( q ) for r ∈ R . It is immediate that the matrix of λ ( u )relative to { , ( t − } is the one in part (i).(ii) and (iii) By Lemma 4.23 C ( ⊳ q ) = { k | k q = q } . Clearly thisformula implies C ( ⊳ ) = A p and C ( ⊳ q ) = { , p − } for q = 0. Letus write R = R/I ( q ) and denote by R ( k ) the C p -module ( R, ( ⊳ q ) k ).By general principles for every k ∈ C ( ⊳ q ), Hom R ( R, R ( k ) ) consists ofmappings λ ( u ) , u ∈ R . Pick λ (1) and observe that for every suitable k the matrices of λ (1) in the basis { , t − } are as given in (ii) and (iii),respectively. (cid:3) The last step of the proof of Theorems 4.16 and 4.3 is-
Lemma 4.29. (i)
There are p + 1 nontrivial distinct Hopf algebras in Ext [ ⊳ ] ( k C p , k G ) ; (ii) There are two nontrivial distinct Hopf algebras in
Ext [ ⊳ q ] ( k C p , k G ) for q = 1 , ζ if either p > or e ≥ , and four otherwise. Proof: (i) One can see easily that b G C p ( ⊳ ) = h e ∗ i and N ( b G ( ⊳ )) = pe ∗ , hence b G C p ( ⊳ ) /N ( b G ( ⊳ )) = h e ∗ i . By Lemma 4.25(ii) X ( ⊳ ) = h e ∗ , β i . Pick a γ ∈ A ( ⊳ ) as in Lemma 4.28. By Lemma 4.25 thereholds e ∗ .γ − = ae ∗ and β.γ − = a β . This type of action occured inProposition 4.1 whose argument yields p + 1 nontrivial A ( ⊳ )-orbits.Turning to G ( ⊳ )-orbits, pick a λ k = diag(1 , k ) from the preceedinglemma. Since e ∗ , β are fixed by t we have e ∗ .ω k = ( φ k − .e ∗ ) .λ k = k − e ∗ and β.ω k = ( φ k − .β ) .λ k = k − β . This shows that G ( ⊳ )-orbits coincidewith A ( ⊳ ) ones, and the proof is complete. (ii) A straighforward calculation gives b G C p ( ⊳ q ) = h pe ∗ i . For calcula-tion of N ( b G ( ⊳ q )) we employ (4.15) which gives readily that φ p ( t ) .e ∗ = [ p − X r =0 (1 + q (cid:18) r (cid:19) p e − )] e ∗ + ( p − X r =0 r ) e ∗ As P p − r =0 r = (cid:0) p (cid:1) and pe ∗ = 0 we conclude φ p ( t ) .e ∗ = ( p + q ( P p − r =0 (cid:0) r (cid:1) ) p e − ) e ∗ . Similarly one can derive φ p ( t ) .e ∗ = q ( p − X r =0 r ) p e − e ∗ + pe ∗ = 0Next we note that an elementary calculation gives P p − r =0 (cid:0) r (cid:1) = (cid:0) p (cid:1) . Letus put c ( p ) = p + q (cid:0) p (cid:1) p e − . We observe that if p >
3, then c ( p ) ≡ p (mod p e ). For p = 3 and either e ≥ e = 2 and q = 1, c (3) = 3 u ,where u is a unit in Z p e . In the exceptional case e = 2 and q = 2, c (3) = 9. This translates into φ p ( t ) .e ∗ = pue ∗ for all p, e, q , except forthe exceptional case where φ ( t ) .e ∗ = 0. We conclude that N ( b G ( ⊳ q )) = h pe ∗ i in the regular case and it is zero, otherwise. In consequence X ( ⊳ ) = h β i for all p, e X ( ⊳ ζ ) = h β i if p > e ≥ X ( ⊳ ) = h e ∗ , β i if p = 2 = e. By Lemmas 4.25(ii), 4.28(i) β.φ − = a β for every φ ∈ A ( ⊳ q ). Itfollows that there are two nontrivial A ( ⊳ q )-orbits in X ( ⊳ q ) in the regularcase and also for X ( ⊳ ) in all cases, namely { cqβ | c ∈ Z p • for q =1 , ζ } . Using Lemma 4.28(iii) it is immediate that β.ω p − = β . Thatsays G ( ⊳ q )-orbits coinside with A ( ⊳ q )-orbits. In the exceptional case3 e ∗ .φ − = a (3 e ∗ ) and 3 e ∗ .ω = − e ∗ . It follows that A ( ⊳ ) and G ( ⊳ )act on X ( ⊳ ) by ( c , c ) .φ − = ( ac , a c ) and ( c , c ) .ω = ( − c , c ) withthe usual identification X ( ⊳ ) ≃ Z . By the argument of Proposition4.1 there are four nontrivial A ( ⊳ )-orbits. One can check directly thatthe mapping ( c , c ) ( − c , c ) preserves the orbits, completing theproof. (cid:3) Some old classification results revisited
The first result concerns the G. Kac’s 8-dimensional Hopf algebra[8, 19] which we denote by H . Theorem 5.1.
