Forms of Hopf Algebras and Galois Theory 3

FORMS OF HOPF ALGEBRAS AND

GALOIS THEORY

BODO PAREIGIS

The theory of Hopf algebras is closely connected with various ap-

plications, in particular to algebraic and formal groups. Although the

rst o ccurence of Hopf algebras was in algebraic top ology, they are now

found in areas as remote as combinatorics and analysis. Their struc-

ture has b een studied in great detail and many of their prop erties are

well understo o d. We are interested in a systematic treatmentofHopf

algebras with the techniques of forms and descent.

The rst three paragraphs of this pap er givea survey of the present

state of the theory of forms of Hopf algebras and of Hopf Galois theory

esp ecially for separable extensions. It includes many illustrating exam-

ples some of which cannot b e found in detail in the literature. The last

two paragraphs are devoted to some new or partial results on the same

eld. There we formulate some of the op en questions whichshould

be interesting ob jects for further study.We assume throughout most

of the pap er that k is a base eld and do not touch up on the recent

b eautiful results of Hopf Galois theory for rings of integers in algebraic

numb er elds as develop ed in [C1].

1. Hopf algebra forms

As a rst example of the o ccurence of a Hopf algebra let us consider

the units functor. In the sequel let k be a commutative, asso ciativering

with unit. Later on it will b e a eld, in particular the eld of rationals

or reals. Let k -Alg denote the category of commutative k -algebras and

Grp the category of groups. Then there is the imp ortantfunctor

U : k -Alg !Grp;

the units functor, which asso ciates with each k -algebra its group of

invertible elements or units. This functor is representable by the k -

algebra k [x; x ]=k Z, the group-ring of the in nite cyclic group Z,

Date :February 24, 1990.

1991 Mathematics Subject Classi cation. Primary 16A10. 1

2 BODO PAREIGIS

i.e.

U (A) k - Alg(k [x; x ];A):

The multiplication of the units group U U ! U induces a commu-

tative diagram

U (A) U (A) U (A)

? ?

1 1 1

k - Alg(k [x; x ] k [x; x ];A) k -Alg (k [x; x ];A)

with vertical arrows isomorphisms. By the Yoneda Lemma the sec-

ond horizontal arrow induces a comultiplication on the representing

k -algebra

1 1 1 1 1

: k [x; x ] ! k [x; x ] k [x; x ]=k [x; x ] k [x; x ];

de ned by(x)= x x. Observe that the tensor pro duct of com-

mutative algebras is the copro duct in k - Alg. Also the inverse inv: U

! U and the neutral element f:g! U de ne corresp onding maps on

the representing algebra. All in all we obtain the structure of a Hopf

algebra on the algebra k [x; x ].

De nition 1.1. A k -algebra H together with k -algebra homomor-

phisms

: H ! H H;

: H ! H;

: H ! k

is called a Hopf algebra,ifH together with and is a coalgebra and

if satis es the following commutative diagrams

- -

H k H

H H H H

1(1 )

Here the map r : H H ! H denotes the multiplication r(a b):=

ab of the algebra H and : k ! H is de ned by (a):=a 1 ,the

canonical map from k into H .

The map : H ! H H is called the diagonal or comultiplication

on H .Itisawkward to write the images as tensors in the usual way,

esp ecially if comp osites of such maps o ccur. The following simpli ed

notation has b een intro duced bySweedler. For a linear map f : A

! B C we de ne a a := f (a) or in the sp ecial case of a Hopf

(B ) (C )

algebra (H; ;)we write h h := (h). The advantage of this

(1) (2)

notation is, that it can b e extended to bilinear maps along the following

FORMS OF HOPF ALGEBRAS AND GALOIS THEORY 3

example. If g : B C ! D is a bilinear map with induced mapg ^ : B C

! D on the tensor pro duct, then we can de ne g (a ;a ):=

(B ) (C )

g^(f (a)) and thus use the "comp onents" a and a as if they were

(B ) (C )

well-de ned ordinary elements, which can b e used as arguments in

bilinear maps.

Similar to the Hopf algebra k [x; x ]each commutative (as an alge-

bra) Hopf algebra H represents a functor

~ ~

H : k -Alg !Grp; H (A):=k -Alg (H; A)

where the multiplication on H is given bythecommutativediagram

~ ~ ~

H (A) H (A) H (A)

? ?

k -Alg (H H; A) k - Alg(H; A)

So the group-ring k Z has b een seen to b e a Hopf algebra with the

diagonal (g )= g g for every element g in the group Z. This holds

not only for the group Z. Every group-ring kG is a Hopf algebra

with the same comultiplication, even for non-commutative groups G.

The non-commutative group rings, however, do not anymore represent

group valued functors on k -Alg . They are sp ecial instances of formal

groups.

Another concrete example of a group valued functor is

C : k -Alg !Grp;

2 2

the circle group, de ned by C (A):=f(a; b) 2 A Aja + b =1g.

The group structure is given by(a; b) (c; d):=(ac bd; ad + bc).

The representing Hopf algebra is the "trigonometric algebra" H =

2 2

k [c; s]=(c + s 1). The diagonal is de ned by

(s) = c s + s c:

The most interesting observation is this. Let A be a commutative k -

algebra with 2 invertible and containing i = 1. Then the assignment

1 1

U (A) 3 a 7! 2 C (A) (a + a ); (a a )

2 2i

de nes a functorial isomorphism of groups. If 2 ;i 2 k then U and

C are isomorphic group valued functors, hence they have isomorphic

representing Hopf algebras

1 2 2

k [x; x ] k [c; s]=(c + s 1): =

4 BODO PAREIGIS

If i=2 k then the two group valued functors are not isomorphic, neither

1 2 2

are their representing Hopf algebras k [x; x ]and k [c; s]=(c + s 1).

