Some applications of Frobenius algebras to Hopf algebras

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010 SOME APPLICATIONS OF FROBENIUS ALGEBRAS TO HOPF ALGEBRAS

MARTIN LORENZ

ABSTRACT. This expository article presents a unified ring theoretic approach, based on thetheory of Frobenius algebras, to a variety of results on Hopfalgebras. These include a theoremof S. Zhu on the degrees of irreducible representations, theso-called class equation, the de-termination of the semisimplicity locus of the Grothendieck ring, the spectrum of the adjointclass and a non-vanishing result for the adjoint character.

INTRODUCTION

0.1. It is a well-known fact that all finite-dimensional Hopfalgebras over a field are Frobe-nius algebras. More generally, working over a commutative base ringR with trivial Picardgroup, any HopfR-algebra that is finitely generated projective overR is a FrobeniusR-algebra [22]. This article explores the Frobenius property, and some consequences thereof,for Hopf algebras and for certain algebras that are closely related to Hopf algebras withoutgenerally being Hopf algebras themselves: the Grothendieck ring G0(H) of a split semisim-ple Hopf algebraH and the representation algebraR(H) ⊆ H∗. Our principal goal is toquickly derive various consequences from the fact that the latter algebras are Frobenius, oreven symmetric, thereby giving a unified ring theoretic approach to a variety of known resultson Hopf algebras.

0.2. The first part of this article, consisting of four sections, is entirely devoted to Frobeniusand symmetric algebras over commutative rings; its sole purpose is to deploy the requisitering theoretical tools. The content of these sections is classical over fields and the case ofgeneral commutative base rings is easily derived along the same lines. In the interest ofreadability, we have opted for a self-contained development. The technical core of this partare the construction of certain central idempotents in Propositions 4 and 5 and the descriptionof the separability locus of a Frobenius algebra in Proposition 6. The essence of the latterproposition goes back to D. G. Higman [11].

0.3. Applications to Hopf algebras are given in Part 2. We start by considering Hopf algebrasthat are finitely generated projective over a commutative ring. After reviewing some standardfacts, due to Larson-Sweedler [16], Pareigis [22], and Oberst-Schneider [21], concerning theFrobenius property of Hopf algebras, we spell out the content of the aforementioned Propo-sitions 4 and 5 in this context (Proposition 15). A generalization, due to Rumynin [24], of

2000Mathematics Subject Classification.16L60, 16W30, 20C15.Key words and phrases.Frobenius algebra, symmetric algebra, group algebra, Hopfalgebra, separable algebra,

character, Grothendieck ring, integrality, irreducible representation, adjoint representation, rank .Research of the author supported in part by NSA Grant H98230-09-1-0026.

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2 MARTIN LORENZ

Frobenius’ classical theorem on the degrees of irreduciblecomplex representations of finitegroups follows in a few lines from this result (Corollary 16).

The second section of Part 2 focuses on semisimple Hopf algebrasH over fields, specif-ically their Grothendieck ringsG0(H). As an application of Proposition 15, we derive aresult of S. Zhu [27] on the degrees of certain irreducible representations ofH (Theorem 18).The celebrated class equation for semisimple Hopf algebrasis presented as an application ofProposition 4 in Theorem 19. The proof given here follows theoutline of our earlier proofin [18], with a clearer separation of the purely ring theoretical underpinnings. Other appli-cations concern a new proof of an integrality result, originally due to Sommerhauser [26],for the eigenvalues of the “adjoint class” (Proposition 20), the determination of the semisim-plicity locus ofG0(H) (Proposition 22), and a non-vanishing result for the adjoint character(Proposition 23).

For the sake of simplicity, we have limited ourselves for themost part to base fields ofcharacteristic0. In some cases, this restriction can be removed with the aid of p-modularsystems. For example, as has been observed by Etingof and Gelaki [8], any bi-semisimpleHopf algebra over an algebraically closed field of positive characteristic, along with all itsirreducible representations, can be lifted to characteristic 0. We plan to address this aspectmore fully in a sequel to this article.

In closing I wish to thank the referee for a careful reading ofthis article and for valuablecomments.

Part 1. RING THEORY

Throughout this first part of the paper,R will denote a commutative ring andA will be anR-algebra that is finitely generated projective overR.

1. PRELIMINARIES ON FROBENIUS AND SYMMETRIC ALGEBRAS

1.1. Definitions.

1.1.1. Frobenius algebras.PutA∨ = HomR(A,R); this is an(A,A)-bimodule via

(afb)(x) = f(bxa) (a, b, x ∈ A, f ∈ A∨) . (1)

The algebraA is called Frobenius if A ∼= A∨ as leftA-modules. This is equivalent toA ∼= A∨ as rightA-modules. Indeed, using the standard isomorphismAA

∼−→ (A∨∨)A =

HomR(AA∨, R)A given bya 7→ (f 7→ f(a)), one deduces fromAA∨ ∼= AA that

AA∼= HomR(AA

∨, R)A ∼= HomR(AA,R)A = A∨A . (2)

The converse is analogous.More precisely, any isomorphismL : AA

∼−→ AA

∨ has the form

L(a) = aλ (a ∈ A) ,

where we have putλ = L(1) ∈ A∨. The linear formλ is called aFrobenius homomorphism.Tracing1 ∈ A through (2) we obtain1 7→ (aλ 7→ (aλ)(1) = λ(a)) 7→ (a 7→ λ(a)) = λ.Hence, the resulting isomorphismR : AA

∼−→ A∨

A is explicitly given by

R(a) = λa (a ∈ A) .

FROBENIUS ALGEBRAS 3

In particular,λ is also a free generator ofA∨ as rightA-module. The automorphismα ∈AutR-alg(A) that is given byλa = α(a)λ for a ∈ A is called theNakayama automorphism.

1.1.2. Symmetric algebras.If A ∼= A∨ as (A,A)-bimodules then the algebraA is calledsymmetric. Any (A,A)-bimodule isomorphismA ∼= A∨ restricts to an isomorphism ofHochschild cohomology modulesH0(A,A) ∼= H0(A,A∨). Here,H0(A,A) = Z(A) isthe center ofA, andH0(A,A∨) = A∨

trace consists of alltrace formsonA, that is,R-linearformsf ∈ A∨ vanishing on all Lie commutators[x, y] = xy − yx for x, y ∈ A. Thus, ifA issymmetric then we obtain an isomorphism ofZ(A)-modules

Z(A)∼

−→ A∨

trace . (3)

1.2. Bilinear forms.

1.2.1. Nonsingularity. LetBil(A;R) denote theR-module consisting of allR-bilinear formsβ : A×A → R. Puttingrβ(a) = β(a, . ) for a ∈ A, we obtain an isomorphism ofR-modules

r : Bil(A;R)∼

−→ HomR(A,A∨) , β 7→ rβ . (4)

The bilinear formβ is called left nonsingular ifrβ is an isomorphism. Inasmuch asA andA∨

are locally isomorphic projectives overR, it suffices to assume thatrβ is surjective; see [6,Cor. 4.4(a)].

Similarly, there is an isomorphism

l : Bil(A;R)∼

−→ HomR(A,A∨) , β 7→ lβ (5)

with lβ(a) = β( . , a), andβ is called right nonsingular iflβ is an isomorphism. Right and leftnonsingularity are in fact equivalent. After localization, this follows from [14, Prop. XIII.6.1].In the following, we will therefore call such forms simplynonsingular.

1.2.2. Dual bases.Fix a nonsingular bilinear formβ : A × A → R. SinceA is finitelygenerated projective overR, we have a canonical isomorphismEndR(A) ∼= A ⊗R A∨; see[2, II.4.2]. Thus, the isomorphismlβ in (5) yields an isomorphismEndR(A)

∼−→ A ⊗R A.