There is a unique semisimple, nontrivial -dimensionalHopf algebra. LASSIFICATION OF HOPF ALGEBRAS 41
Proof:
It is easy to see that every Hopf algebra H as in the Theorem isisomorphic to k ⊕ M ( k ) as algebra where M ( k ) is the algebra of 2 × H ∗ we conclude that H ∗ has exactly4 characters, hence G ( H ) has order 4. Thus H is almost abelian, hence H ∈ Ext( k C , k G ( H ) ). By Theorem 3.12(II) the number of nontrivialisotypes in Ext [ ⊳ ] ( k C , k G ) equals to the number of nontrivial A ( ⊳ )-orbits in H c ( ⊳ ) for every action ⊳ of C on G . By Corollary 2.4 thatnumber coincides with the number of nontrivial A ( ⊳ )-orbits in X ( ⊳ ).For every cyclic group C n , Alt( C n ) is trivial. Hence, were G = C we would have X ( ⊳ ) = b G C /N ( b G ) by Lemma 3.16(ii) and therefore X ( ⊳ ) does not have nontrivial orbits. We take up the remaining case G = G ( H ) = C × C . Let { x , x } be a basis for G and { x ∗ , x ∗ } itsdual. There is only one equivalence class of actions on G . We choose theaction x ⊳t = x , x ⊳t = x . A routine verification gives b G C = N ( b G ) = h x ∗ x ∗ i . Thus by Lemma 3.16 X ( ⊳ ) ≃ a ( Z N ( ⊳ )) and by Proposition2.5(3) we have a ( Z N ( ⊳ )) = Alt N ( G ). Further, it is immediate thatAlt N ( G ) = Alt( G ) and the latter consists of one nonzero element.This shows that X ( ⊳ ) has one nontrivial A ( ⊳ )-orbit, and the proof iscomplete. (cid:3) With a small additional effort one can give a presentation of H bygenerators and relations. For two vectors a = x j x j , b = x k x k we letdet( a, b ) = j k − j k . Proposition 5.2. H is generated as algebra by x ∗ , x ∗ , t subject to therelations x ∗ = x ∗ = t = 1 tx ∗ t − = x ∗ , tx ∗ t − = x ∗ The coalgebra structure is specified by ∆( t ) = ( X a,b ∈ G ι − det( a,b ) p a ⊗ p b ) t ⊗ t, where ι = − . In addition the equations S ( x ∗ i ) = x ∗ i , i = 1 , , S ( t ) = t and ǫ ( x ∗ ) = ǫ ( x ∗ ) = ǫ ( t ) = 1 determine the antipode and augmentation. Proof:
Since H is a special cocentral extensions H = k b G k C asalgebra. With t a generator of C the algebra relations follow immedi-ately. By (1.8)∆( t ) = ( X a,b ∈ G τ ( t, a, b ) p a ⊗ p b ) t ⊗ t where τ ( t, a, b ) ∈ X ( ⊳ ). As X ( ⊳ ) hasonly one nonzero element, the latter provided by Proposition 2.5(3ii),we have τ ( t, a, b ) = s , δg . A straightforward calculation gives τ ( t, a, b ) = ι − det( a,b ) .We find the antipode by using [23, Prop. 4.7]. In our case, i.e.for a special cocentral extension, the formula specializes to S ( p a t ) = τ − ( t, a − , a ) p a − ⊳t t − . Since a = 1 and τ ( t, a, a ) = 1, we obtain S ( t ) = P a S ( p a t ) = P a p a⊳t t = t . The rest of the Proposition is self-evident. (cid:3) A. Masuoka [19] presents H by a different set of generators andrelations. The two are related by replacing t with z = gx ∗ t . The set { x ∗ , x ∗ , z } generates H and one can derive all relations of [19, Thm.2.13], with one exception, viz. S ( z ) = ( − ǫ + x ∗ + x ∗ + x ∗ x ∗ ) z . Weleave the details to the reader.We take up the problem of classifying isotypes of Hopf algebras H of dimension 2 n with G ( H ) = Z n × Z n for an odd n . Put differentlywe want to determine the isotypes of Ext( k C , k Z n × Z n ). We let G = Z n × Z n Following the general procedure we split up the argument into steps.