FORMS OF HOPF ALGEBRAS AND

GALOIS THEORY

BODO PAREIGIS

The theory of Hopf algebras is closely connected with various applications, in particular to algebraic and formal groups. Although the first occurence of Hopf algebras was in algebraic topology, they are now found in areas as remote as combinatorics and analysis. Their structure has been studied in great detail and many of their properties are well understood. We are interested in a systematic treatment of Hopf algebras with the techniques of forms and descent.

The first three paragraphs of this paper give a survey of the present state of the theory of forms of Hopf algebras and of Hopf Galois theory especially for separable extensions. It includes many illustrating examples some of which cannot be found in detail in the literature. The last two paragraphs are devoted to some new or partial results on the same field. There we formulate some of the open questions which should be interesting objects for further study. We assume throughout most of the paper that k is a base field and do not touch upon the recent beautiful results of Hopf Galois theory for rings of integers in algebraic number fields as developed in [C1].

1. Hopf algebra forms

As a first example of the occurence of a Hopf algebra let us consider the units functor. In the sequel let k be a commutative, associative ring with unit. Later on it will be a field, in particular the field of rationals or reals. Let k-Alg denote the category of commutative k-algebras and Grp the category of groups. Then there is the important functor

U: k-Alg ⟶ Grp,

the units functor, which associates with each k-algebra its group of invertible elements or units. This functor is representable by the kalgebra k[x, x^{−1}] = kZ, the group-ring of the infinite cyclic group Z, i.e.

Date: February 24, 1990.

1991 Mathematics Subject Classification. Primary 16A10.

U(A) ≅ k-Alg(k[x, x^{−1}], A).

The multiplication of the units group U × U ⟶ U induces a commutative diagram

[figure: U(A) × U(A); U(A); k-Alg(k[x, x^{−1}] ⊗ k[x, x^{−1}], A); k-Alg(k[x, x^{−1}], A)]

with vertical arrows isomorphisms. By the Yoneda Lemma the second horizontal arrow induces a comultiplication on the representing k-algebra

Δ: k[x, x^{−1}] ⟶ k[x, x^{−1}] ∐ k[x, x^{−1}] = k[x, x^{−1}] ⊗ k[x, x^{−1}],

defined by Δ(x) = x ⊗ x. Observe that the tensor product of commutative algebras is the coproduct in k-Alg. Also the inverse inv: U ⟶ U and the neutral element {.} ⟶ U define corresponding maps on the representing algebra. All in all we obtain the structure of a Hopf algebra on the algebra k[x, x^{−1}]. Definition 1.1. A k-algebra H together with k-algebra homomorphisms

\begin{aligned}Δ & : H ⟶ H ⊗ H, \\ σ & : H ⟶ H, \\ ϵ & : H ⟶ k \end{aligned}

is called a Hopf algebra, if H together with Δ and ϵ is a coalgebra and if σ satisfies the following commutative diagrams

[figure: H; k; ϵ; H; η; Δ; ∇; H ⊗ H; H ⊗ H; σ ⊗ 1(1 ⊗ σ)]

Here the map ∇: H ⊗H ⟶ H denotes the multiplication ∇(a⊗b) := ab of the algebra H and η: k ⟶ H is defined by η(a) := a · 1_{H}, the canonical map from k into H.

The map Δ: H ⟶ H ⊗ H is called the diagonal or comultiplication on H. It is awkward to write the images as tensors in the usual way, especially if composites of such maps occur. The following simplified notation has been introduced by Sweedler. For a linear map f: A ⟶ B⊗C we define ∑ a_{(B)}⊗a_{(C)} := f(a) or in the special case of a Hopf algebra (H, Δ, ϵ) we write ∑ h_{(1)} ⊗h_{(2)} := Δ(h). The advantage of this notation is, that it can be extended to bilinear maps along the following example. If g: B×C ⟶ D is a bilinear map with induced map ĝ: B⊗C ⟶ D on the tensor product, then we can define ∑ g(a_{(B)}, a_{(C)}) := ĝ(f(a)) and thus use the ”components” a_{(B)} and a_{(C)} as if they were well-defined ordinary elements, which can be used as arguments in bilinear maps.

Similar to the Hopf algebra k[x, x^{−1}] each commutative (as an algebra) Hopf algebra H represents a functor

H̃: k-Alg ⟶ Grp,  H̃(A) := k-Alg(H, A)

where the multiplication on H̃ is given by the commutative diagram

[figure: H̃(A) × H̃(A); H̃(A); k-Alg(H ⊗ H, A); k-Alg(H, A)]

So the group-ring kZ has been seen to be a Hopf algebra with the diagonal Δ(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 special instances of formal groups.

Another concrete example of a group valued functor is

C: k-Alg ⟶ Grp,

the circle group, defined by C(A) := {(a, b) ∈ A × A|a^{2} + b^{2} = 1}. The group structure is given by (a, b) ∗ (c, d) := (ac − bd, ad + bc). The representing Hopf algebra is the ”trigonometric algebra” H = k[c, s]/(c^{2} + s^{2} − 1). The diagonal is defined by

\begin{aligned}Δ(c) & = & c ⊗ c − s ⊗ s, \\ Δ(s) & = & c ⊗ s + s ⊗ c. \end{aligned}

The most interesting observation is this. Let A be a commutative kalgebra with 2 invertible and containing i = \sqrt{−1}. Then the assignment

U(A) ∋ a ↦ (\frac{1}{2}(a + a^{−1}), \frac{1}{2i}(a − a^{−1})) ∈ C(A)

defines a functorial isomorphism of groups. If 2^{−1}, i ∈ k then U and C are isomorphic group valued functors, hence they have isomorphic representing Hopf algebras

k[x, x^{−1}] ≅ k[c, s]/(c^{2} + s^{2} − 1).

