The group of strong Galois objects associated to a cocommutative Hopf quasigroup
J.N. Alonso Álvarez, J.M. Fernández Vilaboa, R. González Rodríguez
aa r X i v : . [ m a t h . R A ] F e b The group of strong Galois ob jects associatedto a cocommutative Hopf quasigroup
J.N. Alonso Álvarez , J.M. Fernández Vilaboa and R. González Rodríguez Departamento de Matemáticas, Universidad de Vigo, Campus Universitario Lagoas-Marcosende, E-36280 Vigo, Spain (e-mail: [email protected]) Departamento de Álxebra, Universidad de Santiago de Compostela, E-15771 Santiago de Compostela,Spain (e-mail: [email protected]) Departamento de Matemática Aplicada II, Universidad de Vigo, Campus Universitario Lagoas-Marco-sende, E-36310 Vigo, Spain (e-mail: [email protected])
Abstract
Let H be a cocommutative faithfully flat Hopf quasigroup in a strict symmetric monoidal category withequalizers. In this paper we introduce the notion of (strong) Galois H -object and we prove that the set ofisomorphism classes of (strong) Galois H -objects is a (group) monoid which coincides, in the Hopf algebra setting,with the Galois group of H -Galois objects introduced by Chase and Sweedler. MSC 2010:
Keywords:
Monoidal category, unital magma, Hopf quasigroup, (strong) Galois H -object, Galoisgroup, normal basis. Introduction
Let R be a commutative ring with unit. The notion of Galois H -object for a commutative, cocom-mutative Hopf R -algebra H , which is a finitely generated projective R -module, is due to Chase andSweedler [7]. As was pointed by Beattie [4], although the discussion of Galois H -objects in [7] is limitedto commutative algebras, the main properties can be easily extended to non commutative algebras. Oneof more relevant is the following: if H is cocommutative, the isomorphism classes of Galois H -objectsform a group denoted by Gal ( R, H ) . The product in Gal ( R, H ) is defined by the kernel of a suitablemorphism and the class of H is the identity element. This construction can be extended to symmetricclosed categories with equalizers and coequalizers working with monoids instead of algebras and some ofthe more important properties and exact sequences involving the group Gal ( R, H ) were obtained in thiscategorical setting ([10], [8], [9]).An interesting generalization of Hopf algebras are Hopf quasigroups introduced by Klim and Majidin [11] in order to understand the structure and relevant properties of the algebraic -sphere. They arenot associative but the lack of this property is compensated by some axioms involving the antipode.The concept of Hopf quasigroup is a particular instance of the notion of unital coassociative H -bialgebraintroduced in [13] and includes the example of an enveloping algebra U ( L ) of a Malcev algebra (see [11])as well as the notion of quasigroup algebra RL of an I.P. loop L . Then, quasigroups unify I.P. loops andMalcev algebras in the same way that Hopf algebras unified groups and Lie algebras.In this paper we are interested to answer the following question: is it possible to extend the constructionof Gal ( R, H ) to the situation where H is a cocommutative Hopf quasigroup? in other words, can weconstruct in a non-associative setting a group of Galois H -objects? The main obstacle to define the groupis the lack of associativity because we must work with unital magmas, i.e. objects where there exists a non-associative product with unit. As we can see in the first section of this paper, Hopf quasi groups areexamples of these algebraic structures.The paper is organized as follows. We begin introducing the notion of right H -comodule magma,where H is a Hopf quasigroup, and defining the product of right H -comodule magmas. In the secondsection we introduce the notions of Galois H -object and strong Galois H -objects proving that, with theproduct defined in the first section for comodule magmas, the set of isomorphism classes forms a monoid,in the case of Galois H -objects, and a group when we work with strong Galois H -objects. In this pointit appears the main difference between our Galois H -objects and the ones associated to a Hopf algebrabecause in the Hopf algebra setting the inverse of the class of a Galois H -object A is the class of theopposite Galois H -object A op , while in the quasigroup context this property fails. We only have thefollowing: the product of A and A op is isomorphic to H only as comodules. To obtain an isomorphism ofmagmas we need to work with strong Galois H -objects. Then, the strong condition appears in a naturalway and we want to point out that in the classical case of Galois H -objects associated to a Hopf algebra H all of them are strong. Finally, in the last section, we study the connections between Galois H -objectsand invertible comodules with geometric normal basis.Throughout this paper C denotes a strict symmetric monoidal category with equalizers where ⊗ denotesthe tensor product, K the unit object and c the symmetry isomorphism. We denote the class of objectsof C by |C| and for each object M ∈ |C| , the identity morphism by id M : M → M . For simplicity ofnotation, given objects M , N and P in C and a morphism f : M → N , we write P ⊗ f for id P ⊗ f and f ⊗ P for f ⊗ id P . We will say that A ∈ |C| is flat if the functor A ⊗ − : C → C preserves equalizers. Ifmoreover A ⊗ − reflects isomorphisms we say that A is faithfully flat.By a unital magma in C we understand a triple A = ( A, η A , µ A ) where A is an object in C and η A : K → A (unit), µ A : A ⊗ A → A (product) are morphisms in C such that µ A ◦ ( A ⊗ η A ) = id A = µ A ◦ ( η A ⊗ A ) .If µ A is associative, that is, µ A ◦ ( A ⊗ µ A ) = µ A ◦ ( µ A ⊗ A ) , the unital magma will be called a monoid in C .For any unital magma A with A we will denote the opposite unital magma ( A, η A = η A , µ A = µ A ◦ c A,A ) .Given two unital magmas (monoids) A = ( A, η A , µ A ) and B = ( B, η B , µ B ) , f : A → B is a morphismof unital magmas (monoids) if µ B ◦ ( f ⊗ f ) = f ◦ µ A and f ◦ η A = η B . By duality, a counital comagmain C is a triple D = ( D, ε D , δ D ) where D is an object in C and ε D : D → K (counit), δ D : D → D ⊗ D (coproduct) are morphisms in C such that ( ε D ⊗ D ) ◦ δ D = id D = ( D ⊗ ε D ) ◦ δ D . If δ D is coassociative,that is, ( δ D ⊗ D ) ◦ δ D = ( D ⊗ δ D ) ◦ δ D , the counital comagma will be called a comonoid. If D = ( D, ε D , δ D ) and E = ( E, ε E , δ E ) are counital comagmas (comonoids), f : D → E is morphism of counital magmas(comonoids) if ( f ⊗ f ) ◦ δ D = δ E ◦ f and ε E ◦ f = ε D . Finally note that if A , B are unital magmas (monoids) in C , the object A ⊗ B is a unital magma(monoid) in C where η A ⊗ B = η A ⊗ η B and µ A ⊗ B = ( µ A ⊗ µ B ) ◦ ( A ⊗ c B,A ⊗ B ) . With A e we will denotethe unital magma A ⊗ A . In a dual way, if D , E are counital comagmas (comonoids) in C , D ⊗ E is acounital comagma (comonoid) in C where ε D ⊗ E = ε D ⊗ ε E and δ D ⊗ E = ( D ⊗ c D,E ⊗ E ) ◦ ( δ D ⊗ δ E ) . Comodule magmas for Hopf quasigroups
This first section is devoted to the study of the notion of H -comodule magma associated to a Hopfquasigroup H . We will show that, as in the Hopf algebra setting, it is possible to define a product usingsuitable equalizers which induces a monoidal structure in the category of flat H -comodule magmas.The notion of Hopf quasigroup was introduced in [11] and the following is its monoidal version. Definition 1.1.
A Hopf quasigroup H in C is a unital magma ( H, η H , µ H ) and a comonoid ( H, ε H , δ H ) such that the following axioms hold:(a1) ε H and δ H are morphisms of unital magmas.(a2) There exists λ H : H → H in C (called the antipode of H ) such that:(a2-1) µ H ◦ ( λ H ⊗ µ H ) ◦ ( δ H ⊗ H ) = ε H ⊗ H = µ H ◦ ( H ⊗ µ H ) ◦ ( H ⊗ λ H ⊗ H ) ◦ ( δ H ⊗ H ) . (a2-2) µ H ◦ ( µ H ⊗ H ) ◦ ( H ⊗ λ H ⊗ H ) ◦ ( H ⊗ δ H ) = H ⊗ ε H = µ H ◦ ( µ H ⊗ λ H ) ◦ ( H ⊗ δ H ) . If H is a Hopf quasigroup, the antipode is unique, antimultiplicative, anticomultiplicative and leavesthe unit and the counit invariable: λ H ◦ µ H = µ H ◦ ( λ H ⊗ λ H ) ◦ c H,H , δ H ◦ λ H = c H,H ◦ ( λ H ⊗ λ H ) ◦ δ H , (1) λ H ◦ η H = η H , ε H ◦ λ H = ε H . (2)([11], Proposition 4.2 and [14], Proposition 1). Note that by (a2), µ H ◦ ( λ H ⊗ id H ) ◦ δ H = µ H ◦ ( id H ⊗ λ H ) ◦ δ H = ε H ⊗ η H . (3)A Hopf quasigroup H is cocommutative if c H,H ◦ δ H = δ H . In this case, as in the Hopf algebra setting,we have that λ H ◦ λ H = id H (see Proposition 4.3 of [11]).Let H and B be Hopf quasigroups. We say that f : H → B is a morphism of Hopf quasigroups if it isa morphism of unital magmas and comonoids. In this case λ B ◦ f = f ◦ λ H (see Proposition 1.5 of [1]). Examples 1.2.
The notion of Hopf quasigroup was introduced in [11] and it can be interpreted as thelinearization of the concept of quasigroup. A quasigroup is a set Q together with a product such thatfor any two elements u, v ∈ Q the equations ux = v , xu = v and uv = x have unique solutions in Q . Aquasigroup L which contains an element e L such that ue L = u = e L u for every u ∈ L is called a loop.A loop L is said to be a loop with the inverse property (for brevity an I.P. loop) if and only if, to everyelement u ∈ L , there corresponds an element u − ∈ L such that the equations u − ( uv ) = v = ( vu ) u − hold for every v ∈ L .If L is an I.P. loop, it is easy to show (see [5]) that for all u ∈ L the element u − is unique and u − u = e L = uu − . Moreover, for all u , v ∈ L , the equality ( uv ) − = v − u − holds.Let R be a commutative ring and L and I.P. loop. Then, by Proposition 4.7 of [11], we know that RL = M u ∈ L Ru is a cocommutative Hopf quasigroup with product given by the linear extension of the one defined in L and δ RL ( u ) = u ⊗ u, ε RL ( u ) = 1 R , λ RL ( u ) = u − on the basis elements.Now we briefly describe another example of Hopf quasigroup constructed working with Malcev algebras(see [12] for details). Consider a commutative and associative ring K with and in K . A Malcevalgebra ( M, [ , ]) over K is a free module in K − M od with a bilinear anticommutative operation [ , ]on M satisfying that [ J ( a, b, c ) , a ] = J ( a, b, [ a, c ]) , where J ( a, b, c ) = [[ a, b ] , c ] − [[ a, c ] , b ] − [ a, [ b, c ]] is theJacobian in a, b, c . Denote by U ( M ) the not necessarily associative algebra defined as the quotient of K { M } , the free non-associative algebra on a basis of M , by the ideal I ( M ) generated by the set { ab − ba − [ a, b ] , ( a, x, y ) + ( x, a, y ) , ( x, a, y ) + ( x, y, a ) : a, b ∈ M, x, y ∈ K { M }} , where ( x, y, z ) = ( xy ) z − x ( yz ) is the usual additive associator.By Proposition 4.1 of [12] and Proposition 4.8 of [11], the diagonal map δ U ( M ) : U ( M ) → U ( M ) ⊗ U ( M ) defined by δ U ( M ) ( x ) = 1 ⊗ x + x ⊗ for all x ∈ M, and the map ε U ( M ) : U ( M ) → K defined by ε U ( M ) ( x ) = 0 for all x ∈ M , both extended to U ( M ) as morphisms of unital magmas; together with themap λ U ( M ) : U ( M ) → U ( M ) , defined by λ U ( M ) ( x ) = − x for all x ∈ M and extended to U ( M ) as anantimultiplicative morphism, provide a cocommutative Hopf quasigroup structure on U ( M ) . Definition 1.3.
