Hopf Galois Structures and Classification in Small Degree

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Introduction

The concept of Hopf algebra was introduced by Heinz Hopf in 1941 in the context of algebraic topology and has later proved useful in many areas of mathematics. In particular Hopf algebras play an important role in studying a class of nite elds extensions, called Hopf Galois extensions.

If we consider nite extensions that are also separable, we can think of a Hopf Galois extension as a generalization of a Galois extension; indeed an equivalent condition for a nite and separable elds extension L/K to be Galois with group G is that the vector spaces homomorphism j : L ⊗ KG → EndK(L)given by

(l ⊗ g)(m) = lg(m)is a bijection. The group algebra KG is a particular example of Hopf algebra; roughly speaking, a elds extension is Hopf Galois if the above condition holds replacing KG with a Hopf algebra H.

The rst signicant results on Hopf Galois extensions in the separable case were obtained in 1987 by Cornelius Greither and Bodo Pareigis [GP87]; previ-ously, in the late 1960s, Moss E. Sweedler and Stephen U. Chase had used Hopf algebras with the goal of studying the automorphisms of purely inseparable elds extensions [CS69], but it turned out that, for a generic purely inseparable extension L/K, a Hopf algebra H of dimension [L : K] is too small to enclose the structure of automorphisms of L/K (see [Cha76]).

In this thesis we will consider the separable case, discussing many of the results obtain by Greither and Pareigis together with some recent development of the theory.

The concept of Hopf Galois extension recovers many cases of extensions that are not Galois. There are some dierences with the classical theory; the rst big dierence is that a given extension L/K can have many dierent Hopf Galois structures, and this can happen also if L/K is a Galois extension. How many are these structures and how do they look like? If we consider a nite and separable extension L/K with normal closure E, G = Gal(E/K) and G0 _{= Gal(E/L)}_{, a}

remarkable theorem by Greither and Pareigis shows that there is a bijective correspondence between the Hopf Galois structures of L/K and the regular subgroups of Perm(G/G0₎_{normalized by G; so in order to investigate the Hopf}

Galois structures we can study such particular groups. Moreover in classical Galois theory a central result is the correspondence theorem, that is, there is an inclusion-reversing bijective correspondence beetwen the subgroups of the Galois group and the intermediate elds. A Hopf Galois version of this theo-rem has been proved rst by Sweedler and Chase (again in [CS69]): there is an inclusion-reversing injection between the set of sub-Hopf algebras of H and

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the set of intermediate elds. Greither and Pareigis show how for a special class of Hopf Galois separable extensions, namely the almost classically Galois extensions, we get surjectivity in Sweedler and Chase's theorem. We propose a reformulation of this result for separable extensions, due to Crespo, Rio and Vela [CRV16b], that is more similar to the classical correspondence theorem.

For extensions of high degree it becomes dicult to nd regular subgroups of Perm(G/G0₎_{normalized by G (and thus Hopf Galois structures). A}

fonda-mental result by Byott [Byo96] (improving previous work by Childs, [Chi89]) allows us to reverse the relation between N and G; in this way the Hopf Galois structures on L/K correspond to the embeddings of G in the normalizer of N, Hol(N ), where N is a group of order n = [L : K]. Hol(N) is much smaller than Perm(G/G0₎ _{and, letting N run through a system of representatives of}

isomorphism classes of groups of order n, Byott's theorem can be used as an algorithmic procedure to check if a given extension is Hopf Galois. In this way, following the work of Crespo, Rio and Vela [CRV16a], we can classify the small degree extensions; we will consider the cases n = 4, 5, 6 and we will also briey comment on the case n = p, p prime. Apart from particular cases the classi-cation for degree n ≥ 8 is still a open problem.

The thesis is organized as follows. In Chapter 1 we show some basic results about Hopf algebras (mostly from [Swe69]) and we dene the Hopf Galois exten-sions following [Chi00]. In Chapter 2 we present the aforementioned theorem by Greither and Pareigis and we discuss the Hopf Galois version of the correspon-dence theorem and its reformulation [CRV16b]. We then introduce the almost classically Galois extensions, showing that for these extensions the map of the correspondence theorem is a bijection; moreover there exists another reformu-lation of this theorem for almost classically Galois extensions. Main sources for this chapter are [Chi00] and [GP87]. In Chapter 3 we present the translation result of Byott (again following [Chi00]) and, as an application, we show the classication of the Hopf Galois structures for extensions of degree n = 4, 5, 6 and we study the intermediate extensions of these. As a corollary we present the smallest degree example of Hopf Galois extension that is not almost classically Galois [CRV16a].

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Chapter 1

Basics

1.1 All we need about Hopf Algebras

1.1.1 Algebras and coalgebras

The denition of algebra over a eld K can be formulated in terms of commu-tative diagrams, in this way:

Denition 1.1. A K-algebra is a triple (A, µ, ι) where A is a K-vector space, µ : A ⊗_KA → Ais a linear map, called multiplication, such that the following diagram A ⊗ A ⊗ A A ⊗ A A ⊗ A A µ⊗id id⊗µ µ µ

commutes and ι : K → A is a linear map, called unity, such that the following diagrams K ⊗ A A ⊗ A A ι⊗id s µ A ⊗ K A ⊗ A A id⊗ι s µ

commute (s is the scalar multiplication).

Note that linearity encodes distributivity and compatibilty of µ with respect to the sum of A and to s, respectively; the rst diagram accounts for associa-tivity of µ and the last two diagrams ensure that 1A= ι(1K)is a unity for µ.

In the sequel we will often write A for (A, µ, ι), when no confusion can arise; we will also write ab for µ(a ⊗ b). Unadorned tensors ⊗ will always denote tensors over K.

The advantage of dening algebras as above is that we can dene coalgebras turning all arrows the other way around, as follow:

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Denition 1.2. A K-coalgebra is a triple (C, ∆, ε) where C is a K-vector space, ∆ : C → C ⊗ C is a linear map, called comultiplication, such that the following diagram C ⊗ C ⊗ C C ⊗ C C ⊗ C C ∆⊗id id⊗∆ ∆ ∆

commutes and ε : C → K is a linear map, called counit, such that the following diagrams K ⊗ C C ⊗ C C ε⊗id ∆ t C ⊗ K C ⊗ C C id⊗ε ∆ t0

commute (t, t0 _{are respectively the maps c 7→ 1 ⊗ c, c 7→ c ⊗ 1).}

The property that the rst diagram commutes (resp. the last two diagrams commute) is called coassociativity (resp. counitary).

Example 1.3. Let S be a set. We denote with KS the vector space over K with basis S. If we dene

∆ : s 7→ s ⊗ s, ε : s 7→ 1 for s ∈ S and we extend linearly, we get that KS is a K-coalgebra.

Generally the expression for ∆ is not so simple as above and it is convenient to introduce the so-called Sweedler notation. If c is an element of C, we can write ∆(c) = n X i=0 ai⊗ bi

for some ai, bi∈ C. In Sweedler notation, instead of a, b, we use the composed

symbols c(1), c(2) to denote the rst and second factor of the comultiplication

and instead of the index i we write (c) with the convention that we are summing up all the terms we need for a given c, so

∆(c) =X

(c)

c(1)⊗ c(2).

For example, the counitary property can be expressed in Sweedler notation by c =P (c)ε(c(1))c(2)=P(c)c(1)ε(c(2)). By coassociativity we have: X (c) c(1)⊗ X (c(2)) c(2)(1)⊗ c(2)(2) =X (c) X (c(1)) c(1)(1)⊗ c(1)(2) ⊗ c(2)

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Denition 1.4. Let (A, µA, ιA)and (B, µB, ιB)be K-algebras. A linear map

f : A → B is an algebra homomorphism if the diagrams

A B A ⊗ A B ⊗ B f µA f ⊗f µB A B K f ιB ιA commute.

Denition 1.5. Let (C, ∆C, εC) and (D, ∆D, εD) be K-coalgebras. A linear

map g : C → D is a coalgebra homomorphism if the diagrams

C D C ⊗ C D ⊗ D g ∆C ∆D g⊗g C D K g εC εD commute.

Remark 1.6. It is easily seen that K is both an algebra and a coalgebra over itself. That being said, if A and B are K-algebras, so is A ⊗ B equipped with

µA⊗B: A ⊗ B ⊗ A ⊗ B idA⊗τ ⊗idB −−−−−−−−→ A ⊗ A ⊗ B ⊗ B µA⊗µB −−−−−→ A ⊗ B ιA⊗B : K ∆K −−→ K ⊗ K ιA⊗ιB −−−−→ A ⊗ B

where τ is the switch map that is, the linear extension of the map a ⊗ b 7→ b ⊗ a. So µA⊗B is the componentwise multiplication. In the same way, if C and D are

K-coalgebras, so is C ⊗ D with ∆C⊗D : C ⊗ D ∆C⊗∆D −−−−−−→ C ⊗ C ⊗ D ⊗ D idC⊗τ ⊗idD −−−−−−−−→ C ⊗ D ⊗ C ⊗ D εC⊗D : C ⊗ D εA⊗εB −−−−→ K ⊗ K µK −−→ K.

Note also that, writing ∆C⊗D(c ⊗ d) in Sweedler notation, we get P(c⊗d)(c ⊗

d)(1)⊗ (c ⊗ d)(2)=P(c),(d)(c(1)⊗ d(1)) ⊗ (c(2)⊗ d(2)).

Denition 1.7. A K-bialgebra is a quintuple (H, µ, ι, ∆, ε) where (H, µ, ι) is a K-algebra, (H, ∆, ε) is a K-coalgebra and either of the following conditions hold:

1) ∆and ε are algebra homomorphisms; 2) µand ι are coalgebra homomorphisms.

We will write H for (H, µ, ι, ∆, ε), when the context is clear.

Conditions 1) and 2) are equivalent: drawing the diagrams for the algebras homomorphism property of ∆ and ε, and writing explicity the maps µH⊗H,

ιH⊗H as in Remark 1.6, one gets the coalgebras homomorphism property for µ

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Denition 1.8. If H, H0 _{are K-bialgebras, f : H → H}0 _{is a bialgebras}

homo-morphism if it is both an algebra and a coalgebra homo-morphism.