If k is a eld of characterictic 6= 2 and i=2 k ,thenU and C are

non-isomorphic but they induce isomorphic functors U j and C j if

k (i) k (i)

restricted to the k (i)-algebras. Let K = k (i) and let A be a K -algebra.

Then wehave

1 1

K -Alg (K k [x; x ];A) k -Alg (k [x; x ];A)

U j (A)

C j (A)

2 2

k -Alg (k [c; s]=(c + s 1);A)

2 2

K -Alg (K k [c; s]=(c + s 1);A)

2 2 1

K k [c; s]=(c + s 1) as K -Hopf algebras, hence K k [x; x ]

where the tensor pro duct is always taken over the base ring k .Observe

that a cancellation prop erty cannot b e exp ected in this case.

1 2 2

In particular, the Q-Hopf algebras Q[x; x ] and Q[c; s]=(c + s 1)

1 2 2

and the R-Hopf algebras R[x; x ]andR[c; s]=(c + s 1) are not

2 2 1

C[c; s]=(c + s 1) isomorphic, but the C-Hopf algebras C[x; x ]

are. This is an example for the next de nition.

De nition 1.2. Let G and G b e group valued functors on k - Alg. Let

K b e a faithfully at commutative k -algebra. If the restrictions to

G j ,thenG and K -Alg are isomorphic group valued functors: Gj

K K

G are called K -forms of each other as groups.

Let H and H b e Hopf algebras over the commutative ring k . Let

K H as K b e a faithfully at commutative k -algebra. If K H

K -Hopf algebras, then H and H are called K -forms of eachotheras

Hopf algebras

0 0

WesaythatG and G resp. H and H are forms of each other if

there exists a faithfully at k -algebra K such that they are K -forms of

each other.

So for G and G to b e K -forms of each other we need an isomorphism

of set valued functors : Gj ! G j such that

K K

0 0

Gj Gj G j G j

K K K K

? ?

Gj G j

K K

commutes.

There maybemany di erent Hopf algebras H which are forms for H

with resp ect to some faithfully at extension K . In particular the rich-

ness of Hopf algebras over Q should b e higher than over C. Granted

there may b e Hopf algebras de ned over C, which do not come ab out

FORMS OF HOPF ALGEBRAS AND GALOIS THEORY 5

by a base ring extension from Q, but e.g. semisimple co commutative

Hopf algebras over C are always de ned over Q. This is a consequence

of a more general structure theorem of Milnor, Mo ore and Cartier on

co commutative Hopf algebras over algebraicly closed elds. Our inter-

ests are in this richness of Hopf algebras over "small" elds. One can

show for example that over the eld R of reals the circle functor C is

the only non-trivial form of the units functor U .

There is a description of K -forms for quite general algebraic struc-

tures given by the theory of faithfully at descent. We apply it to the

case of Hopf algebras. Let H b e a Hopf algebra over k . The group of

automorphisms of this Hopf algebra will b e denoted by k - Hopf-Aut(H ).

After a base ring extension by k ! K we get a Hopf algebra K H over

K with group of automorphisms K - Hopf-Aut(K H ). Every change

of the base ring extension, i.e. every homomorphism of commutative k -

algebras K ! L induces a group homomorphism K - Hopf-Aut(K H )

! L- Hopf- Aut(L H ). Thus wehave a functor Aut (H ): k - Alg !Grp

de ned by Aut(H )(K ):=K -Hopf-Aut(K H ). Every group valued

functor on the category k -Alg of commutative k -algebras has an asso-

ciated Amitsur cohomology H (K=k; Aut(H )). It is not necessary to

know the precise de nition of these cohomology groups to apply the

following theorem.

Theorem 1.3. Let H beak -Hopf algebra. Then there is a bijektion

between the set of (isomorphism classes of ) K -forms of H and the

Amitsur cohomology group H (K=k; Aut (H )).

Pro ofs of this may b e found in various forms in [G], [H], or [KO].

Actually this theorem holds in greater generality and the pro of is quite

technical and involved.

In view of this theorem the main problem of calculating forms is to

determine the set of Hopf algebra automorphisms of a Hopf algebra.

In fact wedonothave to calculate the cohomology group, since bya

twofold application of this theorem { going from certain forms to the

cohomology group and then from the same cohomology group backto

some other forms { we will eliminate the explicit computation of the

cohomology.

In the case of group-rings kG of nitely generated groups G the

automorphism group k -Hopf-Aut(kG) can b e calculated, in particular

for cyclic groups C of order n. We assume that the automorphism

group F of G is nite. Then one can showthatk -Hopf-Aut(kG)is

) of the trivial F - isomorphic to the automorphism group G al-Aut(E

Galois extension E of k . This Galois extension can b e describ ed by

the ring E =(kF ) , the dual space of the group ring kF ,onwhich F k

6 BODO PAREIGIS

acts by automorphisms in sucha way, that the ring extension (kF ) =k

is an F -Galois extension in the sense of [CHR]. Actually this leads to a

functorial isomorphism Aut(kG) G al- Aut(E ), so that the Amitsur

cohomology groups of these two group functors also coincide.

Weformulate one of the most interesting consequences of these con-

siderations.

Theorem 1.4. [HP] Let k beacommutative ring with 2 not a zero

divisor in k and Pic (k )=0, the two-torsion of the Picardgroup.