If i ∉ k then the two group valued functors are not isomorphic, neither are their representing Hopf algebras k[x, x^{−1}] and k[c, s]/(c^{2} + s^{2} − 1).

If k is a field of characterictic ≠ 2 and i ∉ k, then U and C are non-isomorphic but they induce isomorphic functors U|_{k(i)} and C|_{k(i)} if restricted to the k(i)-algebras. Let K = k(i) and let A be a K-algebra. Then we have

K-Alg(K ⊗ k[x, x^{−1}], A) ≅ k-Alg(k[x, x^{−1}], A)
≅ U|_{K}(A)
≅ C|_{K}(A)
≅ k-Alg(k[c, s]/(c^{2} + s^{2} − 1), A)
≅ K-Alg(K ⊗ k[c, s]/(c^{2} + s^{2} − 1), A)

hence K ⊗ k[x, x^{−1}] ≅ K ⊗ k[c, s]/(c^{2} + s^{2} − 1) as K-Hopf algebras, where the tensor product is always taken over the base ring k. Observe that a cancellation property cannot be expected in this case.

In particular, the Q-Hopf algebras Q[x, x^{−1}] and Q[c, s]/(c^{2} +s^{2} −1) and the R-Hopf algebras R[x, x^{−1}] and R[c, s]/(c^{2} + s^{2} − 1) are not isomorphic, but the C-Hopf algebras C[x, x^{−1}] ≅ C[c, s]/(c^{2} + s^{2} − 1) are. This is an example for the next definition. Definition 1.2. Let G and G′ be group valued functors on k-Alg. Let K be a faithfully flat commutative k-algebra. If the restrictions to K-Alg are isomorphic group valued functors: G|_{K} ≅ G′|_{K}, then G and G′ are called K-forms of each other as groups.

Let H and H′ be Hopf algebras over the commutative ring k. Let K be a faithfully flat commutative k-algebra. If K ⊗ H ≅ K ⊗ H′ as K-Hopf algebras, then H and H′ are called K-forms of each other as Hopf algebras

We say that G and G′ resp. H and H′ are forms of each other if there exists a faithfully flat k-algebra K such that they are K-forms of each other.

So for G and G′ to be K-forms of each other we need an isomorphism of set valued functors α: G|_{K} ⟶ G′|_{K} such that

[figure: G|_{K} × G|_{K}; G′|_{K} × G′|_{K}; α × α; G|_{K}; G′|_{K}; α]

commutes.

There may be many different Hopf algebras H′ which are forms for H with respect to some faithfully flat extension K. In particular the richness of Hopf algebras over Q should be higher than over C. Granted there may be Hopf algebras defined over C, which do not come about by a base ring extension from Q, but e.g. semisimple cocommutative Hopf algebras over C are always defined over Q. This is a consequence of a more general structure theorem of Milnor, Moore and Cartier on cocommutative Hopf algebras over algebraicly closed fields. Our interests are in this richness of Hopf algebras over ”small” fields. One can show for example that over the field 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 structures given by the theory of faithfully flat descent. We apply it to the case of Hopf algebras. Let H be a Hopf algebra over k. The group of automorphisms of this Hopf algebra will be 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 kalgebras K ⟶ L induces a group homomorphism K-Hopf-Aut(K ⊗H) ⟶ L-Hopf-Aut(L⊗H). Thus we have a functor Aut(H): k-Alg ⟶ Grp defined by Aut(H)(K) := K-Hopf-Aut(K ⊗ H). Every group valued functor on the category k-Alg of commutative k-algebras has an associated Amitsur cohomology H^{n}(K/k, Aut(H)). It is not necessary to know the precise definition of these cohomology groups to apply the following theorem.

Theorem 1.3. Let H be a k-Hopf algebra. Then there is a bijektion between the set of (isomorphism classes of) K-forms of H and the Amitsur cohomology group H^{1}(K/k, Aut(H)).

Proofs of this may be found in various forms in [G], [H], or [KO]. Actually this theorem holds in greater generality and the proof 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 we do not have to calculate the cohomology group, since by a twofold application of this theorem – going from certain forms to the cohomology group and then from the same cohomology group back to some other forms – we will eliminate the explicit computation of the cohomology.

In the case of group-rings kG of finitely generated groups G the automorphism group k-Hopf-Aut(kG) can be calculated, in particular for cyclic groups C_{n} of order n. We assume that the automorphism group F of G is finite. Then one can show that k-Hopf-Aut(kG) is isomorphic to the automorphism group Gal-Aut(E_{k}^{F}) of the trivial F - Galois extension E_{k}^{F} of k. This Galois extension can be described by the ring E_{k}^{F} = (kF)^{∗}, the dual space of the group ring kF, on which F acts by automorphisms in such a 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) ≅ Gal-Aut(E^{F}), so that the Amitsur cohomology groups of these two group functors also coincide.

We formulate one of the most interesting consequences of these considerations.