Writing the image ofIdA ∈ EndR(A) as∑n

i=1 xi ⊗ yi ∈ A⊗R A, we have

a =∑

i

β(a, yi)xi for all a ∈ A. (6)

Conversely, assume thatβ : A × A → R is such that there are elements{xi}n1 , {yi}n1 ⊆ A

satisfying (6). Then anyf ∈ A∨ satisfies

f =∑

i

lβ(yi)f(xi) = lβ(∑

i

f(xi)yi) . (7)

This shows thatlβ : A → A∨ is surjective, and henceβ is nonsingular. To summarize, a bilin-ear formβ ∈ Bil(A;R) is nonsingular if and only if there exist “dual bases”{xi}

n1 , {yi}

n1 ⊆ A

satisfying (6).Note also that, for a nonsingularR-bilinear formβ : A×A → R and elements{xi}n1 , {yi}

n1 ⊆

A, condition (6) is equivalent to

b =∑

i

β(xi, b)yi for all b ∈ A. (8)

Indeed, both (6) and (8) are equivalent toβ(a, b) =∑

i β(a, yi)β(xi, b) for all a, b ∈ A.

4 MARTIN LORENZ

1.2.3. Associative bilinear forms.An R-bilinear formβ : A × A → R is calledassocia-tive if β(xy, z) = β(x, yz) for all x, y, z ∈ A. Let Bilassoc(A;R) denote theR-submoduleof Bil(A;R) consisting of all such forms. Under the isomorphism (4),Bilassoc(A;R) corre-sponds to the submoduleHom(AA, A

∨A) ⊆ HomR(A,A

∨). Similarly, (5) yields an isomor-phism ofR-modulesBilassoc(A;R) ∼= Hom(AA,AA

∨). Therefore:

The algebraA is Frobenius if and only if there exists a nonsingular associativeR-bilinear formβ : A×A → R.

1.2.4. Symmetric forms.The formβ is calledsymmetricif β(x, y) = β(y, x) for all x, y ∈ A.The isomorphismsr andl in (4), (5) agree on the submodule consisting of all symmetric bilin-ear forms, and they yield an isomorphism between theR-module consisting of all associativesymmetric bilinear forms onA and the submoduleHom(AAA,AA

∨A) ⊆ HomR(A,A

∨)consisting of all(A,A)-bimodule mapsA → A∨. Thus:

The algebraA is symmetric if and only if there exists a nonsingular associativesymmetricR-bilinear formβ : A×A → R.

1.2.5. Change of bilinear form.Given two nonsingular formsβ, γ ∈ Bilassoc(A;R), we ob-tain an isomorphism of leftA-modulesl−1

β ◦ lγ : AA∼

−→ AA. Since this isomorphism has theform a 7→ au (a ∈ A) for some unitu ∈ A, we see that

γ( . , . ) = β( . , . u) .

If β andγ are both symmetric thenl−1β ◦lγ is an isomorphism of(A,A)-bimodules, and hence

u ∈ Z(A), the center ofA.

2. CHARACTERS

Throughout this section,M will denote a leftA-module that is assumed to be finitelygenerated projective overR. Fora ∈ A, we letaM ∈ EndR(M) denote the endomorphismgiven by the action ofa onM :

aM (m) = am (a ∈ A,m ∈ M) .

2.1. Trace and rank. The trace map

Tr: EndR(M) ∼= M ⊗R M∨ → R ;

it is defined via evaluation ofM∨ = HomR(M,R) onM ; see [2, II.4.3]. The image of thetrace map complements the annihilatorannR M = {r ∈ R | rm = 0 ∀m ∈ M}:

Im(Tr)⊕ annR M = R ; (9)

see [5, Proposition I.1.9]. TheHattori-Stallings rankof M is defined by

rankRM = Tr(1M ) ∈ R .

If M is free of rankn overR thenrankR M = n · 1. The following lemma is standard andeasy.


Lemma 1. TheR-algebraEndR(M) is symmetric, with nonsingular associative symmetricR-bilinear formEndR(M) × EndR(M) → R , (x, y) 7→ Tr(xy) . IdentifyingEndR(M)withM ⊗R M∨ and writing1M =

∑

mi ⊗ fi , dual bases for this form are given by

{xi,j = mi ⊗ fj} , {yi,j = mj ⊗ fi} .

2.2. The character ofM . Thecharacterof M is the trace formχM ∈ A∨trace that is defined

byχM (a) = Tr(aM ) (a ∈ A) .

If e = e2 ∈ A is an idempotent theneM = 1eM ⊕ 0(1−e)M , and so

χM (e) = rankR eM . (10)

Now assume thatA is Frobenius with associative nonsingular bilinear formβ, and let{xi}

n1 , {yi}

n1 ⊆ A be dual bases forβ as in (6). Then the preimage ofχM ∈ A∨

trace ⊆ A∨

under the isomorphismlβ : AA∼

−→ AA∨ in (5) is the element

z(M) = zβ(M) :=∑

i

χM (xi)yi ∈ A ; (11)

see (7). SoχM ( . ) = β( . , z(M)) . (12)

In particular,z(M) is independent of the choice of dual bases{xi}, {yi}.If β is symmetric thenz(M) ∈ Z(A) by (3).

2.3. The regular character. The left regular representation ofA is defined byA → EndR(A),a 7→ (aA : x 7→ ax). Similarly, the right regular representation is given byA → EndR(A),a 7→ (Aa : x 7→ xa).

If A is Frobenius, with associative nonsingular bilinear formβ and dual bases{xi}n1 , {yi}n1 ⊆

A for β, then equations (6) and (8) give the following expression

aA =∑

i

xi ⊗ β(a . , yi) =∑

i

yi ⊗ β(xi, a . ) ∈ A⊗R A∨ ∼= EndR(A) .

Similarly, Aa =∑

i xi ⊗ β( . a, yi) =∑

i yi ⊗ β(xi, . a). Taking traces, we obtain

Tr(aA) =∑

i

β(xi, ayi) =∑

i

β(xia, yi) = Tr(Aa) . (13)

This trace is called theregular characterof A; it will be denoted byχreg ∈ A∨. SinceTr(aA) =

∑

i β(axi, yi) andTr(Aa) =∑

i β(xi, yia), we have

χreg = β( . , z) = β(z, . ) with z = zβ :=∑

i

xiyi . (14)

Thus, the elementz is associated to the regular character as in (12).

Example 2. We compute the regular character for the algebraEndR(M) using the trace formand the dual bases{xi,j}, {yi,j} from Lemma 1. The elementz in (14) evaluates to

z =∑

i,j

(mi ⊗ fj)(mj ⊗ fi) =∑

i,j

mifj(mj)⊗ fi = (rankR M)1 .

Therefore, the regular character ofEndR(M) is equal toTr( . z) = (rankR M)Tr .

6 MARTIN LORENZ

2.4. Central characters. Assume thatEndA(M) ∼= R asR-algebras. Then, for eachx ∈Z(A), we havexM = ωM (x)1M with ωM (x) ∈ R. This yields anR-algebra homomorphism

ωM : Z(A) → R ,

called thecentral characterof M . SinceTr(xM ) = ωM(x)Tr(1M ), we have

χM (x) = ωM(x) rankR M (x ∈ Z(A)) . (15)

Now assume thatA is Frobenius with associative nonsingular bilinear formβ, and letz(M) = zβ(M) ∈ A be as in (11). Then

xz(M) = ωM (x)z(M) (x ∈ Z(A)) ; (16)

this follows from the computationβ(a, xz(M)) = β(ax, z(M)) = χM (ax) = ωM (x)χM (a)= ωM (x)β(a, z(M)) = β(a, ωM (x)z(M)) for a ∈ A.

If β is symmetric thenz(M) ∈ Z(A) and we can define theindex1 of M by

[A : M ]β := ωM(z(M)) ∈ R . (17)

2.5. Integrality. LetA be a FrobeniusR-algebra, with associative nonsingular bilinear formβ and dual bases{xi}n1 , {yi}

n1 ⊆ A. Assume that we are given a subringS ⊆ R. An S-

subalgebraB ⊆ A will be called aweakS-form of (A, β) if the following conditions aresatisfied:

(i) B is a finitely generatedS-module, and(ii)

∑

i xi ⊗ yi ∈ A⊗R A belongs to the image ofB ⊗S B in A⊗R A.