(2)

Then

a) the Hopf algebraformsofk Z are

2 2

H = k [c; s]=(s asc bc + u);

b) the Hopf algebraformsofkC are

2 2

H = k [c; s]=(s asc bc + u; (c +1)(c 2); (c +1)(s a));

c) the Hopf algebraformsofkC are

2 2

H = k [c; s]=(s asc bc + u; c(ac 2s));

d) the Hopf algebraformsofkC are

2 2

H = k [c; s]=(s ascbc +u; (c2)(c1)(c+1)(c+2); (c1)(c+1)(sc2a)):

In al l cases a; b; u 2 k satisfy a +4b = u and u is a unit in k .The

Hopf algebra structureinallcases is de nedby

1 2

(s) = u (abc c +2b(c s + s c)+as s);

Wegive an indication of the wayhow this result is obtained. In all

cases of the theorem the group F is the cyclic group with twoelements.

The theory of C -Galois extensions ( = quadratic Galois extensions) is

well known. Actually every quadratic Galois extension of k is a form of

k k of k as will b e seen the trivial quadratic Galois extension (kC )

G al- Aut(k k ) b elow. Since the automorphism groups Aut(kG)

coincide, the rst Amitsur cohomology groups describing the forms

coincide, to o. So there is a bijective corresp ondence b etween the forms

of the group-rings in the theorem and the quadratic Galois extensions of

k [see Thm. 4.1]. This corresp ondence was used to explicitly calculate

the forms given in the theorem.

FORMS OF HOPF ALGEBRAS AND GALOIS THEORY 7

2. Hopf Galois extensions

A di erent class of "forms" is obtained if one considers the following

cancellation problem.

De nition 2.1. Let G : k -Alg !Grp b e a group valued functor. Then

the multiplication of G on itself G G ! G makes G a G-set valued

functor. Here we de ne the functor G G by(G G)(K ):=G(K )

G(K ), so that the multipication of each group G(K ) de nes a functorial

homomorphism G G ! G, brie y the multiplication on G and the

G-set structure is de ned "comp onentwise".

Let X : k -Alg !Set b e another functor whichisalsoaG-set valued

functor by X G ! X .LetK b e a faithfully at commutativering

extension of k . If the restrictions Gj and X j to K -Alg are isomorphic

K K

as Gj -set valued functors, then G and X are called K -forms of each

other as G-set valued functors.

So for G and X to b e K -forms of each other we need an isomorphism

of set valued functors : Gj ! X j suchthat

K K

Gj Gj X j Gj

K K K K

? ?

Gj X j

K K

commutes.

A Hopf algebraic description of this is somewhat more complicated.

The notion of a G-set and of forms of a G-set translated to the repre-

senting ob jects of the representable functors G and X gives the follow-

ing de nition.

De nition 2.2. Let H b e a commutative Hopf algebra and A be a

commutative algebra. A is called an H -comodule algebra if there is an

algebra map : A ! A H such that the diagrams

- -

A A H A A H

? ? ? ?

and - -

A H A H H

A A k

commute.

Let H b e a commutative Hopf algebra and A beacommutative

H -como dule algebra. Let K b e a faithfully at commutativering

K A as K H -como dule algebras, then extension of k .IfK H

A is called a K -form of H .

8 BODO PAREIGIS

Closely connected with K -forms of G-set valued functors is the no-

tion of a principal homogeneous space.

De nition 2.3. If G is a group and X is a set, then a G-set X is

called homogeneous, if for eachpairx; y 2 X there exists a g 2 G

such that xg = y .AG-set X is a principal homogeneous G-set if X is

homogeneous and xg = x for any x 2 X implies g = e.

It is easy to verify, that a G-set X is a principal homogeneous space

i the map ' : X G 3 (x; g ) 7! (x; xg ) 2 X X is bijective. This

holds also in the case X = ;.IfX 6= ; then X and G are isomorphic as

G-sets. These statements are easily translated into terms of functors.

The map ' : X G ! X X which is de ned for any G-set valued

functor X induces the algebra homomorphism : A A 3 s t 7!

st t 2 A H on the representing ob jects. ' is an isomor-

(A) (H )

phism i is.

Prop osition 2.4. Let G bearepresentable group valued functor and X

bearepresentable G-set valued functor on k -Alg . Let the representing

algebra A of X be faithful ly at. Then G and X are K -forms of each

other as G-sets for some faithful ly at commutative k -algebra K i X

is a principal homogeneous spaceover G.

Proof. We rst remark the following. Let X and Y b e representable

functors, let f : X ! Y b e a natural transformation, and K b e faith-

fully at. Assume that f j : X j ! Y j is an isomorphism. Then

K K K

f is an isomorphism. This is due to the fact that the corresp onding

statement holds for the representing algebras.

Now let there b e a natural isomorphism of Gj -set valued functors

: Gj ! X j . Then since (X Y )j = X j Y j the following

K K K K K

diagram commutes

(G G)j (G G)j

K K

? ?

- :

(X G)j (X X )j

K K

Since Gj is a principal homogeneous space over Gj "comp onen-

K K

twise", the top morphism is an isomorphism. So are the twovertical

arrows. Thus the b ottom arrow is an isomorphism. By the ab ove

argumentwegetthat ' : X G ! X X is an isomorphism.

Conversely if ' : X G ! X X is an isomorphism, then in par-

ticular the induced k-algebra homomorphism : A A 3 s t 7!

st t 2 A H of the representing algebras is an isomor-

(A) (H )

phism. (Here weusetheSweedler notation in context with a bilinear

FORMS OF HOPF ALGEBRAS AND GALOIS THEORY 9

map.) This is even an isomorphism of A-algebras. So wegetforany

A-algebra B

Gj (B ) k - Alg(H ;B) A-Alg (A H ;B)

= =

X j (B ): k -Alg (A; B ) A-Alg(A A; B )

= = =

It is now easy to verify that this is an isomorphism of Gj -set valued

functors.