Theorem 1.4. [HP] Let k be a commutative ring with 2 not a zero divisor in k and Pic_{(2)}(k) = 0, the two-torsion of the Picard group. Then

a) the Hopf algebra forms of kZ are

H = k[c, s]/(s^{2} − asc − bc^{2} + u),

b) the Hopf algebra forms of kC_{3} are

H = k[c, s]/(s^{2} − asc − bc^{2} + u, (c + 1)(c − 2), (c + 1)(s − a)),

c) the Hopf algebra forms of kC_{4} are

H = k[c, s]/(s^{2} − asc − bc^{2} + u, c(ac − 2s)),

d) the Hopf algebra forms of kC_{6} are

H = k[c, s]/(s^{2}−asc−bc^{2}+u, (c−2)(c−1)(c+1)(c+2), (c−1)(c+1)(sc−2a)).

In all cases a, b, u ∈ k satisfy a^{2} + 4b = u and u is a unit in k. The Hopf algebra structure in all cases is defined by

\begin{aligned}Δ(c) & = u^{−1}((a^{2} + 2b)c ⊗ c − a(c ⊗ s + s ⊗ c) + 2s ⊗ s), \\ Δ(s) & = u^{−1}(−abc ⊗ c + 2b(c ⊗ s + s ⊗ c) + as ⊗ s), \\ ϵ(c) & = 2, ϵ(s) = a, σ(c) = c, σ(s) = ac − s. \end{aligned}

We give an indication of the way how this result is obtained. In all cases of the theorem the group F is the cyclic group with two elements. The theory of C_{2}-Galois extensions ( = quadratic Galois extensions) is well known. Actually every quadratic Galois extension of k is a form of the trivial quadratic Galois extension (kC_{2})^{∗} ≅ k×k of k as will be seen below. Since the automorphism groups Aut(kG) ≅ Gal-Aut(k × k) coincide, the first Amitsur cohomology groups describing the forms coincide, too. So there is a bijective correspondence between the forms of the group-rings in the theorem and the quadratic Galois extensions of k [see Thm. 4.1]. This correspondence was used to explicitly calculate the forms given in the theorem.

2. Hopf Galois extensions

A different class of ”forms” is obtained if one considers the following cancellation problem. Definition 2.1. Let G: k-Alg ⟶ Grp be a group valued functor. Then the multiplication of G on itself G × G ⟶ G makes G a G-set valued functor. Here we define the functor G × G by (G × G)(K) := G(K) × G(K), so that the multipication of each group G(K) defines a functorial homomorphism G × G ⟶ G, briefly the multiplication on G and the G-set structure is defined ”componentwise”.

Let X: k-Alg ⟶ Set be another functor which is also a G-set valued functor by X × G ⟶ X. Let K be a faithfully flat commutative ring extension of k. If the restrictions G|_{K} and X|_{K} to K-Alg are isomorphic as G|_{K}-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 be K-forms of each other we need an isomorphism of set valued functors α: G|_{K} ⟶ X|_{K} such that

[figure: G|_{K} × G|_{K}; X|_{K} × G|_{K}; α × 1; G|_{K}; X|_{K}; α]

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 representing objects of the representable functors G and X gives the following definition. Definition 2.2. Let H^{∗} be a commutative Hopf algebra and A be a commutative algebra. A is called an H^{∗}-comodule algebra if there is an algebra map χ: A ⟶ A ⊗ H^{∗} such that the diagrams

[figure: A; A ⊗ H^{∗}; χ; A ⊗ H^{∗}; A ⊗ H^{∗} ⊗ H^{∗}; χ ⊗ 1; χ; 1⊗Δ; and; A; A ⊗ H^{∗}; χ; A; A ⊗ k; ∼; id; 1⊗ϵ]

commute.

Let H^{∗} be a commutative Hopf algebra and A be a commutative H^{∗}-comodule algebra. Let K be a faithfully flat commutative ring extension of k. If K ⊗H^{∗} ≅ K ⊗A as K ⊗H^{∗}-comodule algebras, then A is called a K-form of H^{∗}.

Closely connected with K-forms of G-set valued functors is the notion of a principal homogeneous space. Definition 2.3. If G is a group and X is a set, then a G-set X is called homogeneous, if for each pair x, y ∈ X there exists a g ∈ G such that xg = y. A G-set X is a principal homogeneous G-set if X is homogeneous and xg = x for any x ∈ X implies g = e.

It is easy to verify, that a G-set X is a principal homogeneous space iff the map φ: X × G ∋ (x, g) ↦ (x, xg) ∈ X × X is bijective. This holds also in the case X = ∅. If X ≠ ∅ then X and G are isomorphic as G-sets. These statements are easily translated into terms of functors.

The map φ: X × G ⟶ X × X which is defined for any G-set valued functor X induces the algebra homomorphism ψ : A ⊗ A ∋ s ⊗ t ↦ ∑ st_{(A)} ⊗ t_{(H^{∗})} ∈ A ⊗ H^{∗} on the representing objects. φ is an isomorphism iff ψ is. Proposition 2.4. Let G be a representable group valued functor and X be a representable G-set valued functor on k-Alg. Let the representing algebra A of X be faithfully flat. Then G and X are K-forms of each other as G-sets for some faithfully flat commutative k-algebra K iff X is a principal homogeneous space over G.

Proof. We first remark the following. Let X and Y be representable functors, let f: X ⟶ Y be a natural transformation, and K be faithfully flat. Assume that f|_{K}: X|_{K} ⟶ Y |_{K} is an isomorphism. Then f is an isomorphism. This is due to the fact that the corresponding statement holds for the representing algebras.