Example 1.9. Given any group G, the group algebra KG is a K-bialgebra. As seen in example 1.3, KG is a coalgebra with ∆ : σ 7→ σ ⊗ σ and ε : σ 7→ 1 for σ ∈ G, extended linearly. So we only have to check that these maps are algebra homomorphisms, i.e. ∆ ◦ µ = µ_KG⊗KG◦ (∆ ⊗ ∆) , ∆ ◦ ι = ι_KG⊗KG , ε ◦ µ = µ_K◦ (ε ⊗ ε) , ε ◦ ι = ιK . But µKG⊗KG ∆⊗∆(σ⊗τ ) = µKG⊗KG(σ⊗σ⊗τ ⊗τ ) = στ ⊗στ = ∆ µ(σ⊗τ ) , where σ, τ ∈ G, and by linearity of all maps the property holds for all elements in KG; furthermore ιKG⊗KG(k) = k ι(1) ⊗ ι(1) = ∆ ι(k), where k ∈ K. The

property for ε is straightforward.

1.1.2 Hopf algebras

Denition 1.10. Let H be a K-bialgebra. A linear map λ : H → H is an antipode for H if satises the following conditions:

µ ◦ (id ⊗ λ) ◦ ∆ = ι ◦ ε µ ◦ (λ ⊗ id) ◦ ∆ = ι ◦ ε A K-Hopf algebra is a K-bialgebra with an antipode. Denition 1.11. A Hopf algebra H is said to be:

• commutative if HAlg= (H, µ, ι)is a commutative algebra, that is µ◦τ =

µ;

• cocommutative if HCoAlg= (H, ∆, ε)is a cocommutative coalgebra, that

is τ ◦ ∆ = ∆;

• abelian if both the above conditions hold.

Example 1.12. Let us consider the group algebra KG again. We dene λ : σ 7→ σ−1 for σ ∈ G and, as usual, extend linearly; we want to check that λ is an antipode:

µ id ⊗ λ ∆(σ) = σσ−1= 1 = ι ε(σ) = σ−1σ = µ λ ⊗ id ∆(σ) for σ ∈ G and by linearity of all maps the condition holds for all elements in KG. Thus KG is a Hopf algebra and it is also cocommutative:

τ∆ X σ∈G kσσ = τ X σ∈G kσ ∆(σ) = τ X σ∈G kσ(σ ⊗ σ)

by linearity of τ the previous expression equals X σ∈G kστ (σ ⊗ σ) = X σ∈G kσ(σ ⊗ σ) = ∆ X σ∈G kσσ

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We now introduce the convolution product and we will then show its con-nection with the antipode.

Let C be a K-coalgebra and A a K-algebra. We can dene an internal multi-plication ? on the vector space HomK(C, A), called convolution, as

(f, g) 7→ f ? g = µ ◦ (f ⊗ g) ◦ ∆

for f, g ∈ HomK(C, A). Note that f ? g is a map from C to A and all maps

involved in its denition are linear, so the denition above is well-posed. In Sweedler notation we have (f ? g)(c) = P(c)f (c1)g(c2).

If f, g, h ∈ HomK(C, A), by coassociativity of ∆, we have:

(f ? g) ? h(c) = X

(f ? g)(c(1))h(c(2)) =

X

f (c(1))g(c(2))h(c(3))

= Xf (c(1))(g ? h)(c(2)) = f ? (g ? h)(c) ,

that is, ? is associative; moreover ι ◦ ε ∈ HomK(C, A)is a unity:

f ? (ι ◦ ε)(c) = X f (c(1))(ι ◦ ε)(c(2)) = X f (c(1))ε(c(2)) · 1A= f (c) (ι ◦ ε) ? f(c) = X(ι ◦ ε)(c(1))f (c(2)) = X ε(c(1)) · 1Af (c(2)) = f (c).

Therefore we can state the following:

Proposition 1.13. With the above notation (HomK(C, A), ?, ι ◦ ε) is a

K-algebra.

Applying the above proposition for H a Hopf algebra, A = HAlg, C =

HCoAlg, we have that (EndK(H), ?, ι ◦ ε)is a K-algebra and we get

1) the antipode λ is an inverse for idH in (EndK(H), ?, ι ◦ ε):

id ? λ = µ ◦ (id ⊗ λ) ◦ ∆ = ι ◦ ε = µ ◦ (λ ⊗ id) ◦ ∆ = λ ? id ; 2) the antipode is necessarily unique: it is right and left inverse for id in the

algebra (EndK(H), ?, ι ◦ ε).

Denition 1.14. A Hopf algebra homomorphism is a bialgebra homomorphism. If H, H0 _{are Hopf algebras, the set of K-Hopf algebra homomorphisms of H, H}0

is denoted by HomHopf_K(H, H0).

We did not require that Hopf algebra homomorphism preserve the antipode becouse if f : H → H0 _{is a bialgebra homomorphism and H, H}0 _{are Hopf}

algebras with antipode λ, λ0_{respectively, it follows that f ◦ λ = λ}0_{◦ f}_{. To prove}

this, one can show that for any f ∈ HomK(H, H0)the following equalities

(λ0◦ f ) ? f = ι0◦ ε f ? (f ◦ λ) = ι0◦ ε hold (the maps with superscript0 _{are those of H}0_{), thus}

f ◦ λ = (ι0◦ ε) ? (f ◦ λ) = (λ0_{◦ f ) ? f}_{? (f ◦ λ) = (λ}0_{◦ f ? (f ◦ λ)} = (λ0◦ f ) ? (ι0◦ ε) = (λ0◦ f ) .

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Proposition 1.15. Let H be a Hopf algebra over K; then:

1) λ ◦ µ = µ ◦ (λ ⊗ λ), meaning that λ(ab) = λ(b)λ(a) for every a, b ∈ H; 2) τ ◦ (λ ⊗ λ) ◦ ∆ = ∆ ◦ λ, meaning that ∆(λ(a)) = P_(a)λ(a(2)) ⊗ λ(a(1)),

for a ∈ H.

Proof. H ⊗ H is a coalgebra, by remark 1.6, and so HomK(H ⊗ H, H) is an

algebra, by Proposition 1.13. We dene f, g : H ⊗ H → H as f (a ⊗ b) = λ(ab) ,

g(a ⊗ b) = λ(b)λ(a) ,

and it is easy to check that f, g ∈ HomK(H ⊗ H, H). Since ι is a linear map,

for a, b ∈ H one has ιεH⊗H(a ⊗ b) = εH⊗H(a ⊗ b)1H; so we omit 1H and write

simply εH⊗H for ιεH⊗H. We prove 1) showing that f ? µ = εH⊗H = µ ? g; in

this case f = f ? εH⊗H= f ? (µ ? g) = (f ? µ) ? g = εH⊗H? g = g. We have:

(f ? µ)(a ⊗ b) = X (a⊗b) f ((a ⊗ b)(1))µ((a ⊗ b)(2)) = X (a),(b) f (a(1)⊗ b(1))µ(a(2)⊗ b(2)) = X (ab) λ((ab)(1))((ab)(2))

= (λ ? id)(ab) = ε(ab) = ε(a)ε(b) = εH⊗H(a ⊗ b) ,

where the third equality holds becouse ∆ is an algebra homomorphism, so a(i)b(i)= (ab)(i). On the other hand,

(µ ? g)(a ⊗ b) = X (a),(b) µ(a(1)⊗ b(1))g(a(2)⊗ b(2)) = X (a),(b) a(1)b(1)λ(b(2))λ(a(2)) = X (a) a(1) X (b) b(1)λ(b(2))λ(a(2)) =X (a) a(1)ε(b)λ(a(2)) = X (a) a(1)λ(a(2))ε(b) = ε(a)ε(b) = εH⊗H(a ⊗ b) .

A similar argument holds for 2) taking ∆ in place of µ, f = ∆ ◦ λ and g = τ ◦ (λ ⊗ λ) ◦ ∆.

Remark 1.16. In [Swe69, Proposition 4.0.1] it is shown that in a Hopf algebra H we also have the properties λ(1) = 1 and ε ◦ λ = ε, and that if H is com-mutative or cocomcom-mutative then λ ◦ λ = id. Note that the rst two properties togheter with 1) and 2) of the above proposition imply that λ is both an algebra antimorphism and a coalgebra antimorphism.

An important class of examples of Hopf algebras is given by the dual Hopf algebras. We start with the dual of a K-coalgebra C; if we take A = K in Proposition 1.13, we immediately get that C∗ _{= Hom}

K(C, K) is a K-algebra

with multiplication µC∗ = ? and unit ι_C∗ = ι_K◦ ε. Explicitly, for f,g ∈ C∗,

c ∈ C, k ∈ K:

µC∗(f ⊗ g)(c) = (f ? g)(c) =

X

f (c(1))g(c(2))

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This argument cannot be applied to algebras: given a K-algebra A its dual A∗ = HomK(A, K) in general is not a coalgebra. We can bypass this problem

assuming dimKA < ∞; in this case we can dene two maps ∆A∗, ε_A∗ in such a

way that we have a coalgebra structure. More precisely ∆A∗: A∗ −→ (A ⊗ A)∗' A∗⊗ A∗

f 7−→ f ◦ µ εA∗: A∗ −→ K

f 7−→ f (1A) .

Note that (A ⊗ A)∗_{is isomorphic to A}∗_{⊗ A}∗_{becouse A is nite-dimensional; in}

general for an innite-dimensional vector space A we have A∗_{⊗ A}∗

( (A ⊗ A)∗. It is easy to check coassociativity and counitary of the above maps.

Remark 1.17. If C is cocommutative then C∗ _{is commutative: the rst}

con-dition is equivalent to P c(1) ⊗ c(2) = P c(2) ⊗ c(1) ∀c ∈ C, while the

sec-ond csec-ondition is the same as P f(c(1))g(c(2)) = P g(c(1))f (c(2)) ∀f, g ∈ C∗

and ∀c ∈ C. Therefore, the implication holds because of commutativity of K. We have also a dual statement, i.e. if A is commutative then A∗ is cocommutative. By denition of ∆∗_{, for f ∈ A}∗_{, a, b ∈ A we have that}

∆∗(f )(a ⊗ b) = f (µ(a ⊗ b)) = f (µ(b ⊗ a)) = ∆∗(f )(b ⊗ a); in Sweedler notation ∆∗(f )(b ⊗ a) =P f(1)(b)f(2)(a) = τ (∆∗(f ))(a ⊗ b), again by commutativity of

K.