The translation of the notion of principal homogeneous spaces into

terms of Hopf algebras has a most interesting variation. Let A be

an H -como dule algebra. Assume nowthatH is nitely generated

and pro jectiveasa k -mo dule and that A is faithfully at. The dual

H := Hom (H ;k) is a nitely generated pro jectivecocommutative

Hopf algebra which acts on A by h t = t h(t ). Then the

(A) (H )

following holds:

Theorem and De nition 2.5. Under the above assumptions the fol-

lowing areequivalent:

a) A is a Hopf Galois extension of k with Hopf algebra H (or

simply H -Galois).

b) : A A 3 s t 7! st t 2 A H is an isomorphism.

(A) (H )

c) There is a faithful ly at extension K of k with K A K H

as K H -comodule algebras.

d) : H A 3 h s 7! (t 7! s(h t)) 2 End (A) is an

isomorphism and A is nitely generated faithful projective as a

k -module.

e) k is the xring

A := fs 2 Aj8h 2 H : h s = (h)sg

of A under the action of H and the rings A and A#H are

Morita equivalent.

Proof. a): A Hopf Galois extension is de ned to b e one of the equivalent

conditions b) - e). b) implies c) with K = A. The equivalence of

b) and c) is the preceding Prop osition. The equivalence b etween b)

and d) is a simple calculation with dual bases for H and H and use

of faithful atness. e) is essentially a translation of d) into terms of

Morita equivalences. Detailed pro ofs of this can b e found in [P].

There are various di erent generalizations of Galois extensions. Non-

commutative algebras with Hopf algebras acting on them havebeen

investigated. Commutative algebras with nite groups acting on them

10 BODO PAREIGIS

have b een studied in [CHR]. The de nition used here has b een intro-

duced in [CS] and is also describ ed in [S1]. Sp ecial instances of Galois

extensions are encluded in this general concept.

Let k bea eldand H = kG the group (Hopf ) algebra of a nite

group. Let K b e a eld extension of k whichis H -Galois. Then G

acts by automorphisms on K .Furthermore wehave k = fs 2 K j8g 2

G : g (s)=sg = K . Since [K : k ]=jGj we get that K is a "classical"

Galois extension of k with Galois group G.Conversely if K is a "clas-

sical" Galois extension of k with Galois group G then by Dedekind's

lemma and d) of the ab oveTheoremK is Hopf Galois with Hopf alge-

bra H = kG.

Jacobson's extension [J] of Galois theory to purely inseparable eld

extensions can b e incorp orated into the general framework of Hopf Ga-

lois theory in the following way. Jacobson uses restricted Lie algebras

acting by derivations on purely inseparable eld extensions of exp onent

one. The restricted universal enveloping algebras of the restricted Lie

algebras are Hopf algebras and the action extends to a Hopf Galois ac-

tion on the same extension. Details and an extension to a larger class

of purely inseparable eld extensions can b e found e.g. in [S2] and [W].

The question arises which parts of the "classical" Galois theory can

b e transferred to Hopf Galois theory. The de nition of a Hopf subalge-

bra H H causes some problems on the coalgebra side. If we always

assume, however, that H is a direct summand of H as a k -mo dule,

these problems can b e resolved. The fundamental theorem of Galois

theory can b e extended to

Theorem 2.6. [CS] Let K be Hopf Galois with Hopf algebra H .For

H a Hopf subalgebraofH let

0 0

Fix(H ):=fx 2 K j8h 2 H : h x = (h)xg:

Then

0 0

Fix : fH H jH Hopf subalgebra g!fLjk L K subalgebra g

is injective and inclusion-reversing.

Wesay, that the fundamental theorem of Galois theory hold in its

strong form, if the map Fix is bijective. This, however, is not the

case in general, as we will see b elow. There is another deviation from

the "classical" Galois theory. The Hopf algebra acting on a Galois

extension K of k is not uniquely determined. Examples havebeen

known for inseparable eld extensions.

FORMS OF HOPF ALGEBRAS AND GALOIS THEORY 11

3. Separable Field Extensions

We give an example of a separable eld extension which is not Galois

in the classical sense, but which is Hopf Galois with two di erentHopf

2) and k = Q.Itiswell known that this is not algebras. Let K = Q(

a "classical" Galois extension. Let

2 2

H = Q[c; s]=(c + s 1;cs)

with the coalgebra structure as given in part I. Abbreviate ! := 2.

Then the op eration of H on K is given by

2 3

1 ! ! !

c 1 0 ! 0

0 ! 0 ! : s

If K = k ( 2) were a classical Galois extension for example over the

base eld Q(i) then the Galois group is cyclic with generator e.Here

the generator e has b een replaced bythetwo op erators c and s which

op erate k -linearly and according to the rules

c(xy )= c(x)c(y ) s(x)s(y ) and s(xy )= s(x)c(y )+ c(x)s(y ):

Similarities with the trigonometric equalities are intended. If one ex-

tends the base eld from Q to Q(i) then (Q(i) Q( 2)) : Q(i) b ecomes

a classical Galois extension and the Hopf algebra H is extended to the

2 ) the group ring QC . By further extending the base eld to Q(i;

p p

4 4

2) Q( 2) b ecomes isomorphic to the dual of the ring extension Q(i;

extended group algebra (Q(i; 2)C ) . This isomorphism is compati-

ble with the como dule algebra structure. So we see that the original

H -como dule algebra K is a form of the trivial H -como dule algebra

H .