Let H be a Hopf quasigroup and let A be a unital magma (monoid) with a right coaction ρ A : A → A ⊗ H . We will say that A = ( A, ρ A ) is a right H -comodule magma (monoid) if ( A, ρ A ) is aright H -comodule (i.e. ( ρ A ⊗ H ) ◦ ρ A = ( A ⊗ δ H ) ◦ ρ A , ( A ⊗ ε H ) ◦ ρ A = id A ), and the following identities(b1) ρ A ◦ η A = η A ⊗ η H , (b2) ρ A ◦ µ A = µ A ⊗ H ◦ ( ρ A ⊗ ρ A ) , hold.Obviously, if H is a Hopf quasigroup, the pair H = ( H, δ H ) is an example of right H -comodule magma.Let A , B be right H -comodule magmas (monoids). A morphism of right H -comodule magmas(monoids) f : A → B is a morphism f : A → B in C of unital magmas (monoids) and right H -comodules,that is ( f ⊗ H ) ◦ ρ A = ρ B ◦ f . Remark 1.4.
Note that, if H is cocommutative, every endomorphism α : H → H of right H -comodulemagmas is an isomorphism. Indeed: First note that by the comodule condition an the cocommutativityof H we have α = (( ε H ◦ α ) ⊗ H ) ◦ δ H = ( H ⊗ ( ε H ◦ α )) ◦ δ H and then α ′ = ( H ⊗ ( ε H ◦ α ◦ λ H )) ◦ δ H isthe inverse of α because by the properties of H : α ′ ◦ α = α ◦ α ′ = ( H ⊗ ((( ε H ◦ α ) ⊗ ( ε H ◦ α ◦ λ H )) ◦ δ H )) ◦ δ H = ( H ⊗ ( ε H ◦ α ◦ µ H ◦ ( H ⊗ λ H ) ◦ δ H )) ◦ δ H = id H . Proposition 1.5.
Let H be a Hopf quasigroup and A , B right H -comodule magmas. The pairs A ⊗ B =( A ⊗ B, ρ A ⊗ B = ( A ⊗ c H,B ) ◦ ( ρ A ⊗ B )) , A ⊗ B = ( A ⊗ B, ρ A ⊗ B = A ⊗ ρ B ) are right H -comodule magmas.Moreover A ⊗ B and B ⊗ A are isomorphic right H -comodule magmas.Proof. We give the proof only for A ⊗ B . The calculus for A ⊗ B are analogous and we left to the reader.First note that the object A ⊗ B is a unital magma in C . On the other hand, the pair ( A ⊗ B, ρ A ⊗ B ) is a right H -comodule because trivially ( A ⊗ B ⊗ ε H ) ◦ ρ A ⊗ B = id A ⊗ B and using the naturality of c weobtain that ( ρ A ⊗ B ⊗ H ) ◦ ρ A ⊗ B = ( A ⊗ δ H ) ◦ ρ A ⊗ B . Moreover, ρ A ⊗ B ◦ η A ⊗ B = η A ⊗ B ⊗ η H and also bythe naturality of c we have ρ A ⊗ B ◦ µ A ⊗ B = ( µ A ⊗ B ⊗ µ H ) ◦ ( A ⊗ B ⊗ c H,A ⊗ B ⊗ H ) ◦ ( ρ A ⊗ B ⊗ ρ A ⊗ B ) . Finally, c A,B is an isomorphism of right H -comodule magmas between A ⊗ B and B ⊗ A becauseby the naturally of c we obtain that c A,B ◦ η A ⊗ B = η B ⊗ A , µ B ⊗ A ◦ ( c A,B ⊗ c A,B ) = c A,B ◦ µ A ⊗ B and ρ B ⊗ A ◦ c A,B = ( c A,B ⊗ H ) ◦ ρ A ⊗ B . (cid:3) Proposition 1.6.
Let H be a cocommutative Hopf quasigroup and A a right H -comodule magma. Then A = ( A, ρ A = ( A ⊗ λ H ) ◦ ρ A ) is a right H -comodule magma.Proof. Trivially ( A ⊗ ε H ) ◦ ρ A = id A . Using that H is cocommutative and (1) we obtain ( ρ A ⊗ H ) ◦ ρ A =( A ⊗ δ H ) ◦ ρ A . Moreover by (b1) of Definition 1.3 and (2), the identity ρ A ◦ η A = η A ⊗ η H holds. Finally,by the naturality of c , (b2) of Definition 1.3 and (1) the equality ρ A ◦ µ A = µ A ⊗ H ◦ ( ρ A ⊗ ρ A ) followseasily. (cid:3) Proposition 1.7.
Let H be a Hopf quasigroup and A , B right H -comodule magmas. The object A • B defined by the equalizer diagram ✲ ✲✲ A • B A ⊗ B A ⊗ B ⊗ H,i A • B ρ A ⊗ B ρ A ⊗ B where ρ A ⊗ B and ρ A ⊗ B are the morphisms defined in Proposition 1.5, is a unital magma where η A • B and µ A • B are the factorizations through i A • B of the morphisms η A ⊗ B and µ A ⊗ B ◦ ( i A • B ⊗ i A • B ) respectively.Moreover, if H is flat and the coaction ρ A • B : A • B → A • B ⊗ H is the factorization of ρ A ⊗ B ◦ i A • B through i A • B ⊗ H , the pair A • B = ( A • B, ρ A • B ) is a right H -comodule magma.Proof. Trivially ρ A ⊗ B ◦ η A ⊗ B = η A ⊗ η B ⊗ η H = ρ A ⊗ B ◦ η A ⊗ B . Therefore, there exists a unique morphism η A • B : K → A • B such that i A • B ◦ η A • B = η A ⊗ B . On the other hand, using the properties of ρ A and ρ B and the naturality of c we have ρ A ⊗ B ◦ µ A ⊗ B ◦ ( i A • B ⊗ i A • B )= ( µ A ⊗ B ⊗ µ H ) ◦ ( A ⊗ B ⊗ c H,A ⊗ B ⊗ H ) ◦ (( ρ A ⊗ B ◦ i A • B ) ⊗ ( ρ A ⊗ B ◦ i A • B ))= ( µ A ⊗ B ⊗ µ H ) ◦ ( A ⊗ B ⊗ c H,A ⊗ B ⊗ H ) ◦ (( ρ A ⊗ B ◦ i A • B ) ⊗ ( ρ A ⊗ B ◦ i A • B ))= ρ A ⊗ B ◦ µ A ⊗ B ◦ ( i A • B ⊗ i A • B ) . Then, there exists a unique morphism µ A • B : A • B ⊗ A • B → A • B such that i A • B ◦ µ A • B = µ A ⊗ B ◦ ( i A • B ⊗ i A • B ) . Moreover, µ A • B ◦ ( η A • B ⊗ A • B ) = id A • B = µ A • B ◦ ( A • B ⊗ η A • B ) because i A • B ◦ µ A • B ◦ ( η A • B ⊗ A • B ) = i A • B = i A • B ◦ µ A • B ◦ ( A • B ⊗ η A • B ) . Therefore, A • B is a unital magma.Moreover, ✲ ✲✲ A • B ⊗ H A ⊗ B ⊗ H A ⊗ B ⊗ H ⊗ H i A • B ⊗ H ρ A ⊗ B ⊗ Hρ A ⊗ B ⊗ H is an equalizer diagram, because − ⊗ H preserves equalizers, and by the the properties of ρ A and ρ B andthe naturally of c we obtain ( ρ A ⊗ B ⊗ H ) ◦ ρ A ⊗ B ◦ i A • B = ( ρ A ⊗ B ⊗ H ) ◦ ρ A ⊗ B ◦ i A • B . As a consequence,there exists a unique morphism ρ A • B : A • B → A • B ⊗ H such that ( i A • B ⊗ H ) ◦ ρ A • B = ρ A ⊗ B ◦ i A • B . Then, the pair ( A • B, ρ A • B ) is a right H -comodule because ( i A • B ⊗ ε H ) ◦ ρ A • B = i A • B andalso ((( i A • B ⊗ H ) ◦ ρ A • B ) ⊗ H ) ◦ ρ A • B = ( i A • B ⊗ δ H ) ◦ ρ A • B . Finally, (b1) and (b2) of Definition1.3 follow, by a similar reasoning, from ( i A • B ⊗ H ) ◦ ρ A • B ◦ η A • B = ( i A • B ⊗ H ) ◦ ( η A • B ⊗ η H ) and ( i A • B ⊗ H ) ◦ ρ A • B ◦ µ A • B = ( i A • B ⊗ H ) ◦ µ A • B ⊗ H ◦ ( ρ A • B ⊗ ρ A • B ) . (cid:3) Proposition 1.8.
Let H be a flat Hopf quasigroup and f : A → B , g : T → D morphisms of right H -comodule magmas. Then the morphism f • g : A • T → B • D , obtained as the factorization of ( f ⊗ g ) ◦ i A • T : A • T → B ⊗ D through the equalizer i B • D , is a morphism of right H -comodule magmasbetween A • T and B • D . Moreover, if f and g are isomorphisms, so is f • g .Proof. Using that f and g are comodule morphisms we obtain ρ B ⊗ D ◦ ( f ⊗ g ) ◦ i A • T = ρ B ⊗ D ◦ ( f ⊗ g ) ◦ i A • T and as a consequence there exist a unique morphism ( f • g ) : A • T → B • D such that i B • D ◦ ( f • g ) =( f ⊗ g ) ◦ i A • T . The morphism f • g is a morphism of unital magmas because i B • D ◦ η B • D = i B • D ◦ ( f • g ) ◦ η A • T and for the product the equality i B • D ◦ µ B • D ◦ (( f • g ) ⊗ ( f • g )) = i B • D ◦ ( f • g ) ◦ µ A • T holds. Also, itis a comodule morphism because ( i B • D ⊗ H ) ◦ ρ B • D ◦ ( f • g ) = ( i B • D ⊗ H ) ◦ (( f • g ) ⊗ H ) ◦ ρ A • T . Finally, it is easy to show that, if f and g are isomorphisms, f • g is an isomorphism with inverse f − • g − . (cid:3) Proposition 1.9.