Proposition 1.18. Let C (resp. A) be a K-coalgebra with dimKC < ∞(resp.

a K-algebra with dimKA < ∞). Then (C, ∆, ε) ' (C∗∗, ∆∗∗, ε∗∗)as coalgebras

(resp.(A, µ, ι) ' (A∗∗_{, µ}∗∗_{, ι}∗∗₎_{as algebras).}

Proof. We only prove the statement for C (a similar argument holds for A). We already know that there is a vector space isomorphism

ϕ : C −→ C∗∗ c 7−→ ϕc

where, for f ∈ C∗_{, ϕ}

c(f ) = f (c). We now check that ϕ is also a coalgebra

isomorphism, i.e., for c ∈ C, ∆∗∗_{(ϕ(c)) = (ϕ ⊗ ϕ)∆(c)}_{and ε}∗∗_{(ϕ)(c) = ε(c)}_.

Regarding the last equality

ε∗∗(ϕ)(c) = ε∗∗(ϕc) = ϕc(1C∗) = ϕ_c ι∗(1_K) = ι∗(1_K)(c) = 1_Kε(c) = ε(c) ,

while for the rst one we have ∆∗∗ _{ϕ(c) = ∆}∗∗_(ϕ

c), (ϕ ⊗ ϕ)∆(c) = P ϕc(1)⊗ ϕc(2) and, for f, g ∈ C ∗_, ∆∗∗(ϕc)(f ⊗ g) = ϕc µ∗(f ⊗ g) = µ∗(f ⊗ g)(c) = X f (c(1))g(c(2)) =Xϕc(1)(f )ϕc(2)(g) = X ϕc(1)⊗ ϕc(2) (f ⊗ g) .

Remark 1.19. The above proposition yields the converse implications of the statements in remark 1.17, thus C is cocommutative i C∗ _{is commutative and}

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If V, W are nite-dimensional vector spaces and f : V → W is a linear map, we write f† _{for the transpose map, that is, f}†_{: W}∗_{→ V}∗_{, f}†_{(φ)(v) = φ(f (v))}_.

Proposition 1.20. Let C, D be K-coalgebras and A, B K-algebras. If f : C → D is a coalgebras homomorphism, then f† : D∗ → C∗ _{is an algebras}

homo-morphism; in the same way, if g : A → B is an algebras homomorphism, then g†: B∗→ A∗ _{is a coalgebras homomorphism.}

Proof. To show that f† _{is an algebra homomorphism we have to check that f}†_◦

ιD∗= ι_C∗and f†◦µ_D∗ = µ_C∗◦(f†⊗f†). Since f is a coalgebra homomorphism,

εD◦ f = εC. So, for k ∈ K, c ∈ C,

(f†◦ ιD∗)(k)(c) = f†(kε_D)(c) = kε_D(f (c)) = kε_C(c) = ι_C∗(k)(c) .

Now, for φ, ψ ∈ D∗_{, c ∈ C we have:}

(f†◦ µD∗)(φ ◦ ψ)(c) = f†◦ (φ ? ψ)(c) = (φ ? ψ)(f (c)) = µ(φ ⊗ ψ)∆_D(f (c)) = µ(φ ⊗ ψ) Xf (c(1)) ⊗ f (c(2)) = X φf (c(1))ψf (c(2)) = (φ ◦ f ) ? (ψ ◦ f )(c) = µC∗(f†(φ) ⊗ f†(ψ))(c) = µC∗(f†⊗ f†)(φ ⊗ ψ)(c) ,

where for going from the rst to the second line we have used that f is a coalgebra homomorphism. To prove that g† _{is a coalgebras homomorphism one}

can proceed in a similar way.

Remark 1.21. Note that if (C, ∆, ε) is a K-coalgebra, thanks to our denition of multiplication and unity on the dual (C∗_{, µ}∗_{, ι}∗₎_{, we have µ}∗ _{= ∆}† _{and ι}∗_{= ε}†

(identifying (H ⊗ H)∗ _{with H}∗_{⊗ H}∗ _{and K with K}∗_).

If H = (µ, ι, ∆, ε) is a K-bialgebra and dimKH < ∞, for what seen above,

H∗ = (µ∗, ι∗, ∆∗, ε∗) is both an algebra and a coalgebra. We can actually say more: H∗ _{is itself a bialgebra; since ∆ and are algebra homomorphisms, by}

Proposition 1.20 ∆†_,† _{are coalgebras homomorphisms and thus, by Remark}

1.21, so are µ∗_{, ι}∗_{. More generally, if H is a Hopf algebra, we also have that H}∗

is a Hopf algebra and its antipode is given by λ∗ _{: f 7→ f ◦ λ}_{, where f ∈ H}∗_;

by Remark 1.19 we have that H is commutative i H∗ _{is cocommutative and}

that H is cocommutative i H∗ _{is commutative and nally, as a corollary to}

Proposition 1.18 and Proposition 1.20, H ' H∗∗_{. If f ∈ H}∗ _{and h ∈ H ' H}∗∗_,

we can see f as a functional with argument h ∈ H but also h ∈ H∗∗ _{as a}

functional with argument f; sometimes, to avoid this choice, we will write hf, hi in place of f(h), h(f), where h·, ·i is the image of the pairing map

h·, ·i : H∗_{× H} _−→ _K

(f, h) 7−→ hf, hi = f (h)

Example 1.22. Let G be a nite group, and H the Hopf algebra KG. H∗ ₌

HomK(KG, K)is the dual Hopf algebra. Let us look at its structure. We

con-sider the dual basis {eσ: σ ∈ G}, so eσ(τ ) = δσ,τ. Considering multiplication,

we have

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that is, for σ = τ, eσ2(ρ) = δσ,ρ = eσ(ρ), while for σ 6= τ eσ and eτ are

orthogonal. Therefore {eσ : σ ∈ G}are pairwise orthogonal idempotents. The

unity is 1H∗=P σ∈Geσ; for f ∈ H ∗_{, f = P} τ ∈Gaτeτ and we have 1 · f = X σ∈G eσ X aτeτ = X σ,τ aτ(eσ· eτ) = X τ aτeτ= f ,

and similarly f · 1 = f. We look at comultiplication: ∆(eσ)(τ ⊗ ρ) = eσ(τ ρ) = δσ,τ ρ

It is easy to check that ∆(eσ) =Pτ ρ=σeτ⊗ eρ. The counit is such that ε(eσ) =

eσ(1G) = δσ,1G and nally the antipode is such that λ(eσ) = eσ◦ λH = eσ−1.

Furthermore we note that H∗ _{is commutative (becouse of cocommutativity of}

H = KG) and H∗ is cocommutative i G is an abelian group. Let us check this last statement. By linearity of ∆, saying that H∗ _{is cocommutative, is the}

same as saying that {eσ: σ ∈ G}is a set of cocommutative elements. Therefore,

if H∗ _{is cocommutative, then P}

ρτ =σeτ⊗ eρ =Pρτ =σeρ⊗ eτ for all σ ∈ G.

Since on both sides we are summing elements of the basis {eµ⊗ eν : µ, ν ∈ G}

of H∗_{⊗ H}∗_{, the above equality holds only if each addend of the left-hand side}

appears also on the right-hand side. Therefore the set of all τ, ρ such that τρ = σ coincides with the set of all τ, ρ such that ρτ = σ. But this is true for all σ ∈ G, so G is an abelian group. The reverse implication is easy to check.

We have dened comultiplication for KG as ∆ : σ 7→ σ ⊗ σ for all σ ∈ G. More generally, given H a Hopf algebra, we dene a non-zero element h ∈ H to be grouplike if ∆(h) = h ⊗ h.

Proposition 1.23. Let K be a eld and H a Hopf algebra. If h ∈ H is group-like, then

ε(h) = 1 .

Moreover the set G(H) = {h ∈ H : h is grouplike} is a subgroup of the multi-plicative group of units of H.

Proof. Let h be grouplike; it follows easily from counitary that h = t−1_{(ε ⊗}

id)∆(h), where t : c 7→ 1 ⊗ c. So h = t−1(ε ⊗ id)(h ⊗ h) = t−1(ε(h) ⊗ id(h)) = ε(h)h, and applying ε we get ε(h) = ε(h)ε(h). Thus ε(h) is an idempotent of K; since the only idempotents of K are 0 and 1, and it cannot be zero (h 6= 0 and h = hε(h)), necessarily ε(h) = 1. We now check that G(H) is a group; for h, h0∈ G(H) we have

∆(hh0) = ∆(h)∆(h0) = (h ⊗ h)(h0⊗ h0) = hh0⊗ hh0 ,

where we have used that comultipilcation is an algebra homomorphism. So G(H)is closed under multiplication. We also have, using that λ is a coalgebra antimorphism, that for h ∈ G(H)

∆ λ(h) = (λ ⊗ λ)τ ∆(h) = (λ ⊗ λ)(h ⊗ h) = λ(h) ⊗ λ(h) ,

that is, G(H) is closed under antipode and nally we have that λ(h) = h−1_.

Indeed

1 = ιε(h) = µ(id ⊗ λ)∆(h) = µ(id ⊗ λ)(h ⊗ h) = µ h ⊗ λ(h) = hλ(h) , and so G(H) is closed under inverse and its elements are units of H.

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Remark 1.24. f ∈ H∗_{= Hom}

K(H, K)is grouplike i f : H → K is an algebra

homomorphism. Actually, for f ∈ H∗_{, h, h}0 _{∈ H}_{, we have}

∆∗(f )(h ⊗ h0) = f µ(h ⊗ h0) = f (xy) µK(f ⊗ f )(h ⊗ h0) = µK f (h) ⊗ f (h0) = f (h)f (h0)

Note only that, since in the denition of ∆∗_{the map ∆}∗_{(f )}_{has image in K, while}

in the denition of algebra homomorphism the map f ⊗ f has image in K ⊗ K, for f to be grouplike we cannot require that ∆∗_{(f )}_{is the same as f ⊗f. Because}

of this in the dual space we have that f is grouplike if ∆∗_{(f ) = µ}

K(f ⊗ f ).

Proposition 1.25. Distinct grouplike elements are linearly independent on K. Proof. Let h1, . . . , hn be grouplike linearly independent elements of H and h ∈

H grouplike such that h = Pn_i=1kihi, ki∈ K. Applying ∆ we have

h ⊗ h =X

i

ki(hi⊗ hi)

thus, replacing h with Pn i=1kihi

h ⊗ h =X

i,j

kikj(hi⊗ hj) .