Recall from Section 1.2.2 that the element∑

i xi ⊗ yi only depends onβ. Note also that (ii)implies thatBR = A. Indeed, for anya ∈ A, the mapIdA⊗β(a, . ) : A⊗RA → A⊗RR = Asends

∑

i xi ⊗ yi to a by (6), and it sends the image ofB ⊗S B in A⊗R A to BR.

Lemma 3. LetA be a symmetricR-algebra with formβ. Assume thatEndA(M) ∼= R, andlet z(M) ∈ Z(A) be as in(11). If there exists a weakS-form of (A, β) for some subringS ⊆ R, thenχM (z(M)) and [A : M ]β = ωM (z(M)) are integral overS .

Proof. All b ∈ B are integral overS by condition (i). Hence the endomorphismsbM ∈EndR(M) are integral overS, and so are their tracesχM (b); see Prop. 17 in [1, V.1.6] for thelatter. By (ii), it follows thatχM (z(M)) =

∑

i χM (xi)χM (yi) is integral overS. Moreover,the subringS′ = S[χM (B)] ⊆ R is finite overS by (i). Thus, all elements ofBS′ ⊆ A areintegral overS. In particular, this holds forz(M), whenceωM (z(M)) is integral overS. �

2.6. Idempotents.

Proposition 4. Let (A, β) be a symmetricR-algebra. Lete = e2 ∈ A be an idempo-tent and assume that theA-moduleM = Ae satisfiesEndA(M) ∼= R asR-algebras. LetωM : Z(A) → R be the central character ofM and letz(M) ∈ Z(A) be as in(11). Then:

(a) [A : M ]β is invertible inR, with inverseβ(e, 1).(b) e(M) := [A : M ]−1

β z(M) ∈ Z(A) is an idempotent satisfyingωM(e(M)) = 1 and

xe(M) = ωM (x)e(M) (x ∈ Z(A)) .

1The index[A : M ] is also called theSchur elementof M ; see, e.g., [3, Lemma 4.6].


(c) Letz = zβ =∑

i xiyi ∈ Z(A) be as in(14). Then

ωM (z) · rankR M = [A : M ]β · rankR e(M)A .

Proof. Note thateM = eAe ∼= EndA(M) ∼= R. By (10), it follows thatχM (e) = rankR eM =1. Sincexe = ωM(x)e for x ∈ Z(A), we obtain

1 = χM (e) =(12)

β(e, z(M)) = β(z(M)e, 1) = ωM(z(M))β(e, 1) .

This proves (a). In (b),ωM(e(M)) = 1 is clear by definition ofe(M), and (16) gives theidentity xe(M) = ωM(x)e(M) for all x ∈ Z(A). Together, these facts imply thate(M) isan idempotent. Finally, the following computation proves (c):

ωM(z) rankR M =(15)

χM (z) =(12)

β(z, z(M)) =(14)

χreg(z(M))

= ωM (z(M))χreg(e(M)) =(10)

ωM(z(M)) rankR e(M)A .

�

We now specialize the foregoing to separable algebras. For background, see DeMeyer andIngraham [5]. We mention that, by a theorem of Endo and Watanabe [7, Theorem 4.2], anyfaithful separableR-algebra is symmetric.

Proposition 5. Assume that the algebraA is separable and that theA-moduleM is cyclicand satisfiesEndA(M) ∼= R. Lete(M) ∈ Z(A) be the idempotent in Proposition 4(b). Then:

(a) e(M)A ∼= EndR(M) andrankR e(M)A = (rankR M)2 .(b) χreg e(M) = (rankR M)χM , whereχreg is the regular character ofA .

Proof. We first note thatM , being assumed projective overR, is in fact projective overAby [5, Proposition II.2.3]. SinceM is cyclic, we haveM ∼= Ae with e = e2 ∈ A; soProposition 4 applies.

(a) It suffices to show thate(M)A ∼= EndR(M), because the rank ofEndR(M) ∼= M ⊗R

M∨ equals(rankR M)2. SinceM is finitely generated projective and faithful overR, theR-algebraEndR(M) is Azumaya; see [5, Proposition II.4.1]. The Double Centralizer Theorem[5, Proposition II.1.11 and Theorem II.4.3] and our hypothesisEndA(M) ∼= R together implythat the mapA → EndR(M), a 7→ aM , is surjective. LettingI = annAM denote thekernel of this map, we further know by [5, Corollary II.3.7 and Theorem II.3.8] thatI =(I ∩ Z(A))A. Finally, Proposition 4 tells us thatI ∩ Z(A) is generated by1− e(M), whichproves (a).

(b) In view of the isomorphisme(M)A ∼= EndR(M), e(M)a 7→ aM from part (a) andExample 2, we haveχreg(e(M)a) = (rankR M)Tr(aM ) = (rankR M)χM (a) . �

3. SEPARABILITY

TheR-algebraA is assumed to be Frobenius throughout this section. We fix a nonsingularassociativeR-bilinear formβ : A×A → R and dual bases{xi}n1 , {yi}

n1 ⊆ A for β.

8 MARTIN LORENZ

3.1. The Casimir operator. Define a map, called theCasimir operator2 of (A, β), by

c = cβ : A → Z(A) , a 7→∑

i

yiaxi . (18)

In order to check thatc(a) ∈ Z(A) we calculate, fora, b ∈ A,

bc(a) =(8)

∑

i,j

β(xj , byi)yjaxi =∑

i,j

yjaβ(xjb, yi)xi =(6)

c(a)b .

The mapc is independent of the choice of dual bases{xi}, {yi}, because∑

i xi⊗yi ∈ A⊗RAonly depends onβ; see Section 1.2.2.

In caseβ is symmetric,{yi}, {xi} are also dual bases forβ, and hence

c(a) =∑

i

xiayi .

In particular, the elementz = zβ in (14) arises aszβ = c(1) if β is symmetric. We will referto the elementz = zβ as theCasimir element3 of the symmetric algebra(A, β); it dependsonβ only up to a central unit (see 3.2 below).

3.2. The Casimir ideal. Sincec is Z(A)-linear, the imagec(A) of the mapc = cβ in (18)is an ideal ofZ(A). This ideal will be called theCasimir ideal4 of A; it does not depend onthe choice of the bilinear formβ. Indeed, recall from Section 1.2.5 that any two nonsingularformsβ, γ ∈ Bilassoc(A;R) are related byγ( . , a) = β( . , au) for some unitu ∈ A. Hence, if{xi}, {yi} ⊆ A are dual bases forβ then{xi}, {yiu−1} ⊆ A are dual bases forγ. Therefore,

cγ(a) = cβ(u−1a) (a ∈ A) .

If β andγ are both symmetric thenu ∈ Z(A) and socγ(a) = u−1cβ(a).

3.3. The separability locus. For a given Frobenius algebraA, we will now determine theset of all primesp ∈ SpecR such that theQ(R/p)-algebraA ⊗R Q(R/p) is separable or,equivalently, theRp-algebraA ⊗R Rp is separable [5, Theorem II.7.1]. The collection ofthese primes is called the separability locus ofA.

Proposition 6. The separability locus of a FrobeniusR-algebraA is

SpecR \ V (c(A) ∩R) = {p ∈ SpecR | p + c(A) ∩R} .

Proof. The case of a base fieldR is covered by Higman’s Theorem which states that a Frobe-nius algebraA over a fieldR is separable if and only ifc(A) = Z(A) or, equivalently,1 ∈ c(A); see [11, Theorem 1] or [4, 71.6].