Now let there be a natural isomorphism of G|_{K}-set valued functors α: G|_{K} ⟶ X|_{K}. Then since (X × Y)|_{K} = X|_{K} × Y |_{K} the following diagram commutes

[figure: (G × G)|_{K}; (G × G)|_{K}; φ|_{K}; (X × G)|_{K}; (X × X)|_{K}; φ|_{K}; α×1; α×α] .

Since G|_{K} is a principal homogeneous space over G|_{K} ”componentwise”, the top morphism is an isomorphism. So are the two vertical arrows. Thus the bottom arrow is an isomorphism. By the above argument we get that φ: X × G ⟶ X × X is an isomorphism.

Conversely if φ: X × G ⟶ X × X is an isomorphism, then in particular the induced k-algebra homomorphism ψ: A ⊗ A ∋ s ⊗ t ↦ ∑ st_{(A)} ⊗ t_{(H^{∗})} ∈ A ⊗ H^{∗} of the representing algebras is an isomorphism. (Here we use the Sweedler notation in context with a bilinear map.) This is even an isomorphism of A-algebras. So we get for any A-algebra B

G|_{A}(B) ≅ k-Alg(H^{∗}, B) ≅ A-Alg(A ⊗ H^{∗}, B)
≅ A-Alg(A ⊗ A, B) ≅ k-Alg(A, B) ≅ X|_{A}(B).

It is now easy to verify that this is an isomorphism of G|_{A}-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^{∗}-comodule algebra. Assume now that H^{∗} is finitely generated and projective as a k-module and that A is faithfully flat. The dual H := Hom_{k}(H^{∗}, k) is a finitely generated projective cocommutative Hopf algebra which acts on A by h · t = ∑ t_{(A)}h(t_{(H^{∗})}). Then the following holds: Theorem and Definition 2.5. Under the above assumptions the following are equivalent:

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

b) ψ : A⊗A ∋ s⊗t ↦ ∑ st_{(A)}⊗t_{(H^{∗})} ∈ A⊗H^{∗} is an isomorphism.

c) There is a faithfully flat extension K of k with K ⊗A ≅ K ⊗H^{∗} as K ⊗ H^{∗}-comodule algebras.

d) δ : H ⊗ A ∋ h ⊗ s ↦ (t ↦ ∑ s(h · t)) ∈ End_{k}(A) is an isomorphism and A is finitely generated faithful projective as a k-module.

e) k is the fixring

A^{H} := {s ∈ A|∀h ∈ H : h · s = ϵ(h)s}

of A under the action of H and the rings A^{H} and A#H are Morita equivalent.

Proof. a): A Hopf Galois extension is defined to be one of the equivalent conditions b) - e). b) implies c) with K = A. The equivalence of b) and c) is the preceding Proposition. The equivalence between b) and d) is a simple calculation with dual bases for H and H^{∗} and use of faithful flatness. e) is essentially a translation of d) into terms of Morita equivalences. Detailed proofs of this can be found in [P]. □

There are various different generalizations of Galois extensions. Noncommutative algebras with Hopf algebras acting on them have been investigated. Commutative algebras with finite groups acting on them have been studied in [CHR]. The definition used here has been introduced in [CS] and is also described in [S1]. Special instances of Galois extensions are encluded in this general concept.

Let k be a field and H = kG the group (Hopf) algebra of a finite group. Let K be a field extension of k which is H-Galois. Then G acts by automorphisms on K. Furthermore we have k = {s ∈ K|∀g ∈ G: g(s) = s} = K^{G}. Since [K : k] = |G| we get that K is a ”classical” Galois extension of k with Galois group G. Conversely if K is a ”classical” Galois extension of k with Galois group G then by Dedekind’s lemma and d) of the above Theorem K is Hopf Galois with Hopf algebra H = kG.

Jacobson’s extension [J] of Galois theory to purely inseparable field extensions can be incorporated into the general framework of Hopf Galois theory in the following way. Jacobson uses restricted Lie algebras acting by derivations on purely inseparable field extensions of exponent one. The restricted universal enveloping algebras of the restricted Lie algebras are Hopf algebras and the action extends to a Hopf Galois action on the same extension. Details and an extension to a larger class of purely inseparable field extensions can be found e.g. in [S2] and [W].

The question arises which parts of the ”classical” Galois theory can be transferred to Hopf Galois theory. The definition of a Hopf subalgebra 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-module, these problems can be resolved. The fundamental theorem of Galois theory can be extended to

Theorem 2.6. [CS] Let K be Hopf Galois with Hopf algebra H. For H′ a Hopf subalgebra of H let

Fix(H′) := {x ∈ K|∀h ∈ H′ : h · x = ϵ(h)x}.

Then

Fix: {H′ ⊆ H|H′ Hopf subalgebra } ⟶ {L|k ⊆ L ⊆ K subalgebra }

is injective and inclusion-reversing.

We say, 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 below. 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 have been known for inseparable field extensions.

3. Separable Field Extensions

We give an example of a separable field extension which is not Galois in the classical sense, but which is Hopf Galois with two different Hopf algebras. Let K = Q(\sqrt[4]{2}) and k = Q. It is well known that this is not a ”classical” Galois extension. Let

H = Q[c, s]/(c^{2} + s^{2} − 1, cs)

with the coalgebra structure as given in part I. Abbreviate ω := \sqrt[4]{2}. Then the operation of H on K is given by

1  ω  ω^{2} ω^{3}
c  1  0 −ω^{2} 0
s  0 −ω  0 ω^{3}.