By linear independence of h1, . . . , hn on H, we have {hi⊗ hj} are linearly

independent on H ⊗ H, then kikj = 0 ∀i 6= j and k2i = ki ∀i. For these

two conditions to be compatible, there must be at most one i such that ki is

non-zero, hence either h = 0 or h = hi for that value of i.

1.2 Additional structures

1.2.1 Modules and Comodules

Denition 1.26. Let A be a K-algebra. A left module over A is a couple (M, α)where M is a K-vector space and α : A ⊗ M → M is a linear map such that the following diagrams

A ⊗ A ⊗ M A ⊗ M A ⊗ M M id⊗α µ⊗idM α α K ⊗ M A ⊗ M M ι⊗idM s α

commute (s is scalar multiplication).

Denition 1.27. Let C be a K-coalgebra. A right comodule over C is a couple (N, β) where N is a K-vector space and β : N → N ⊗ C is a linear map such that the following diagrams

N N ⊗ C N ⊗ C N ⊗ C ⊗ C β β β⊗id idN⊗∆ N ⊗ C N ⊗ K N idM⊗ε β t0

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commute (t0 _{is the map n 7→ n ⊗ 1).}

We will denote (M, α) (resp. (N, β)) simply with M (resp. N) when no confusion can arise, we will write shortly am or a · m for α(a ⊗ m) and we will sometimes refer to α and β respectively as action and coaction. We adapt the Sweedler notation to coactions in this way: for n ∈ N we write

β(n) =X

(n)

n(0)⊗ n(1) ,

and it should be kept in mind that n(0)∈ N and n(1)∈ C.

If N is a right comodule over C then, as we now show, it is a left module over C∗ _dening

α : C∗⊗ N −→ N

f ⊗ n 7−→ (t0)−1τ (id ⊗ f )β(n) =Xf (n(1))n(0) .

(1.1) So we check that (N, α) satises the properties of a left module over C∗_{, that}

is,

α ◦ (idC∗⊗ α) = α ◦ (µ∗⊗ id_N) ,

α ◦ (ι∗⊗ idN) = s .

We start from the last one; for k ∈ K, n ∈ N: α(ι∗⊗ idN)(k ⊗ n) = α(ιK(k)ε ⊗ n) =

X

kε(n(1))n(0)

= kXε(n(1))n(0)= kn = s(k ⊗ n) .

Now, if f, g ∈ C∗ _{and n ∈ N, we have}

α(id ⊗ α)(f ⊗ g ⊗ n) = α f ⊗ α(g ⊗ n) = α f ⊗X g(n(1))n(0) =Xg(n(1))α(f ⊗ n(0)) =Xg(n(1)) X f (n(0)(1))n(0)(0) =Xg(n(2))f (n(1))n(0) =X Xg(n(1)(0))f (n(1)(1))n(0)= X (f ? g)(n(1))n(0) = α (f ? g) ⊗ n = α(µ∗_{⊗ id)(f ⊗ g ⊗ n) .}

We now prove a similar result starting with an algebra; more precisely if A is an algebra of dimension l over K and M is a left module over A, then M is a right comodule over A∗_{. Let a}

1, . . . , al be a basis for A and a∗1, . . . , a∗l be its

dual basis in A∗_{, and set}

β : M −→ M ⊗ A∗ m 7−→ l X i=1 aim ⊗ a∗i . (1.2)

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We now want to check that

(β ⊗ idA∗) ◦ β = (id_M ⊗ ∆∗) ◦ β ,

(idM ⊗ ε∗) ◦ β = t0 .

We start from the rst identity; for m ∈ M, the left-hand side equals (β ⊗ id)β(m) = (β ⊗ id) X i aim ⊗ a∗i = X i β(aim) ⊗ a∗i =X i (X j aj(aim) ⊗ a∗j) ⊗ a∗i = X i,j aj(aim) ⊗ a∗j⊗ a∗i so that, for b, c ∈ A, (β ⊗ id)β(m)(b ⊗ c) = X i,j aj(aim) ⊗ (a∗j ⊗ a ∗ i)(b ⊗ c) =X i,j aj(aim) ⊗ a∗j(b)a ∗ i(c) = X j a∗_j(b)aj X i a∗_i(c)aim = b(cm) .

For the right-hand side, instead, we have (id ⊗ ∆∗)β(m) = (id ⊗ ∆∗) X i aim ⊗ a∗i = X i aim ⊗ X (a∗ i) a∗_{i (1)}⊗ a∗_{i (2)} = X i,(a∗ i) aim ⊗ a∗_{i (1)}⊗ a∗_{i (2)} so that, for b, c ∈ A, (id ⊗ ∆∗)β(m)(b ⊗ c) = Xaim ⊗ (a∗_{i (1)}⊗ a∗_{i (2)})(b ⊗ c) =Xaima∗i(bc) = (bc)m .

The second identity can be checked in the same way.

Remark 1.28. By the above discussion, given a left module (M, α) over an algebra A, α induces a coaction β by 1.2 making (M, β) a right comodule over A∗, and β induces an action α∗∗by 1.1 making (M, α∗∗)a left module over A∗∗. We have that, up to the isomorphism A ' A∗∗_{, α}∗∗ _{is the same as the original}

action α:

α∗∗(a ⊗ m) = (t0)−1τ Xaim ⊗ ha∗i, ai

=Xha∗i, aiaim = am = α(a ⊗ m) .

The same holds if we start with (N, β) a right module over a coalgebra C, i.e. β induces a module action α on N over C∗ that inturn induces a comodule coaction β∗∗ _{on N over C}∗∗_{, that is the same as the original coaction β, up to}

the isomorphism C ' C∗∗_: β∗∗(n) =Xa∗_i· n ⊗ ai= X X n(0)ha∗i, n(1)i ⊗ ai= X n(0)⊗ n(1)= β(n) .

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Denition 1.29. Let (M, α) and (M0_{, α}0₎ _{be left modules over A. A}

homo-morphism of left modules over A is a linear map f : M → M0 _{such that the}

following diagram M M0 A ⊗ M A ⊗ M f α id⊗f α0 commutes.

Denition 1.30. Let (N, β) and (N0_{, β}0₎_{be right comodules over C. A}

homo-morphism of right comodules over C is a linear map g : N → N0 _{such that the}

following diagram N N0 N ⊗ C N ⊗ C g β β0 g⊗id commutes.

1.2.2 (Co)Module Algebras

If H is a Hopf algebra, we can consider left modules and right comodules over H. If M, M0 are left modules over H we have

• Kis a left H-module via ε: h · k = ε(h)k • M ⊗ M0 _{is a left H-module via ∆:}

h · (m ⊗ m0) = ∆(h)(m ⊗ m0) =Xh(1)· m ⊗ h(2)· m0

Note that if α, α0_{are the actions on M, M}0_{, thanks to ∆, the action on M ⊗M}0

is the tensor product α ⊗ α0_{. It is not true in general: if H is only an algebra}

M ⊗ M0 can be given a module structure but with an action that is not the tensor product of the actions.

If a left module over H is also a K-algebra we can dene an additional structure.

Denition 1.31. Let H be a Hopf algebra and S be both a left module over H and a K-algebra. S is a (left) H-module algebra if for every s, t ∈ S, h ∈ H and k ∈ K

h · st =X

(h)

(h(1)· s)(h(2)· t) ,

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Remark 1.32. If we see S⊗S and K as H-modules then S is a H-module algebra i µS, ιS are H-module homomorphisms. Indeed for µS holds

µS αS⊗S(h ⊗ s ⊗ t) = µS X (h) h(1)· s ⊗ h(2)· t = X (h) (h(1)· s)(h(2)· t) , αS(id ⊗ µS)(h ⊗ s ⊗ t) = αS(h ⊗ st) = h · st ,

and for ιS holds

αK(id ⊗ ιS)(h ⊗ k) = αK(h ⊗ ιS(k)) = ε(h)ιS(k) ,

ιS αS(h ⊗ k) = ιS(h · k) .

There is, as usual, a corresponding denition for comodules. If N, N0 _are

right comodules over H then

• Kis a right H-comodule via ι: βK(k) = k ⊗ ι(1K)

• N ⊗ N0 _{is a right H-comodule via µ:}

βN ⊗N0(n ⊗ n0) =

X

n(0)⊗ n0(0)⊗ µ(n(1)⊗ n0(1))

Seeing now N ⊗ N0 _{and K as right H-comodules, we dene}

Denition 1.33. Let H be a Hopf algebra and S be both a right comodule over H and a K-algebra. S is a (right) H-comodule algebra if µS, ιS are H-comodule

homomorphisms.

Remark 1.34. µS is a H-comodule homomorphism i β ◦ µS = (µS⊗ id) ◦ βS⊗S;

for s, t ∈ S β µS(s ⊗ t) = β(st) , (µS⊗ id) X s(0)⊗ t(0)⊗ µ(s(1)⊗ t(1)) = X s(0)t(0)⊗ s(1)t(1)= β(s)β(t) ,

so µS is a H-comodule homomorphism i β is a K-algebra homomorphism.

Moreover ιS is a H-comodule homomorphism i β ◦ ιS = (ιS ⊗ id) ◦ βK; for

k ∈ K:

β ιS(k) = β kιS(1K) = kβ(1S) = k(1S⊗ 1H) ,

(ιS⊗ id)βK(k) = (ιS⊗ id) k ⊗ ι(1K) = k(ιS(1K) ⊗ ι(1K)) ,

therefore ιS is a H-comodule homomorphism i ιS(1K) = 1S.

If S is a H-module algebra, we already know that it is a H∗_{-comodule; it}

is easy to check that S is also a H∗_{-comodule algebra and that also the reverse}

holds.