Now letR be arbitrary and letp ∈ SpecR be given. PutF = Q(R/p) andAF = A⊗R F .We know thatAF is Frobenius, with formβ = β ⊗R IdF and corresponding dual bases{xi},

2We follow the terminology of Higman’s original article [11]; the mapc is called theGaschutz-Ikeda operatorin [4].

3In [3], the Casimir elementz is referred to as thecentral Casimir element, while∑

ixi ⊗ yi ∈ A ⊗R A is

called the Casimir element. The latter element is referred to as theFrobenius elementin [13].4The Casimir idealc(A) of a symmetric algebraA coincides with theprojective centerof A in the terminology

of [3, Prop. 3.13]. It is also called theHigman idealof A; see [10].


{yi}, where : A → AF , x = x⊗ 1, denotes the canonical map. By Higman’s Theorem, weknow thatAF is separable if and only if1 ∈ c(AF ) = c(A)F . Thus:

TheF -algebraAF is separable if and only if(Ap+ c(A)) ∩R % p.

If p + c(A) ∩R then clearly(Ap+ c(A)) ∩R % p, and henceAF is separable. Conversely,assume thatc(A)∩R ⊆ p. SinceA is integral over its centerZ(A), we havec(A)A∩Z(A) ⊆√

c(A), the radical of the idealc(A); see Lemma 1 in [1, V.1.1]. Therefore,c(A)A ∩ R ⊆√

c(A)∩R ⊆ p. By Going Up [19, 13.8.14], there exists a prime idealP of A with c(A)A ⊆P andP ∩R = p. But then(Ap+ c(A)) ∩R ⊆ P ∩R = p, and henceAF is not separable.This proves the proposition. �

3.4. Norms. Assume that the algebraA is free of rankn overR. Then thenormof an elementa ∈ A is defined by

N(a) = det aA ∈ R ,

where(aA : x 7→ ax) ∈ EndR(A) = Mn(R) is the left regular representation ofA as inSection 2.3. The norm mapN : A → R satisfiesN(ab) = N(a)N(b) andN(r) = rn fora, b ∈ A andr ∈ R. Up to sign,N(a) is the constant term of the characteristic polynomial ofaA. Sincea satisfies this polynomial by the Cayley-Hamilton Theorem, we see thata dividesN(a) in R[a] ⊆ A. Putting

N(c(A)) =∑

a∈c(A)

RN(a) ,

we obtainN(c(A)) ⊆ c(A) ∩R. Moreover, sinceN(r) = rn for r ∈ R, we further concludethat, for anyp ∈ SpecR, we have

p ⊇ N(c(A)) ⇐⇒ p ⊇ c(A) ∩R .

4. ADDITIONAL STRUCTURE: AUGMENTATIONS, INVOLUTIONS, POSITIVITY

4.1. Augmentations and integrals. Let (A, β) be a Frobenius algebra and suppose thatAhas an augmentation, that is, an algebra homomorphism

ε : A → R .

PutΛβ = r−1β (ε) ∈ A, whererβ is as in Section 1.2; soβ(Λβ , . ) = ε. From (8), we obtain

the following expression in terms of dual bases{xi}, {yi} for β:

Λβ =∑

i

ε(yi)xi . (19)

The computationβ(Λβa, . ) = β(Λβ , a . ) = ε(a)ε = ε(a)β(Λβ , . ) for all a ∈ A showsthatΛβa = ε(a)Λβ . Conversely, ift ∈ A satisfiesta = ε(a)t for all a ∈ A thenβ(t, a) =β(ta, 1) = ε(a)β(t, 1), whencet = β(t, 1)Λβ . We put

∫ r

A= {t ∈ A | ta = ε(a)t for all a ∈ A}

and call the elements of∫ r

Aright integrals in A. The foregoing shows that

∫ r

A= RΛβ.

Moreover,rr−1β (ε) = 0 impliesrε = 0 and hencer = 0. Thus:

∫ r

A= RΛβ

∼= R .

10 MARTIN LORENZ

Similarly, one can define theR-module∫ l

Aof left integralsin A and show that

∫ l

A= RΛ′

β∼= R with Λ′

β =∑

i ε(xi)yi = l−1β (ε) .

Define the idealDimεA of R by

DimεA := ε(∫ r

A) = ε(

∫ l

A) = ε(c(A)) = (ε(z)) , (20)

wherec(A) is the Casimir ideal andz = zβ ∈ Z(A) is as in (14). Note that alwaysc(A)∩R ⊆ε(c(A)); so

c(A) ∩R ⊆ DimεA . (21)

If β is symmetric thenrβ = lβ and henceΛβ = Λ′

β and∫ r

A=

∫ l

A=:

∫

A. For further

information on the material in this section, see [13, 6.1].

4.2. Involutions. Let A be a symmetric algebra with symmetric associative bilinearformβ : A×A → R. Suppose further thatA has an involution∗, that is, anR-linear endomorphismof A satisfying(xy)∗ = y∗x∗ andx∗∗ = x for all x, y ∈ A. If A is R-free with basis{xi}n1satisfying

β(xi, x∗

j ) = δi,j , (22)

then we will callA a symmetric∗-algebra. The Casimir operatorc = cβ : A → Z(A) takesthe form

c(a) =∑

i

x∗i axi =∑

i

xiax∗

i ,

and the Casimir elementz = zβ = c(1) is

z =∑

i

x∗i xi =∑

i

xix∗

i . (23)

Lemma 7. Let (A, β, ∗) be a symmetric∗-algebra. Then:

(a) β is ∗-invariant: β(x, y) = β(x∗, y∗) for all x, y ∈ A.(b) The Casimir operatorc is ∗-equivariant: c(a)∗ = c(a∗) for all a ∈ A. In particular,

z∗ = z.(c) If a = a∗ ∈ Z(A) then the matrix of(aA : x 7→ ax) ∈ EndR(A) with respect to the

R-basis{xi} ofA is symmetric.

Proof. Part (a) follows fromβ(xi, x∗j ) = δi,j = β(xj , x∗

i ) = β(x∗i , xj), and (b) follows fromcβ(a)

∗ =∑

(x∗i axi)∗ =

∑

i x∗

i a∗xi = cβ(a

∗).(c) Let (ai,j) ∈ Mn(R) be the matrix ofaA; so axj =

∑

i ai,jxi. We compute usingassociativity, symmetry and∗-invariance ofβ:

ai,j = β(x∗i , axj) = β(ax∗i , xj) = β(xj , ax∗

i ) = β(x∗j , axi) = aj,i .

�


4.3. Positivity. Let (A, β, ∗) be a symmetric∗-algebra withR-basis{xi}n1 satisfying (22).Assume thatR ⊆ R and putR+ = {r ∈ R | r ≥ 0}. If

A+ :=⊕

i

R+xi is closed under multiplication and stable under∗ (24)

then we will then say thatA has apositive structureand callA+ thepositive coneof A.We now consider the endomorphism(zA : x 7→ zx) ∈ EndR(A) for the Casimir element

z = zβ in (23). By Lemma 7, we know that the matrix ofzA with respect to the basis{xi} issymmetric. The following proposition gives further information.

Proposition 8. Let (A, β, ∗) be a symmetric∗-algebra over the ringR ⊆ R, and letz = zβbe the Casimir element.

(a) The matrix ofzA with respect to the basis{xi} is symmetric and positive definite. Inparticular, all eigenvalues ofzA are positive real numbers that are integral overR.

(b) If A has a positive structure and an augmentationε : A → R satisfyingε(a) > 0 forall 0 6= a ∈ A+. Then the largest eigenvalue ofzA is ε(z).

Proof. (a) LetZ = (zi,j) be the matrix ofzA; so zi,j = β(x∗i , zxj). Extending∗ andβ toAR = A⊗R R by linearity, one computes forx =

∑

i ξixi ∈ AR:∑

l

β((xlx)∗, xlx) = β(x∗, zx) = (ξ1, . . . ξn)Z(ξ1, . . . ξn)

tr .