One can show that there is a second Hopf algebra over Q and action

on K = Q( 2 ) such that the setup is a Hopf Galois extension. The

Hopf algebra is

2 2

H = Q[c; s]=(s 2c +2;cs)

with the action

2 3

1 ! ! !

c 1 0 ! 0

0 ! 0 2!: s

The maps c and s are k -linear and satisfy the multiplicative relations

c(xy )= c(x)c(y ) s(x)s(y ) and s(xy )= c(x)s(y )+ s(x)c(y ):

To see that this gives a Hopf Galois extension one has to extend the

base eld to Q( 2 ) and then pro cede as ab ove.

12 BODO PAREIGIS

This is an example of a k -algebra which is a Hopf Galois extension

with two di erent Hopf algebras. We will see further down that this

will happ en very often. Even the "classical" Galois extensions often

have more than one Hopf algebra for which they are Hopf Galois. On

the other hand there are separable eld extensions which are not Hopf

Galois at all. The separable eld extensions which are Hopf Galois can

b e classi ed by the following theorem.

Toformulate the theorem we x the following notation. Let K be a

nite separable eld extension of k . Assume

K = normal closure of K over k;

G =Aut(K=k);

G =Aut(K=K);

S = G=G (left cosets),

B =Perm(S ) (group of p ermutations of S ):

Theorem 3.1. [GP] Under the assumptions made above the fol lowing

areequivalent

a) There is a Hopf k -algebra H such that K=k is H -Galois.

b) Thereisaregular subgroup N B such that the subgroup G

B normalizes N .

The examples given ab ove are of a rather sp ecial typ e whichwecall

"almost classical" Hopf Galois extensions. They are characterized by

the following

Theorem 3.2. [GP] The fol lowing conditions areequivalent:

a) There exists a Galois extension E=k such that E K is a eld

containing K .

b) There exists a Galois extension E=k such that E K = K .

c) G has a normal complement N in G.

d) There exists a regular subgroup N B normalizedbyG and

contained in G.

The last condition of this theorem shows that we are indeed talking

ab out Hopf Galois extensions. These extensions are particularly well

b ehaved b ecause they satisfy the fundamental theorem of Galois theory

in its strong form.

Theorem 3.3. [GP] If K=k is almost classical ly Galois, then there

is a Hopf algebra H such that K=k is H -Galois and the map Fix is

bijective.

The ambiguity of the Hopf algebra acting on a Hopf Galois extension

is exp osed in the following

FORMS OF HOPF ALGEBRAS AND GALOIS THEORY 13

Theorem 3.4. [GP] Any classical Galois extension K=k can be en-

dowed with an H -Galois structure such that the fol lowing variant of

the fundamental theorem holds: Thereisacanonical bijection between

Hopf subalgebras of H and normal intermediate elds k E K .

One of the simplest examples of a classical Galois extension with

this new H -Galois structure is the following. Let be a 3 primitive

ro ot of unity and let ! := 2 . Then K = Q(!; ) is a classical Galois

extension of Q with Galois group S . ItisalsoHopfGaloiswithHopf

algebra

2 2 2

H = Q[c; s; t]=(c(c 1)(c +1); 2c + st + ts 2; cs; sc; ct; tc; s ;t ):

The action of H on K is describ ed by the table

1 !

c 1 0

s 0 ! 0

t 0 0 0.

The action of the three generating elements c; s; t on K sati es

1 1

s(x)t(y )+ t(x)s(y ); c(xy ) = c(x)c(y )+

2 2

s(xy ) = c(x)s(y )+ s(x)c(y )+ t(x)t(y );

t(xy ) = c(x)t(y )+ t(x)c(y )+ s(x)s(y ):

We nish this paragraph on separable eld extensions which are Hopf

Galois by giving a family of examples of separable eld extensions which

are not Hopf Galois: no eld extension K over Q of degree 5 with

automorphism group S of K=k can b e Hopf Galois [GP].

4. Hopf Algebra Forms revisited

Many of the following results have b een obtained in co op eration and

discussions with students and collegues of mine. In particular I grate-

fully acknowledge the co op eration of C. Greither, R. Haggenmuller,

and C. Wenninger.

The techniques to prove Theorem 1.2 can b e used to calculate more

forms of group rings. The advantage in the pro of of Theorem 1.2 was

that all quadratic extensions of a commutative ring can b e explicitly

describ ed if the ring satis es only minor conditions [Sm]. If 2 is not

a zero divisor in k and if Pic (k ) = 0 then all quadratic extensions

(2)

of k are free and can b e describ ed as K = k [x]=(x ax b) where

a +4b = u is a unit in k . The non-trivial automorphism is f (x)=a x.

This information was translated into terms of Hopf algebra forms using

the following

14 BODO PAREIGIS

Theorem 4.1. [HP] Let G be a nitely generatedgroup with nite

automorphism group F = Grp- Aut(G) Then thereisabijection between

Gal(k; F ), the set of ismorphism classes of F -Galois extensions of k ,

and Hopf(kG), the set of Hopf algebra forms of kG. This bijection

associates with each F -Galois extension K of k the Hopf algebra

o n

X X X

8f 2 F : f (c )f (g )= c g : c g 2 KG H =

g g g

Furthermore H is a K -form of kG by the isomorphism

! : H K KG; ! (h a)=ah:

On the other hand it is not trivial to describ e F -Galois extensions of a

eld k . They are not just the classical Galois eld extensions of k .The

simple example of the trivial F -Galois extension k k ::: k is not

a eld. Actually F -Galois extensions are just Hopf Galois extensions

with Hopf algebra kF [CHR, Thm 1.3]. Arbitrary commutativerings

K are admitted as Galois extensions. The action of the group F on the

extension K by di erentelements f; f has to b e "strongly distict", i.e.

for every idemp otent e 2 K there is an x 2 K such that f (x)e 6= f (x)e.