Example 1.35. Let H be a nite-dimensional Hopf algebra. We want to check that H∗ _{is a H-module algebra. H}∗_{is a H}∗_{-comodule via ∆}∗ _{and so H}∗_{is a}

H-module via h·f = P f(1)hh, f(2)i. Now, since ∆∗is a K-algebra homomorphism,

we have that µH∗ and ι_H∗are homomorphism of H∗-comodules and so H∗is a

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Example 1.36. Let L/K be a nite Galois extension with G = Gal(L/K). L is a left module over the Hopf algebra KG, where the action KG ⊗ L → L is given by the Galois action G y L extended linearly to KG, so σ · l = σ(l). For σ, τ ∈ Gand l ∈ L we have (στ)(l) = σ(τ(l)), so L is a KG-module. Since L is also a K-algebra and σ(lm) = σ(l)σ(m), σ(1) = 1 for all σ ∈ G, l, m ∈ L, we have that L is a KG-modulo algebra. Note that by linearity of the action, it is sucient to check the properties only for the elements in G.

Example 1.37. For G nite group and S a nite-dimensional commutative K-algebra, we have that S is a KG-module algebra i G acts as automorphism of S. If G < Aut(S), where Aut(S) is the set of algebra homomorphisms of S, then for σ ∈ G, s, t ∈ S we have that σ(st) = σ(s)σ(t). We can dene a KG-module action on S by KG ⊗ S −→ S X kσσ ⊗ s 7−→ X kσσ(s)

and in this way S becomes a KG-module algebra: X kσσ(st) = X kσσ(st) = X kσσ(s)σ(t) =Xkσ(σ ⊗ σ)(s ⊗ t) = ∆ X kσσ(s ⊗ t) .

It is easy to check the reverse implication. On the other hand, if S is a G-graded algebra then S is KG-comodule algebra. We say that S is a G-graded K-algebra if

S =M

σ∈G

Sσ

with the properties K ⊆ Sid and Sσ· Sτ⊆ Sστ for all σ, τ ∈ G. A G-grading on

S induces a coaction

β : S −→ S ⊗ KG

s 7−→ s ⊗ σ for σ ∈ Sσ;

indeed β is a K-linear map and it is an algebra homomorphism by the property Sσ· Sτ ⊆ Sστ, so S is a KG-comodule algebra.

1.3 Hopf-Galois extensions

From now on, all extensions of elds will be nite and separable. Let L/K be a Galois extension with G = Gal(L/K). By Example 1.36, we know that the Galois action G y L extends linearly to a module-algebra action α : KG → EndK(L). We have that the vector spaces homomorphism

id ⊗ α : L ⊗ KG −→ EndK(L) l ⊗ X σ kσσ 7−→ lα X σ kσσ : m 7→ l X σ kσσ(m)

is a bijection: if (id ⊗ α)(l ⊗ (P kσσ)) = 0, that is, the map l P kσσ = 0in

EndK(L), then by Proposition 1.25 all coecients are zero and so l⊗ P kσσ =

0, i.e. id ⊗ α is injective. Finally the map id ⊗ α is an isomorphism because dimK(L ⊗ KG) = dimK(EndK(L)).

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Remark 1.38. If G < Aut(L), as in Example 1.37 we get a modulo algebra action α which makes L a KG-modulo algebra; moreover, by classical Galois theory, L/LG_{is a Galois extension with group G = Aut(L/L}G₎_{. Thus if id⊗α is}

an isomorphism, then [L : K] = |G|; since |G| = | Aut(L/LG_{)| ≤ | Aut(L/K)| ≤}

[L : K], LG_{= K} _{and G = Gal(L/K).}

The discussion above togheter with Remark 1.38 gives us an equivalent def-inition of Galois extension: for L/K elds extension and G < Aut(L), we have that L/K is Galois with group G = Aut(L/K) i L ⊗ KG ' EndK(L) via

id ⊗ α.

This equivalent denition is particularly interesting because it can be general-ized to the setting of Hopf algebras.

Denition 1.39. Let S be a nite commutative K-algebra and H a cocom-mutative K-Hopf algebra. S is an Hopf Galois extension over K with Hopf algebra H (shortly H-Galois) if it is a left H-module algebra and the vector space homomorphism

j : S ⊗ H −→ EndK(S)

s ⊗ h 7−→ j(s ⊗ h) : t 7→ sh(t) is an isomorphism.

We consider again a Galois extension L/K and dualize the argument above; the action α : KG ⊗ L → L gives us a coaction

β : L −→ L ⊗ KG∗

l 7−→ X

σ∈G

σ(l) ⊗ eσ

(where {eσ : σ ∈ G} is the dual basis of {σ : σ ∈ G}) that makes L a KG∗

-comodule algebra. We have that the vector spaces homomorphism γ : L ⊗ L −→ L ⊗ KG∗

l ⊗ m 7−→X

σ

lσ(m) ⊗ eσ

is a bijection. Let {m1, . . . , mn} be a K-basis for L and we observe that every

element in L ⊗ L can be written as Pili⊗ mi; suppose now γ(Pσli⊗ mi) = 0,

that is Pσ

P

iliσ(mi) ⊗ eσ = 0. Since Piliσ(mi) =P aσimi for aσi ∈ K, we

have Pi,σa σ

imi⊗ eσ = 0 and so aσi = 0 for all i, σ, i.e. Piliσ(mi) = 0 for

all σ. Therefore li = 0 for all i and so γ is injective and it is a bijection for

dimensional reasons.

Remark 1.40. Also in this case we can say that for L/K a elds extension and G a subgroup of Aut(L), L/K is Galois with group G = Aut(L/K) i L ⊗ L ' L ⊗ KG∗ via γ.

Hence we have a generalization also for the dual structure:

Denition 1.41. Let S be a nite commutative K-algebra and H a nite cocommutative K-Hopf algebra, so that the dual Hopf algebra H∗ _{is nite and}

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commutative. S is a H∗_{-Galois object if it is a right H}∗_{-comodule algebra via}

β and the linear map

γ : S ⊗ S −→ S ⊗ H∗ s ⊗ t 7−→ (s ⊗ 1)β(t) is an isomorphism.

The term "H-Galois object" originates from the common use of referring to H-comodules as H-object.

As we could expect, these denitions are equivalent.

Proposition 1.42. Let H be a nite K-Hopf algebra and S a nite commu-tative K-algebra which is also a left H-module algebra. We have that j is an isomorphism i γ is an isomorphism. Proof. Consider S ⊗ H HomK(S, S) HomS(S ⊗ H∗, S) HomS(S ⊗ S, S) j η ν γ†

where η(s⊗h)(t⊗f) = stf(h), ν(g)(s⊗t) = sg(t) are isomorphism and γ†_{is the}

transpose map γ†_{(f )(s ⊗ t) = f (γ(s ⊗ t)) = f ((s ⊗ 1)β(t)) = f (}P (t)st(0)⊗ t(1)). It is a commutative diagram: ν(j(s ⊗ h))(t ⊗ u) = t(j(s ⊗ h))(n) = tsh(u) , γ†(η(s ⊗ h))(t ⊗ u) = η(s ⊗ h) X (u) tu(0)⊗ u(1) = st X (u) n(0)h(u(1)) = sth(u) .

Since γ is an isomorphism i γ† _{is an isomorphism, it follows that j is an}

isomorphism i γ is an isomorphism.

Corollary 1.43. Let S be a nite commutative K-algebra that is also a H-module algebra, with H a nite cocommutative K-Hopf algebra. We have that S is H-Galois i S is a H∗-Galois object.

The vector space EndK(S) is also a K-algebra endowed with composition;

for S ⊗ H the same holds: it is a tensor product of algebras and so it is an algebra. We want to introduce another product.

Denition 1.44. Let S be a H-module algebra. The smash product ] on the K-vector space S ⊗ H is given by

(s ⊗ h)](t ⊗ g) =X

(h)

sh(1)(t) ⊗ h(2)g ,

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It is easy to check that S ⊗ H equipped with ] is a K-algebra. But now we come to the reason we have introduced the smash product: we have that the map j : (S ⊗ H, ]) → (EndK(S), ◦) is a K-algebras homomorphism. For

s, t, u ∈ S and h, g ∈ H: j (s ⊗ h)](t ⊗ g)(u) = j X sh(1)(t) ⊗ h(2)g(u) = X sh(1)(t)h(2)(g(u)) = sh(tg(u)) = j(s ⊗ h)(tg(u)) = j(s ⊗ h) j(t ⊗ g)(u) = j(s ⊗ h) ◦ j(t ⊗ g)(u) . If S is a H-module algebra, the ring of the invariants of the H-action is SH _{= {s ∈ S : h(s) = ε(h)s ∀h ∈ H}}_{; if S is a H-comodule algebra the ring of}

the coinvariants of the H-coaction β is ScoH _{= {s ∈ S : β(s) = s ⊗ 1} H}.

Proposition 1.45. If the Galois map j is an algebras isomorphism, then SH₌

K.

Proof. For k ∈ K we have h(k) = ε(h)k for all h ∈ H, so K ⊆ SH_{. We suppose}

now s ∈ SH_{; for t ∈ S, h ∈ H we have}

(t ⊗ h)](s ⊗ 1) =Xth(1)(s) ⊗ h(2)= X tε(h(1))s ⊗ h(2) = tl Xε(h(1)) ⊗ h(2) = ts 1S⊗ X ε(h(1))h(2) = tl ⊗ h = (s ⊗ 1)](t ⊗ h) .

Therefore s⊗1 commutes with all elements t⊗h and, since j is an isomorphism, also j(s ⊗ 1) commutes with all elements j(t ⊗ h); But j(s ⊗ 1) = s · 1H is the

multiplication by s and it commutes with all maps in EndK(S), so it must be

in K.

Remark 1.46. Let S be a nite commutative K-algebra, H a nite cocommuta-tive K-algebra and suppose that S is a H∗_{-comodule algebra; the vector spaces}

S ⊗ S and S ⊗ H∗ equipped with the componentwise multiplication are K-algebras. We have that γ : S ⊗ S → S ⊗ H∗ _{is an algebras homomorphism; for}

s, t, a, b ∈ S,

γ (s ⊗ t)(a ⊗ b) = (sa ⊗ 1)β(tb) = (sa ⊗ 1)β(t)β(b)

= X (t) sat(0)⊗ t(1) X (b) b(0)⊗ b(1) ,

where the second equality holds because S is a H∗_{-comodule algebra, thus β is}

an algebras homomorphism. Recalling that on S ⊗ H∗ _{we have componentwise}

multiplication, by commutativity of S the last quantity equals X (t) st(0)⊗ t(1) X (b) ab(0)⊗ b(1) = γ(s ⊗ t)γ(a ⊗ b) .

We now introduce the concepts of base change and Galois descent that will be needed in the proof of the Greither and Pareigis theorem.