The sum on the left is positive ifx 6= 0, becauseβ(y∗, y) =∑

η2i for y =∑

i ηixi ∈ AR.This shows thatZ is positive definite. The assertion about the eigenvalues ofZ is a standardfact about positive definite symmetric matrices over the reals.

(b) By hypothesis onA+, the matrix ofaA with respect to the basis{xi} has non-negativeentries for anya ∈ A+. Moreover, the Casimir elementz =

∑

i x∗

ixi belongs toA+, and sozA is non-negative. Now letΛ =

∑

i ε(x∗

i )xi ∈∫

Abe the integral ofA that is associated to

the augmentationε; see (19). ThenΛ is an eigenvector forzA with eigenvalueε(z). Sinceall ε(x∗i ) > 0, it follows thatε(z) is in fact the largest (Frobenius-Perron) eigenvalue ofzA isdimkH; see [9, Chapter XIII, Remark 3 on p. 63/4]. �

Corollary 9. If A is a symmetric∗-algebra over the ringR ⊆ R, then the Casimir elementzis a regular element ofA. Furthermore,A⊗R Q(R) is separable.

Proof. Regularity ofz is clear from Proposition 8(a). Sincez is integral overR, it followsthatzZ(A) ∩ R 6= 0. Therefore,c(A) ∩ R 6= 0 and Proposition 6 gives thatA ⊗R Q(R) isseparable. �

Part 2. HOPF ALGEBRAS

Throughout this part,H will denote a finitely generated projective Hopf algebra over thecommutative ringR (which will be assumed to be a field in Section 6), with unitu, multipli-cationm, counit ε, comultiplication∆, and antipodeS. We will use the Sweedler notation∆h =

∑

h1 ⊗ h2.In addition to the bimodule action ofH on H∨ in (1), we now also have an analogous

bimodule action of the dual algebraH∨ on H = H∨∨. In order to avoid confusion, it is

12 MARTIN LORENZ

customary to indicate the target of the various actions by⇀ or ↼ :

〈a ⇀ f ↼ b, c〉 = 〈f, bca〉 (a, b, c ∈ H, f ∈ H∨) ,

〈e, f ⇀ a ↼ g〉 = 〈gef, a〉 (e, f, g ∈ H∨, a ∈ H) .(25)

Here and for the remainder of this article,〈 . , . 〉 : H∨ × H → R denotes the evaluationpairing.

5. FROBENIUS HOPF ALGEBRAS OVER COMMUTATIVE RINGS

5.1. The following result is due to Larson-Sweedler [16], Pareigis [22], and Oberst-Schneider[21].

Theorem 10. (a) The antipodeS is bijective. Consequently,∫ l

H= S(

∫ r

H).

(b) H is a FrobeniusR-algebra if and only if∫ r

H∼= R. This always holds ifPicR = 1.

Furthermore, ifH is Frobenius then so is the dual algebraH∨.(c) Assume thatH is Frobenius. ThenH is symmetric if and only if

(i) H is unimodular (i.e.,∫ l

H=

∫ r

H), and

(ii) S2 is an inner automorphism ofH.

Proof. Part (a) is [22, Proposition 4] and (c) is [21, 3.3(2)]. For necessity of the condition∫ r

H∼= R in (b), in the more general context of augmented Frobenius algebras, see Section 4.1.

Conversely, if∫ r

H∼= R holds then [22, Theorem 1] asserts that the dual algebraH∨ is Frobe-

nius. This forces∫ r

H∨ to be free of rank1 overR, and henceH is Frobenius by [22, Theorem1]. The statement aboutPicR = 1 is a consequence of the fact that theR-module

∫ r

His

invertible (i.e., locally free of rank1) for any finitely generated projective HopfR-algebraH;see [22, Proposition 3]. �

5.2. We spell out some of the data associated with a FrobeniusHopf algebraH referring thereader to the aforementioned references [16], [22], [21] for complete details.

Fix a generatorΛ ∈∫ r

H. There is a uniqueλ ∈

∫ l

H∨ satisfyingλ ↼ Λ = ε or, equivalently,

〈λ,Λ〉 = 1. Note that this equation implies that∫ l

H∨ = Rλ, because∫ l

H∨ is an invertibleR-module. A nonsingular associative bilinear formβ = βλ for H is given by

β(a, b) = 〈λ, ab〉 (a, b ∈ H) . (26)

Dual bases{xi}, {yi} for β are given by∑

xi ⊗ yi =∑

Λ2 ⊗ S(Λ1):

a =∑

〈λ, aS(Λ1)〉Λ2 =∑

〈λ,Λ2a〉S(Λ1) (a ∈ H) . (27)

By [16, p. 83], the formβ is orthogonalfor the right action↼ of H∨ onH :

β(a, b ↼ f) = β(a ↼ S∨(f), b) (a, b ∈ H, f ∈ H∨) , (28)

whereS∨ = . ◦ S is the antipode ofH∨.


5.3. By (27) the Casimir operator has the form

c = cΛ : H → Z(H) , a 7→∑

S(Λ1)aΛ2 .

In particular,c(1) = 〈ε,Λ〉 ∈ R. Therefore, equality holds in (21):

DimεH = 〈ε,∫ r

H〉 = 〈ε,

∫ l

H〉 = c(H) ∩R . (29)

Proposition 6 now gives the following classical result of Larson and Sweedler [16].

Corollary 11. The separability locus of a Frobenius Hopf algebraH overR is

SpecR \ V (DimεH) .

In particular, H is separable if and only if〈ε,Λ〉 = 1 for some right or left integralΛ ∈ H.

The equality〈ε,Λ〉 = 1 implies thatΛ is an idempotent two-sided integral such that∫ r

H=

∫ l

H= RΛ, because

∫ r

Hand

∫ l

Hare invertibleR-modules.

5.4. LetH be a Frobenius Hopf algebra overR, and letΛ ∈∫ r

Handλ ∈

∫ l

H∨ be as in 5.2;so〈λ,Λ〉 = 1. The isomorphismsrβ andlβ in (4) and (5) for the the bilinear formβ = βλ in(26) will now be denoted byrλ andlλ, respectively:

rλ : HH∼

−→ H∨

H , a 7→ (β(a, . ) = λ ↼ a) ,

lλ : HH∼

−→ HH∨ , a 7→ (β( . , a) = a ⇀ λ) .(30)

Equation (28) states that

rλ(a ↼ S∨(f)) = frλ(a) and lλ(a ↼ f) = S∨(f)lλ(a) . (31)

for a ∈ H, f ∈ H∨. Sincerλ(Λ) = ε is the identity ofH∨, we obtain the followingexpression for the inverse ofrλ:

r−1λ : H∨

H∼

−→ HH , f 7→ (Λ ↼ S∨(f)) . (32)

5.5. In contrast with the Frobenius property, symmetry doesnot generally pass fromH toH∨; see [17, 2.5]. We will callH bi-symmetricif both H andH∨ are symmetric.

Lemma 12. Assume thatH is Frobenius and fix integralsΛ ∈∫ r

Handλ ∈

∫ l

H∨ such that〈λ,Λ〉 = 1, as in Sections 5.2, 5.4. Then:

(a) If H is involutory (i.e.,S2 = 1) then the regular characterχreg of H is given byχreg = 〈ε,Λ〉λ .

(b) H is symmetric and involutory if and only ifH is unimodular and all left and rightintegrals inH∨ belong toH∨

trace.(c) LetH be separable and involutory. ThenH is bi-symmetric. Furthermore,

∫

H∨ = Rχreg and Dimu∨ H∨ = (rankR H) ,

whereu∨ = 〈 . , 1〉 the counit ofH∨.

14 MARTIN LORENZ

Proof. (a) Equations (14), (26) and (27), withz =∑

i xiyi =∑

Λ2S(Λ1) = 〈ε,Λ〉 (usingS2 = 1), giveχreg = 〈λ, . z〉 = 〈ε,Λ〉λ .