(1) A survey of actions.We will assume n = p e · · · p e m m is the prime decomposition of n .We let G ( i ) denote the p i -primary summand of G . Clearly G ( i ) = Z p eii ⊕ Z p eii and G = ⊕ G ( i ). Every G ( i ) is invariant under any auto-morphism of G , in particular under any action of C . Since every p i is odd Z p eii C = Z p eii ǫ ⊕ Z p eii ǫ − where ǫ = t , ǫ − = − t . Idempo-tents ǫ ν induce a splitting G ( i ) = G ( i ) ǫ ⊕ G ( i ) ǫ − into a direct sum ofsubgroups on which t acts as ± id. Therefore for every action ⊳ we canwrite G as(5.1) G = G ⊕ G − ⊕ G , − , where G = ⊕{ G ( i ) | t | G ( i ) = id } , G − = ⊕{ G ( i ) | t | G ( i ) = − id } , and G , − = ⊕{ G ( i ) | t | G ( i ) = ± id } . Every equivalence class of actions is determined by its decomposition(5.1).(2) Classifying groups.First we show that b G C /N ( b G ) = (0). Pick χ b G C . Then N ( χ ) :=(1 + t ) .χ = 2 χ . Since 2 is a unit in Z n , χ ∈ N ( G ), which proves ourassertion. By Lemma 3.16(iii) X ( ⊳ ) = Alt N ( G ). Consider an alternatemapping β : G × G → Z n . It is apparent that β ( g, h ) = 0 whenever g, h lie in different components ( G ( i ) of decomposition (5.1). For g, h ∈ G (1 + t ) .β ( g, h ) = 2 β ( g, h ) and similarly for if g, h ∈ G − . It transpiresthat (1 + t ) .β ( g, h ) = 0 iff β ( g, h ) = 0 for every β : G ν × G ν → Z n , ν =0 , −
1. We conclude that X ( ⊳ ) = 0 if G , − = 0. LASSIFICATION OF HOPF ALGEBRAS 43
The above discussion shows that Alt N ( G ) = Alt N ( G , − ). Let usrenumber the prime divisors of n so that G , − = ⊕ ri =1 G ( i ). We notedabove that G ( i ) = G ( i ) ǫ ⊕ G ( i ) ǫ − and since Z p eii is an indecomposablegroup, G ( i ) ǫ ν ≃ Z p eii . Therefore we can select a basis { a i , b i } of G ( i )with a i , b i generating G ( i ) ǫ , G ( i ) ǫ − , respectively and both of order p e i i . Set a = P a i , b = P b i and observe that a, b generate subgroups G , − ǫ ν , ν = 0 , −
1, respectively. Let us write n ( ⊳ ) = Q { p e i i | t | G ( i ) = ± id } . Set a = P a i , b = P b i and observe that a, b generate subgroups G , − ǫ ν , ν = 0 , −
1, respectively. In addition both subgroups h a i , h b i are cyclic of order n ( ⊳ ), hence G , − ≃ Z n ( ⊳ ) × Z n ( ⊳ ) . It follows thatAlt( G , − ) is cyclic on a generator, say, β defined by β ( a, b ) = 1 Z n ( ⊳ ) .The calculation (1+ t ) .β ( a, b ) = β ( a, b )+ β ( a, − b ) = 0 gives the equal-ity Alt N ( G , − ) = Alt( G , − ). It follows that X ( ⊳ ) = Alt( G , − ) ≃ Z n ( ⊳ ) . We observe that since X ( ⊳ ) ≃ H c ( k C , k G , ⊳ ) that formula im-plies a result of A. Masuoka [22, Thm. 2.1] on Opext( k C , k G ).We summarize Theorem 5.3. (1) If ⊳ is such that G , − = 0 , then Ext [ ⊳ ] ( k C , k G ) has a unique Hopf algebra k [ G ⋊ C ] where G ⋊ C is the semidirectproduct with respect to ⊳ . (2) For ⊳ with a nonzero G , − the isotypes in Ext [ ⊳ ] ( k C , k G ) corre-spond bijectively to the subgroups of Z n ( ⊳ ) . The trivial subgroup of Z n ( ⊳ ) corresponds to a unique trivial Hopf algebra k [ G ⋊ C ] . Proof:
