On semisimple Hopf algebras with few representations of dimension greater than one

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ON SEMISIMPLE HOPF ALGEBRAS WITH FEW

REPRESENTATIONS OF DIMENSION GREATER THAN ONE

V. A. ARTAMONOV

Abstract. In the paper we consider semisimple Hopf algebrasH with the following property: irreducibleH-modules of the same dimension>1 are iso-morphic. Suppose that there exists an irreducibleH-moduleM of dimension

>1 such that its endomorphism ring is a Hopf ideal inH. ThenM is the unique irreducibleH-module of dimension>1.

Introduction

Let H be a semisimple Hopf algebra over an algebraically closed field k. It is assumed that either chark= 0 or chark >dimH. Semisimplicity ofH means that

H is a semisimple left H-module. In that caseH has finite dimension.

Throughout the paper we shall keep to notations from [M]. For example H∗

stands for the dual Hopf algebra with the natural pairingh−,−i : H∗⊗H →k. The algebraH is a left and right H∗-module algebra with respect to the left and right actionsf ⇀ x, x ↼ f off ∈H∗ onxH, defined as follows, [M, Example 4.1.10]: if

∆(x) =X

x

x(1)⊗x(2) (1)

then

f ⇀ x=X

x

x(1)hf, x(2)i, x ↼ f =X

x

hf, x(1)ix(2).

Denote byG=G(H∗) the group of group-like elements in H. Elements ofGare algebra homomorphismsH →k. HenceH as ak-algebra has a semisimple direct decomposition

H = (⊕g∈Gkeg)⊕ ⊕nj=1Mat(dj, k)

, (2)

where{eg, g∈G}is a system of central orthogonal idempotents inH

correspond-ing to one-dimensional direct summands. Ifh∈H andg∈Gthen

heg=egh=hg, hieg. (3)

The counitε:H →khas the form

ε(x) = (

δg,1, x=eg,

0, x∈Mat(di, k).

(2)

Each one-dimensionalH-moduleEg=kug related tog∈Ghas a baseugand

hug=hg, hiug, h∈H. (4)

In this paper we consider the case when

1< d1< d2<· · ·< dn. (5)

It just means that irreducibleH-modules of the same dimension>1 are isomorphic. The main result of the paper is

Theorem 0.1. LetH be a semisimple Hopf algebra with decomposition (2),n>1, such that (5) holds. Suppose that at least one single matrix constituent is a Hopf ideal inH. Thenn= 1.

Throughout the paper we shall use the following notations. By E(i) we shall denote the identity matrix from Mat(di, k) and byE

(i)

rs ∈Mat(di, k) we shall denote

corresponding matrix unit.

The author thanks Prof. N. Andruskiewitsch and Prof. S. Natale for useful and very helpful discussions and comments. The author is grateful to the referee for many useful remarks and suggestions.

Note that the similar classes of Hopf algebras were considered from another point of view in [N2,§3.4] and in [Ma], [N1], [N2], [T], [TY].

1. The category of modules

LetHbe a semisimple Hopf algebra with direct sum decomposition (2) such that (5) is satisfied. The category HMof left H-modules is a monoidal category with

respect to tensor products of modules. SinceH is semisimple the abelian category is completely defined up to an isomorphism by its Grothendieck groupK0(H) =

K0(HM). Recall that the Abelian K-group K0(H) is a free additive Abelian

group with a base consisting of ismorphism clases of irreducible left H-modules

Eg, g ∈ G, and modules Mi, i = 1, . . . , n, of dimension di > 1 corresponding

to matrix constituents in (2). As we have already mentioned one-dimensionalH -modulesEgcorrespond to one-dimensional simple direct summands ofH such that

(4) is satisfied and elements g∈G.

IfM, N are leftH-modules thenM⊗Nis also a leftH-module under the action

h(x⊗y) =X

h

h(1)x⊗h(2)y, h∈H, x∈M, y∈N. (6) Tensor multiplication induces a structure of an associative ring onK0(H).

The next fact is well known.

Proposition 1.1. Let M be an irreducible H-module and Eg a one-dimensional H-module. ThenM⊗Eg andEg⊗M are irreducible H-modules. In particular if f, g∈GthenEf⊗Eg≃Ef g. Under assumption (5)we haveM⊗Eg≃Eg⊗M.