This is the key to the following

Theorem 4.2. Let F be a nite group and k a eld. K=k is an F -

Galois extension if and only if

n times

z }| {

K L ::: L

where L=k is a U -Galois eld extension with U F asubgroup of

index n.

Proof. Let K=k be an F -Galois extension. K is a commutative separa-

L ::: L ble k -algebra by [CHR Thm 1.3] hence is a pro duct K

1 n

of separable eld extensions L =k . The automorphisms in F map the

primitive idemp otents to primitive idemp otents and F op erates transi-

tively on the set primitive idemp otents, since the sum of idemp otents

in an orbit is in the xed eld. For anytwo idemp otents e and e the

i j

automorphism f of F mapping e to e also maps L to L .HenceL

i j i j i

is isomorphic to a sub eld of L . By symmetry all the elds L are

j i

mutually isomorphic. The stabilizer U F of e acts as Galois group

on L =k since it acts strongly distinctly and jU j =[L : k ].

Conversely let U G b e a subgroup and L : k be U -Galois. Let

g ;:::;g b e a set of representatives for G=U = fg U;:::;g U g. Let

1 n 1 n

K = L ::: L with idemp otens e ;:::;e . De ne the action : G

1 n

! S by (g )(i)= j if gg U = g U the regular representation of G

n i j

gg (l )e . Observe that gg U = on G=U .Wede ne g (le ):=g

i i i

(g )(i)

FORMS OF HOPF ALGEBRAS AND GALOIS THEORY 15

gg 2 U . Then the xring of K under g implies u := g

i (g )(i)U g;i

(g )(i)

the action of G is k , for let l e 2 K . Then for all g 2 G wehave

i i

P P

u (l )e = l e .For g := g ug we get gg U = g U , hence

g;i i i i i i i

(g )(i)

1 1

(g )(i)= i and u = g g ug g = u, so that u(l )=l for all u 2 U ,

g;i i i i i

i i

weget gg U = g U hence (g )(i)= j hence l 2 k . For g := g g

i j i j

1 1

and u = g g g g = id, so that l e = l e , hence l = l for all

g;i j i i j j j i j

j i

P P

i; j . This shows l e = e = 2 k .Obviously all elements of k

i i i

remain xed under the action of G so that k = K .Furthermore K

is separable by de nition. To show that G op erates strongly distictly

it suces to nd for every g 2 G; g 6=idand e 2 K and x 2 K such

that g (x)e 6= xe . Assume rst that (g )(i) 6= i. Cho ose x = e . Then

i i i

gg 2 U and g (e )e = e e =0 6= e = e e .If (g )(i)= i then g

i i i i i i i

(g )(i)

u 6= id since gnot =id. Cho ose an l 2 L with u(l ) 6= l and x = le .

Then g (x)e = g (le )e = g gg (l )e u(l )e 6= le = le e = xe . This

i i i i i i i i i i

concludes the pro of.

Observeby the way that kC has no non-trivial forms, since C has

2 2

trivial automorphism group, so the corresp onding Galois extension of

a form must b e k itself. Already the next simplest cases after studying

the forms of k Z, kC , kC ,and kC cause unsatisfactory calculations.

3 4 6

We discuss the case of QC .

The automorphism group of C is C which has exactly one non-

5 4

trivial subgroup C . The C -Galois extensions K of Q canbeofthe

2 4

following forms

1) K is a C -Galois eld extension of Q,

L L where L is a quadratic eld extension of Q, 2) K

Q Q Q Q. 3) K

The problem is now to describ e as explicitly as p ossible all C - resp.

C -Galois eld extensions K of Q, to describ e the action of C on K

4 4

and then calculate the forms according to Theorem 4.1. Asso ciated

with a C -Galois eld extension K is the following form of QC :

4 5

5 3 2

H Q[a]=(a 5pa +(5(p 3q ) 10 q (p 4q ))a)

4 2

q (p 4q ) 2 k Here K is the splitting eld of x + px + q and

necessarily holds if K is a C -Galois eld extension. The diagonal

maps can b e describ ed by

5 5 3 3

((u +v )(a a)(u +v )(a b+b a)+(u+v )(b b)); (a)=

2 2 2

uv (u v )

16 BODO PAREIGIS

where

s s

p p

2 2

p + p 4q p p 4q

u = and v = :

2 2

The case of K L L with quadratic extension L=Q is somewhat

easier. We get

5 3 1

H Q[a]=(a +5p(a + pa)) with (a)= u (1; 1)(a a):

where L = Q(u) is the splitting eld of x p.

Finally the case of K Q Q Q Q leads to the trivial form

QC .

The other simple example is that of forms of QC C . The auto-

2 2

morphism group of C C is the symmetric group S .Nowwehave

2 2 3

to study the di erent cases of S -Galois extensions

1) K is an S -Galois eld extension of Q,

2) K L L where L is a C -Galois eld extension of Q,

L L L where L is a quadratic eld extension of Q, 3) K

4) K Q Q Q Q Q Q.