It remains to compute the orbits of A ( ⊳ ) in Alt( G , − ). Firstoff, every φ ∈ A ( ⊳ ) preserves G , − ǫ ν , whence aφ = ua, bφ = vb for some u, v ∈ Z • n ( ⊳ ) . Therefore ( β.φ )( a, b ) := β ( aφ − , b.φ − ) = u − v − β ( a, b ). This showsthat transwering action of A ( ⊳ ) along the isomorphism β β ( a, b ) :Alt( G , − ) ∼ → Z n ( ⊳ ) we get the action m.φ = u − v − m, m ∈ Z n ( ⊳ ) .It becomes clear that orbits are exactly sets of generators of cyclicsubgroups of Z n ( ⊳ ) , which completes the proof. (cid:3) Appendices
Appendix 1: Crossed product splitting of abelian extensionsProposition 6.1. . Let H be an extension of k F by k G . Then H is acrossed product of k F over k G . Proof:
First observe that H is a Hopf-Galois extension of k G by k F via ρ π = (id ⊗ π )∆ H : H → H ⊗ k F , see e.g. the proof of [24, H is a strongly F -graded algebra. Setting H x = { h ∈ H | ρ π ( h ) = h ⊗ x } we have H = ⊕ x ∈ F H x with H = k G and H x H x − = k G for all x ∈ F . Next for every a ∈ G we constructelements u ( a ) ∈ H x , v ( a ) ∈ H x − such that u ( a ) v ( a ) = p a , p a u ( a ) = u ( a ) , v ( a ) p a = v ( a ) , and u ( a ) v ( b ) = 0 for all a = b. Indeed, were all uv, u ∈ H x , v ∈ H x − lie in span { p b | b = a } , then sowould H x H x − , a contradiction. Therefore for every a ∈ G there are u ∈ H x , v ∈ H x − such that uv = P c b p b , c a = 0. Setting u ( a ) =1 c a p a u, v ( a ) = vp a we get elements satisfying the first three proper-ties stated above. Furthermore, the last property also holds because u ( a ) v ( b ) = p a u ( a ) v ( b ) p b = p a p b u ( a ) v ( b ) = 0. It follows that the ele-ments u x = P a ∈ G u ( a ) , v x = P a ∈ G v ( a ) satisfy u x v x = 1 hence, as H is finite-dimensional, v x u x = 1 as well. Thus u x is a 2-sided unit in H x .Now define γ : k F → H by γ ( x ) = 1 ǫ H ( u x ) u x . One can see imme-diately that γ is a convolution invertible mapping satisfying ρ π ◦ γ = γ ⊗ id , γ (1 F ) = 1 and ǫ H ◦ γ = ǫ F . Thus γ is a section of k F in H ,which completes the proof. (cid:3) Appendix 2: Non-splitting of X ( ⊳ ) as A ( ⊳ ) -module for p = 2We take a closer look at the exact sequence b G C p /N ( b G ) X ( ⊳ ) ։ a ( Z N ( ⊳ )) of Lemma 3.16. We know by Theorem 2.5 that for p > a ( Z N ( ⊳ )) = Alt N ( G ) and the above sequence splits up, that is X ( ⊳ ) ≃ b G C p /N ( b G ) × Alt N ( G ) as A ( ⊳ )-modules. We want to show that this isnot the case for p = 2.Let G be an elementary 2-group of rank n and ⊳ be the trivial action.By the argument of part (2) of Proposition 2.5 our assumptions imply b G C p /N ( b G ) = b G and a ( Z N ( ⊳ )) = Alt( G ). The main result of thisAppendix is Theorem 6.2.