The moduleM⊗Egis identified withM in which elementsh∈H act asg ⇀ h.

An isomorphismM ≃M⊗Egmeans that there exists an invertible linear operator

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x∈M. Note that we can identify the endomorphism algebra EndM with an ideal in H. Namely, if M ≃ Mi then EndM ≃Mat(di, k). If g ∈ Gthen the action g ⇀is an endomorphism ofH. Because of the assumption (5) the ideal EndM is stable under the actiong ⇀. Thusg ⇀ h= ΦM(g)hΦM(g)−1 inH. Note that we

can always assume that ΦM(1) = 1 and ΦM g−1

= ΦM(g)−1 for allg∈G.

We denote byS the antipode ofH.

Proposition 1.2. Let M be an irreducible H-module of dimension > 1. The map ΦM : G → PGL(M) is a projective representation of the group G if M is an irreducible H-module. Moreover [ΦM(g), S(ΦM(v))] = 1 in PGL(M) for all g, v∈G.

Proof. Sinceg ⇀(v ⇀ h) =gv ⇀ h, conjugations by ΦM(g)ΦM(v) and by ΦM(gv)

coincide inGL(M). Hence ΦM(gv)−1ΦM(g)ΦM(v) is a scalar matrix and therefore

ΦM is a projective representation ofG in M. By the assumption (5) the algebra

EndM is an ideal which is stable not only under the actiong ⇀whereg∈Gbut also under the action↼ g and under the antipodeSsinceS is an involution of the algebraH.

Similarly Eg⊗M can be identified with the vector space M in which h ∈ H

acts as

h ↼ g=S(S(g)⇀ S(h)) =S ΦM g−1

S(h)ΦM(g)

=S(ΦM(g))hS ΦM g−1

.

In order to prove the last statement it is necessary to recall that the actions↼, ⇀

commute.

Proposition 1.3. LetM, N be irreducible H-modules andg∈G. Then

HomH(M⊗N, Eg)≃HomH(M⊗N, Eε).

Proof. Let φ∈Hom(M ⊗N, Eg). By Proposition1.1 there is an isomorphism of H-modulesω :M ⊗N →M ⊗N⊗Eg−1. Hence φinduces by Proposition1.1 a homomorphism ofH-modules

M ⊗N ω−1 //M ⊗N⊗Eg1

φ⊗1

/

/Eg⊗Eg1 //E

ε.

Denote by ˜φthis composition. It is easy to see that the correspondence φ7→φ˜is an isomorphism between HomH(M⊗N, Eg) and HomH(M⊗N, Eε). Corollary 1.4. If M, N are irreducible H-modules of dimensions > 1, then the multiplicity of each one-dimensional H-module Eg, g ∈ G, in M ⊗N does not

depend upong.

If M is an irreducible left H-module then H∗ = Hom

k(M, k) is again an

irre-ducible leftH-module. So we have

Proposition 1.5. Let Mi, be the unique irreducible left H-module of dimension di >1.Then Mi∗≃Mi. In particular there exists a non-degenerate bilinear func-tion hx, yi on Mi such that hhx, yi = hx, S(h)yi for all x, y ∈ Mi and for any

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Ife∗= (e1, . . . , edi) is a base ofMias a vector space then we get an identification

of EndMi with the full matrix algebra Mat(di, k) which is an ideal inH. Denote

byUi the square matrix of size di whose (α, β)th entryuαβ=heα, eβi. It means

thatUi is the Gram matrix of the bilinear functionhx, yi.

Theorem 1.6. The bilinear functionhx, yiis (skew-)symmetric. Ifx∈Mat(di, k) from decomposition (2), thenS(x) =UitxUi−1.Using a canonical form of a Gram matrix of a (skew-)symmetric bilinear function we can always choose a base inMi such that Ui is either an identity matrix in the symmetric case or the matrix

Ui=

0 −E

E 0

,

in skew-symmetric (hyperbolic) case where E is the identity matrix of size di

2. In

both casesUi−1=±tU i.