Let H be a flat Hopf quasigroup and A , B right H -comodule magmas. Then A • B and B • A are isomorphic as right H -comodule magmas.Proof. First note that by the naturally of c and the properties of the equaliser morphism i A • B we havethat ρ B ⊗ A ◦ c A,B ◦ i A • B = ρ B ⊗ A ◦ c A,B ◦ i A • B and then there exist a morphism τ A,B : A • B → B • A such that i B • A ◦ τ A,B = c A,B ◦ i A • B . Also there exists an unique morphism τ B,A : B • A → A • B such that i A • B ◦ τ B,A = c B,A ◦ i B • A . Then i A • B ◦ τ B,A ◦ τ A,B = c B,A ◦ c A,B ◦ i A • B = i A • B and similarly i B • A ◦ τ A,B ◦ τ B,A = i B • A . Thus τ A,B is an isomorphism with inverse τ B,A . Moreover, i B • A ◦ τ A,B ◦ η A • B = c A,B ◦ i A • B ◦ η A • B = η B ⊗ A = i B • A ◦ η B • A and i B • A ◦ τ A,B ◦ µ A • B = µ B ⊗ A ◦ (( c A,B ◦ i A • B ) ⊗ ( c A,B ◦ i A • B )) = µ B ⊗ A ◦ (( i B • A ◦ τ A,B ) ⊗ ( i B • A ◦ τ A,B ))= i B • A ◦ µ B • A ◦ ( τ A,B ⊗ τ A,B ) . Therefore, τ A,B is a morphism of unital magmas and finally it is a morphism of right H -comodulesbecause (( i B • A ◦ τ A,B ) ⊗ H ) ◦ ρ A • B = ( i B • A ⊗ H ) ◦ ρ B • A ◦ τ A,B . (cid:3) Proposition 1.10.
Let H be a flat Hopf quasigroup and A , B , D right H -comodule magmas such that A and D are flat. Then A • ( B • D ) and ( A • B ) • D are isomorphic as right H -comodule magmas. Proof.
First, note that ✲ ✲✲ A ⊗ B • D A ⊗ B ⊗ D A ⊗ B ⊗ D ⊗ H A ⊗ i B • D A ⊗ ρ B ⊗ D A ⊗ ρ B ⊗ D and ✲ ✲✲ A • B ⊗ D A ⊗ B ⊗ D A ⊗ B ⊗ D ⊗ H i A • B ⊗ D ( A ⊗ B ⊗ c H,D ) ◦ ( ρ A ⊗ B ⊗ D )( A ⊗ B ⊗ c H,D ) ◦ ( ρ A ⊗ B ⊗ D ) are equalizer diagrams because A and D are flat and A ⊗ B ⊗ c H,D an isomorphism. On the other hand,it is easy to show that ( A ⊗ i B • D ⊗ H ) ◦ ρ A ⊗ B • D = ( A ⊗ B ⊗ c H,D ) ◦ ( ρ A ⊗ B ⊗ D ) ◦ ( A ⊗ i B • D ) and ( A ⊗ i B • D ⊗ H ) ◦ ρ A ⊗ B • D = ( A ⊗ B ⊗ c H,D ) ◦ ( ρ A ⊗ B ⊗ D ) ◦ ( A ⊗ i B • D ) . Therefore ( A ⊗ B ⊗ c H,D ) ◦ ( ρ A ⊗ B ⊗ D ) ◦ ( A ⊗ i B • D ) ◦ i A • ( B • D ) = ( A ⊗ B ⊗ c H,D ) ◦ ( ρ A ⊗ B ⊗ D ) ◦ ( A ⊗ i B • D ) ◦ i A • ( B • D ) and as a consequence there exist a unique morphism h : A • ( B • D ) → ( A • B ) ⊗ D such that ( i A • B ⊗ D ) ◦ h = ( A ⊗ i B • D ) ◦ i A • ( B • D ) . (4)The diagram ✲ ✲✲ A • ( B • D ) ( A • B ) ⊗ D A • B ⊗ D ⊗ Hh ρ A • B ⊗ D ρ A • B ⊗ D is an equalizer diagram. Indeed, it is easy to see that ρ A • B ⊗ D ◦ h = ρ A • B ⊗ D ◦ h and, if f : C → A • B ⊗ D is a morphism such that ρ A • B ) ⊗ D ◦ f = ρ A • B ) ⊗ D ◦ f , we have that ( A ⊗ ρ B ⊗ D ) ◦ ( i A • B ⊗ D ) ◦ f = ( A ⊗ ρ B ⊗ D ) ◦ ( i A • B ⊗ D ) ◦ f because ( A ⊗ ρ B ⊗ D ) ◦ ( i A • B ⊗ D ) = ( i A • B ⊗ D ⊗ H ) ◦ ρ A • B ⊗ D and ( A ⊗ ρ B ⊗ D ) ◦ ( i A • B ⊗ D ) = ( i A • B ⊗ D ⊗ H ) ◦ ρ A • B ⊗ D . Then, there exists a unique morphism t : C → A ⊗ B • D such that ( A ⊗ i B • D ) ◦ t = ( i A • B ⊗ D ) ◦ f . Themorphism t factorizes through the equalizer i A • ( B • D ) because ( A ⊗ i B • D ⊗ H ) ◦ ρ A ⊗ B • D ◦ t = ( A ⊗ i B • D ⊗ H ) ◦ ρ A ⊗ B • D ◦ t and then ρ A ⊗ B • D ◦ t = ρ A ⊗ B • D ◦ t holds. Thus, there exists a unique morphism g : C → A • ( B • D ) satisfying the equality i A • ( B • D ) ◦ g = t .As a consequence ( i A • B ⊗ D ) ◦ h ◦ g = ( A ⊗ i B • D ) ◦ i A • ( B • D ) ◦ g = ( A ⊗ i B • D ) ◦ t = ( i A • B ⊗ D ) ◦ f and then h ◦ g = f . Moreover, g is the unique morphism such that h ◦ g = t , because if d : C → A • ( B • D ) satisfies h ◦ d = f , we obtain that i A • ( B • D ) ◦ d = t and therefore d = g .As a cosequence, there exists an isomorphism n A,B,C : A • ( B • D ) → ( A • B ) • D such that i ( A • B ) • D ◦ n A,B,D = h (5) The isomorphism n A,B,C is a morphism of unital magmas because by (4), (5) and the naturality of c we have ( i A • B ⊗ D ) ◦ i ( A • B ) • D ◦ n A,B,D ◦ η A • ( B • D ) = ( i A • B ⊗ D ) ◦ h ◦ η A • ( B • D ) = ( A ⊗ i B • D ) ◦ i A • ( B • D ) ◦ η A • ( B • D ) = η A ⊗ η B ⊗ η D = ( i A • B ⊗ D ) ◦ i ( A • B ) • D ◦ η ( A • B ) • D and ( i A • B ⊗ D ) ◦ i ( A • B ) • D ◦ n A,B,D ◦ µ A • ( B • D ) = ( i A • B ⊗ D ) ◦ h ◦ µ A • ( B • D ) = ( A ⊗ i B • D ) ◦ i A • ( B • D ) ◦ µ A • ( B • D ) = ( A ⊗ i B • D ) ◦ µ A ⊗ ( B • D ) ◦ ( i A • ( B • D ) ⊗ i A • ( B • D ) )= µ A ⊗ B ⊗ D ◦ ((( A ⊗ i B • D ) ◦ i A • ( B • D ) ) ⊗ (( A ⊗ i B • D ) ◦ i A • ( B • D ) ))= µ A ⊗ B ⊗ D ◦ ((( i A • B ⊗ D ) ◦ h ) ⊗ (( i A • B ⊗ D ) ◦ h ))= ( i A • B ⊗ D ) ◦ µ A • B ⊗ D ◦ ( h ⊗ h )= ( i A • B ⊗ D ) ◦ µ A • B ⊗ D ◦ (( i ( A • B ) • D ◦ n A,B,C ) ⊗ ( i ( A • B ) • D ◦ n A,B,C ))= ( i A • B ⊗ D ) ◦ i ( A • B ) • D ◦ µ ( A • B ) • D ◦ ( n A,B,D ⊗ n A,B,D ) Finally, using a similar reasoning, we obtain that n A,B,C is a morphism of right H -comodules because ( i A • B ⊗ D ⊗ H ) ◦ ( i ( A • B ) • D ⊗ H ) ◦ ρ ( A • B ) • D ◦ n A,B,D = ( A ⊗ B ⊗ ρ D ) ◦ ( i A • B ⊗ D ) ◦ i ( A • B ) • D ◦ n A,B,C = ( A ⊗ B ⊗ ρ D ) ◦ ( i A • B ⊗ D ) ◦ h = ( A ⊗ B ⊗ ρ D ) ◦ ( A ⊗ i B • D ) ◦ i A • ( B • D ) = ( A ⊗ i B • D ⊗ H ) ◦ ( A ⊗ ρ B • D ) ◦ i A • ( B • D ) = ( A ⊗ i B • D ⊗ H ) ◦ ( i A • ( B • D ) ⊗ H ) ◦ ρ A • ( B • D ) = ( i A • B ⊗ D ⊗ H ) ◦ ( h ⊗ H ) ◦ ρ A • ( B • D ) = ( i A • B ⊗ D ⊗ H ) ◦ ( i ( A • B ) • D ⊗ H ) ◦ ( n A,B,D ⊗ H ) ◦ ρ A • ( B • D ) . (cid:3) Proposition 1.11.
Let H be a cocommutative Hopf quasigroup and A a right H -comodule magma. Then ✲ ✲✲ A A ⊗ H A ⊗ H ⊗ H,ρ A ρ A ⊗ H ρ A ⊗ H (6) is an equalizer diagram. If H is flat A • H and A are isomorphic as right H -comodule magmas.Proof. We will begin by showing that (6) is an equalizer diagram. Indeed, if H is cocommutative we havethat ρ A ⊗ H ◦ ρ A = ( A ⊗ ( c H,H ◦ δ H )) ◦ ρ A = ρ A ⊗ H ◦ ρ A . Moreover, if there exists a morphism f : D → A ⊗ H such that ρ A ⊗ H ◦ f = ρ A ⊗ H ◦ f , we have that ρ A ◦ ( A ⊗ ε H ) ◦ f = f and if g : D → A is a morphism suchthat ρ A ◦ g = f this gives g = ( A ⊗ ε H ) ◦ f . Therefore there is a unique isomorphism r A : A • H → A satisfying ρ A ◦ r A = i A • H .In the second step we show that r A is a morphism of right H -comodule magmas. Trivially, r A ◦ η A • H = η A because ρ A ◦ r A ◦ η A • H = i A • H ◦ η A • H = η A ⊗ η H = ρ A ◦ η A . Also µ A ◦ ( r A ⊗ r A ) = r A ◦ µ A • H because ρ A ◦ µ A ◦ ( r A ⊗ r A ) = ρ A ◦ r A ◦ µ A • H and as a consequence r A is a morphism of unital magmas. Finally,the H -comodule condition follows from (( ρ A ◦ r A ) ⊗ H ) ◦ ρ A • H = ( ρ A ⊗ H ) ◦ ρ A ◦ r A . (cid:3) Remark 1.12.
Note that, under the conditions of the previous proposition, the coaction for A • H is i A • B . On the other hand, Proposition 1.11 gives that ✲ ✲✲ H H ⊗ H A ⊗ H ⊗ H,δ H ρ H ⊗ H ρ H ⊗ H is an equalizer diagram. Proposition 1.13.