In the rst and second case we get

H = Q[a]=(a(a + ua + v ))

3 3

where K is the splitting eld of x + ux + v irreducible. If D = 4u

27v is the discriminantthen

[ 2u(3va a u(a c + c a)) (a)=

2 2

+ (4u b b +9c c +6u(b c + c b)) 9v (a b + b a)]

where

4u 4

a + a +3 b =

v v

and

4u 4u

3 2

c = a +2a a 4u:

v v

In the third case of K L L L weget

2 2

H = Q[a]=((a 1)(a u))

where L is the splitting eld of x u and the diagonal is

2 3 3 3 3

(u u 1)a a a a + a a + a a : (a)=

u u

Q Q Q Q Q Q leads to the trivial form The case of K

Q(C C ).

2 2

FORMS OF HOPF ALGEBRAS AND GALOIS THEORY 17

Problems 4.3. The generators of Hopf algebra forms and their diag-

onals are rather arbitrary. It often turns out that either the diagonal

or the ideal to b e factored out can b e chosen to b e relatively simple,

but not b oth. Is there a canonical choice of the generators of a Hopf

algebra form? Is there a way to determine the minimal number of

generators? Can one describ e the "cyclic" Hopf algebra forms? This

seems to b e of interest for the representation theory of Hopf algebras,

like cyclic groups are for the representation theory of groups.

Another problem area arises from the following considerations. Let

kC b e the group algebra of a nite cyclic group and let k bea eld

with char(k ) = n. Then the group algebra is semisimple byMaschke's

theorem. Let K b e a eld extension of k or a commutative separable

algebra. Then every K -form H of kC is again semisimple, since a

nilp otent ideal of H would remain nilp otentinK H KC in b oth

cases, but KC is still semisimple. There may b e forms which are

even b etter in their representation prop erties as the example of part I.

shows.

2 2

The Hopf algebra H = R[c; s]=(c + s 1;cs)isaC-form of RC .

R R R R as R-algebras. Thus H is It is easy to see that H

absolutely semisimple, i.e. all its simple mo dules are one-dimensional

over the base eld. RC ,however, is not absolutely semisimple. It

R R C as an algebra, so it has a two- decomp oses as RC

dimensional simple mo dule.

So there is the problem of determining which group algebras have

absolutly semisimple forms and to describ e all those forms. If every

semisimple group algebra had an absolutely semisimpleformthiswould

mean, that one do es not need to extend the base eld of a group alge-

bra kG to obtain total splitting, but that the splitting can already b e

obtained over the base ring for a suitable form H .Sincewe are not talk-

ing ab out algebra forms but ab out Hopf algebra forms the p ossibility

of tensoring H -mo dules over the base eld | an imp ortanttechnique

for representation theory | is preserved.

Theorem 4.4. If k is a eld of characteristic not dividing n,then

the Hopf algebra kC has a uniquely determined absolutely semisimple

=(kC ) . Hopf algebra form k

Proof. Any absolutely semisimple form of kC has underlying algebra

I I

k . But k is a Hopf algebra i I is a nite group. After base eld

extension the group structure of I remains unchanged, so there can b e

Added in pro of: A complete answer to this question for the case of group rings

has b een given recently by the author in Twistedgroup rings submitted to Comm. Alg.

18 BODO PAREIGIS

at most one group structure on I and at most one Hopf algebra struc-

I I

ture on k so that k is a form of kC . So an absolutely semisimple

form of kC is a Hopf algebra k with a uniquely determined com-

becomes mutative group G of order n.Weshow G C so that k

the absolutely semisimpleformof kC . It suces to show this over a

eld k containing an n-th primitiverootofunity. But then kC splits

completely and the statementiswell known.

This unique absolutely semisimpleformof kC is asso ciated with

an F -Galois extension K of k with F Aut(C ). It turns out that

k [x]=(' (x)) is an F -Galois extension and asso ciated to k . ' (x)

n n

is the n-th cyclotomic p olynomial. In general k [x]=(' (x)) will not

b e a eld extension of k . According to Theorem 4.2 and with some

additional calculations one can see that k [x]=(' (x)) k ( ) :::

n n

k ( ).

Problems 4.5. It would b e interesting to know which group algebras

over Q have absolutely semisimple forms. The Hopf algebra QS is

itself absolutely semisimple. There are also examples of groups G whose

group algebras have no absolutely semisimple forms.

5. Separable Hopf Galois Extensions

Problems 5.1. In part I I I wehave seen examples of separable eld

extensions K=k which are Hopf Galois. All the examples were in fact

"almost classically" Galois. A 16-dimensional example of a Hopf Galois

extension which is not "almost classically" Galois is given in [GP]. M.

Takeuchi has checked that all that all Hopf Galois extensions of dimen-

sion less than 8 are "almost classically" Galois. The obvious question

is, are there prop er Hopf Galois extensions of dimension less than 16?

Questions ab out the corresp ondence b etween "normal" Hopf subalge-

bras and Hopf Galois sub elds have b een addresses in [C2]. Manyof

those questions are still op en. Childs also addresses the question of

the uniqueness of the Hopf algebra H w.r.t. which a separable eld

extension is Hopf Galois. He obtains results for "classical" Galois eld

extensions. He shows that the Hopf algebra H is never unique if G is

cyclic of o dd prime p ower order. H is never unique for non-ab elian G.

This needs a di erent pro of, however, than given in [C2]. Childs also

shows that H is unique if G is cyclic of prime order, a result whichwe

will extend b elow.

Assume that wehave the same setup K=k; K=k;G;G ;S;B as in I I I.

In [C2] the following result is shown.