Proof. Without loss of generality we can assume that h, S(h) ∈ Mat(di, k) have

entrieshζβ, rγλ, respectively. Then

hheα, eβi=

X

ζ

heζ, eβihαζ=

X

ζ

uζβhαζ;

heα, S(h)eβi=

X

γ

rβγheα, eγi=

X

γ

rβγuαγ.

The equalityhhx, yi=hx, S(h)yimeans thattU

i·th=S(h)tUi.

Finally since the antipodeS has order 2, the matrix Ui and the formhx, yiare

(skew)-symmetric [LR]. So we can replacetU

i byUi.

We now study some properties of irreducible H-modules. The first one is a well-known, see [DNR, Lemma 7.5.10, p. 322]. Recall thatMi is an irreducible H-module of dimensiondi, 1≤i≤n.

Lemma 1.7. If 1≤i, j≤n, thendim HomH(Mi⊗Mj, Eε) =δij.

SetA=⊕g∈GEg.

Proposition 1.8. There is a direct sum decomposition Mi⊗Mj =δijA⊕ ⊕nt=1mtijMt,

mtij = dimkHomH(Mi⊗Mj, Mt)>0.

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Hence

didj=δij|G|+ n

X

t=1

mtijdt. (8)

In particular |G|6d2 1.

Proof. Apply Lemma1.7, Corollary1.4and compare dimensions of modules in (7). This gives (7), (8). We prove the last claim. Each coefficientmipqestimated by (8)

as

06mipq6 1 di

(5)

In particular ifp=q then 06d2

p− |G|.

Proposition1.5implies thatA⊗Ms=Ms⊗A=|G|Ms. Using the associativity

conditions in the Grothendieck ring we obtain

Theorem 1.9. The multiplicitiesmt

ij satisfy the equation (8)and the equations

msij =mijs, δijδls|G|+ n

X

t=1

mtijmlts=δjsδli|G|+ n

X

t=1

mtjsmlit

for alli, j, s, l= 1, . . . , n. In particular ms

ij =mijs =m j si and

δijδls|G|+ n

X

t=1

mjtimlts=δjsδli|G|+ n

X

t=1

mjstmlit.

Corollary 1.10. If i, j, p= 1, . . . , n, thenmpij 6dmin(i,j,p).

Proof. Since mpij =mi jp =m

j

pi we can assume that p>i, j. Then by (8) we get mpijdp6didj or

mpij 6didj dp

6min(di, dj) =dmin(i,j)=dmin(i,j,p).

2. Special matrices

Consider special elements

Ti=

di

X

α,β=1

Eαβ(i)⊗Eαβ(i), Ri = 1

di di

X

α,β=1

E(αβi)⊗E(βαi),

Di = (1S)Ri

(9)

in Mat(di, k)⊗2. These elements will be used in the rest of the paper.

Direct calculations prove the first equality in

Proposition 2.1. LetF, H ∈Mat(di, k). Then F⊗E(i)

Ti E(i)H

= E(i) tF

Ti E(i)tH

,

(F⊗H)Ri =Ri(HF).

HereE(i) stands for the identity matrix of the size d

i. Moreover

(1⊗S)Ti= (S1)Ti, Di= (1S)Ri= (S1)Ri.

In particular forF, H as above

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Proof. The second equality was proved in [ACh, Proposition 1.1].

In order to prove the nest two equalities we shall use the first equality, (skew-) symmetry ofUi and the propertyUi−1=±Ui. Hence

(1⊗S)Ti=diE(i)UiRiE(i)U−1

i

=di

Ui⊗E(i)

RiU−1

i ⊗E(i)

= (S⊗1)Ti;

(1⊗S)Ri= 1

dt

E(i)U

i

TiE(i)U−1

i

= 1

dt

Ui⊗E(i)

TiU−1

i ⊗E

(i)= (

S⊗1)Ri =Di.

In order to prove the last equality apply (1⊗S) to (9) and use previous equalities.

3. From module category to Hopf algebra

In this section we shall apply previous results to a presentation of an explicit form of comultiplication and antipode inH. Let H have the direct sum decomposition (2) such that (5) holds.