Let H be a cocommutative Hopf quasigroup and M ag f ( C , H ) be the category whoseobjects are flat H -comodule magmas and whose arrows are the morphism of H -comodule magmas. Then M ag f ( C , H ) is a symmetric monoidal category.Proof. The category
M ag f ( C , H ) is a monoidal category with the tensor product defined by the product ” • ” introduced in Proposition 1.7, with unit H , with associative constraints a A , B , D = n − A,B,D , where n A,B,D is the isomorphism defined in Proposition 1.10, and right unit constraints and left unit constraints r A = r A , l A = r A ◦ τ H,A respectively, where r A is the isomorphism defined in Proposition 1.11 and τ H,A the one defined in Proposition 1.9. It is easy but tedious, and we leave the details to the reader, toshow that associative constraints and right and left unit constraints are natural and satisfy the PentagonAxiom and the Triangle Axiom. Finally the tensor product of two morphisms is defined by Proposition1.8 and, of course, the symmetry isomorphism is the transformation τ defined in Proposition 1.9. (cid:3) The group of strong Galois objects
The aim of this section is to introduce the notion of strong Galois H -object for a cocommutative Hopfquasigroup H . We will prove that the set of isomorphism classes of strong H -Galois objects is a groupthat becomes the classical Galois group when H is a cocommutative Hopf algebra. Definition 2.1.
Let H be a Hopf quasigroup and A a right H -comodule magma. We will say that A isa Galois H -object if(c1) A is faithfully flat.(c2) The canonical morphism γ A = ( µ A ⊗ H ) ◦ ( A ⊗ ρ A ) : A ⊗ A → A ⊗ H is an isomorphism.If moreover, f A = γ − A ◦ ( η A ⊗ H ) : H → A e is a morphism of unital magmas, we will say that A is astrong Galois H -object.A morphism between to (strong) Galois H -objects is a morphism of right H -comodule magmas.Note that if A is a strong Galois H -object and B is a Galois H -object isomorphic to A as Galois H -objects then B is also a strong Galois H -object because, if g : A → B is the isomorphism, we have γ B ◦ ( g ⊗ g ) = ( g ⊗ H ) ◦ γ A and it follows that f B = ( g ⊗ g ) ◦ f A . Then, f B is a morphism of unitalmagmas and B is strong. Example 2.2. If H is a faithfully flat Hopf quasigroup, H is a strong Galois H -object because γ H =( µ H ⊗ H ) ◦ ( H ⊗ δ H ) is an isomorphism with inverse γ − H = (( µ H ◦ ( H ⊗ λ H )) ⊗ H ) ◦ ( H ⊗ δ H ) and f H = ( λ H ⊗ H ) ◦ δ H : H → H e is a morphism of unital magmas. Remark 2.3. If H is a Hopf algebra and A is a right H -comodule monoid, we say that A is a Galois H -object when A is faithfully flat and the canonical morphism γ A is an isomorphism. In this settingevery Galois H -object is a strong Galois H -object because γ A ◦ µ A e ◦ (( γ − A ◦ ( η A ⊗ H )) ⊗ ( γ − A ◦ ( η A ⊗ H )))= ( µ A ⊗ H ) ◦ ( A ⊗ µ A ⊗ H ) ◦ ( A ⊗ ( γ A ◦ γ − A ◦ ( η A ⊗ H )) ⊗ ρ A ) ◦ ( c H,A ⊗ A ) ◦ ( H ⊗ ( γ − A ◦ ( η A ⊗ H )))= ( A ⊗ µ H ) ◦ ( c H,A ⊗ H ) ◦ ( H ⊗ ( γ A ◦ γ − A ◦ ( η A ⊗ H )))= η A ⊗ µ H = γ A ◦ γ − A ◦ ( η A ⊗ µ H ) where the equalities follow by (b2) of Definition 1.3, the naturality of c and the associativity of µ A . Proposition 2.4.
Let H be a Hopf quasigroup and A a Galois H -object. Then ✲ ✲✲ K A ⊗ Hη A ρ A A ⊗ η H A (7) is an equalizer diagram.Proof. First note that ✲ ✲✲
A A ⊗ A ⊗ AA ⊗ η A A ⊗ A ⊗ η A A ⊗ η A ⊗ AA ⊗ A is an equalizer diagram. Then, using that A s faithfully flat, so is ✲ ✲✲ K A ⊗ Aη A A ⊗ η A η A ⊗ AA On the other hand, γ A ◦ ( A ⊗ η A ) = A ⊗ η H , γ A ◦ ( η A ⊗ A ) = ρ A . Therefore, if γ A is an isomorphism,(7) is an equalizer diagram. (cid:3) Lemma 2.5.
Let H be a Hopf quasigroup and let A a Galois H -object. The following equalities hold: (i) ρ A ⊗ A ◦ γ − A = ( γ − A ⊗ H ) ◦ ( A ⊗ δ H ) . (ii) ρ A ⊗ A ◦ γ − A = ( γ − A ⊗ H ) ◦ ( A ⊗ c H,H ) ◦ ( A ⊗ µ H ⊗ H ) ◦ ( ρ A ⊗ (( λ H ⊗ H ) ◦ δ H )) . Proof.
The proof for (i) follows from the identity ( γ A ⊗ H ) ◦ ρ A ⊗ A = ( A ⊗ δ H ) ◦ γ A . To obtain (ii), firstwe prove that ( A ⊗ µ H ⊗ H ) ◦ ( ρ A ⊗ (( λ H ⊗ H ) ◦ δ H )) ◦ γ A = ( A ⊗ c H,H ) ◦ ( γ A ⊗ H ) ◦ ρ A ⊗ A . (8)Indeed, ( A ⊗ µ H ⊗ H ) ◦ ( ρ A ⊗ (( λ H ⊗ H ) ◦ δ H )) ◦ γ A = ( A ⊗ ( µ H ◦ ( µ H ⊗ λ H ) ◦ ( H ⊗ δ H )) ⊗ H ) ◦ ( µ A ⊗ H ⊗ δ H ) ◦ ( A ⊗ c H,A ⊗ H ) ◦ ( ρ A ⊗ ρ A )= ( µ A ⊗ H ⊗ (( ε H ⊗ H ) ◦ δ H )) ◦ ( A ⊗ c H,A ⊗ H ) ◦ ( ρ A ⊗ ρ A )= ( µ A ⊗ H ⊗ H ) ◦ ( A ⊗ c H,A ⊗ H ) ◦ ( ρ A ⊗ ρ A )= ( µ A ⊗ c H,H ) ◦ ( ρ A ⊗ A ⊗ H ) ◦ ρ A ⊗ A = ( A ⊗ c H,H ) ◦ ( γ A ⊗ H ) ◦ ρ A ⊗ A . where the first equality follows by the comodule condition for A , the second one by (a2-2) of Definition1.1, the third one by the counit condition, the fourth and the last ones by the symmetry of c and thenaturality of the braiding.Then, by (8) we obtain ρ A ⊗ A ◦ γ − A = ( γ − A ⊗ H ) ◦ ( A ⊗ ( c H,H ◦ c H,H )) ◦ ( γ A ⊗ H ) ◦ ρ A ⊗ A ◦ γ − A = ( γ − A ⊗ H ) ◦ ( A ⊗ c H,H ) ◦ ( A ⊗ µ H ⊗ H ) ◦ ( ρ A ⊗ (( λ H ⊗ H ) ◦ δ H )) and (ii) holds. (cid:3) Proposition 2.6.
Let H be a cocommutative faithfully flat Hopf quasigroup. The following assertionshold: (i) If A and B are Galois H -objects so is A • B . (ii) If A and B are strong Galois H -objects so is A • B . Proof.
First we prove (i). Let A and B be Galois H -objects. By Proposition 1.7 we know that A • B is a unital magma where η A • B and µ A • B are the factorizations through i A • B of the morphisms η A ⊗ B and µ A ⊗ B ◦ ( i A • B ⊗ i A • B ) respectively. Moreover, using that H is flat we have that A • B is a right H -comodule magma where the coaction ρ A • B : A • B → A • B ⊗ H is the factorization of ρ A ⊗ B ◦ i A • B (or ρ A ⊗ B ◦ i A • B ) through i A • B ⊗ H .The objects A and B are faithfully flat and then so is A ⊗ B . Therefore ✲ ✲✲ A ⊗ B ⊗ A • B A ⊗ B ⊗ A ⊗ B A ⊗ B ⊗ A ⊗ B ⊗ H A ⊗ B ⊗ i A • B A ⊗ B ⊗ ρ A ⊗ B A ⊗ B ⊗ ρ A ⊗ B is an equalizer diagram. On the other hand, if H is cocommutative ✲ ✲✲ A ⊗ B ⊗ H A ⊗ B ⊗ H ⊗ H A ⊗ B ⊗ H ⊗ H ⊗ H A ⊗ B ⊗ δ H A ⊗ B ⊗ ρ H ⊗ H A ⊗ B ⊗ ρ H ⊗ H is an equalizer diagram (see Remark 1.12).Let Γ A ⊗ B : A ⊗ B ⊗ A ⊗ B → A ⊗ B ⊗ H ⊗ H be the morphism defined by Γ A ⊗ B = ( A ⊗ c H,B ⊗ H ) ◦ ( γ A ⊗ γ B ) ◦ ( A ⊗ c B,A ⊗ B ) . Trivially Γ A ⊗ B is an isomorphism wit inverse Γ − A ⊗ B = ( A ⊗ c A,B ⊗ B ) ◦ ( γ − A ⊗ γ − B ) ◦ ( A ⊗ c B,H ⊗ H ) and satifies ( A ⊗ B ⊗ ρ H ⊗ H ) ◦ Γ A ⊗ B ◦ ( A ⊗ B ⊗ i A • B )= ( µ A ⊗ B ⊗ ρ H ⊗ H ) ◦ ( A ⊗ B ⊗ ((( ρ A ⊗ B ◦ i A • B ) ⊗ H ) ◦ ρ A • B ))= ( µ A ⊗ B ⊗ ρ H ⊗ H ) ◦ ( A ⊗ B ⊗ (( i A • B ⊗ H ⊗ H ) ◦ ( ρ A • B ⊗ H ) ◦ ρ A • B ))= ( µ A ⊗ B ⊗ H ⊗ ( c H,H ◦ δ H )) ◦ ( A ⊗ B ⊗ (( i A • B ⊗ δ H ) ◦ ρ A • B ))= ( µ A ⊗ B ⊗ H ⊗ δ H ) ◦ ( A ⊗ B ⊗ (( i A • B ⊗ H ⊗ H ) ◦ ( ρ A • B ⊗ H ) ◦ ρ A • B ))= ( A ⊗ B ⊗ ρ H ⊗ H ) ◦ Γ A ⊗ B ◦ ( A ⊗ B ⊗ i A • B ) where the first equality follows by the naturality of c and the properties of ρ A • B , the second and thethird ones by the comodule structure of A • B , the fourth one by the cocommutativity of H and the lastone was obtained repeating the same calculus with ρ H ⊗ H .As a consequence, there exists a unique morphism h : A ⊗ B ⊗ A • B → A ⊗ B ⊗ H such that ( A ⊗ B ⊗ δ H ) ◦ h = Γ A ⊗ B ◦ ( A ⊗ B ⊗ i A • B ) . (9)On the other hand, in an analogous way the morphism Γ − A ⊗ B ◦ ( A ⊗ B ⊗ δ H ) : A ⊗ B ⊗ H → A ⊗ B ⊗ A ⊗ B factorizes through through the equalizer A ⊗ B ⊗ i A • B because by (i) of Lemma 2.5, the naturality andsymmetry of c and the cocommutativity of H we have ( A ⊗ B ⊗ ρ A ⊗ B ) ◦ Γ − A ⊗ B ◦ ( A ⊗ B ⊗ δ H )= ( A ⊗ c A,B ⊗ c H,B ) ◦ ( A ⊗ A ⊗ c H,B ⊗ B ) ◦ ((( A ⊗ ρ A ) ◦ γ − A ) ⊗ γ − B ) ◦ ( A ⊗ c B,H ⊗ H ) ◦ ( A ⊗ B ⊗ δ H )= ( A ⊗ c A,B ⊗ c H,B ) ◦ ( A ⊗ A ⊗ c H,B ⊗ B ) ◦ ((( γ − A ⊗ H ) ◦ ( A ⊗ δ H )) ⊗ γ − B ) ◦ ( A ⊗ c B,H ⊗ H ) ◦ ( A ⊗ B ⊗ δ H )= ( A ⊗ c A,B ⊗ B ⊗ H ) ◦ ( γ − A ⊗ (( γ − B ⊗ H ) ◦ ( B ⊗ δ H ))) ◦ ( A ⊗ c B,H ⊗ H ) ◦ ( A ⊗ B ⊗ δ H )= ( A ⊗ B ⊗ ρ A ⊗ B ) ◦ Γ − A ⊗ B ◦ ( A ⊗ B ⊗ δ H ) Thus, let g be the unique morphism such that ( A ⊗ B ⊗ i A • B ) ◦ g = Γ − A ⊗ B ◦ ( A ⊗ B ⊗ δ H ) . (10)By (9) and (10) ( A ⊗ B ⊗ δ H ) ◦ h ◦ g = Γ A ⊗ B ◦ ( A ⊗ B ⊗ i A • B ) ◦ g = Γ A ⊗ B ◦ Γ − A ⊗ B ◦ ( A ⊗ B ⊗ δ H ) = A ⊗ B ⊗ δ H , ( A ⊗ B ⊗ i A • B ) ◦ g ◦ h = Γ − A ⊗ B ◦ ( A ⊗ B ⊗ δ H ) ◦ h = Γ − A ⊗ B ◦ Γ A ⊗ B ◦ ( A ⊗ B ⊗ i A • B ) = A ⊗ B ⊗ i A • B and then we obtain that h is an isomorphism with inverse g . As a consequence A • B is faithfully flatbecause A , B and H are faithfully flat.The morphism Γ − A ⊗ B ◦ ( i A • B ⊗ δ H ) : A • B ⊗ H → A ⊗ B ⊗ A ⊗ B admits a factorization α A,B : A • B ⊗ H → A ⊗ B ⊗ A • B through the equalizer A ⊗ B ⊗ i A • B because as we saw in the previous lines Γ − A ⊗ B ◦ ( A ⊗ B ⊗ δ H ) admits a factorization through A ⊗ B ⊗ i A • B .Now consider the equalizer diagram ✲ ✲✲ A • B ⊗ A • B A ⊗ B ⊗ A • B A ⊗ B ⊗ H ⊗ A • B i A • B ⊗ A • B ρ A ⊗ B ⊗ A • Bρ A ⊗ B ⊗ A • B We have that ( ρ A ⊗ B ⊗ i A • B ) ◦ α A,B = ( ρ A ⊗ B ⊗ A ⊗ B ) ◦ ( A ⊗ c A,B ⊗ B ) ◦ ( γ − A ⊗ γ − B ) ◦ ( A ⊗ c B,H ⊗ H ) ◦ ( i A • B ⊗ δ H )= ( A ⊗ (( B ⊗ c A,H ) ◦ ( c A,B ⊗ H ) ◦ ( A ⊗ c H,B )) ⊗ B ) ◦ ((( ρ A ⊗ H ◦ γ − A ) ⊗ γ − B )) ◦ ( A ⊗ c B,H ⊗ H ) ◦ ( i A • B ⊗ δ H )= ( A ⊗ (( B ⊗ c A,H ) ◦ ( c A,B ⊗ H ) ◦ ( A ⊗ c H,B )) ⊗ B ) ◦ ((( γ − A ⊗ H ) ◦ ( A ⊗ c H,H ) ◦ ( A ⊗ µ H ⊗ H ) ◦ ( ρ A ⊗ (( λ H ⊗ H ) ◦ δ H ))) ⊗ γ − B ) ◦ ( A ⊗ c B,H ⊗ H ) ◦ ( i A • B ⊗ δ H )= ( A ⊗ (( B ⊗ c A,H ) ◦ ( c A,B ⊗ H ) ◦ ( A ⊗ c H,B )) ⊗ B ) ◦ ( γ − A ⊗ H ⊗ γ − B ) ◦ ( A ⊗ (( c H,H ⊗ B ) ◦ ( µ H ⊗ c B,H ) ◦ ( H ⊗ c B,H ⊗ H ) ◦ ( c B,H ⊗ (( λ H ⊗ H ) ◦ δ H ))) ⊗ H ) ◦ (( ρ A ⊗ B ◦ i A • B ) ⊗ δ H )= ( A ⊗ (( B ⊗ c A,H ) ◦ ( c A,B ⊗ H ) ◦ ( A ⊗ c H,B )) ⊗ B ) ◦ ( γ − A ⊗ H ⊗ B ⊗ B ) ◦ ( A ⊗ (( c H,H ⊗ γ − B ) ◦ ( H ⊗ c B,H ⊗ H ) ◦ ( c B,H ⊗ H ⊗ H ) ◦ ( B ⊗ (( µ H ◦ ( H ⊗ λ H )) ⊗ H ) ⊗ H ))) ◦ (( ρ A ⊗ B ◦ i A • B ) ⊗ (( δ H ⊗ H ) ◦ δ H ))= ( A ⊗ (( c H,B ⊗ A ) ◦ ( H ⊗ c A,B ) ◦ ( c A,H ⊗ B )) ⊗ B ) ◦ ( γ − A ⊗ H ⊗ γ − B ) ◦ ( A ⊗ (( H ⊗ c B,H ⊗ H ) ◦ ( c B,H ⊗ ( c H,H ◦ c H,H )) ◦ ( B ⊗ c H,H ⊗ H ))) ◦ ( A ⊗ B ⊗ µ H ⊗ ( c H,H ◦ δ H )) ◦ (( ρ A ⊗ B ◦ i A • B ) ⊗ (( λ H ⊗ H ) ◦ δ H ))= ( A ⊗ (( B ⊗ c A,H ⊗ B ) ◦ ( c A,B ⊗ c B,H )) ◦ ( γ − A ⊗ γ − B ⊗ H ) ◦ ( A ⊗ c B,H ⊗ c H,H ) ◦ ( A ⊗ B ⊗ ( c H,H ◦ ( µ H ⊗ H )) ⊗ H ) ◦ ((( A ⊗ ρ B ) ◦ i A • B ) ⊗ (( λ H ⊗ ( c H,H ◦ δ H )) ◦ δ H ))= ( A ⊗ (( B ⊗ c A,H ⊗ B ) ◦ ( c A,B ⊗ c B,H ))) ◦ ( γ − A ⊗ (( γ − B ⊗ H ) ◦ ( B ⊗ c H,H ) ◦ ( B ⊗ µ H ⊗ H ) ◦ ( ρ B ⊗ (( λ H ⊗ H ) ◦ δ H )))) ◦ ( A ⊗ c B,H ⊗ H ) ◦ ( i A • B ⊗ ( c H,H ◦ δ H ))= ( A ⊗ (( B ⊗ c A,H ) ◦ ( c A,B ⊗ H ) ◦ ( A ⊗ ρ B )) ⊗ B ) ◦ ( γ − A ⊗ γ − B ) ◦ ( A ⊗ c B,H ⊗ H ) ◦ ( i A • B ⊗ δ H )= ( ρ A ⊗ B ⊗ i A • B ) ◦ α A,B where the first equality follows by the definition, the second, the fourth and the fifth ones by the nat-urality and symmetry of c , the third and the nineth ones by (ii) of Lemma 2.5, the sixth one by thecocommutativity of H and, finally, the eighth and the tenth ones by the naturality of c .Then, there exists a unique morphism β A,B : A • B ⊗ H → A • B ⊗ A • B such that ( i A • B ⊗ A • B ) ◦ β A,B = α A,B (11)and then ( i A • B ⊗ i A • B ) ◦ β A,B = Γ − A ⊗ B ◦ ( i A • B ⊗ δ H ) (12)The morphism β A,B satisfies ( i A • B ⊗ H ) ◦ γ A • B ◦ β A,B = ( µ A ⊗ B ⊗ H ) ◦ ( i A • B ⊗ (( i A • B ⊗ H ) ◦ ρ A • B )) ◦ β A,B = ( µ A ⊗ B ⊗ H ) ◦ ( i A • B ⊗ (( A ⊗ ρ B ) ◦ i A • B )) ◦ β A,B = ( µ A ⊗ γ B ) ◦ ( A ⊗ c B,A ⊗ B ) ◦ Γ − A ⊗ B ◦ ( i A • B ⊗ δ H ) = ((( A ⊗ ε H ) ◦ γ A ◦ γ − A ) ⊗ B ⊗ H ) ◦ ( A ⊗ c B,H ⊗ H ) ◦ ( i A • B ⊗ δ H )= i A • B ⊗ H and by the cocommutativity of H we have ( i A • B ⊗ i A • B ) ◦ β A,B ◦ γ A • B = Γ − A ⊗ B ◦ ( i A • B ⊗ δ H ) ◦ ( µ A • B ⊗ H ) ◦ ( A • B ⊗ ρ A • B )= Γ − A ⊗ B ◦ ( µ A ⊗ B ⊗ δ H ) ◦ ( i A • B ⊗ (( A ⊗ ρ B ) ◦ i A • B ))= Γ − A ⊗ B ◦ ( A ⊗ B ⊗ c H,H ) ◦ ( µ A ⊗ γ B ⊗ H ) ◦ ( A ⊗ c B,A ⊗ ρ B ) ◦ ( i A • B ⊗ i A • B )= ( A ⊗ c A,B ⊗ B ) ◦ ( γ − A ⊗ B ⊗ B ) ◦ ( A ⊗ c B,H ⊗ B ) ◦ ( µ A ⊗ B ⊗ ( c B,H ◦ ρ B )) ◦ ( A ⊗ c B,A ⊗ B ) ◦ ( i A • B ⊗ i A • B )= ( A ⊗ c A,B ⊗ B ) ◦ ( γ − A ⊗ B ⊗ B ) ◦ ( µ A ⊗ c B,H ⊗ B ) ◦ ( A ⊗ c B,A ⊗ H ⊗ B ) ◦ ( i A • B ⊗ (( ρ A ⊗ B ) ◦ i A • B ))= ((( A ⊗ c A,B ) ◦ (( γ − A ◦ γ A ) ⊗ B )) ⊗ B ) ◦ ( A ⊗ c B,A ⊗ B ) ◦ ( i A • B ⊗ i A • B )= i A • B ⊗ i A • B Taking into account that H is flat and that A • B is faithfully flat we obtain that β A,B is the inverse ofthe canonical morphism γ A • B .Now we assume that A and B are strong Galois H -objects. To prove that A • B is a strong Galois H -object we only need to show that f A • B : H → ( A • B ) e is a morphism of unital magmas. If f A and f B are morphisms of unital magmas, by the properties of i A • B and the naturality of c we have ( i A • B ⊗ i A • B ) ◦ f A • B ◦ η H = ( A ⊗ c A,B ⊗ B ) ◦ (( f A ◦ η H ) ⊗ ( f B ◦ η H ))= η A ⊗ η B ⊗ η A ⊗ η B = ( i A • B ⊗ i A • B ) ◦ η ( A • B ) e and ( i A • B ⊗ i A • B ) ◦ µ ( A • B ) e ◦ ( f A • B ⊗ f A • B ) = ( A ⊗ c A,B ⊗ B ) ◦ (( µ A e ◦ ( f A ⊗ f A )) ⊗ ( µ B e ◦ ( f B ⊗ f B )) ◦ δ H ⊗ H = ( A ⊗ c A,B ⊗ B ) ◦ ( f A ⊗ f B ) ◦ δ H ◦ µ H = ( i A • B ⊗ i A • B ) ◦ f A • B ◦ µ H . Therefore, f A • B ◦ η H = η ( A • B ) e and µ ( A • B ) e ◦ ( f A • B ⊗ f A • B ) = f A • B ◦ µ H . (cid:3) Proposition 2.7.