If K = k(\sqrt[4]{2}) were a classical Galois extension for example over the base field Q(i) then the Galois group is cyclic with generator e. Here the generator e has been replaced by the two operators c and s which operate 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 extends the base field from Q to Q(i) then (Q(i)⊗Q(\sqrt[4]{2})) : Q(i) becomes a classical Galois extension and the Hopf algebra H is extended to the group ring QC_{4}. By further extending the base field to Q(i, \sqrt[4]{2}) the ring extension Q(i, \sqrt[4]{2})⊗Q(\sqrt[4]{2}) becomes isomorphic to the dual of the extended group algebra (Q(i, \sqrt[4]{2})C_{4})^{∗}. This isomorphism is compatible with the comodule algebra structure. So we see that the original H^{∗}-comodule algebra K is a form of the trivial H^{∗}-comodule algebra H^{∗}.

One can show that there is a second Hopf algebra over Q and action on K = Q(\sqrt[4]{2}) such that the setup is a Hopf Galois extension. The Hopf algebra is

H = Q[c, s]/(s^{2} − 2c^{2} + 2, cs)

with the action

1 ω ω^{2}  ω^{3}
c  1 0 −ω^{2}  0
s  0 ω^{3}  0 −2ω.

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

c(xy) = c(x)c(y) − \frac{1}{2}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 field to Q(\sqrt{−2}) and then procede as above.

This is an example of a k-algebra which is a Hopf Galois extension with two different Hopf algebras. We will see further down that this will happen 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 field extensions which are not Hopf Galois at all. The separable field extensions which are Hopf Galois can be classified by the following theorem.

To formulate the theorem we fix the following notation. Let K be a finite separable field extension of k. Assume

K̃ = normal closure of K over k,
G = Aut(K̃/k),
G′ = Aut(K̃/K),
S = G/G′ (left cosets),
B = Perm(S) (group of permutations of S).

Theorem 3.1. [GP] Under the assumptions made above the following are equivalent

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

b) There is a regular subgroup N ⊆ B such that the subgroup G ⊆ B normalizes N.

The examples given above are of a rather special type which we call ”almost classical” Hopf Galois extensions. They are characterized by the following Theorem 3.2. [GP] The following conditions are equivalent:

a) There exists a Galois extension E/k such that E ⊗ K is a field containing K̃.

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

c) G′ has a normal complement N in G.

d) There exists a regular subgroup N ⊆ B normalized by G and contained in G.

The last condition of this theorem shows that we are indeed talking about Hopf Galois extensions. These extensions are particularly well behaved because they satisfy the fundamental theorem of Galois theory in its strong form. Theorem 3.3. [GP] If K/k is almost classically 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 exposed in the following Theorem 3.4. [GP] Any classical Galois extension K/k can be endowed with an H-Galois structure such that the following variant of the fundamental theorem holds: There is a canonical bijection between Hopf subalgebras of H and normal intermediate fields k ⊆ E ⊆ K.

One of the simplest examples of a classical Galois extension with this new H-Galois structure is the following. Let ζ be a 3rd primitive root of unity and let ω := \sqrt[3]{2}. Then K = Q(ω, ζ) is a classical Galois extension of Q with Galois group S_{3}. It is also Hopf Galois with Hopf algebra

H = Q[c, s, t]/(c(c − 1)(c + 1), 2c^{2} + st + ts − 2, cs, sc, ct, tc, s^{2}, t^{2}).

The action of H on K is described by the table

1 ω ζ
c  1 0 ζ^{2}
s  0 ω^{2} 0
t  0 0 0.

The action of the three generating elements c, s, t on K satifies

\begin{aligned}c(xy) & = c(x)c(y) + \frac{1}{2}s(x)t(y) + \frac{1}{2}t(x)s(y), \\ s(xy) & = c(x)s(y) + s(x)c(y) + \frac{1}{2}t(x)t(y), \\ t(xy) & = c(x)t(y) + t(x)c(y) + s(x)s(y). \end{aligned}

We finish this paragraph on separable field extensions which are Hopf Galois by giving a family of examples of separable field extensions which are not Hopf Galois: no field extension K over Q of degree 5 with automorphism group S_{5} of K̃/k can be Hopf Galois [GP].

4. Hopf Algebra Forms revisited

Many of the following results have been obtained in cooperation and discussions with students and collegues of mine. In particular I gratefully acknowledge the cooperation of C. Greither, R. Haggenmüller, and C. Wenninger.

The techniques to prove Theorem 1.2 can be used to calculate more forms of group rings. The advantage in the proof of Theorem 1.2 was that all quadratic extensions of a commutative ring can be explicitly described if the ring satisfies only minor conditions [Sm]. If 2 is not a zero divisor in k and if Pic_{(2)}(k) = 0 then all quadratic extensions of k are free and can be described as K = k[x]/(x^{2} − ax − b) where a^{2}+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 Theorem 4.1. [HP] Let G be a finitely generated group with finite automorphism group F = Grp-Aut(G) Then there is a bijection 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

H = {∑ c_{g}g ∈ KG ∀f ∈ F: ∑ f(c_{g})f(g) = ∑ c_{g}g} .

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

ω: H ⊗ K ≅ KG, ω(h ⊗ a) = ah.