Denote by πg, g ∈G,and by πi, 1 6i6n, the projections ofH ontoEg and

onto Mat(di, k), respectively. Then the isomorphism Ef⊗Eg ≃Ef g in

Proposi-tion1.1means that

(πf⊗πg)∆(h) =hf g, hi(ef⊗eg);

(πg⊗πi)∆(h) =eg⊗(πi(h)↼ g),

(πi⊗πg)∆(h) = (g ⇀ πi(h))⊗eg.

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Applying Proposition1.8we obtain

(πi⊗πj)∆(πg(h)) = 0, 16i6=j6n, g∈G. (11)

Using (10), (11) we see that

∆(eg) =

X

f∈G

ef⊗ef−1g+

X

i=1,...,n

Dg,i, (12)

where {Dg,i, g G,1 6 i 6 n}, is a system of orthogonal idempotents in Mat(di, k)⊗2. Moreover

Dg,iMat(di, k)Mat(di, k)Mat(d2

i, k)≃Endk(Mi⊗Mi).

Lemma 3.1. Let g ∈ G, and Di from (9), i = 1, . . . , n. Then the H-module Dg,i(MiMi)is isomorphic toEg and

Dg,i= 1 g−1D

i.

Proof. Leth∈H andx, y∈Mi. Applying (3), (4), (6) and (12) we obtain hDg,i(xy) = ∆(h)Dg,i(xy)

= ∆(h)∆(eg) (x⊗y) = ∆(heg) (x⊗y) =hg, hiDg,i(x⊗y).

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By (7) and previous considerations it suffices to show that the element (1⊗

g−1

Di is an idempotent and

∆(h) 1⊗ g−1⇀D

i=hg, hi 1⊗ g−1⇀Di.

Direct calculations show that

D2 i = 1 d2 i X α,β,γ,ξ

Eαβ(i)Eγξ(i)⊗SE(βαi)SEξγ(i)

= 1

d2

i

X

α,β,γ,ξ

Eαβ(i)Eγξ(i)⊗SE(ξγi)Eβα(i)

= 1

d2idi

X

α,β=γ,ξ

Eαξ(i)⊗SEξα(i)=Di.

Sinceg ⇀is an algebra automorphism ofH by Proposition2.1we have

∆(h) 1⊗ g−1⇀

SR

i=

X

h

h(1)⊗h(2) g−1⇀

SR

i

=X

h

h(1)⊗ g−1⇀

g ⇀ h(2)

SR

i

=X

h

h(1)⊗ g−1⇀

S RiS g ⇀ h(2)

= 1⊗ g−1⇀

S "

X

h

h1)⊗1 R

i 1⊗ S g ⇀ h(2)

#

= 1⊗ g−1⇀

S "

Ri 1X

h

h(1)S g ⇀ h(2) !#

Finally

X

h

h(1)S g ⇀ h(2)

=X

h

h(1)S h(2)hg, h(3)i

=X

h

h(1)S h(2)hg, h(3)i= X

h

ε h(1)hg, h(2)i =hg,X

h

ε h(1)h(2)i=hg, hi

Applying (10) for anyx∈Mat(dr, k) we get

∆(x) =X

g∈G

[(g ⇀ x)⊗eg+eg⊗(x ↼ g)] + n

X

i,j=1 ∆r

ij(x), (13)

where ∆r

ij(x)∈Mat(di, k)⊗Mat(dj, k).

Note that ∆r

ij(x) = (πi⊗πj)∆(x). It means that

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is an algebra homomorphism not necessarily preserving the unit element.

Lemma 3.2. Let x∈Mat(di, k), g∈Gandτ is the twistτ(a⊗b) =b⊗afor all a∈Mat(dq, k), b∈Mat(dp, k). Then

∆ipq(x) = (S⊗S) ∆iqp

τ

(S(x)), (14)

S(eg) =eg−1, µ(1⊗S) ∆ipq(x) =µ(S⊗1) ∆ipq(x) = 0,

µ(S⊗1)Dg,i=µ(1S)Dg,i=δg,1E(i). (15)

Here µ:H⊗H →H is the multiplication map.

Proof. The antipodeS is an algebra and coalgebra involution. Hence each ideal Mat(di, k) in H isS-stable and hence (14) holds. The equalities (15) were proved

in [A].