Let G be a -elementary group of rank n > . Thesequence of A ( triv ) -modules b G → X ( triv ) → Alt( G ) does not split. Proof:
Will be given in steps. To simplify notation we write X and A for X (triv) and A (triv).(1) Let S be a copy of Alt( G ) in Z ( G, k • ) constructed in Proposition2.5(2). Clearly S ⊂ Z N (triv) and complements B N (triv). Passing on LASSIFICATION OF HOPF ALGEBRAS 45 to X the image of S , denoted by S , forms a complement to b G . Fix abasis { x i | ≤ i ≤ n } of G and let { x ∗ i | ≤ i ≤ n } be its dual in b G .Observe that Φ : B N (triv) → b G acts in the present case by Φ( δf ) = f .Let b i : G × G → k • be the bimultiplicative map defined by b i ( x i , x i ) = − , b i ( x k , x l ) = 1 for ( k, l ) = ( i, i ) . Lemma 6.3. (1) Φ( b i ) = x ∗ i for all i ;(2) Alt( G ) ⊂ ker Φ. Proof: (1) Recall B ( G, k • ) is the subgroup of all symmetric functionsof Z ( G, k • ), hence b i ∈ B ( G, k • ) and therefore b i = δf i for some f i : G → k • . Then b i ( x j , x j ) = δf i ( x j , x j ) = f i ( x j ) f i ( x j ) f i ( x j ) − = f i ( x j ) . We note that as b i = ǫ , b i lies in B N ( G, k • ), hence f i ∈ b G and as f i ( x j ) = ( − δ ij f i = x ∗ i . This proves (1).(2) Elements of Alt( G ) are symmetric functions, henceAlt( G ) ⊂ B ( G, k • ). By part (1) for every α = δf ∈ Alt( G ) Φ( α ) = f = ǫ as α ( x, x ) = 1. (cid:3) (2) Let b G ∧ b G be the exterior square of b G . There is a well-knownidentification Alt( G ) = b G ∧ b G . In the additive notation b G ∧ b G has astandard basis x ∗ i ∧ x ∗ j where x ∗ i ∧ x ∗ j ( x k , x l ) = δ ik δ jl . Passing on to S we write s x ∗ i ∧ x ∗ j as s h i,j i which by the definition of s α is given by s h i,j i ( x k , x l ) = ( , if { k, l } = { i, j } and k < l , else . We note the equality s h i,j i = s h j,i i . Pick φ ∈ A and let φ ∗ : b G → b G be the transpose of φ , i.e. ( χ.φ ∗ )( g ) = χ ( g.φ ) , χ ∈ b G, g ∈ G . If M ( φ ) is the matrix of φ in the basis { x k } then M ( φ ∗ ) = M ( φ ) tr is thematrix of φ ∗ in the dual basis. Therefore the matrix of the mapping b φ ,( χ. b φ )( g ) = χ ( g.φ − ) induced by φ in b G is M ( φ − ) tr . Next we describeaction of A in X Lemma 6.4.
Suppose φ ∈ A and M ( φ − ) = ( a kl ) . φ acts in X asfollows (6.1) s h i,j i .φ = s x ∗ i ∧ x ∗ j .φ + n X k =1 a ki a kj x ∗ k . Proof:
One can see easily that the mapping a is A -linear therefore a ( s h i,j i .φ ) = x ∗ i ∧ x ∗ j .φ . We also know a ( s α ) = α for every α ∈ Alt( G ) which gives(6.2) s h i,j i .φ = s x ∗ i ∧ x ∗ j .φ + b, . where b := b ker Φ ∈ B N (triv) / ker Φ. By Lemma 6.3 the set { b k } forms a basis for B N (triv) / ker Φ, hence b = n P k =1 c k b k , c k ∈ Z . Since s α ( x k , x k ) = 0 for every α and k , evaluating (6.2) at ( x k , x k ) yields c k = s h i,j i .φ ( x k , x k ) = s h i,j i ( x k φ − , x k φ − )= s h i,j i ( X i a ki x i , X j a kj x j ) = a ki a kj A reference to Lemma 6.3(1) completes the proof. (cid:3)
It is well known that A is generated by transvections, linear mappings t pq : x p → x p + x q , x r → x r , r = p . Since t − pq = t pq and the matrix of t ∗ pq is M ( t pq ) tr we have readily x ∗ k .t pq = x k , k = q,x ∗ q .t pq = x ∗ q + x ∗ p . We see that t pq induces the transvection t qp in b G . In consequence wehave Lemma 6.5.