On the other hand it is not trivial to describe F -Galois extensions of a field k. They are not just the classical Galois field extensions of k. The simple example of the trivial F -Galois extension k^{F} ≅ k ×…×k is not a field. Actually F -Galois extensions are just Hopf Galois extensions with Hopf algebra kF [CHR, Thm 1.3]. Arbitrary commutative rings K are admitted as Galois extensions. The action of the group F on the extension K by different elements f, f′ has to be ”strongly distict”, i.e. for every idempotent e ∈ K there is an x ∈ K such that f(x)e ≠ f′(x)e. This is the key to the following Theorem 4.2. Let F be a finite group and k a field. K/k is an F - Galois extension if and only if

K ≅ \overbrace{L × … × L}^{n times}

where L/k is a U-Galois field extension with U ⊆ F a subgroup of index n.

Proof. Let K/k be an F -Galois extension. K is a commutative separable k-algebra by [CHR Thm 1.3] hence is a product K ≅ L_{1} × … × L_{n} of separable field extensions L_{i}/k. The automorphisms in F map the primitive idempotents to primitive idempotents and F operates transitively on the set primitive idempotents, since the sum of idempotents in an orbit is in the fixed field. For any two idempotents e_{i} and e_{j} the automorphism f of F mapping e_{i} to e_{j} also maps L_{i} to L_{j}. Hence L_{i} is isomorphic to a subfield of L_{j}. By symmetry all the fields L_{i} are mutually isomorphic. The stabilizer U ⊆ F of e_{1} acts as Galois group on L_{1}/k since it acts strongly distinctly and |U| = [L : k].

Conversely let U ⊆ G be a subgroup and L : k be U-Galois. Let g_{1}, …, g_{n} be a set of representatives for G/U = {g_{1}U, …, g_{n}U}. Let K = L × … × L with idempotens e_{1}, …, e_{n}. Define the action σ: G ⟶ S_{n} by σ(g)(i) = j if gg_{i}U = g_{j}U the regular representation of G on G/U. We define g(le_{i}) := g_{σ(g)(i)}^{−1}gg_{i}(l)e_{σ(g)(i)}. Observe that gg_{i}U = g_{σ(g)(i)U} implies u_{g,i} := g_{σ(g)(i)}^{−1}gg_{i} ∈ U. Then the fixring of K under the action of G is k, for let ∑ l_{i}e_{i} ∈ K^{G}. Then for all g ∈ G we have ∑ u_{g,i}(l_{i})e_{σ(g)(i)} = ∑ l_{i}e_{i}. For g := g_{i}ug_{i}^{−1} we get gg_{i}U = g_{i}U, hence σ(g)(i) = i and u_{g,i} = g_{i}^{−1}g_{i}ug_{i}^{−1}g_{i} = u, so that u(l_{i}) = l_{i} for all u ∈ U, hence l_{i} ∈ k. For g := g_{j}g_{i}^{−1} we get gg_{i}U = g_{j}U hence σ(g)(i) = j and u_{g,i} = g_{j}^{−1}g_{j}g_{i}^{−1}g_{i} = id, so that l_{i}e_{j} = l_{j}e_{j}, hence l_{i} = l_{j} for all i, j. This shows ∑ l_{i}e_{i} = λ ∑ e_{i} = λ ∈ k. Obviously all elements of k remain fixed under the action of G so that k = K^{G}. Furthermore K is separable by definition. To show that G operates strongly distictly it suffices to find for every g ∈ G, g ≠ id and e_{i} ∈ K and x ∈ K such that g(x)e_{i} ≠ xe_{i}. Assume first that σ(g)(i) ≠ i. Choose x = e_{i}. Then g(e_{i})e_{i} = e_{σ(g)(i)}e_{i} = 0 ≠ e_{i} = e_{i}e_{i}. If σ(g)(i) = i then g_{i}^{−1}gg_{i} ∈ U and u ≠ id since gnot = id. Choose an l ∈ L with u(l) ≠ l and x = le_{i}. Then g(x)e_{i} = g(le_{i})e_{i} = g_{i}^{−1}gg_{i}(l)e_{i}u(l)e_{i} ≠ le_{i} = le_{i}e_{i} = xe_{i}. This concludes the proof. □

Observe by the way that kC_{2} has no non-trivial forms, since C_{2} has trivial automorphism group, so the corresponding Galois extension of a form must be k itself. Already the next simplest cases after studying the forms of kZ, kC_{3}, kC_{4}, and kC_{6} cause unsatisfactory calculations. We discuss the case of QC_{5}.

The automorphism group of C_{5} is C_{4} which has exactly one nontrivial subgroup C_{2}. The C_{4}-Galois extensions K of Q can be of the following forms

1) K is a C_{4}-Galois field extension of Q,

2) K ≅ L × L where L is a quadratic field extension of Q,

3) K ≅ Q × Q × Q × Q.

The problem is now to describe as explicitly as possible all C_{2}- resp. C_{4}-Galois field extensions K of Q, to describe the action of C_{4} on K and then calculate the forms according to Theorem 4.1. Associated with a C_{4}-Galois field extension K is the following form of QC_{5}:

H ≅ Q[a]/(a^{5} − 5pa^{3} + (5(p^{2} − 3q) − 10\sqrt{q(p^{2} − 4q)})a)

Here K is the splitting field of x^{4} + px^{2} + q and \sqrt{q(p^{2} − 4q)} ∈ k necessarily holds if K is a C_{4}-Galois field extension. The diagonal maps can be described by

Δ(a) = \frac{1}{uv(u^{2} − v^{2})^{2}} ((u^{5}+v^{5})(a⊗a)−(u^{3}+v^{3})(a⊗b+b⊗a)+(u+v)(b⊗b)),

where

u = \sqrt{\frac{−p + \sqrt{p^{2} − 4q}}{2}} and v = \sqrt{\frac{−p − \sqrt{p^{2} − 4q}}{2}} .