Note that Dg,i and ∆t

ij(Et) are in fact projections of Mi ⊗Mj onto Eg by

Lemma3.1and ontomt

ijMt, respectively. HenceDg,iand ∆tij(Et) are matrices in

Mat(didj, k) whose ranks are equal to dimEg= 1 and tomtijdimMt, respectively.

In particular, (14) impliesmi

pq=miqp for alli, p, q.

Since ∆ is an algebra homomorphism for all i, j, p, q, r, s = 1, . . . , m and x ∈ Mat(dr, x), y∈Mat(ds, k) we have

Dg,i·Dh,j=δghδijDg,i, Dg,i·r

pq(x) = ∆rpq(x)·∆g,i= 0,

∆r

ij(x)·∆spq(y) =δipδjqδrs∆rij(xy).

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4. Coassociativity conditions

In this section we shall consider coassociativity conditions of comultiplication ∆ from (12), (13). Multiplying the equalities from [A, §2, Proposition 2.1] by ∆r

ij(E(r))⊗1 from the left, byE(j)⊗∆rij(E(r)) from the right and the last equation

by ∆t

jp(E(t))⊗E(q)from the left and by E(j)⊗∆rpq(E(r)) from the right,

respec-tively, we obtain using (16) that for anyi, j, p, q= 1, . . . , nand all x∈Mat(di, k)

the following five groups of identities are satisfied. The first group:

[1⊗(g ⇀)]Dt,i=Dtg−1,i; [(↼ g)⊗1]Dt,i=Dg1t,i;

[1⊗(↼ g)]Dt,i= [(g ⇀)1]Dt,i.

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The second group:

[(↼ g)⊗1] ∆i

pq(x) = ∆ipq(x ↼ g) ;

[1⊗(g ⇀)] ∆i

pq(x) = ∆ipq(g ⇀ x) ;

[1⊗(↼ g)] ∆ipq(x) = [(g ⇀)⊗1] ∆ipq(x).

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The third group:

1⊗∆i pj

D

i=

∆jip⊗1Dj;

(DiE(i)) [xDi] = [Dix]E(i)Di.

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The fourth group: h

∆r ij(E

(r))

E(j)i[x⊗Dj] = ∆r

ij⊗1

∆i

rj(x)

(E(i)⊗Dj);

DjE(i)(1rji)∆ijr(x)= [Djx]hE(j)rji(E(r))i.

(20)

The last fifth group: h

∆tjp(E(t))⊗E(q)

i

1⊗∆rpq

∆ijr(x)

=

∆tjp⊗1

∆itq(x)

h

E(j)⊗∆rpq(E(r))

i

.

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Also the equations (16) and (15) are satisfied.

We shall apply the identities (17) — (21) to the study of the comultiplication and the antipode ofH.

Proposition 4.1. Let

∆irj(x) =

X

τ

Aτ(x)⊗A′τ(x), ∆ijr(x) =

X

τ

Bτ′(x)⊗Bτ(x), (22)

whereAτ, Bτ ∈Mat(dr, k)and A′τ, Bτ′ ∈Mat(dj, k). Then the equalities (20)are equivalent to

∆r ij

E(r) x⊗E(j)=X

τ

∆r

ij(Aτ(x))

h

E(i)⊗S(A′τ(x))i;

E(j)⊗x∆r ij

E(r)=X

τ

h

S(B′τ(x))⊗E(i)

i ∆r

ji(Bτ(x))

for allx∈Mat(di, k).

Proof. In the notations (22) using Proposition2.1we obtain in the right hand part of the first equation in (20)

∆r

ij⊗1

∆i

rj(x)

E(i)⊗Dj =

" X

τ

∆r

ij(Aτ(x))

E(i)⊗S(Aτ′(x))⊗E(j)

# h

E(i)⊗Dji. By Proposition2.1 the first equation in (20) has the form

X

α,β

h ∆rij

E(r) x⊗Eαβ(j)⊗SEαβ(j)i

= X

τ,α,β

∆r

ij(Aτ(x))

(10)

which is equivalent to

X

α,β

∆r ij

E(r) x⊗Eαβ(j) = X

τ,α,β

∆r

ij(Aτ(x))

h

E(i)⊗S(Aτ′(x))Eαβ(j)i

for any α, β. Identifying α= β and taking a sum over all αwe obtain the first required equality. The second equality is obtained in a similar way from the second

equality in (20).