Let H be a cocommutative Hopf quasigroup and A a Galois H -object. Then the right H -comodule magma A defined in Proposition 1.6 is a Galois H -object. Moreover, if A is strong so is A .Proof. To prove that A is a Galois H -object we only need to show that γ A is an isomorphism. We beginby proving the following identity: ( A ⊗ ( µ H ◦ c H,H ◦ ( λ H ⊗ H ))) ◦ ( ρ A ⊗ H ) ◦ γ A = γ A ◦ c A,A . (13)Indeed: ( A ⊗ ( µ H ◦ c H,H ◦ ( λ H ⊗ H ))) ◦ ( ρ A ⊗ H ) ◦ γ A = ( µ A ⊗ ( µ H ◦ ( H ⊗ µ H ) ◦ ( c H,H ⊗ H ) ◦ ( H ⊗ c H,H ) ◦ ( c H,H ⊗ H ) ◦ ( λ H ⊗ λ H ⊗ H ) ◦ ( H ⊗ δ H ))) ◦ ( A ⊗ c H,A ⊗ H ) ◦ ( ρ A ⊗ ρ A )= ( µ A ⊗ ( µ H ◦ ( λ H ⊗ µ H ) ◦ ( δ H ⊗ H ) ◦ c H,H ◦ ( λ H ⊗ H ))) ◦ ( A ⊗ c H,A ⊗ H ) ◦ ( ρ A ⊗ ρ A )= ( µ A ⊗ H ) ◦ ( A ⊗ c H,A ) ◦ ( ρ A ⊗ A )= γ A ◦ c A,A where the first equality follows by (b2) of Definition 1.3, (1) and the naturality of c , the second one bythe cocommutativity of H and the naturality of c , the third one by (a2-1) of Definition 1.1 and the lastone by the symmetry and naturality of c .Define the morphism γ ′ A : A ⊗ H → A ⊗ A by γ ′ A = c A,A ◦ γ − A ◦ ( A ⊗ ( µ H ◦ c H,H )) ◦ ( ρ A ⊗ H ) . (14)Then, by (13), the naturality of c , the cocommutativity of H and (a2-2) of Definition 1.1, we have thefollowing: γ A ◦ γ ′ A = γ A ◦ c A,A ◦ γ − A ◦ ( A ⊗ ( µ H ◦ c H,H )) ◦ ( ρ A ⊗ H )= ( A ⊗ ( µ H ◦ c H,H ◦ ( λ H ⊗ H ))) ◦ ( ρ A ⊗ ( µ H ◦ c H,H )) ◦ ( ρ A ⊗ H )= ( A ⊗ ( µ H ◦ ( µ H ⊗ H ) ◦ ( H ⊗ c H,H ) ◦ ( c H,H ⊗ H ) ◦ ( λ H ⊗ c H,H ) ◦ ( δ H ⊗ H ))) ◦ ( ρ A ⊗ H )= ( A ⊗ (( µ H ◦ ( µ H ⊗ H ) ◦ ( H ⊗ λ H ⊗ H ) ◦ ( H ⊗ δ H )) ◦ c H,H )) ◦ ( ρ A ⊗ H )= id A ⊗ H . Moreover, by a similar reasoning but using (a2-1) of Definition 1.1 instead of (a2-2) we obtain γ ′ A ◦ γ A = c A,A ◦ γ − A ◦ (( µ A ◦ c A,A ) ⊗ ( µ H ◦ ( H ⊗ µ H ) ◦ ( H ⊗ λ H ⊗ H ) ◦ ( δ H ⊗ H ) ◦ c H,H )) ◦ ( A ⊗ c H,A ⊗ H ) ◦ ( ρ A ⊗ ρ A )= c A,A ◦ γ − A ◦ γ A ◦ c A,A = id A ⊗ A . Therefore, γ A is an isomorphism and A a Galois H -object.Finally, it is easy to show that f A = c A,A ◦ f A . Then, if f A is a morphism of unital magmas, so is f A .Thus if A is strong, A is strong. (cid:3) Proposition 2.8.
Let H be a cocommutative flat Hopf quasigroup and A a Galois H -object. Then A • A isisomorphic to H as right H -comodules. Moreover, if A is strong, the previous isomorphism is a morphismof right H -comodule magmas.Proof. First note that, by Proposition 2.4, we know that (7) is an equalizer diagram and then so is ✲ ✲✲
H A ⊗ H ⊗ Hη A ⊗ H ρ A ⊗ HA ⊗ η H ⊗ HA ⊗ H because H is flat. For the morphism γ A ◦ i A • A : A • A : → A ⊗ H we have the following: ( ρ A ⊗ H ) ◦ γ A ◦ i A • A = ( µ A ⊗ H ⊗ λ H ) ◦ ( ρ A ⊗ (( A ⊗ δ H ) ◦ ρ A )) ◦ i A • A = ( µ A ⊗ H ⊗ λ H ) ◦ ( A ⊗ H ⊗ (( A ⊗ δ H ) ◦ ρ A )) ◦ ( A ⊗ ( c A,H ◦ ( A ⊗ λ H ) ◦ ρ A )) ◦ i A • A = ( µ A ⊗ µ H ⊗ λ H ) ◦ ( A ⊗ (( A ⊗ H ⊗ δ H ) ◦ ( A ⊗ c H,H ) ◦ ( A ⊗ (( H ⊗ λ H ) ◦ δ H )))) ◦ ( A ⊗ ρ A ) ◦ i A • A = ( µ A ⊗ (( µ H ◦ (( λ H ⊗ H ) ◦ δ H ) ⊗ λ H ) ◦ δ H )) ◦ ( A ⊗ ρ A ) ◦ i A • A = ( A ⊗ η H ⊗ H ) ◦ γ A ◦ i A • A where the first equality follows by the naturality of c and (b2) of Definition 1.3, the second one because ρ A ⊗ A ◦ i A • A = ρ A ⊗ A ◦ i A • A , the third one relies on the symmetry and the naturality of c , the fourth onefollows by (1) and the last one by (3).Therefore, there exists an unique morphism h A : A • A → H such that ( η A ⊗ H ) ◦ h A = γ A ◦ i A • A . (15)The morphism h A is a right comodule morphism because by the cocommutativity of H , (1) and thecomodule properties of A , we have η A ⊗ (( h ⊗ H ) ◦ ρ A • A )= (( γ A ◦ i A • A ) ⊗ H ) ◦ ρ A • A = (( µ A ◦ c A,A ) ⊗ (( λ H ⊗ λ H ) ◦ δ H )) ◦ ( A ⊗ ρ A ) ◦ i A • A = (( µ A ◦ c A,A ) ⊗ (( λ H ⊗ λ H ) ◦ c H,H ◦ δ H )) ◦ ( A ⊗ ρ A ) ◦ i A • A = (( µ A ◦ c A,A ) ⊗ ( δ H ◦ λ H )) ◦ ( A ⊗ ρ A ) ◦ i A • A = ( A ⊗ δ H ) ◦ γ A ◦ i A • A = η A ⊗ ( δ H ◦ h ) .and using that η A ⊗ H ⊗ H is an equalizer morphism we obtain ( h ⊗ H ) ◦ ρ A • A = δ H ◦ h. On the other hand, for f A : H → A ⊗ A we have the following ( γ A ⊗ H ) ◦ ρ A ⊗ A ◦ f A = ( µ A ⊗ λ H ⊗ H ) ◦ ( c A,A ⊗ H ⊗ H ) ◦ ( A ⊗ ρ A ⊗ H ) ◦ ( A ⊗ c H,A ) ◦ ( ρ A ⊗ A ) ◦ c A,A ◦ f A = ( µ A ⊗ λ H ⊗ H ) ◦ ( A ⊗ c H,A ⊗ H ) ◦ ( ρ A ⊗ ρ A ) ◦ f A = ( µ A ⊗ (( λ H ⊗ H ) ◦ γ − H ◦ γ H )) ◦ ( A ⊗ c H,A ⊗ H ) ◦ ( ρ A ⊗ ρ A ) ◦ f A = ( A ⊗ ( λ H ◦ µ H ) ⊗ H ) ◦ ( ρ A ⊗ (( λ H ⊗ H ) ◦ δ H )) ◦ γ A ◦ f A = η A ⊗ ((( λ H ◦ λ H ) ⊗ H ) ◦ δ H )= η A ⊗ δ H where the first equality follows because f A = c A,A ◦ f A , the second one by the symmetry and thenaturality of c . In the third one we used that H is a Galois H -object and the fourth and the sixthones are a consequence of (b1) of Definition 1.3. Finally, in the fifth one we applied that A is a Galois H -object, and the last one relies on the cocommutativity of H . Also ( γ A ⊗ H ) ◦ ρ A ⊗ A ◦ f A = (( µ A ◦ c A,A ) ⊗ (( λ H ⊗ λ H ) ◦ δ H )) ◦ ( A ⊗ ρ A ) ◦ c A,A ◦ f A = ( µ A ⊗ (( λ H ⊗ λ H ) ◦ δ H )) ◦ ( A ⊗ c H,A ) ◦ ( ρ A ⊗ A ) ◦ f A = ( µ A ⊗ (( ε H ⊗ (( λ H ⊗ λ H ) ◦ δ H )) ◦ c H,H ◦ (( µ H ◦ ( µ H ⊗ λ H ) ◦ ( H ⊗ δ H )) ⊗ H ))) ◦ ( A ⊗ c H,A ⊗ δ H ) ◦ ( ρ A ⊗ ρ A ) ◦ f A = ( A ⊗ (( λ H ⊗ λ H ) ◦ δ H )) ◦ ( A ⊗ µ H ) ◦ (( ρ A ◦ η A ) ⊗ λ H )= η A ⊗ δ H , where the first equality follows by (b1) of Definition 1.3 and the comodule properties of A , the secondone by the naturality of c , the third one by (a2-2) of Definition 1.1 and the counit properties, the fourthone by (b2) of Definition 1.3 and in the last one we used that A is a Galois H -object, (b1) of Definition1.3 and the cocommutativity of H .Then, ρ A ⊗ A ◦ f A = ρ A ⊗ A ◦ f A and, as a consequence, there exists a unique morphism h ′ A : H → A • A such that i A • A ◦ h ′ A = f A . (16)Therefore, by (15) and (16) we have i A • A ◦ h ′ A ◦ h A = f A ◦ h A = γ − A ◦ ( η A ⊗ H ) ◦ h A = γ − A ◦ γ A ◦ i A • A = i A • A and ( η A ⊗ H ) ◦ h A ◦ h ′ A = γ A ◦ i A • A ◦ h ′ A = γ A ◦ f A = γ A ◦ γ − A ◦ ( η A ⊗ H ) = ( η A ⊗ H ) . Then, h ′ A ◦ h A = id A • A and h A ◦ h ′ A = id H and h is an isomorphism.Finally, assume that A is strong. By (16) and the equality f A = c A,A ◦ f A we obtain that h ′ A is amorphism of unital magmas. Then h A is a morphism of unital magmas and the proof is finished. (cid:3) Remark 2.9.