Action of transvections on the standard basis of
Alt( G ) is given by x ∗ i ∧ x ∗ j .t pq = x ∗ i ∧ x ∗ j if q = i, j or ( p, q ) = ( i, j ) , ( j, i ) x ∗ i ∧ x ∗ j .t pi = x ∗ i ∧ x ∗ j + x ∗ p ∧ x ∗ j , p = jx ∗ i ∧ x ∗ j .t pj = x ∗ i ∧ x ∗ j + x ∗ i ∧ x ∗ p , p = i. (cid:3) With the help of Lemma 6.4 we deduce
Lemma 6.6.
Action of transvections on generators of S is given by s h i,j i .t pq = s h i,j i , if q = i, js h i,j i .t ij = s h i,j i + x ∗ i s h i,j i .t ji = s h i,j i + x ∗ j s h i,j i .t pi = s h i,j i + s h p,j i s h i,j i .t pj = s h i,j i + s h i,p i . Proof:
In view of Lemmas 6.4 and 6.5 we need only to calculate the b G - components. If ( p, q ) = ( i, j ) , ( j, i ), then for the entries of M ( t pq )there holds a ki = 0 or a kj = 0 for every k . In M ( t ij ) , M ( t ji ) we have a ki a kj = 1 only for k = i, j , respectively. (cid:3) LASSIFICATION OF HOPF ALGEBRAS 47 (3) End of the Proof. Suppose there is an A -linear section ζ :Alt( G ) → X splitting a . Say(6.3) ζ ( x ∗ i ∧ x ∗ j ) = χ ij + s h i,j i , χ ij ∈ b G. Then there holds(6.4) ζ ( x ∗ i ∧ x ∗ j .t pq ) = ( χ ij + s h i,j i ) .t pq for all p, q. Let us expand χ ij in the basis { x ∗ k } , χ ij = X k c ijk x ∗ k . Observe the equality χ ij .t pq = χ ij + c ijq x ∗ p . Next specialize (6.4) to p = i, q = j or p = j, q = i . Then Lemmas 6.5 and 6.6 give c ijj x ∗ i + x ∗ i = 0and c iji x ∗ j + x ∗ j = 0, respectively. We see that c iji = c ijj = 1, that is χ ij = x ∗ i + x ∗ j + P k = i,j c ijk x ∗ k . Note that if n = 2 we have shown that Z ( x ∗ + x ∗ + s h , i ) is an A -complement to b G . Suppose n >
2. Forevery q = i, j we have by (6.4) and Lemmas 6.5 and 6.6 the equality χ ij + s h i,j i = χ ij .t iq + s h i,j i .t iq Using χ ij .t iq = χ ij + c ijq x ∗ i and s h i,j i .t iq = s h i,j i we conclude c ijq = 0.Thus χ ij = x ∗ i + x ∗ j for all i, j .Next pick p = i, j , and apply (6.4). We have ζ ( x ∗ i ∧ x ∗ j + x ∗ p ∧ x ∗ j ) = ( x ∗ i + x ∗ j + s h i,j i ) .t pi which in turn gives the equality x ∗ i + x ∗ j + s h i,j i + x ∗ p + x ∗ j + s h p,j i = x ∗ i + x ∗ p + x ∗ j + s h i,j i + s h p,j i , hence x ∗ j = 0, a contradiction.On the evidence we have so far we propose Conjecture . Suppose G = m Y i =1 C n i p ei , e < · · · e m . Let N ( G, p ) bethe number of almost abelian Hopf algebras of dimension | G | p . Thefunction N ( G, p ) is a polynomial over Z of degree ≤ e m for all p ≥ e + · · · + e m . References [1] N. Andruskiewitsch, Notes on extensions of Hopf algebras,
Can. J. Math. (1)(1996), 3-42.[2] F. R. Beyl and J. Tappe, Group Extensions, Representations, and the SchurMultiplicator, Lecture Notes in Mathematics , Springer-Verlag, 1982.[3] R.J. Blatttner, M. Cohen and S. Montgomery, Crossed Product and InnerActions of Hopf Algebras, Trans. Amer. Math. Soc (2)(1986),671-711. [4] N. Bourbaki, Elements of Mathematics, Algebra I, Springer-Verlag, 1989.[5] N.P. Byott, Cleft extensions of Hopf algebras,