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1.3.1 Base change

Let L/K be a H-Galois elds extension and F a nite elds extension of K; we want to show that (F ⊗ L)/F is an (F ⊗ H)-Galois extension. It is easy to check that F ⊗ H is a F -Hopf algebra; we have also that F ⊗ L is a F -algebra with component-wise product, and the action induced by H y L makes F ⊗ L a (F ⊗ H)-module algebra: (a ⊗ h) (b ⊗ l)(c ⊗ m) = abc ⊗ h(lm) = abc ⊗ X (h) h(1)(l)h(2)(m) = a X (h) (b ⊗ h(1)(l))(c ⊗ h(2)(m)) = a X (1⊗h) (1 ⊗ h(1))(b ⊗ l)(1 ⊗ h(2))(c ⊗ m) ,

for a, b, c ∈ F , h ∈ H and l, m ∈ L, and

(a ⊗ h)(1 ⊗ 1) = a ⊗ h(1) = a ⊗ 1 · εH(h) = aεH(h) ⊗ 1 = ε(a ⊗ h) ⊗ 1 .

We have to check that the Galois map j0 _{: (F ⊗ L) ⊗}

F(F ⊗ H) → EndF(F ⊗ L)

is an isomorphism, but, recalling that j is an isomorphism, this follows easily from the commutative diagram

(F ⊗ L) ⊗F(F ⊗ H) EndF(F ⊗ L)

F ⊗ (L ⊗ H) F ⊗ EndK(L) . j0

' '

id⊗j

Let now L be a eld which is also a H-module algebra over K and F a nite extension of elds of K. If (F ⊗ L)/F is (F ⊗ H)-Galois with action induced by H y L, then L/K is H-Galois; here we have only to check that j is an isomoprhism. Looking at the above diagram we have that id ⊗ j is an isomorphism, therefore, by atness of F , so is j.

1.3.2 Galois descent

Let L/K be a Galois nite extension. We have:

• if A is a K-vector space, then A ⊗ L is a L-vector space;

• if f : A → B is a homomorphism of K-vector spaces, then (id ⊗ f) : L ⊗ A → L ⊗ B, (id ⊗ f)(a ⊗ l) = af(l) is a homomorphism of L-vector spaces.

Note that the same holds replacing vector spaces with algebras or Hopf algebras. Descent theory studies the reverse process, giving conditions under which one can say that

• if A is a L-vector space, then A ' L ⊗ A0 as L-vector spaces, with A0

K-vector space,

• if f : L ⊗ A0 → L ⊗ B0 is a L-vector spaces homomorphism, then f =

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(and make analogous assertions for algebras or Hopf algebras).

We will use (a special case of) Morita's theorem to get some information on descent. We write∗Mfor the category of left ∗-modules.

Theorem 1.47 (Morita). Let K be a eld, S a nite K-algebra and E = EndK(S). The covariant functors

S ⊗K∗ :K M−→EM

HomE(S, E) ⊗E∗ :EM−→KM

dene an equivalence of categories betweenKMandEM.

Remark 1.48. A general statement and a proof of the above result can be found in [CR81, Theorem 3.54].

As an immediate consequence we get that if M is a S-module then M de-scends (i.e. M ' S ⊗KM0, with M0K-module) i the S-action on M extends

to an action of EndR(S)on M.

Proposition 1.49. Let S be a nite commutative K-algebra and assume that S is H-Galois, so that EndK(S) ' S ⊗ H is an algebras homomorphism. We have

that for any M ∈E M, M is isomorphic to S ⊗ MH via the map s ⊗ m 7→ ms.

Proof. By Morita's theorem we have, for M a left E-module, M ' S ⊗K⊗ HomE(S, E) ⊗EM .

It is easy to check that HomE(S, E) ⊗EM ' HomE(S, M ); since E ' S ⊗ H as

algebras (with smash product), HomE(S, M ) ' HomS⊗H(S, M )and so

M ' S ⊗KHomS⊗H(S, M ) .

Now we show that

ϕ : HomS⊗H(S, M ) −→ MH

φ 7−→ φ(1)

is an isomorphism. First we note that φ is uniquely determined by φ(1): for any s ∈ S we have φ(s) = φ((s ⊗ 1)1) = (s ⊗ 1)φ(1), where the rst equality is due to the action of S ⊗ H over S. Now, for h ∈ H,

hφ(1) = (1 ⊗ h)φ(1) = φ((1 ⊗ h)1) = φ(ε(h)1) = ε(h)φ(1)

and so φ(1) ∈ MH_{. We have that ϕ is well dened and injective; it can be seen}

easily that ϕ is an homomorphism and surjectivity follows since, for m ∈ MH_,

mis the image of φ : s 7→ sm.

By the proposition above follows that for H-Galois extensions, the base change inverse functor is (∗)H_.

Let L/K be a Galois extension with group G, so we have that (EndK(L), ◦) '

(L ⊗ KG, ]) as rings and therefore the EndK(L)-modules are the (L ⊗

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Denition 1.50. If L/K is a Galois extension with group G and A is a L-vector space, we say that A is a G-compatible L-vector space if it is a KG-module and the structure map of A is G-equivariant, that is, s ◦ (g·) = (g·) ◦ s for every g ∈ G, where s : L ⊗ A → A is the scalar multiplication and G acts on L ⊗ A diagonally.

This denition is justied by:

Proposition 1.51. We have the following equivalences:

1) Ais a (L ⊗ KG)-module i A is a G-compatible L-vector space;

2) f : A → B is a (L ⊗ KG)-modules homomorphism i f is a L-linear map G-equivariant.

Proof. 1) Suppose that A is a (L ⊗ KG)-module, so φ : L ⊗ KG −→ End(A)

l ⊗ σ 7−→ ϕlσ: a 7→ lσ(a)

is a rings homomorphism. Restricting this module action to {l ⊗ 1}l∈L and

{1⊗σ}σ∈Gwe get a L-module structure and a KG-module structure on A given

by φ(l ⊗ 1) = ϕl and φ(1 ⊗ σ) = ϕσ, respectively. Since φ((1 ⊗ σ)](l ⊗ 1)) =

φ(1 ⊗ σ) ◦ φ(l ⊗ 1), i.e. ϕσ(l)σ = ϕσ◦ ϕl, for all a ∈ A

σ(la) = ϕσ(ϕl(a)) = ϕσ(l)σ(a) = σ(l)σ(a) .

Note that the above condition is exactly the G-equivariance of the scalar mul-tiplication of A, so A is a G-compatible L-vector space. On the contrary, let A be a G-compatible L-vector space, so that we have the modules ac-tions φL : L → End(A), φL(l)(a) = ϕl(a) = la and φG : KG → End(A),

φG(σ)(a) = ϕσ(a) = σ(a). As we now show, the diagonal G-action on L ⊗ A

allows us to extend the L-module action over A to a (L ⊗ KG)-module ac-tion over A. Firstly we note that any l ⊗ σ ∈ L ⊗ KG can be written as l ⊗ σ = (1 ⊗ σ)](σ−1(l) ⊗ 1); thus we dene

φ : L ⊗ KG −→ End(A) l ⊗ 1 7−→ ϕl

1 ⊗ σ 7−→ ϕσ

l ⊗ σ 7−→ ϕlσ:= ϕσ◦ ϕσ−1_(l) ,

so that ϕlσ(a) = ϕσ(ϕσ−1_(l)(a)) = σ(σ−1(l)a) = lσ(a). We now check that φ is

a ring homomorphism: for l, m ∈ L, σ, τ ∈ G, a ∈ A we have

φ (l ⊗ σ)](m ⊗ τ )(a) = φ(lσ(m) ⊗ στ )(a) = ϕlσ(m)στ(a) = lσ(m)στ (a)

φ(l ⊗ σ) ◦ φ(m ⊗ τ )(a) = (φlσ◦ φmτ)(a) = φlσ(mτ (a)) = lσ(mτ (a)) ,

and, by compatibility of the G-action, lσ(m)στ(a) = lσ(mτ(a)). Thus φ is a rings homomorphism.

2)Let f be a (L ⊗ KG)-modules homomorphism between A and B. A and B, endowed with the structure dened in the proof of 1), are G-compatible L-vector spaces. Therefore f is an homomorphism of G-compatible L-vector spaces i it is both L-linear and G-equivariant.

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Since a (L⊗KG)-module is a G-compatible L-vector space, for A a L-vector space we have:

Adescends ⇔ the L-action on A extends to a (L ⊗ KG)-action on A ⇔ Gacts on A compatibly with the L-vector space structure. In this case, by Morita's theorem and Proposition 1.49, A ' L ⊗ AG_{, where A}G

is a K-vector space. The functors of Morita's theorem can be applied also to maps; for A, B (L ⊗ KG)-modules and f : A → B a L-linear map, we have

f descends ⇔ f is a (L ⊗ KG)-modules homomorphism ⇔ f is G-equivariant,

and in this case f = idL⊗ f0, where f0: AG → BG is a K-linear map.

Thus if A is a (L ⊗ KG)-module and we have a commutative diagram in-volving only (L ⊗ KG)-modules and (L ⊗ KG)-modules homomorphisms and which denes a property for A, then we have the same commutative diagram for the xed spaces through the functor (∗)G_{and so the property holds also for}

xed space AG_.

Similarly to L-vector spaces, if A is a L-algebra (or a L-Hopf algebra), we say that A is a G-compatible L-(Hopf)algebra if A is a KG-module and the structure maps of A are G-equivariant (here G acts diagonally on A ⊗ A). So if A is a G-compatible (Hopf) algebra over L then the structure maps of A (and properties of those structure maps denible by commutative diagrams) are G-equivariant, and they induce structure maps on AG with the same property. Therefore if A is a G-compatible L-(Hopf) algebra, then AG _{is a K-(Hopf)}

algebra.

Let now H be a Hopf algebra over L and A a L-algebra which is also a H-module algebra, i.e. the properties h(ab) = µ(∆(h)(a ⊗ b)) and h(1) = ε(h) · 1 hold. Both properties can be expressed by commutative diagrams, so if H and A are a G-compatible Hopf algebra and a G-compatible algebra respectively and the module algebra action is G-equivariant, then it descends and applying the functor (∗)G _{to the commutative diagrams which dene the properties of}

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Chapter 2

Hopf Galois theory

2.1 Hopf Galois structures for elds extensions

We have already dened what a Hopf Galois extension is and we have seen how it generalizes the concept of Galois extension. In contrast to the classical theory, a elds extension can be Hopf Galois for dierent Hopf algebras H. In this section we will determine all possible Hopf Galois structures for a given elds extension. Remember that we always consider nite and separable extension.