5. Hopf ideals

The next statement follows from the definition of a Hopf ideal and Theorem1.9.

Proposition 5.1. For a matrix constituentMat(dt, k)inH from direct decompo-sition (2)the following are equivalent:

(i) Mat(dt, k)is a Hopf ideal in H;

(ii) ∆t

ij= 0 for i, j6=t;

(iii) mt

ij = 0 for i, j6=t;

(iv) if i, j6=tthen there are H-module isomorphisms

Mi⊗Mj≃δij(⊕g∈GEg)⊕⊕l6=tmlijMl Mj⊗Mt≃djMt≃Mt⊗Mj,

Mt⊗Mt≃(⊕g∈GEg)⊕⊕nl=1mlttMl.

Proposition 5.2. If 16i, j, p6n then the following are equivalent:

(i) ∆i pj6= 0,

(ii) ∆pji6= 0,

(iii) ∆jip6= 0.

Proof. Let ∆i

pj 6= 0. Then mipj 6= 0 by Proposition 5.1. But mipj = m p ji =m

j ip.

Hence ∆pji,∆jip6= 0.

Proposition 5.3. Assume thatMat(dt, k)is a Hopf ideal in H. Ifq6=t, then

∆t tq(E

(t)) =E(t)E(q), t qt(E

(t)) =E(q)E(t),

andq < t. Thust=n.

Proof. Suppose that there exists an index j 6=t such that ∆jtq 6= 0. By

Proposi-tion5.2we have ∆t

qj6= 0 which contradicts Proposition5.1. Hence E(t)⊗E(q)=X

j

∆jtqE(j)= ∆t tq

E(t).

The case ∆t

qt is similar.

The equalities mean thatmt

(11)

Suppose thatt < n. By (8) we getdtdt+1=Pimit,t+1di.Butmit,t+1=mtt+1,i6=

0 implies i = t since Mat(dt, k) is a Hopf ideal. Hence dtdt+1 = mtt,t+1dt and mt

t,t+1=dt+1. On the other from previous considerations it follows thatmtt,t+1=

0.

Consider in each matrix ideal Mat(di, k) the bilinear form hx, yi such that

hEαβ(i), Eγτ(i)i = δαγδβτ. Then the dual space Mat(di, k)∗ can be identified with

itself via this bilinear function.

Proposition 5.4.LetMat(dn, k)be a Hopf ideal inH. Suppose thatx∈Mat(dn, k) andq < n. Then

∆n nq(x) =

dq

X

i,j=1

E(ijq)⇀ x⊗Eij(q), ∆n qn(x) =

dq

X

i,j=1

E(ijq)⊗x ↼ Eij(q). Proof. Let

∆n nq(x) =

dq

X

i,j=1

aij(x)⊗Eij(q), aij(x)∈Mat(dn, k).

ThenErs(q)⇀ x=Pdqi,j=1aij(x)hE

(q)

rs, Eij(q)i=ars(x). The second case is similar.

Proposition 5.5. Leti < n andx∈Mat(di, k). Then

∆i nn(x) =

di dn dn X p,q=1 h

Eqp(n)↼ S txi

⊗SEpq(n)

=di ↼ S tx⊗1Dn.

Proof. The first identity in (19) withp=j=nhas the form

1⊗∆i nn

D

i= (∆nin⊗1)Dn. (23)

Applying Proposition2.1 and Proposition 5.4 in the left and in the right hands sides of (23) we get the following

1⊗∆i nn   1 di di X a,b=1

Eab(i)⊗SEba(i)

 = 1 di di X a,b=1

E(abi)⊗∆i nn

h

SEba(i)i;

(∆nin⊗1)

" 1 dn dn X p,q=1

Epq(n)⊗SEqp(n)

# = 1 dn " dn X p,q=1

∆nin(Epq(n))⊗S

Eqp(n)

#

= 1

dn

X

p,q=1,...,dn r,s=1,...,di

Ers(i)⊗E(pqn)↼ Ers(i)⊗SEqp(n).