Note that, in the Hopf algebra setting, for any Galois H -object A , the morphism h A obtained in the previous proposition is a morphism of monoids because this property can be deducedfrom the associativity of the product defined in A . In the Hopf quasigroup world this proof does notwork because A is a magma. Theorem 2.10.
Let H be a cocommutative faithfully flat Hopf quasigroup. The set of isomorphismclasses of Galois H -objects is a commutative monoid. Moreover, the set of isomorphism classes of strongGalois H -objects is a commutative group.Proof. Let
Gal C ( H ) be the set of isomorphism classes of Galois H -objects. For a Galois H -object A wedenote its class in Gal C ( H ) by [ A ] . By by Propositions 2.6 and 1.8, the product [ A ] . [ B ] = [ A • B ] (17)is well-defined. By Propositions 1.10, 1.9 and 1.11 we obtain that Gal C ( H ) is a commutative monoidwith unit [ H ] . If we denote by
Gal s C ( H ) the set of isomorphism classes of strong Galois H -objects, with the productdefined in (17) for Galois H -objects, Gal s C ( H ) is a commutative group because by (ii) of Proposition 2.6the product of strong Galois H -objects is a strong Galois H -object, by Example 2.2 we know that H isa strong Galois H -object and by Propositions 2.7 and 2.8, the inverse of [ A ] in Gal s C ( H ) is [ A ] . (cid:3) Definition 2.11.
Let H be a cocommutative faithfully flat Hopf quasigroup. If A is a (strong) Galois H -object, we will say that A has a normal basis if ( A, ρ A ) is isomorphic to ( H, δ H ) as right H -comodules.We denote by n A the H -comodule isomorphism between A and H .Obviously, N C ( H ) , the set of isomorphism classes of Galois H -objects with normal basis, is a submonoidof Gal C ( H ) because H = ( H, δ H ) is a Galois H -object with normal basis and if A , B are Galois H -objectswith normal basis and associated isomorphisms n A , n B respectively, then A • B is a Galois H -object withnormal basis and associated H -comodule isomorphism n A • B = r H ◦ n A • n B where n A • n B is defined asin Proposition 1.8 and r H is the isomorphism defined in Proposition 1.11. Moreover, for a strong Galois H -object with normal basis A , with associated isomorphism n A , we have that A = ( A, ρ A ) is also a strongGalois H -object with normal basis, where n A = λ H ◦ n A , and then, if we denote by N s C ( H ) the set ofisomorphism classes of strong Galois H -objects with normal basis, N s C ( H ) is a subgroup of Gal s C ( H ) .Note that, if H is a Hopf algebra we have that Gal s C ( H ) = Gal C ( H ) and N s C ( H ) = N C ( H ) . Therefore,in the associative setting we recover the classical group of Galois H -objects. Remark 2.12.
In this remark we use some classical results of algebraic K -theory (see [3] for the details).Let G ( C , H ) and G s ( C , H ) be the categories of Galois H -objects and strong Galois H -objects, respectively.Then, by Proposition 1.13 these categories are symmetric monoidal and then they are categories withproduct. The Grothendieck group of G ( C , H ) is the abelian group generated by the isomorphisms classesof objects A of G ( C , H ) module the relations [ A • B ] = [ A ] . [ B ] . This group will be denoted by K G ( C , H ) and, by the general theory of Grothendieck groups, we know that for A , B in G ( C , H ) , [ A ] = [ B ] in K G ( C , H ) if and only if there exists a D in G ( C , H ) such that A • D is isomorphic in G ( C , H ) to D • D . The unit of K G ( C , H ) is [ H ] . In a similar way we can define K G s ( C , H ) , but in this case K G s ( C , H ) = Gal s C ( H ) because the set of isomorphism classes of objects of G s ( C , H ) is a group.The inclusion functor i : G s ( C , H ) → G ( C , H ) is a product preserving functor and then we have a groupmorphism K i : Gal s C ( H ) → K G ( C , H ) . If [ A ] ∈ Ker ( K i ) we have that [ A ] = [ H ] in K G ( C , H ) . Thenthere exists D in G ( C , H ) such that A • D ∼ = H • D ∼ = D in G ( C , H ) . As a consequence A • D • D ∼ = D • D in G ( C , H ) . Then, By Proposition 2.8, A ∼ = H as right H -comodules. Therefore A is a strong Galois H -object with normal basis and Ker ( K i ) is a subgroup of N s C ( H ) .The full subcategory H = { H } of G s ( C , H ) is cofinal because, for all A in G s ( C , H ) , A • A ∼ = H asright H -comodule magmas. Therefore, the Whitehead group of G s ( C , H ) is isomorphic to the Whiteheadgroup of H . Therefore, K G s ( C , H ) ∼ = Aut G s ( C ,H ) ( H ) . The group
Aut G s ( C ,H ) ( H ) admits a good explanation in terms of grouplike elements of a suitable Hopfquasigroup if H is finite, that is, if there exists an object H ∗ in C and an adjunction H ⊗ − ⊣ H ∗ ⊗ − .For this adjunction we will denote with a H : id C → H ∗ ⊗ H ⊗ − and b H : H ⊗ H ∗ ⊗ − → id C the unitand the counit respectively. The object H ∗ will be called the dual of H .A Hopf coquasigroup D in C is a monoid ( D, η D , µ D ) and a counital comagma ( D, ε D , δ D ) such thatthe following axioms hold:(d1) ε D and δ D are morphisms of monoids.(d2) There exists λ D : D → D in C (called the antipode of D ) such that:(d2-1) ( µ D ⊗ D ) ◦ ( λ D ⊗ δ D ) ◦ δ D = η D ⊗ D = ( µ D ⊗ D ) ◦ ( D ⊗ (( λ D ⊗ D ) ◦ δ D )) ◦ δ D . (d2-2) ( D ⊗ µ D ) ◦ ( δ D ⊗ λ D ) ◦ δ D = D ⊗ η D = ( D ⊗ µ D ) ◦ ((( D ⊗ λ D ) ◦ δ D ) ⊗ D ) ◦ δ D . As in the case of quasigroups, the antipode is unique, antimultiplicative, anticomultiplicative, leavesthe unit and the counit invariable and satisfies (3).If D is a Hopf coquasigroup we define G ( D ) as the set of morphisms h : K → D such that δ D ◦ h = h ⊗ h and ε D ◦ h = id K . If D is commutative, G ( D ) with the convolution h ∗ g = µ D ◦ ( h ⊗ g ) is a commutativegroup, called the group of grouplike morphisms of D . Note that the unit element of G ( D ) is η D and theinverse of h ∈ G ( D ) is h − = λ D ◦ h .It is easy to show that, if H is a finite cocommutative Hopf quasigroup, its dual H ∗ is a commutativefinite Hopf coquasigroup where: η H ∗ = ( H ∗ ⊗ ε H ) ◦ a H , µ H ∗ = ( H ∗ ⊗ b H ) ◦ ( H ∗ ⊗ H ⊗ b H ⊗ H ∗ ) ◦ ( H ∗ ⊗ δ H ⊗ H ∗ ⊗ H ∗ ) ◦ ( a H ⊗ H ∗ ⊗ H ∗ )) ,ε H ∗ = b H ◦ ( η H ⊗ H ∗ ) , δ H ∗ = ( H ∗ ⊗ H ∗ ⊗ ( b H ◦ ( µ H ⊗ H ∗ ))) ◦ ( H ∗ ⊗ a H ⊗ H ⊗ H ∗ ) ◦ ( a H ⊗ H ∗ )) and the antipode is ( H ∗ ⊗ b H ) ◦ ( H ∗ ⊗ λ H ⊗ H ∗ ) ◦ ( a H ⊗ H ∗ ) .The groups G ( H ∗ ) and Aut G s ( C ,H ) ( H ) are isomorphic. The proof is equal to the one given in Propo-sition 3.7 of [9]. If α ∈ Aut G s ( C ,H ) ( H ) , the morphism z α = ( H ∗ ⊗ ( ε H ◦ α )) ◦ a H is in G ( H ∗ ) . Then,we define the map Aut G s ( C ,H ) ( H ) → G ( H ∗ ) by z ( α ) = z α . On the other hand, if h ∈ G ( H ∗ ) , then x h = ( H ⊗ b H ) ◦ ( δ H ⊗ h ) : H → H is a morphism of Galois H -objects and then, by Remark 1.4, it is anisomorphism, that is x h ∈ Aut G s ( C ,H ) ( H ) . The map x : G ( H ∗ ) → Aut G s ( C ,H ) ( H ) defined by x ( h ) = x h isthe inverse of z . Therefore, K G s ( C , H ) ∼ = G ( H ∗ ) . Finally, N s ( C , H ) is the subcategory of G s ( C , H ) whose objects are the strong Galois H -objects withnormal basis, note that H = { H } it is also cofinal in N s ( C , H ) and then K N s ( C , H ) ∼ = G ( H ∗ ) . Invertible comodules with geometric normal basis
This section is devoted to study the connections between Galois H -objects and invertible comoduleswith geometric normal basis. First of all, we introduce the notion of invertible comodule with geometricnormal basis which is a generalization to the non associative setting of the one defined by Caenepeel in[6]. Definition 3.1.
Let H be a cocommutative faithfully flat Hopf quasigroup. A right H -comodule M =( M, ρ M ) is called invertible with geometric normal basis if there exists a faithfully flat unital magma S and an isomorphism h M : S ⊗ M → S ⊗ H of right H -comodules such that h M is almost lineal, that is h M = ( µ S ⊗ H ) ◦ ( S ⊗ ( h M ◦ ( η S ⊗ M ))) (18)A morphism between two invertible right H -comodules with normal basis is a morphism of right H -comodules.Note that, if S is a monoid, h M is a morphism of left S -modules, for ϕ S ⊗ M = µ S ⊗ M and ϕ S ⊗ H = µ S ⊗ H , if and only if (18) holds. Then in the Hopf algebra setting this definition is the one introducedby Caenepeel in [6]. Example 3.2.
Let H be a cocommutative faithfully flat Hopf quasigroup and let A = ( A, ρ A ) be aGalois H -object. Then A = ( A, ρ A ) is an invertible right H -comodule with geometric normal basisbecause h A = γ A is an isomorphism of right H -comodules and trivially γ A is almost lineal. In particular, H = ( H, δ H ) is an example of invertible right H -comodule with geometric normal basis. Proposition 3.3.
Let H be a cocommutative faithfully flat Hopf quasigroup and M , N be invertible right H -comodules with geometric normal basis. Then the right H -comodule M • N = ( M • N, ρ M • N ) , where M • N and ρ M • N are defined as in Proposition 1.7, is a right H -comodule with geometric normal basis. Proof.