2.1.1 A special case

We start with a characterization of the Hopf Galois structures for a special type of extensions.

Denition 2.1. Let X be a nite set and Perm(X) its permutation group. A subgroup N < Perm(X) is regular if two of the following conditions hold:

• |N | = |X|;

• the action N y X is transitive; • StabN(x) = idN for all x ∈ X.

The action N y X yields a map ·x : N → X for xed x ∈ X; we have that the second condition holds i for all x ∈ X the map ·x is surjective, while the third condition holds i for all x ∈ X the map ·x is injective. So we have that any two of the above conditions imply the third, and so N is regular i the map ·xis a bijection for every x.

Let E be a eld, X a nite set. We will write XE for the E-vector space Map(X, E) = {f : X → E}. An orthogonal basis for XE is given by {ux: x ∈

X}, where:

ux: X −→ E

y 7−→ δx,y .

Since E is a eld we can see XE as E-algebra with componentwise multiplica-tion; in this way we have that all ux's are idempotents.

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Remark 2.2. Note that if f ∈ XE is idempotent, f = Px∈Xaxuxand we have X x∈X axux= f = f2= X x,y∈X axayuxuy= X x∈X a2_xux,

that is, each ax ∈ E must be idempotent, i.e. ax = 0 or ax = 1, and so

f =P

y∈Yuy for Y ⊆ X.

We call ux, x ∈ X, primitive idempotents.

Theorem 2.3. Let E be a eld, X a nite set. We have:

1) if XE/E is Hopf Galois with Hopf-algebra H, then H is a group ring EN, where N is (identied with) a regular subgroup of Perm(X);

2) if N is a regular subgroup of Perm(X), then XE/E is EN-Galois. Proof. 1) Step 1.(H = EN with N the set of grouplike elements of H) Recalling that an extension is H-Galois i is an H∗_{-Galois object, we have the following}

E-vector spaces isomorphisms (where n = |X|): E × · · · × E | {z } n2_times ' Map(X × X, E) ' XE ⊗EXE ' XE ⊗EH∗' H∗× · · · × H∗ | {z } ntimes

It is easy to check that they are all isomorphisms of E-algebras as well. Thanks to the nite-dimensional assumption the decomposition of a semisimple algebra in simple algebras is unique, thus we get that H∗ _{' E × · · · × E} _{as algebras.}

Since a basis for (E × · · · × E)∗ _{is given by π}

i for i = 1, . . . , n (where πi is the

projection on the i-th coordinate), a basis for H∗∗ _{is given by}

νi: H∗ '

−→ E × · · · × E πi

−→ E ,

for i = 1, . . . , n. We know that the νi's are algebras homomorphisms and so,

by Remark 1.24, they are grouplike elements of H∗∗_{; but H}∗∗ _{' H}_{, therefore}

we can identify N = {νi : i = 1, . . . , n}with a basis for H made by grouplike

elements. So H = hNiE; moreover if h = P aiνi ∈ H is grouplike and h /∈ N,

by Proposition 1.25 h, ν1, . . . , νnmust be linearly indipendents, which is absurd.

Thus N is the set of all grouplike elements of H and by Proposition 1.23, it is a group.

Step 2.(N is a a subgroup of Perm(X)) By our assumption H = EN acts as a modulo algebra on XE. Let us look at the N-action on {ux : x ∈ X}, a

basis of orthogonal idempotents of XE:

ν(ux)ν(ux) = µ (∆(ν))(ux⊗ ux) = ν(uxux) = ν(ux)

ν(ux)ν(uy) = ν(uxuy) = 0for x 6= y .

So ν maps primitive idempotents of XE in orthogonal idempotents of XE: we want to show that these are primitive, that is for every x there exists y such that ν(ux) = uy. It is easy to check that 1XE =P_xux, moreover ν(1XE) =

ε(ν)1XE = 1XE (where we have used the Proposition 1.23); combining these

two equalities: 1XE= ν(1XE) = ν X x∈X ux = X x∈X ν(ux) .

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Now, each ν(ux)is not zero (because ux= 1N(ux) = νν−1(ux), and so ν(ux) 6=

0) and idempotent, therefore ν(ux) =Py∈Yuy for a nonempty set Y ⊆ X; on

the other hand X x∈X ux= 1XE = X x∈X ν(ux) = X x∈X X y∈Y uy ,

so |Y | = 1 and ν(ux) = uy. Thanks to this action we have the embedding

N −→ Perm(X)

ν 7−→ ν : x 7→ yif ν(ux) = uy .

Step 3.(N is regular) Since dimE(EN ) = dimE(XE), |X| = |N|. We now

show that the action N y X is transitive. Suppose, on the contrary, that N ux= {uy: y ∈ Y }for Y ( X and let z be an element in X \ Y ; we consider

the elements exz in EndE(XE) dened as exz(ux) = uz and exz(uy) = 0 for

y 6= x. The Galois map

j : XE ⊗ EN −→ EndE(XE)

ut⊗ νi7−→ j(ut⊗ νi) : ux7→ utνi(ux)

is bijective (and νi(ux) = uy for some y ∈ Y ). So exz ∈ EndE(XE)has to be in

the image of j, and it must be j(uz⊗ νi). But j(uz⊗ νi)(ux) = uzuy for some

y ∈ Y and, since z ∈ X \Y , it has to be zero for every x ∈ X. Thus exz ∈ Im(j)/ ,

against we just obtained. Therefore the action N y X is transitive and N is regular.

2)For x, z ∈ X let us dene exz as above; so {exz : x, z ∈ X}is an E-basis

for EndE(XE). If N is regular, then the action N y X is transitive and so

there exist ν ∈ N such that ν(x) = z, thus j(uz⊗ ν) = exz. This means that

the Galois map is surjective and, since |N| = |X|, it is a bijection.

2.1.2 Greither and Pareigis's Theorem

We want to characterize the Hopf Galois structures H for a elds extension L/K. In order to do this we shift the problem on another extension, obtained from L/K by base change; in this way we will be able to recover the special case of the previous subsection.

We will refer to the following setup as (F): E G0 L X K G L/K elds extension E normal closure of L/K G = Gal(E/K) G0= Gal(E/L) X = G/G0 set of cosets

Remark 2.4. There exists a more general version of the results we will discuss below, where instead of the normal closure E, one can consider a eld containing E. This can be found in [GP87].

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Suppose that L/K is H-Hopf Galois, so that j : L ⊗ H → EndK(L)is an

isomorphism; by the discussion of Subsection 1.3.1, we have that (E ⊗ L)/E is a (E ⊗ H)-Galois extension and j0 _{: (E ⊗ L) ⊗}

E(E ⊗ H) → EndE(E ⊗ L)is an

isomorphism. The advantage of using the base change is that we can express E ⊗ L and E ⊗ H in a convenient way, as we now see.

Remark 2.5. We will simply say module for Kmodule. (E ⊗ L) is a G-module by G-action on the rst component. Also XE = Map(G/G0_{, E)} _{is a}

G-module with action given by σ · f = σf : τ 7→ σf(σ−1_{τ )}_{, for f ∈ XE,}

σ, τ ∈ G.

Proposition 2.6. In the setup (F) we have that φ : E ⊗ L −→ XE

e ⊗ l 7−→ φ(e ⊗ l) : σ 7→ eσ(l) is a E-algebras and G-modules isomorphism.

Proof. The map φ is well-dened: if σ = τ, then σ = τρ0 _{for some ρ}0 _{∈ G}0

and, since L = EG0_{, σ(l) = τ(ρ}0_{(l)) = τ (l)}_{. It is easy to check that φ is a}

E-algebras homomorphism, so we proceed with checking that it is a G-module homomorphism, i.e. for every τ ∈ G, φ ◦ (τ·) = (τ·) ◦ φ. We have

τ · φ(e ⊗ l)(σ) = τ φ(e ⊗ l)(τ−1_{σ) = τ eτ}−1_(σ(l)) = τ (e)τ τ−1(σ(l)) = τ (e)σ(l) = φ τ (e) ⊗ l(σ) = φ τ · (e ⊗ l)(σ) .

Let now {li}i=1,...,n be a K-basis for L; thus {1 ⊗ li}i=1,...,n is a E-basis for

E ⊗ Land so we can write α = P ei⊗ li for every α ∈ E ⊗ L. If α ∈ Ker(φ),

then φ(α) = 0, that is, Pieiσ(li) = 0 for all σ ∈ G. Therefore ei is zero for

every i and φ is injective. For dimensional reasons it has to be bijective. In the sequel we will need the left translation map:

λ : G −→ Perm(X) σ 7−→ λσ: τ 7→ στ ,

which allows us to think of G as a subgroup of Perm(X). Lemma 2.7. The translation map λ is injective.

Proof. We have

Ker(λ) = {σ ∈ G : λσ= idX} = {σ ∈ G : λσ(τ ) = τ ∀τ ∈ X}

⊆ {σ ∈ G : λσ(1) = 1} = {σ ∈ G : σ = 1} = G0 .

Moreover M = Ker(λ) C G, so EM_/K_{is a normal extension; since M ⊆ G}0_{⊆ G}_,

EG_{= K ⊆ E}G0 _{= L ⊆ E}M_{. The latter inclusion, together with E}M_/K _{being a}

normal extension, implies, since E is the normal closure of L/K, that E = EM_,

and so M is trivial.

If L/K is a H-Hopf Galois extension we will sometimes refer to the module algebra action H ⊗ L → L as Hopf action.

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Proposition 2.8. Assume the setup (F) and that L/K is H-Galois; the base change action

α : (E ⊗ H) ⊗E(E ⊗ L) −→ E ⊗ L

is equivalent to an action

α0: EN ⊗EXE −→ XE

which correspondes to a regular embedding N ,→ Perm(X) such that the image of N in Perm(X) is normalized by λ(G), where λ is the left translation. Proof. Step 1.(the actions are equivalent) The base change gives a Hopf action α : (E ⊗ H) ⊗E (E ⊗ L) → (E ⊗ L); moreover the Galois action G y E

gives a G-modules structure on E ⊗ L and E ⊗ H acting on rst components. It is immediate to check that E ⊗ L is a G-compatible E-algebra, E ⊗ H is a G-compatible E-Hopf algebra and that the action α is G-equivariant. By Proposition 2.6, E ⊗ L ' XE; moreover, thanks to Theorem 2.3, 1), E ⊗ H ' EN where N is a regular subgroup of Perm(X), and the Hopf action α is isomorphics to a Hopf action

α0: EN ⊗EXE → XE .