The case of K ≅ L × L with quadratic extension L/Q is somewhat easier. We get

H ≅ Q[a]/(a^{5} + 5p(a^{3} + pa)) with Δ(a) = u^{−1}(1, −1)(a ⊗ a).

where L = Q(u) is the splitting field of x^{2} − p.

Finally the case of K ≅ Q × Q × Q × Q leads to the trivial form QC_{5}.

The other simple example is that of forms of QC_{2} × C_{2}. The automorphism group of C_{2} × C_{2} is the symmetric group S_{3}. Now we have to study the different cases of S_{3}-Galois extensions

1) K is an S_{3}-Galois field extension of Q,

2) K ≅ L × L where L is a C_{3}-Galois field extension of Q,

3) K ≅ L × L × L where L is a quadratic field extension of Q,

4) K ≅ Q × Q × Q × Q × Q × Q. In the first and second case we get

H = Q[a]/(a(a^{3} + ua + v))

where K is the splitting field of x^{3} + ux + v irreducible. If D = −4u^{3} − 27v^{2} is the discriminant then

\begin{aligned}Δ(a) = \frac{1}{D}[ & −2u(3va ⊗ a − u(a ⊗ c + c ⊗ a)) \\ & +\frac{v}{2}(4u^{2}b ⊗ b + 9c ⊗ c + 6u(b ⊗ c + c ⊗ b)) − 9v^{2}(a ⊗ b + b ⊗ a)] \end{aligned}

where

b = \frac{4}{v}a^{3} + \frac{4u}{v}a + 3

and

c = −\frac{4u}{v}a^{3} + 2a^{2} − \frac{4u^{2}}{v} a − 4u.

In the third case of K ≅ L × L × L we get

H = Q[a]/((a^{2} − 1)(a^{2} − u))

where L is the splitting field of x^{2} − u and the diagonal is

Δ(a) = \frac{1}{u^{2} − u} [(u^{2} − u − 1)a ⊗ a − a^{3} ⊗ a^{3} + a^{3} ⊗ a + a ⊗ a^{3}] .

The case of K ≅ Q × Q × Q × Q × Q × Q leads to the trivial form Q(C_{2} × C_{2}).

Problems 4.3. The generators of Hopf algebra forms and their diagonals are rather arbitrary. It often turns out that either the diagonal or the ideal to be factored out can be chosen to be relatively simple, but not both. Is there a canonical choice of the generators of a Hopf algebra form? ^{1} Is there a way to determine the minimal number of generators? Can one describe the ”cyclic” Hopf algebra forms? This seems to be 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_{n} be the group algebra of a finite cyclic group and let k be a field with char(k) / − n. Then the group algebra is semisimple by Maschke’s theorem. Let K be a field extension of k or a commutative separable algebra. Then every K-form H of kC_{n} is again semisimple, since a nilpotent ideal of H would remain nilpotent in K ⊗ H ≅ KC_{n} in both cases, but KC_{n} is still semisimple. There may be forms which are even better in their representation properties as the example of part I. shows.

The Hopf algebra H = R[c, s]/(c^{2} + s^{2} − 1, cs) is a C-form of RC_{4}. It is easy to see that H ≅ R × R × R × R as R-algebras. Thus H is absolutely semisimple, i.e. all its simple modules are one-dimensional over the base field. RC_{4}, however, is not absolutely semisimple. It decomposes as RC_{4} ≅ R × R × C as an algebra, so it has a twodimensional simple module.

So there is the problem of determining which group algebras have absolutly semisimple forms and to describe all those forms. If every semisimple group algebra had an absolutely semisimple form this would mean, that one does not need to extend the base field of a group algebra kG to obtain total splitting, but that the splitting can already be obtained over the base ring for a suitable form H. Since we are not talking about algebra forms but about Hopf algebra forms the possibility of tensoring H-modules over the base field — an important technique for representation theory — is preserved. Theorem 4.4. If k is a field of characteristic not dividing n, then the Hopf algebra kC_{n} has a uniquely determined absolutely semisimple Hopf algebra form k^{C_{n}} = (kC_{n})^{∗}.

Proof. Any absolutely semisimple form of kC_{n} has underlying algebra k^{I}. But k^{I} is a Hopf algebra iff I is a finite group. After base field extension the group structure of I remains unchanged, so there can be at most one group structure on I and at most one Hopf algebra structure on k^{I} so that k^{I} is a form of kC_{n}. So an absolutely semisimple form of kC_{n} is a Hopf algebra k^{G} with a uniquely determined commutative group G of order n. We show G ≅ C_{n} so that k^{C_{n}} becomes the absolutely semisimple form of kC_{n}. It suffices to show this over a field k containing an n-th primitive root of unity. But then kC_{n} splits completely and the statement is well known. □

1Added in proof: A complete answer to this question for the case of group rings has been given recently by the author in Twisted group rings submitted to Comm.

Alg.