(b) First assume thatH is symmetric and involutory. ThenH is unimodular by Theo-rem 10(c). By [21, 3.3(1)] we further know that the Nakayama automorphism ofH is equalto S2, and hence it is the identity. Thus,λ ↼ a = a ⇀ λ for all a ∈ H, which says thatλ isa trace form. Hence,

∫ l

H∨ ⊆ H∨trace . SinceH∨

trace is stable under the antipodeS∨ of H∨, italso contains

∫ r

H∨ . The converse follows by retracing these steps.(c) Now letH be separable and involutory. By Corollary 11 and the subsequent remark,H

is unimodular and we may chooseΛ ∈∫

Hsuch that〈ε,Λ〉 = 1. Part (a) gives

∫ l

H∨ = Rχreg .The computation

〈S∨(χreg), a〉 = Tr(S(a)A) = Tr(S ◦ Aa ◦ S−1) = Tr(Aa) =

(13)〈χreg, a〉

for a ∈ A shows thatS∨(χreg) = χreg . Therefore, we also have∫ r

H∨ = Rχreg . In viewof Theorem 10(c), this shows thatH is bi-symmetric. Finally, since〈χreg, 1〉 = rankR H ,equation (29) yieldsDimu∨ H∨ = (rankRH). �

As was observed in the proof of (b), the mapsrλ andlλ in (30) coincide if and only ifλ isa trace form. In this case, we will denote the(H,H)-bimodule isomorphismrλ = lλ by bλ:

bλ : HHH∼

−→ HH∨H , a 7→ (λ ↼ a = a ⇀ λ) . (33)

We also remark that the formulaDimu∨ H∨ = (rankR H) in (c) is a special case of thefollowing formula which holds for any involutoryH ; see [21, 3.6]:

DimεH ·Dimu∨ H∨ = (rankR H) . (34)

5.6. We letC(H) = {a ∈ H |

∑

a1 ⊗ a2 =∑

a2 ⊗ a1}

denote theR-subalgebra ofH consisting of all cocommutative elements. Thus,C(H∨) =H∨

trace . Recall from Lemma 12(b) that all integrals inH∨ are cocommutative ifH is sym-metric and involutory.

Lemma 13. Let H be bi-symmetric and involutory. Fix a generatorλ ∈∫

H∨ . Then the(H,H)-bimodule isomorphismbλ in (33) restricts to an isomorphisms

Z(H)∼

−→ H∨

trace and C(H)∼

−→ Z(H∨) .

Proof. The isomorphismZ(H)∼

−→ H∨trace = C(H∨) is (3). By the same token, fixing a

(necessarily cocommutative) generatorΛ ∈∫

Hsuch that〈λ,Λ〉 = 1, we obtain thatbΛ is

an (H∨,H∨)-bimodule isomorphismH∨ ∼−→ H∨∨ = H that restricts to an isomorphism

Z(H∨)∼

−→ C(H). By equation (32) we have

bΛ(S∨(f)) = b

−1λ (f) (35)

for f ∈ H∨. SinceZ(H∨) is stable under the antipodeS∨ of H∨, we conclude thatb−1λ

restricts to an isomorphismZ(H∨)∼−→ C(H), and hencebλ restricts to an isomorphism

C(H)∼−→ Z(H∨). �


5.7. Turning to modules now, we review some standard constructions and facts. For any twoleft H-modulesM andN , the tensor productM ⊗R N becomes anH-module via∆ , andHomR(M,N) carries the followingH-module structure:

(aϕ)(m) =∑

a1ϕ(S(a2)m)

for a ∈ H,m ∈ M,ϕ ∈ HomR(M,N) . In particular, viewingR asH-module viaε, theH-action on the dualM∨ = HomR(M,R) takes the following form:

〈af,m〉 = 〈f,S(a)m〉

for a ∈ H,m ∈ M,f ∈ M∨ . TheH-invariants inHomR(M,N) are exactly theH-modulemaps:

HomR(M,N)H = {ϕ ∈ HomR(M,N) | aϕ = 〈ε, a〉ϕ ∀a ∈ A} = HomH(M,N) ;

see, e.g., [28, Lemma 1]. Moreover, it is easily checked thatthe canonical map

N ⊗R M∨ → HomR(M,N) , n⊗ f 7→ (m 7→ 〈f,m〉n)

is a homomorphism ofH-modules. This map is an isomorphism ifM or N is finitely gener-ated projective overR; see [2, II.4.2].

Finally, we consider the trace mapTr: EndR(M) ∼= M ⊗R M∨ → R of Section 2.1.

Lemma 14. Let H be involutory and letM be a leftH-module that is finitely generatedgenerated projective overR. Then the trace mapTr is anH-module map.

Proof. In view of the foregoing, it suffices to checkH-equivariance of the evaluation mapM ⊗R M∨ → R. Using the identity

∑

S(a2)a1 = 〈ε, a〉 for a ∈ H (from S2 = 1), wecompute

a · (m⊗ f) =∑

a1m⊗ a2f 7→∑

〈a2f, a1m〉 =∑

〈f,S(a2)a1m〉 = 〈ε, a〉〈f,m〉 ,

as desired.�

5.8. We now focus on modules over a separable involutory HopfalgebraH. In particular,we will compute the image of the central idempotentse(M) from Proposition 4 under theisomorphismZ(H)

∼−→ H∨

trace in Lemma 13 and theβ-index [H : M ]β of (17). Recall thatH is bi-symmetric by Lemma 12(c).

Proposition 15. Assume thatH is separable and involutory. FixΛ ∈∫

H, λ ∈

∫

H∨ such that〈λ,Λ〉 = 1 and letβ denote the form(26). Then, for every cyclic leftH-moduleM that isfinitely generated projective overR and satisfiesEndH(M) ∼= R ,

(a) rankR M is invertible inR ;(b) [H : M ]β = 〈ε,Λ〉(rankR M)−1 is invertible inR ;(c) bλ(e(M)) = [H : M ]−1

β χM . In particular, bλ(e(M)) = (rankRM)χM holds forλ = χreg .

Proof. (a) Our hypothesisEndH(M) ∼= R implies thatM is faithful asR-module. Hence,the trace mapTr: EndR(M) ∼= M ⊗R M∨ → R is surjective by (9), and it is is anH-module map by Lemma 14. Moreover, for eachϕ ∈ EndR(M), we haveΛϕ = rϕ1M forsomerϕ ∈ R, sinceΛEndR(M) ⊆ EndH(M) ∼= R. Therefore,Tr(Λϕ) = rϕ rankR M

16 MARTIN LORENZ

andTr(Λϕ) = ΛTr(ϕ) = 〈ε,Λ〉Tr(ϕ) . Choosingϕ with Tr(ϕ) = 1, we obtain fromCorollary 11 thatTr(Λϕ) is a unit inR. Hence so isrankR M , proving (a).

(b) By Propositions 4(c) and 5(a), we have

ωM (z) rankR M = [H : M ]β (rankR M)2 ,

wherez =∑

xiyi = 〈ε,Λ〉 is as in the proof of Lemma 12(a). In view on part (a), the aboveequality amounts to the asserted formula for[H : M ]β . Finally, invertibility of [H : M ]β isProposition 4(a) (and it also follows from Corollary 11).

(c) Proposition 5(b) givesχreg ↼ e(M) = (rankRM)χM , which is the asserted formulafor bλ(e(M)) with λ = χreg . For generalλ, we have〈ε,Λ〉bλ(e(M)) = χreg ↼ e(M) byLemma 12(a). The formula forbλ(e(M)) now follows from (b). �

5.9. Assume that, for some subringS ⊆ R, there is anS-subalgebraB ⊆ H such that

(i) B is finitely generated asS-module, and(ii) ((S ⊗ 1H) ◦∆)(Λ) =

∑

S(Λ1)⊗ Λ2 ∈ H ⊗H belongs to the image ofB ⊗S B inH ⊗H .