FORMS OF HOPF ALGEBRAS AND GALOIS THEORY 19

Prop osition 5.2. G normalizes the regular subgroup N of B i G is

a subgroup of the holomorph Hol(N )=N Aut(N ).

We extend Theorem 2 of [C2] as follows:

Theorem 5.3. Let K=k beaseparable eld extension of degree [K :

k ]= p a prime. The fol lowing areequivalent:

1) K=k is Hopf Galois.

2) K=k is almost classical ly Galois.

3) G is solvable.

If any (and al l) of these conditions hold then the Hopf algebra H is

unique for K=k H -Galois.

Proof. If K=k is H -Galois then there is a regular subgroup N of S

Aut(N ), the holomorph of N . Since suchthat G Hol(N )=N

N C the holomorph Hol(N ) C C , hence G is solvable.

= =

p p p1

Let G be solvable. Since K = k (a) with a a zero of an irreducible

p olynomial f of degree p, G is a subgroup of S ,hencep=jGj and

p j= Gj. Belowweshow that for a solvable group G there is a chain

of normal subgroups of G (!)

e / ::: / G /G /G = G

2 1 0

(Z=p Z) with G =G . Consider the sequence of sub elds

i i i+1

k = k k ::: k k = K;

0 1 m1 m

and k =k normal. with k =k Galois with Galois group (Z=p Z)

i+1 i+1 i i

We get a 2 k and a=2 k , otherwise K k since k =k

m m1 m1 m1

normal. Then the minimal p olynomial f of a is irreducible over k

by the last lemma. So all the k -generating elements 1;a;:::;a of

K are linearly indep endentover k .Thus K and k are linearly

m1 m1

disjointand p = p. Since p = [K : k ]andp = p =[k : k ]we

m m m m1

get k K k K = k = K . So by Theorem 3.2 the eld

m1 m1 m

extension K=k is almost classically Galois.

To see that the Hopf algebra H together with the Galois op eration is

uniquely determined, observethat p=jGj and G N Aut(N ) and N

the only Sylow p-subgroup of Hol(N ) imply that the Sylow p-subgroup

A with a subgroup A of G is N , which is unique. Thus G = N

Aut(N ). Consequently N = G B is uniquely determined and so is

H by Theorem 3.1.

To nish the pro of of the theorem weprove the following lemmas.

Lemma 5.4. Let G be a nite solvable group. Then thereisasequence

e / ::: / G /G /G = G

2 1 0

20 BODO PAREIGIS

with G =G (Z=p Z) , p prime and G /G normal subgroups.

i i+1 i i i

Proof. by induction. We only indicate how to construct G from G .

i+1 i

Let M/G b e a normal subgroup of prime index p . De ne G :=

i i i+1

gM g . It has all the required prop erties.

g 2G

Lemma 5.5. Let K=k be a normal separable nite eld extension. Let

f 2 k [x] beseparable and irreducible of degree p a prime. Then either

f is irreducible over K or f completely splits into linear factors.

Proof. Let a ;:::;a b e the zeros of f in the algebraic closure and let G

1 p

b e the automorphism group of K (a ;:::;a )=k . G op erates transitively

1 p

on the zeros, since f is irreducible. Let N b e the xgroup of K .Since

K=k is normal, we get that N/G is a normal subgroup. N decomp oses

fa ;:::;a g into orbits of equal cardinality since G op erates transitively

1 p

and N is normal. So either N op erates transitively or trivially. Hence

f is irreducible or splits completely.

FORMS OF HOPF ALGEBRAS AND GALOIS THEORY 21

References

[C1] L. Childs: Taming Wild Extensions with Hopf Algebras.

Trans. Amer. Math. So c. 304, p. 111-140. 1987.

[C2] L. Childs: On the Hopf Galois Theory for Separable Field Extensions. to

app ear Comm. Alg.

[CHR] S. Chase, D.K. Harrison, A. Rosenberg: Galois Theory and Galois Coho-

mology of Commutative Rings. Mem. Am. Math. So c. 52 (1965).

[CS] S. Chase, M. Sweedler: Hopf Algebras and Galois Theory.LNinMath.97.

Springer 1966.

[GP] C. Greither, B. Pareigis: Hopf Galois Theory for Separable Field Extensions.

J. Algebra 106, p. 239-258, 1987.

[G] A. Grothendieck:Technique de descente I. Seminaire Bourbaki, Exp. 190.

1959/60.

[H] R. Haggenmuller : Ub er Invarianten separabler Galoiserweiterungen kommuta-

tiver Ringe. Dissertation Universitat Munc hen 1979.

[J] N. Jacobson: An Extension of Galois Theory to Non-Normal and Non-

Separable Fields. Am. J. Math. 66, p. 1-29. 1944.

[KO] M.-A. Knus, M. Ojanguren: Theorie de la Descente et Algebres d'Azumaya.

LN in Math. 389, Springer 1974.

[P] B. Pareigis: Descent Theory applied to Galois Theory.Technical Rep ort

Univ. California, San Diego, 1986.

[Sm] C. Small: The Group of Quadratic Extensions. J. Pure Appl. Alg. 2, p.83-105,

395. 1972.

[S1] M. Sweedler: Hopf Algebras. Benjamin { New York 1969.

[S2] M. Sweedler: Structure of Purely Inseparable Extensions. Ann. Math. 87,

p. 401-411. 1968.

[W] C. H. Wenninger: Hopf-Galois-Theorie einer Klasse rein inseparabler

Korp ererweiterungen. Diplom Thesis Universitat Munc hen 1984.

Mathematisches Institut der Universitat Munchen, Germany

E-mail address : [email protected]