Hence (23) is equivalent to

∆inn

h

SE(bai)i= di

dn dn

X

p,q=1

(12)

Ifx=P

a,bxabS

Eba(i)∈Mat(di, k) then

∆i nn(x) =

di dn

X

a,b,γλ

h

Eγλ(n)↼xabEab(n)

i

⊗SEλγ(n)

= di

dn

X

a,b,γλ

h

Eγλ(n)↼ S txi

⊗SE(λγn)

Proposition 5.6. The equations (20)withr=i=n, j < nhold inH if and only if E(j)⇀ x=x=x ↼ E(j) for allxMat(d

n, k).

Proof. Note first that ∆n nq E(n)

=E(n)E(q)by Proposition5.3. Using Proposi-tions5.4,4.1and2.1we can rewite the first equation in (20) withr=i=n, j < n

in the form

x⊗E(j)=X

α,β

∆n nj

Eαβ(j)⇀ x hE(n)⊗SEαβ(j)i

= X

α,β,γ,λ

Eγλ(j)∗Eαβ(j)⇀ x⊗Eγλ(j)SEαβ(j)

= X

α,β,γ,λ

h

Eγλ(j)∗SEαβ(j)⇀ xi⊗Eγλ(j)Eαβ(j)

= X

α,β,γ

h

Eγα(j)∗SEαβ(j)⇀ xi⊗Eγβ(j)

=X

β,γ

h

δγβE(j)⇀ x

i

⊗Eγβ(j)=E(j)⇀ x⊗E(j)

(24)

for anyx∈Mat(dn, k). Thus (24) means thatE(q)⇀ x=xfor allx.

Similarly the second equality x ↼ E(q) is equivalent to the second one from

(20).

Proof of Theorem 0.1. Suppose that n > 1. Then Mat(dn, k) is the unique

component which is a Hopf ideal, Proposition5.3.

Leti < n. By Proposition5.5 and Proposition5.6we have

∆i

nn(E(i)) = di dn

dn

X

p,q=1

Eqp(n)⊗hStE(i)⇀ SEpq(n)i

= di

dn dn

X

p,q=1

Eqp(n)⊗

h

E(i)⇀ SE(pqn)

i = di

dn dn

X

p,q=1

Eqp(n)⊗S

Epq(n)

=diDn.

Suppose now thatn>2.Thene1·E(1)= 0 inH and therefore Dn·1

nn

E(1)= 0.

On the other handDn·1

nn E(1)

(13)

6. The case n= 1 with ∆1 11= 0

We shall remind some results in the case n= 1. For simplicity we put E(1) =

E, Eij(1)=Eij, d1=d, Φ1= Φ, U=U1.

Theorem 6.1. [A, ACh] Suppose that n= 1 and chark is not a divisor of 2d2. Let G=G(H∗) be the group of group-like elements of the dual Hopf algebra H∗. Assume that∆1

11= 0in (13), or equivalently |G|=d2. Taking an isomorphic copy

ofH we can assume that the comultiplication in H has the form:

∆(eg) =

X

h∈G

eh⊗eh−1g+ 1

d d

X

i,j=1

Eij⊗ g−1⇀ S(Eji)

;

∆(x) =X

g∈G

[(g ⇀ x)⊗eg+eg⊗(x ↼ g)] ;

ε(eg) =δg,1, ε(x) = 0; S(x) =U txU−1

(25)

for all x∈ Mat(d, k), where U ∈ GL(d, k) is a (skew-)symmetric matrix. There exists a faithful irreducible projective representationΦ :G→PGL(d, k)such that

g ⇀ x= Φ(g)xΦ(g)−1, x ↼ g=S(Φ(g))xS(Φ(g))−1 (26)

for anyx∈Mat(d, k)and

[Φ(g), S(Φ(v))] = 1 (27)

inPGL(d, k)for allg, v∈G.

It is shown in [F, J2] that G ≃A×A is a direct product of two copies of an Abelian groupA of orderd. The next theorem shows the converse. If we start with a irreducible projective representation of a groupGfrom Theorem6.1we can obtain Hopf algebra structure.