Adapting the teminology of Section 2.5, we will callA aweakR-formof (H,Λ). The follow-ing corollary is a consequence of Proposition 15(b) and Lemma 3; it is due to Rumynin [24]over fields of characteristic0.

Corollary 16. LetH be separable and involutory. Assume that, for some generating integralΛ ∈

∫

H, there is a weakS-form for (H,Λ) for some subringS ⊆ R. Then, for every left

H-moduleM that is finitely generated projective overR and satisfiesEndH(M) ∼= R , theindex[H : M ]β = 〈ε,Λ〉(rankR M)−1 is integral overS.

Example 17. The group algebraRG of a finite groupG has generating (right and left) integralΛ =

∑

g∈G g. The corresponding integralλ ∈∫

(RG)∨ with 〈λ,Λ〉 = 1 is the trace form givenby 〈λ,

∑

g∈G rgg〉 = r1. Note that〈ε,Λ〉 = |G| 1. Thus, Corollary 11 tells us thatRG issemisimple if and only if|G| 1 is a unit inR; this is Maschke’s classical theorem. AssumingcharR = 0, a weakZ-form for (RG,Λ) is given by the integral group ringB = ZG.Therefore, Corollary 16 yields the following version of Frobenius’ Theorem: The rank ofeveryR-freeRG-moduleM such thatEndRG(M) ∼= R divides the order ofG.

6. GROTHENDIECK RINGS OF SEMISIMPLEHOPF ALGEBRAS

From now on, we will focus on the case whereR = k is a field. Throughout, we will assumethatH is a split semisimple Hopf algebra overk. In particular,H is finite-dimensional overkand hence Frobenius. We will write⊗ = ⊗k andk-linear duals will now be denoted by( . )∗.Finally, IrrH will denote a full set of non-isomorphic irreducibleH-modules.

6.1. The Grothendieck ring. We review some standard material; for details, see [17].

6.1.1. TheGrothendieck ringG0(H) of H is the abelian group that is generated by theisomorphism classes[V ] of finite-dimensional leftH-modulesV modulo the relations[V ] =[U ] + [W ] for each short exact sequence0 → U → V → W → 0. Multiplication inG0(H)


is given by[V ] · [W ] = [V ⊗W ] . The subset{[V ] | V ∈ IrrH} ⊆ G0(H) forms aZ-basisof G0(H), and the positive cone

G0(H)+ :=⊕

V ∈IrrH

Z+[V ] = {[V ] | V a finite-dimensionalH-module}

is closed under multiplication. The Grothendieck ringG0(H) has thedimension augmenta-tion,

dim: G0(H) → Z , [V ] 7→ dimk V ,

and an involution given by[V ]∗ = [V ∗], where the dualV ∗ = Homk(V, k) hasH-action asin Section 5.7. The basis{[V ] | V ∈ IrrH} is stable under the involution∗ , and hence so isthe positive coneG0(H)+.

6.1.2. The Grothendieck ringG0(H) is a symmetric∗-algebra overZ. A suitable bilinearform β is given by

β([V ], [W ]) = dimkHomH(V,W ∗) . (36)

Using the standard isomorphism(W ⊗ V ∗)H ∼= HomH(V,W ), where( . )H denotes thespace ofH-invariants, this form is easily seen to beZ-bilinear, associative, symmetric, and∗-invariant. DualZ-bases ofG0(H) are provided by{[V ] | V ∈ IrrH} and{[V ∗] | V ∈IrrH}:

β([V ′], [V ∗]) = δ[V ′],[V ] (V, V ′ ∈ IrrH) .

The integral inG0(H) that is associated to the dimension augmentation ofG0(H) as inSection§4.1 is the class[H] of the regular representation ofH: β([H], [V ]) = dimk V . Thus,

∫

G0(H) = Z [H] . (37)

6.1.3. Thecharacter map

χ : G0(H) → H∗ , [V ] 7→ χV

is a ring homomorphism that respects augmentations:

G0(H)

dim��

χ// H∗

u∗

��Z // k

Moreover,χV ∗ = S∗(χV ) = χV ◦ S .

Thus, ifH is involutory thenχ also commutes with the standard involutions onG0(H) andH∗. The class of the regular representation[H] ∈

∫

G0(H) is mapped to the regular characterχreg ∈ H∗. If H is involutory thenχreg is a nonzero integral ofH∗; see Lemma 12. Thus, inthis case, we have kχ(∫

G0(H)) = kχreg =∫

H∗ .

6.1.4. Thek-algebraR(H) := G0(H)⊗Z k is called therepresentation algebraof H. Themap[V ]⊗ 1 7→ χV gives an algebra embeddingR(H) → H∗ whose image is the subalgebraH∗

trace = (H/[H,H])∗ of all trace forms onH:

R(H)∼

−→ H∗

trace ⊆ H∗ .

18 MARTIN LORENZ

6.2. As an application of Proposition 15, we prove the following elegant generalization ofFrobenius’ Theorem (see Example 17) in characteristic0 due to S. Zhu [27, Theorem 8].

Theorem 18. LetH be a split semisimple Hopf algebra over a fieldk of characteristic0 andlet V ∈ IrrH be such thatχV ∈ Z(H∗). Thendimk V dividesdimkH.

Proof. By [15, Theorem 4]H is involutory and cosemisimple. LetΛ ∈ C(H) denote thecharacter of the regular representation ofH∗; this is an integral ofH by Lemma 12 and,clearly,〈ε,Λ〉 = dimkH. Letλ ∈

∫

H∗ be such that〈λ,Λ〉 = 1 and consider the isomorphismbλ : HHH

∼−→ HH∨

H in (33). By Proposition 15, we havebλ(e(V )) = dimk VdimkH χV and (35)

givesdimkHdimk V e(V ) = bΛ(S

∗(χV )) .

Therefore, it suffices to show thatbΛ(S∗(χV )) is integral overZ.By hypothesis,S∗(χV ) ∈ Z(H∗). Furthermore,S∗(χV ) ∈ χ(G0(H)) is integral overZ. HenceS∗(χV ) ∈ Z(H∗)cl, the integral closure ofZ in Z(H∗). Passing to an al-

gebraic closure ofk, as we may, we can assume thatH∗ and Z(H∗) are split semisim-ple. Thus,Z(H∗) =

⊕

M∈IrrH∗ ke(M) andZ(H∗)cl =⊕

M∈IrrH∗ Oe(M), where wehave putO := {algebraic integers ink}. Proposition 15(c), withH∗ in place ofH, givesbΛ(e(M)) = (dimkM)χM . Thus,

bΛ(Z(H∗)cl) ⊆ χ(G0(H∗))O ⊆ C(H) .

Finally, all elements ofG0(H∗) are integral overZ, and hence the same holds for the elements

of χ(G0(H∗))O. In particular,bΛ(S∗(χV )) is integral overZ, as desired. �

6.3. The class equation.We now derive the celebrated class equation, due to Kac [12, The-orem 2] and Zhu [28, Theorem 1], from Proposition 4. Frobenius’ Theorem (Example 17) incharacteristic0 also follows from this result applied toH = (kG)∗. We also mentioned herethat the class equation was used by Schneider in [25] to provethe following strong versionof the Frobenius property for quasitriangular semisimple Hopf algebrasH over a fieldk ofcharacteristic0: if V is an absolutely irreducibleH-module, then(dimk V )2 dividesdimkH.

Theorem 19 (Class equation). Let H be a split semisimple Hopf algebra over a fieldk ofcharacteristic0. Thendimk(H∗ ⊗R(H) M) dividesdimkH∗ for every absolutely irreducibleR(H)-moduleM .

Proof. Inasmuch asR(H) is semisimple by Corollary 9, we haveM ∼= R(H)e for someidempotente = e2 ∈ R(H) with eR(H)e ∼= k. Thus,H∗ ⊗R(H) M ∼= H∗e and the assertionof the theorem is equivalent to the statement thatdimkH∗e dividesdimkH∗.