This unique absolutely semisimple form of kC_{n} is associated with an F -Galois extension K of k with F ≅ Aut(C_{n}). It turns out that k[x]/(φ_{n}(x)) is an F -Galois extension and associated to k^{C_{n}}. φ_{n}(x) is the n-th cyclotomic polynomial. In general k[x]/(φ_{n}(x)) will not be a field extension of k. According to Theorem 4.2 and with some additional calculations one can see that k[x]/(φ_{n}(x)) ≅ k(ζ_{n}) × … × k(ζ_{n}). Problems 4.5. It would be interesting to know which group algebras over Q have absolutely semisimple forms. The Hopf algebra QS_{n} 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 III we have seen examples of separable field 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 dimension less than 8 are ”almost classically” Galois. The obvious question is, are there proper Hopf Galois extensions of dimension less than 16? Questions about the correspondence between ”normal” Hopf subalgebras and Hopf Galois subfields have been addresses in [C2]. Many of those questions are still open. Childs also addresses the question of the uniqueness of the Hopf algebra H w.r.t. which a separable field extension is Hopf Galois. He obtains results for ”classical” Galois field extensions. He shows that the Hopf algebra H is never unique if G is cyclic of odd prime power order. H is never unique for non-abelian G. This needs a different proof, however, than given in [C2]. Childs also shows that H is unique if G is cyclic of prime order, a result which we will extend below.

Assume that we have the same setup K/k, K̃/k, G, G′, S, B as in III. In [C2] the following result is shown.

Proposition 5.2. G normalizes the regular subgroup N of B iff 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 be a separable field extension of degree [K : k] = p a prime. The following are equivalent:

1) K/k is Hopf Galois.

2) K/k is almost classically Galois.

3) G is solvable. If any (and all) 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_{p} such that G ⊆ Hol(N) = N × Aut(N), the holomorph of N. Since N ≅ C_{p} the holomorph Hol(N) ≅ C_{p} × C_{p−1}, hence G is solvable.

Let G be solvable. Since K = k(a) with a a zero of an irreducible polynomial f of degree p, G is a subgroup of S_{p}, hence p/|G| and p^{2} / − |G|. Below we show that for a solvable group G there is a chain of normal subgroups of G (!)

e ◁ … ◁ G_{2} ◁ G_{1} ◁ G_{0} = G

with G_{i}/G_{i+1} ≅ (Z/p_{i}Z)^{e_{i}}. Consider the sequence of subfields

k = k_{0} ⊂ k_{1} ⊂ … ⊂ k_{m−1} ⊂ k_{m} = K̃,

with k_{i+1}/k_{i} Galois with Galois group (Z/p_{i}Z)^{e_{i}} and k_{i+1}/k normal. We get a ∈ k_{m} and a ∉ k_{m−1}, otherwise K̃ ⊆ k_{m−1} since k_{m−1}/k normal. Then the minimal polynomial f of a is irreducible over k_{m−1} by the last lemma. So all the k-generating elements 1, a, …, a^{p−1} of K are linearly independent over k_{m−1}. Thus K and k_{m−1} are linearly disjoint and p_{m} = p. Since p^{2} / − [K̃ : k] and p = p_{m} = [k_{m} : k_{m−1}] we get k_{m−1} ⊗ K ≅ k_{m−1} · K = k_{m} = K̃. So by Theorem 3.2 the field extension K/k is almost classically Galois.

To see that the Hopf algebra H together with the Galois operation is uniquely determined, observe that p/|G| and G ⊆ N × Aut(N) and N the only Sylow p-subgroup of Hol(N) imply that the Sylow p-subgroup of G is N, which is unique. Thus G = N × A with a subgroup A ⊆ Aut(N). Consequently N = G_{p} ⊆ B is uniquely determined and so is H by Theorem 3.1. □

To finish the proof of the theorem we prove the following lemmas. Lemma 5.4. Let G be a finite solvable group. Then there is a sequence

e ◁ … ◁ G_{2} ◁ G_{1} ◁ G_{0} = G

with G_{i}/G_{i+1} ≅ (Z/p_{i}Z)^{e_{i}}, p_{i} prime and G_{i} ◁ G normal subgroups.

Proof. by induction. We only indicate how to construct G_{i+1} from G_{i}. Let M ◁ G_{i} be a normal subgroup of prime index p_{i}. Define G_{i+1} := ⋂_{g∈G} gMg^{−1}. It has all the required properties. □

Lemma 5.5. Let K/k be a normal separable finite field extension. Let f ∈ k[x] be separable and irreducible of degree p a prime. Then either f is irreducible over K or f completely splits into linear factors.

Proof. Let a_{1}, …, a_{p} be the zeros of f in the algebraic closure and let G be the automorphism group of K(a_{1}, …, a_{p})/k. G operates transitively on the zeros, since f is irreducible. Let N be the fixgroup of K. Since K/k is normal, we get that N ◁G is a normal subgroup. N decomposes {a_{1}, …, a_{p}} into orbits of equal cardinality since G operates transitively and N is normal. So either N operates transitively or trivially. Hence f is irreducible or splits completely. □

References

[C1] L. Childs: Taming Wild Extensions with Hopf Algebras. Trans. Amer. Math. Soc. 304, p. 111-140. 1987.

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

[CHR] S. Chase, D.K. Harrison, A. Rosenberg: Galois Theory and Galois Cohomology of Commutative Rings. Mem. Am. Math. Soc. 52 (1965).

[CS] S. Chase, M. Sweedler: Hopf Algebras and Galois Theory. LN in Math. 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. Haggenmüller: Über Invarianten separabler Galoiserweiterungen kommutativer Ringe. Dissertation Universität München 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 Algèbres d’Azumaya. LN in Math. 389, Springer 1974.

[P] B. Pareigis: Descent Theory applied to Galois Theory. Technical Report 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 Körpererweiterungen. Diplom Thesis Universität München 1984.

Mathematisches Institut der Universität München, Germany

  E-mail address: pareigis@rz.mathematik.uni-muenchen.de