Let S , R and h M , h N be the faithfully flat unital magmas and the isomorphisms of right H -comodules associated to M and N respectively. Then T = S ⊗ R is faithfully flat. On the other hand, ✲ ✲✲ T ⊗ M • N T ⊗ M ⊗ N T ⊗ M ⊗ N ⊗ H T ⊗ i M • N T ⊗ ρ M ⊗ N T ⊗ ρ M ⊗ N and ✲ ✲✲ T ⊗ H T ⊗ H ⊗ H T ⊗ H ⊗ H ⊗ H T ⊗ δ H T ⊗ ρ H ⊗ H T ⊗ ρ H ⊗ H are equalizer diagrams and for the morphism g M ⊗ N = ( S ⊗ c R,H ⊗ H ) ◦ ( h M ⊗ h N ) ◦ ( S ⊗ c R,M ⊗ N ) : S ⊗ R ⊗ M ⊗ N → S ⊗ R ⊗ H ⊗ H we have that ( S ⊗ R ⊗ ρ H ⊗ H ) ◦ g M ⊗ N ◦ ( S ⊗ R ⊗ i M • N )= ( S ⊗ c H,R ⊗ c H,H ) ◦ ( S ⊗ H ⊗ c H,R ⊗ H ) ◦ ((( S ⊗ δ H ) ◦ h M ) ⊗ h N ) ◦ ( S ⊗ c R,M ⊗ N ) ◦ ( S ⊗ R ⊗ i M • N )= ( S ⊗ c H,R ⊗ c H,H ) ◦ ( S ⊗ H ⊗ c H,R ⊗ H ) ◦ ((( h M ⊗ H ) ◦ ( S ⊗ ρ M )) ⊗ h N ) ◦ ( S ⊗ c R,M ⊗ N ) ◦ ( S ⊗ R ⊗ i M • N )= ((( S ⊗ c H,R ⊗ H ) ◦ ( h M ⊗ h N )) ⊗ H ) ◦ ( S ⊗ c R,M ⊗ N ⊗ H ) ◦ ( S ⊗ R ⊗ (( M ⊗ c H,N ) ◦ ( ρ M ⊗ N ) ◦ i M • N ))= ((( S ⊗ c H,R ⊗ H ) ◦ ( h M ⊗ h N )) ⊗ H ) ◦ ( S ⊗ c M,R ⊗ (( M ⊗ ρ N ) ◦ i M • N ))= ( S ⊗ R ⊗ ρ H ⊗ H ) ◦ g M ⊗ N ◦ ( S ⊗ R ⊗ i M • N ) , where the first and the third equalities follow by the naturality of c , the second and the fifth ones by thecomodule morphism condition for h M and h N respectively and finally the fourth one by the propertiesof i M • N .Therefore, there exists a unique morphism h M • N : T ⊗ M • N → T ⊗ H such that ( T ⊗ δ H ) ◦ h M • N = g M ⊗ N ◦ ( T ⊗ i M • N ) . (19)Moreover, if we define the morphism g ′ M ⊗ N = ( S ⊗ c M,R ⊗ N ) ◦ ( h − M ⊗ h − N ) ◦ ( S ⊗ c R,H ⊗ H ) : S ⊗ R ⊗ H ⊗ H → S ⊗ R ⊗ M ⊗ N by the naturality of c , the comodule morphism condition for h − M and h − N and the cocommutativity of H ,the following equalities hold ( S ⊗ R ⊗ ρ M ⊗ N ) ◦ g ′ M ⊗ N ◦ ( S ⊗ R ⊗ δ H )= ( S ⊗ c M,R ⊗ c H,N ) ◦ ( S ⊗ M ⊗ c H,R ⊗ N ) ◦ ((( S ⊗ ρ M ) ◦ h − M ) ⊗ h − N ) ◦ ( S ⊗ c H,R ⊗ H ) ◦ ( S ⊗ R ⊗ δ H )= ( S ⊗ c M,R ⊗ c H,N ) ◦ ( S ⊗ M ⊗ c H,R ⊗ N ) ◦ ((( h − M ⊗ H ) ◦ ( S ⊗ δ H )) ⊗ h − N ) ◦ ( S ⊗ c H,R ⊗ H ) ◦ ( S ⊗ R ⊗ δ H )= ( g ′ M ⊗ N ⊗ H ) ◦ ( S ⊗ R ⊗ (( H ⊗ δ H ) ◦ δ H ))= ( S ⊗ R ⊗ ρ M ⊗ N ) ◦ g ′ M ⊗ N ◦ ( S ⊗ R ⊗ δ H ) As a consequence, there exists a unique morphism h ′ M • N : T ⊗ H → T ⊗ M • N such that ( T ⊗ i M • N ) ◦ h ′ M • N = g ′ M ⊗ N ◦ ( T ⊗ δ H ) . (20)Thus, by (19) and (20) h M • N ◦ h ′ M • N = ( T ⊗ (( ε H ⊗ H ) ◦ δ H )) ◦ h M • N ◦ h ′ M • N = ( T ⊗ ε H ⊗ H ) ◦ g M ⊗ N ◦ ( T ⊗ i M • N ) ◦ h ′ M • N = ( T ⊗ ε H ⊗ H ) ◦ g M ⊗ N ◦ g ′ M ⊗ N ◦ ( T ⊗ δ H ) = id T ⊗ H and ( T ⊗ i M • N ) ◦ h ′ M • N ◦ h M • N = g ′ M ⊗ N ◦ ( T ⊗ δ H ) ◦ h M • N = g ′ M ⊗ N ◦ g M ⊗ N ◦ ( T ⊗ i M • N ) = T ⊗ i M • N and then h M • N is an isomorphism with inverse h − M • N = h ′ M • N .The morphism h M • N is a morphism of right H -comodules because ( h M • N ⊗ H ) ◦ ( T ⊗ ρ M • N )= ( T ⊗ (( H ⊗ ε H ) ◦ δ H ) ⊗ H ) ◦ ( h M • N ⊗ H ) ◦ ( T ⊗ ρ M • N )= ( T ⊗ H ⊗ ε H ⊗ H ) ◦ ( g M ⊗ N ⊗ H ) ◦ ( T ⊗ (( i M • N ⊗ H ) ◦ ρ M • N ))= ( T ⊗ H ⊗ ε H ⊗ H ) ◦ ( g M ⊗ N ⊗ H ) ◦ ( T ⊗ (( M ⊗ ρ N ) ◦ i M • N ))= ( T ⊗ H ⊗ (( H ⊗ ε H ) ◦ δ H )) ◦ g M ⊗ N ◦ ( T ⊗ i M • N )= ( T ⊗ δ H ) ◦ h M • N where the first equality follows by the counit property, the second and the last ones by (19), the thirdone the properties of ρ M • N and the fourth one by the comodule condition for h N .Finally, we will prove that h M • N is almost lineal. Indeed: ( µ T ⊗ H ) ◦ ( T ⊗ ( h M • N ◦ ( η T ⊗ M • N )))= ( µ T ⊗ (( ε H ⊗ H ) ◦ δ H )) ◦ ( T ⊗ ( h M • N ◦ ( η T ⊗ M • N )))= ( µ S ⊗ R ⊗ ε H ⊗ H ) ◦ ( S ⊗ R ⊗ ( g M ⊗ N ◦ ( η S ⊗ η R ⊗ i M • N )))= ( S ⊗ ε H ⊗ R ⊗ H ) ◦ ((( µ S ⊗ H ) ◦ ( S ⊗ ( h M ◦ ( η S ⊗ M )))) ⊗ (( µ R ⊗ H ) ◦ ( R ⊗ ( h N ◦ ( η R ⊗ N ))))) ◦ ( S ⊗ c R,M ⊗ N ) ◦ ( S ⊗ R ⊗ i M • N )= ( T ⊗ ε H ⊗ H ) ◦ g M ⊗ N ◦ ( T ⊗ i M • N )= ( T ⊗ (( ε H ⊗ H ) ◦ δ H )) ◦ h M • N = h M • N In the last equalities, the first and the sixth ones follow by the properties of the counit, the second andthe fifth ones by (19), the third one is a consequence of the naturality of c and the fourth one relies onthe almost lineal condition for h M and h N . (cid:3) As a direct consequence of this proposition we have the following theorem.
Theorem 3.4.
Let H be a cocommutative faithfully flat Hopf quasigroup. If we denote by P gnb ( K, H ) the category whose objects are the invertible right H -comodules with geometric normal basis and whosemorphisms are the morphisms of right H -comodules between them, P gnb ( K, H ) with the product definedin the previous proposition is a symmetric monoidal category where the unit object is H and the symmetryisomorphisms, the left, right an associative constraints are defined as in Proposition 1.13. Moreover, theset of isomorphism classes in P gnb ( K, H ) is a monoid that we will denote by P ic gnb ( K, H ) . Remark 3.5.
There is a monoid morphism ω : Gal C ( H ) → P gnb ( K, H ) defined by ω ([ A ]) = [ A ] . If ω ([ A ]) = [ H ] we have that A ∼ = H as right H -comodules. Then [ A ] ∈ N C ( H ) . Also, if [ A ] ∈ Gal s C ( H ) and ω ([ A ]) = [ H ] , [ A ] ∈ N s C ( H ) . Acknowledgements
The authors were supported by Ministerio de Economía y Competitividad (Spain) and by Feder founds.Project MTM2013-43687-P: Homología, homotopía e invariantes categóricos en grupos y álgebras noasociativas.
References [1] J.N. Alonso Álvarez, J.M. Fernández Vilaboa, R. González Rodríguez and C. Soneira Calvo, Projections and Yetter-Drinfeld modules over Hopf (co)quasigroups,
J. of Algebra (2015), 153-199.[2] J.N. Alonso Álvarez, J.M. Fernández Vilaboa, R. González Rodríguez and C. Soneira Calvo, Cleft comodules over Hopfquasigroups, Commun. Contemp. Math. 17, N. 6, 1550007 (20 pages) (2015).[3] H. Bass, Topics in algebraic K-theory, Tata Institute of Fundamental Research, Bombay, 1969.[4] M. Beattie, A direct sum decomposition for the Brauer group of H -module algebras, J. Algebra (1976), 686-693.[5] R.H. Bruck, Contributions to the theory of loops, Trans. Amer. Math. Soc. (1946), 245-354.[6] S. Caenepeel, A variation of the Sweedler’s complex and the group of Galois objects of an infinite Hopf algebra, Comm.in Algebra (1996), 2991-3015. [7] S. U. Chase, M.E. Sweedler, Hopf algebras and Galois theory, Lecture Notes in Mathematics, Springer-Verlag,Berlin-New York 1969.[8] J.M. Fernández Vilaboa, Grupos de Brauer y de Galois de un álgebra de Hopf de una categoría cerrada, Álxebra, Comm.in Algebra (1996), 3413-3435.[10] M.P. López López, Objetos de Galois sobre un álgebra de Hopf finita, Álxebra, J. of Algebra (2010), 3067-3110.[12] J.M. Pérez-Izquierdo, I.P. Shestakov, An envelope for Malcev algebras,
J. Algebra (2004), 379–393.[13] J.M. Pérez-Izquierdo, Algebras, hyperalgebras, nonassociative bialgebras and loops,
Adv. in Math. (2007), 834-876.[14] E. Villanueva, M.P. López, The antipode and the (co)invariants of a finite Hopf (co)quasigroup,
Appl. Cat. Struct.21