The bilinear formβ in (36) can be written asβ([V ], [W ]) = τ([V ][W ]), whereτ : G0(H) →Z is the trace form given by

τ([V ]) = dimk V H .

Now letΛ ∈∫

Hdenote the regular character of the dual Hopf algebraH∗, as in the proof of

Theorem 18. Thus,〈e,Λ〉 =

(10)dimk eH∗ =

(13)dimkH∗e . (38)


Being an integral ofH, Λ annihilates allV ∈ IrrH \ {kε} and so〈χV ,Λ〉 = 0. On the otherhand,〈χkε ,Λ〉 = 〈ε,Λ〉 = dimkH∗. This shows thatΛ

∣

∣

R(H)= dimkH∗ · τ ′, where we have

put τ ′ = τ ⊗Z Idk : R(H) → k. Therefore, (38) becomes

τ ′(e) =dimkH∗e

dimkH∗.

Now, τ ′(e) = β′(e, 1), whereβ′ = β ⊗Z Idk, and by Proposition 4(a), we haveβ′(e, 1)−1 =[R(H) : M ]β′ . Thus,

[R(H) : M ]β′ =dimkH∗

dimkH∗e=

dimkH∗

dimk(H∗ ⊗R(H) M). (39)

Finally, Lemma 3 withA = G0(H) andA′ = R(H) tells us that this rational number isintegral overZ. Hence it is an integer, proving the theorem. �

6.4. The adjoint class.

6.4.1. The adjoint representation.The left adjoint representation ofH is given by

ad: H −→ EndkH , ad(h)(k) =∑

h1kS(h2)

for h, k ∈ H. There is anH-isomorphism

Had∼=

⊕

V ∈IrrH

V ⊗ V ∗ . (40)

This follows from standardH-isomorphismV ⊗ V ∗ ∼= Endk(V ) (see Section 5.7) combinedwith the Artin-Wedderburn isomorphism,H ∼=

⊕

V ∈IrrH Endk(V ), which is equivariant forthe adjointH-action onH.

6.4.2. The adjoint class.Equation (40) gives the following description of the Casimir elementz = zβ of the symmetricZ-algebraG0(H):

z =∑

V ∈IrrH

[V ][V ∗] = [Had] ∈ Z(G0(H)) . (41)

Therefore, we will refer to the Casimir elementz as theadjoint classof H.We now consider the left regular action ofz onG0(H), that is, the endomorphismzG0(H) ∈

EndZ(G0(H)) that is given by

zG0(H) : G0(H) → G0(H) , x 7→ zx .

By Proposition 8, we know that the eigenvalues ofzG0(H) are positive real algebraic integersand that the largest eigenvalue isdim(z) = dimkH. The following proposition gives moreprecise information; the result was obtained by Sommerhauser [26, 3.11] using a differentmethod.

Proposition 20. LetH be a split semisimple Hopf algebra over a fieldk of characteristic0.Then the eigenvalues ofzG0(H) are positive integers≤ dimkH. If G0(H) or H is commuta-tive then all eigenvalues ofzG0(H) dividedimkH.

20 MARTIN LORENZ

Proof. We may pass to the algebraic closure ofk; this changes neitherG0(H) nor z. Thenthe representation algebraR(H) = G0(H) ⊗Z k is split semisimple by Corollary 9. Sincez ∈ Z(R(H)), the eigenvalues ofzG0(H) are exactly theωM(z) ∈ k, whereM runs over theirreducibleR(H)-modules andωM denotes the central character ofM as in 2.4. We know byPropositions 4(c) and 5(a) and equation (39) that

ωM (z) = dimkM ·dimkH∗

dimk(H∗ ⊗R(H) M),

and this is a positive integer by Theorem 19. SinceM ⊆ H∗ ⊗R(H) M , we haveωM (z) ≤dimkH. If G0(H) is commutative thendimkM = 1, and henceωM (z) dividesdimkH.If H is commutative thenR(H) = H∗, and soωM (z) = dimkH. (Alternatively, if H iscommutative thenHad

∼= kdimkHε andz = [Had] = (dimkH)1 .) �

We mention that the Grothendieck ringG0(H) is commutative whenever the Hopf algebraH is almost commutative. In particular, this holds for all quasi-triangular Hopf algebras; seeMontgomery [20, Section 10.1].

Example 21. Let H = kG be the group algebra of the finite groupG over a splitting fieldkof characteristic0. The representation algebraR(H) ∼=

⊕

V ∈IrrH kχV ⊆ H∗ is isomorphicto the algebra ofk-valued class functions onG, that is, functionsG → k that are constant onconjugacy classes ofG. For any finite-dimensionalkG-moduleV , the character valuesχV (g)(g ∈ G) are the eigenvalues of the endomorphisms[V ]G0(H) ∈ EndZ(G0(H)). Specializingto the adjoint representationV = Had we obtain the eigenvalues ofzG0(H): they are theintegersχHad

(g) = |CG(g)| with g ∈ G.

6.5. The semisimple locus ofG0(H). LetH be a split semisimple Hopf algebra over a fieldk. Recall thatG0(H)⊗Z Q is semisimple by Corollary 9. We will now describe the primespfor which the algebraG0(H)⊗Z Fp is semisimple.

Proposition 22. LetH be a split semisimple Hopf algebra over a fieldk. Then:

(a) If p dividesdimkH thenG0(H)⊗Z Fp is not semisimple.(b) Assume thatchar k = 0. ThenG0(H)⊗Z Fp is semisimple for allp > dimkH.(c) Assume thatchar k = 0 and thatG0(H) or H is commutative. ThenG0(H)⊗Z Fp is

semisimple if and only ifp does not dividedimkH.

Proof. Semisimplicity is equivalent to separability overFp; see [23, 10.7 Corollary b]. There-fore, we may apply Proposition 6. In detail, consider the Casimir operator that is associatedwith the bilinear formβ of Section 6.1.2,

c : G0(H) → Z(G0(H)) , x 7→∑

V ∈IrrH

[V ∗]x[V ] .

By Proposition 6,G0(H)⊗Z Fp is semisimple if and only if(p) + Z ∩ Im c.(a) Consider the dimension augmentationdim: G0(H) → Z, [V ] 7→ dimk V . The com-

positedim ◦c is equal todimkH · dim . Hence,Z ∩ Im c ⊆ Im(dim ◦c) ⊆ (dimkH) holdsin Z, which implies (a).

(b) In view of Proposition 20, our hypothesis onp implies that the normN(z) = det zG0(H)

is not divisible byp. Sincez = c(1), it follows that(p) + N(Im c), and hence(p) + Z∩Im c;see Section 3.4.


(c) Necessity of the condition onp follows from (a) and sufficiency follows from Proposi-tion 20 as in (b). �

6.6. Traces of group-like elements.LetH be a split semisimple Hopf algebra over a fieldkand letχad ∈ R(H) ⊆ H∗ denote the character of the adjoint representation. Equation (41)gives

χad =∑

V ∈IrrH

χV ∗χV .

Proposition 23. LetH be a split semisimple Hopf algebra over a fieldk. If R(H) is semisim-ple thenχad(g) 6= 0 for every group-like elementg ∈ H.

Proof. By Proposition 6, semisimplicity ofR(H) is equivalent to surjectivity of the Casimiroperatorc : R(H) → Z(R(H)), χ 7→

∑

V ∈IrrH χV ∗χχV . Fixing χ with∑

V χV ∗χχV = 1we obtain

1 =∑

V

χV ∗(g)χ(g)χV (g) = χ(g)χad(g) ,

which shows thatχad(g) 6= 0. �

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DEPARTMENT OFMATHEMATICS, TEMPLE UNIVERSITY, PHILADELPHIA , PA 19122E-mail address: [email protected]: www.math.temple.edu/˜lorenz

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Some applications of Frobenius algebras to Hopf algebras