J. Algebra (1993),405-429.[6] M. Hall, Jr., “The Theory of Groups”, The Macmillan Company, New York,1959.[7] I. Hofstetter, Extensions of Hopf algebras and their cohomological description,
J. Algebra (1994), 264-298.[8] G.I. Kac and V.G. Paljutkin, Finite ring groups,
Trans. Moscow Math. Soc. (1966), 251-294.[9] G.I. Kac, Certain arithmetic properties of ring groups, Functional Anal. Appl. (1972), 158-160.[10] Y. Kashina, Classification of semisimple Hopf algebras of dimension 16, J.Algebra (2000), 617-663.[11] Y. Kashina, On semisimple Hopf Algebras of Dimension 2 m , Algebras andRepresentation Theory (2003), 393-425.[12] Y. Kashina, G. Mason and S. Montgomery, Computing the Frobenius-Schurindicator for abelian extensions of Hopf algebras, J. Algebra (2002), 888-913.[13] T. Kobayashi and A. Masuoka, A result extended from groups to Hopf alge-bras,
Tsukuba J. Math (1997), 55-58.[14] R.G. Larson and D.E. Radford, Semisimple cosemisimple Hopf algebras, Amer. J. Math. (1988), 187-195.[15] S. MacLane, “Homology”, Die Grundlehren der Mathematischen Wis-senschaften , Springer-Verlag, 1963.[16] M. Mastnak, Hopf algebra extensions arising from semi-direct products ofgroups,
J. Algebra (2002), 413-434.[17] A. Masuoka and Y. Doi, Generalization of cleft comodule algebras,
Comm.Algebra (1992), 3703-3721.[18] A. Masuoka, Self-dual Hopf algebras of dimension p obtained by extensions, J. Algebra (1995), 791-806.[19] A. Masuoka, Semisimple Hopf algebras of dimension 6,8,
Israel J. Math. , (1995), 361-373.[20] A.Masuoka, The p n theorem for semisimple Hopf algebras, Proc. Amer. Math.Soc. , (1996), 735-737.[21] A. Masuoka, Some further classification results on semisimple Hopf algebras, Comm. Algebra (1996), 307-329.[22] A. Masuoka, Calculations of some groups of Hopf algebra extensions,
J. Al-gebra (1997), 568-588.[23] A. Masuoka, Extensions of Hopf algebras (Lecture Notes, University of Cor-doba, 1997) Notas Mat.No. 41/99, FaMAF Uni. Nacional de Cordoba, 1999.[24] S. Montgomery, Hopf Algebras and their Actions on Rings, in:
CMBS Reg.Conf. Ser.Math. , AMS, 1993.[25] C. Nastasescu and F. Van Oystaeyen, On strongly graded rings and crossedproducts, Comm. Algebra (1982), 2085-2106.[26] W.D. Nichols and M.B. Zoeller, A Hopf algebra freeness theorem, Amer. J.Math (1989), 381-385.[27] H.-J. Schneider, Some remarks on exact sequences of quantum groups,
Comm.Algebra (9) (1993), 3337-3358. LASSIFICATION OF HOPF ALGEBRAS 49 [28] H.-J. Schneider, A normal basis and transitivity of crossed products for Hopfalgebras,
J. Algebra (1992), 289-312.[29] D. Stefan, The set of Types of n -dimensional semisimple and cosemisimpleHopf algebras is finite, J. Algebra (1997), 571-580.[30] K. Yamazaki, On projective representations and ring extensions of finitegroups,
J. Fac. Science Univ. Tokyo, Sect. I (1964), 147-195.[31] Y. Zhu, Hopf algebras of prime dimension, Intenat. Math. Res. Notices (1994), 53-59. DePaul University, Chicago, IL 60614
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