Theorem 6.2. [ACh] Suppose that G is Abelian group of order d2 with direct

decompositionG≃A×A for some Abelian groupAof orderd. The groupGhas a faithful irreducible projective representation Φof degree d. There exists a (skew-) symmetric matrix U ∈GL(d, k)such that (26), (27) holds whereS(x) =UtxU−1

for any x∈ Mat(d, k). Then an algebra H with direct decomposition (2) admits Hopf algebra structure defined in (26),(25).

Each element g∈G can be identified with an elementg∈H∗ such that if

a= X

h inG

αheh+x∈H, αh∈k, x∈Mat(d, k),

thenhg, ai=αg.The identification is an isomorphism of the groupGand the group G(H∗)of group-like elements in H∗.

If d >2, then these Hopf algebras are neither commutative nor cocommutative.

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we expand these results of an arbitrarynusing the language of faithful projective representations of the groupG=G(H∗).

Note that by [J1], [J2] all projective faithful irreducible representations of the groupG≃A×Aas in Theorems6.1,6.2have dimensiondand obtained one from the other by a group automorphism ofG. Any of these representations is monomial and it is induced by a one-dimensional representation ofA.

References

[A] Artamonov V.A. On semisimple finite dimensional Hopf algebras, Math. Sbornik, 198(2007), N 9, 3–28.15,4,6.1

[ACh] Artamonov V.A., Chubarov I.A., Properties of some semisimple Hopf algebras, Contemp. Math. 483, Algebras, representations and applications, A conference in honour of Ivan Shestakov’s 60th birthday, August 26–September 1, 2007, Maresias, Brazil. Edited by: Vyacheslav Futorny, Victor Kac, Iryna Kashuba and E. Zelmanov, Amer. Math. Soc., 2009, 23–36.2,6.1,6.2

[DNR] D˘asc˘alescu S., N˘ast˘asescu C., Raianu S¸., Hopf algebras, an introduction, Pure and Applied Mathematics, 235: a series of Monographs and Textbooks, Marcel Dekker, Inc, NY, Basel, 2001.6

[F] Frucht R., ¨Uber die Darstekkung endlicher abeischer Gruppen durch Kollineationen, J. reine angew. Math., 166(1932), 16–29.6

[HN] Hilgemann M., Ng S. H., Hopf algebras of dimension 2p2, J. London Math. Soc., 80(2009), 295–310.6

[J] Jmud E., On isomorphic irreducible projective representations of finite groups, Zapiski Math. Otdeleniya fiziko-matemt. faculteta Kharkov university and Kharkov math. Soc., 26(1960), N 4, 333–371.

[J1] Jmud E., Simplectic geometry on finite abelian groups, Mat. Sbornik, 86 (1971), 9–34.6 [J2] Jmud E., Simplectic geometry and projective representations of finite abelian groups, Mat.

Sbornik, 87 (1972), 3–17.6,6

[LR] Larson R., Radford D., Finite-dimensional cosemisimple Hopf algebras in characteristic zero are semisimple. J. Algebra, 117(1988), 267–289.6

[Ma] Masuoka A., Some further classification results on semisimple Hopf algebras, Commun. Algebra, 24 (1996), 307–329.5,6

[M] Montgomery S.: Hopf Algebras and Their Actions on Rings, Regional Conf. Ser. Math. Amer. Math. Soc., Providence RI, 1993.(document)

[N0] Natale S., On semisimple Hopf algebras of dimensionpq2, J. Algebra, 221(1999), 242–278. 6

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[N2] Natale S., Semisolvability of semisimple Hopf algebras of low dimension, Memoirs of AMS vol. 186, 2007.5

[T] Tambara D., Representations of tensor categories with fusion rules of self-duality for finite abelian groups, Israel J.Math., 118(2000), 29-60.5

[TY] Tambara D., Yamagami S., Tensor categories with fusion rules of self-duality for finite abelian groups, J.Algebra 209(1998), 692-707.5

V. A. Artamonov Department of Algebra,

Faculty of Mechanics and Mathematics, Moscow State University

artamon@mech.math.msu.su

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