Step 2.(two actions on XE) Remember that {uσ : σ ∈ G/G0}is a basis of

orthogonal idempotents of XE. Note that XE = Map(X, E) = HomE(EX, E),

and G acts both on E and EX, so G acts on HomE(EX, E) = XE:

σ(f )(y) = σ(f (σ−1(y))) ,

for σ ∈ G, f ∈ HomE(EX, E)and y ∈ EX. Let us look at the action of G on

the elements basis uσ:

σ(uτ)(ρ) = σ(uτ(σ−1ρ)) = uτ(σ−1ρ) = uστ(ρ) = uλσ(τ )(ρ) ,

so σ(uτ) = uλσ(τ ), that is, the G-action on the uσ's corresponds to the left

translation. As described in the proof of Theorem 2.3, N acts on the basis {uσ : σ ∈ G/G0} by ν(uσ) = uν(σ), that is, the N-action on the uσ's

corre-sponds to an embedding N ,→ Perm(X).

Step 3.(compatibility of all actions) On E ⊗ L we have the actions of G and E ⊗ H and these actions are compatibile, i.e., as we already said, α is G-equivariant; identifying EN with E ⊗ H, on XE we have the actions of G and EN. Recall that E ⊗ L ' XE via φ which is a G-modules isomorphism, so also α0 is G-equivariant.

Step 4.(Conclusion) We now show that the action of G on EN is given by G × EN −→ EN

(σ, eν) 7−→ σ(e) λσνλσ−1 ,

and, in particular, that G acts on N by conjugation in Perm(X) by elements in λ(G). Since EN is a G-compatible E-Hopf algebra, G respects the Hopf algebra structure of EN; so, if ∆ is the comultiplication of EN,

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Recall that N is the set of grouplike elements of EN, thus ∆(σ(ν)) = σ(ν)⊗σ(ν) and G acts on N. Let us look now at this action; by G-equivariance we have

σ(ν(uτ)) = σ(ν)σ(uτ) ,

and writing explicitly the actions of N and G on the uρ's, we get

σ(ν(uτ)) = σ(uν(τ )) = uλσ(ν(τ )),

σ(ν)σ(uτ) = (σ(ν))(uλσ(τ )) = u(σ(ν))(uλσ (τ )).

Hence λσ(ν(τ )) = (σ(ν))(uλσ(τ )), that is, (σ(ν))(τ) = (λσνλσ−1)(τ ), and σ(ν) =

λσνλσ−1, as desired.

We have just seen that for a given extension L/K a Hopf Galois structure identies a regular subgroup normalized by λ(G). Thanks to the Greither and Pareigis Theorem we can say more, i.e. also the opposite holds.

Theorem 2.9 (Greither-Pareigis). If we are in the setup (F), then there is a bijective correspondence between Hopf Galois structures on L/K and regular subgroups of Perm(X) normalized by λ(G).

Proof. By Proposition 2.8, to prove the statement we have only to check that if we have a regular subgroup N of Perm(X) normalized by λ(G), we can nd a unique Hopf Galois structure H on L/K.

Step 1.(the base change action is G-equivariant) Since N is regular, by Theorem 2.3, 2), XE/E is a EN-Galois extension with Hopf action

α : EN ⊗EXE −→ XE .

Note that, by regularity of N, the action of any ν ∈ N on any f = Pσeσuσ ∈

XE is given by ν(f (τ )) = ν X σ eσuσ(τ ) = X σ eσν(uσ)(τ ) = eν−1_(u τ) =X σ eσuσ(ν−1(τ )) = X σ eσuσ(ν−1(τ )) = f (ν−1(τ )) .

We want to show that XE, EN are G-compatible E-vector spaces and α is G-equivariant. We have that G y XE by

σ(f )(τ ) = σ(f (σ−1_{τ )) = σ(f (λ}

σ−1(τ )))

and E y XE by (ef)(τ) = e(f(τ)). So

σ(ef )(τ ) = σ(e)σ(f (λσ−1(τ ))) = σ(e)σ(f )(τ )

and XE is a G-compatible vector space. Similarly, we have that G y EN by σ(eν) = σ(e)σ(ν) = σ(e)λσνλσ−1

and E y EN by e0_{(eν) = (e}0_e)ν_{. Thus}

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and EN is a G-compatible E-vector space. We now check that the E linear map α is G-equivariant:

α σ(eν ⊗ f )(τ ) = σ(eν)σ(f )(τ ) = σ(e)σ(f ) σ(ν)−1_{(τ )} = σ(e)σfλσ−1 σ(ν)−1(τ )

= σ(e)σ f λσ−1λ_σν−1λ_σ−1(τ )

= σ(e)σ f ν−1λσ−1(τ ) ,

where we have used that N is normalized by λ(G). On the other hand, σ α(eν ⊗ f )(τ ) = σ(eνf )(τ ) = σ(e)σ (νf )(λσ−1(τ ))

= σ(e)σ f ν−1(λσ−1(τ )) = σ(e)σ f ν−1λ_σ−1(τ ) .

By Proposition 1.51 XE and EN are (E ⊗ KG)-modules and α, being G-equivariant, is an (E ⊗ KG)-modules homomorphism.

Step 2.(the extension (XE)G_/K _{is Hopf Galois) Since E/K is a Galois}

extension with group G, EndK(E) ' E ⊗ KG; by Morita's theorem, the (E ⊗

KG)-modules homomorphism α corresponds in a unique way to the K-vector spaces homomorphism

αG: (EN )G⊗K(XE)G−→ (XE)G .

One can check that XE is a G-compatible E-algebra and EN is a E-compatible E-Hopf algebra (i.e. G respects the structures of E-algebra and E-Hopf al-gebra); from this it follows that (EN)G _{is a K-Hopf algebra and (XE)}G _is

a K-algebra. Moreover, since α is a module algebra action, so is αG_{. By}

Proposition 1.49 we know that E ⊗K(EN )G ' EN, E ⊗K(XE)G ' XE; so

from the isomorphism j : XE ⊗EEN → EndE(XE), we get the isomorphism

(E ⊗K(XE)G) ⊗E(E ⊗K(EN )G) ' EndE(E ⊗K(XE)G), which, as in

Subsec-tion 1.3.1, is the same as E ⊗K ((XE)G⊗K(EN )G) ' E ⊗K EndK((XE)G).

Now, by atness of E, the map

jG : (XE)G⊗K(EN )G−→ EndK((XE)G)

is an isomorphism and so (XE)G _{is a H-Galois extension of K, where H =}

(EN )G_.

Step 3.(Conclusion) Let us consider the map f : L −→ (XE)G

l 7−→ X

σ∈X

σ(l)uσ ,

which is well-dened: if l ∈ L = EG0_{, σ(l) does not depend on representative}

element chosen for σ; moreover Im(f) ⊆ XE and

τ X σ σ(l)uσ = X σ τ σ(l)uτ σ= X σ σ(l)uσ ,

(34)

so Im(f) ⊆ (XE)G_{. Since the u}

σ's are a basis for XE, we get immediately that

f is injective. To check surjectivity, let P_σeσuσ be in (XE)G, so

X σ eσuσ = τ X σ eσuσ = X σ τ (eσ)uτ σ

and thus τ(eσ) = eτ σ. In particular τ(e1) = eτ and for τ = 1, that is, τ ∈ G0, we

have τ(e1) = e1. Therefore e1 ∈ EG

0

= L and f(e₁) =P σ(e₁)uσ =P eσuσ.

Thus L ' (XE)G _{and L/K is (EN)}G_{-Hopf Galois.}

Application to Galois extensions

Let L/K be a Galois extension with group G; in the notation of Theorem 2.9, E = Land X = G. The translation map

λ : G −→ Perm(G) σ 7−→ λσ: τ 7→ στ ,

embeds G in Perm(G) as a regular subgroup normalized by λ(G). Another way to do this is given by the right translation:

ρ : G −→ Perm(G) σ 7−→ ρσ: τ 7→ τ σ−1 .

It is easy to check that ρ(G) is a regular subgroup of Perm(X), and moreover λσρπλσ−1(τ ) = σσ−1τ π−1= ρ_π(τ ) ,

so λ(G) acts (by conjugacy) on ρ(G) leaving all elements xed and in particular λ(G)normalizes ρ(G).

It holds:

λ(G) = ρ(G) ⇔ Gis a abelian group.

Indeed, if G is abelian, then λσ = ρσ−1. To show the other implication, suppose

λπ= ρσ; thus σ−1 = ρσ(1) = λπ(1) = π. Now, if there exist τ, σ ∈ G such that

στ 6= τ σ, then ρσ(τ ) = τ σ−16= σ−1τ = λσ−1(τ ), against ρσ= λσ−1.

We get that if L/K is a non abelian Galois extension, there are (at least) two dierent Hopf Galois structures.

Proposition 2.10. Let L/K be a Galois extension. The regular subgroup ρ(G) normalized by λ(G) corresponds to the classical Galois structure.

Proof. By Theorem 2.9, we know that N = ρ(G) corresponds to the Hopf Galois structure H = (LN)G_{, where G is identyed with λ(G). By the discussion}

above, G ' λ(G) y ρ(G) = N trivially (that is, leaving all elements xed), so H = LGN = KN. The action KN = H y (GL)G is induced by the action LN y GL; moreover (GL)G ' L, hence, if l ∈ L correponds to P τ(l)uτ ∈

(GL)G, for σ ∈ G we have ρσ X τ τ (l)uτ = X τ τ (l)ρσ(uτ) = X τ τ (l)uτ σ−1 = X τ τ σ(l)(uτ) .

Thus, since Pττ σ(l)(uτ) corresponds to σ(l), the action N = ρ(G) y (GL) G

correponds to the action G y L (and therefore the Hopf action of KN on (GL)G _{correspond to the Hopf action of KG on L).}