Split extensions and actions of bialgebras and Hopf algebras
aa r X i v : . [ m a t h . C T ] A ug SPLIT EXTENSIONS AND ACTIONS OF BIALGEBRAS AND HOPF ALGEBRAS
FLORENCE STERCK
Abstract.
We introduce a notion of split extension of (non-associative) bialgebras. We show that this defi-nition is equivalent to the notion of action of (non-associative) bialgebras. We particularize this equivalenceto (non-associative) Hopf algebras by defining split extensions of (non-associative) Hopf algebras and provingthat they are equivalent to actions of (non-associative) Hopf algebras. Moreover, we prove the validity of theSplit Short Five Lemma for these kinds of split extensions, and we examine some examples.
Introduction
In the category of groups, split extensions have a lot of interesting properties. A split extension of groupsis a diagram of the form(0.1)
G HK k fs where f · s = 1 H and k is the kernel of f . One of the interesting properties of split extensions of groups is thefact that the category of split extensions is equivalent to the category of group actions. An action of a group(G,1) on a group (X,1) is a map ρ : G × X → X : ( g, x ) → g x such that for any g, g ′ ∈ G and x, x ′ ∈ X thefollowing identities hold gg ′ x = g ( g ′ x ) , x = x, g ( xx ′ ) = g x g x ′ . In any semi-abelian category, [16], there is an equivalence between (internal) actions and split extensions([6] and [5]). Unfortunately, this property does not hold in more general categories: for example in the cate-gory of monoids there is no such an equivalence between split extensions and monoid actions. Nevertheless,it was proved in [7], that there exists a restricted equivalence for the “Schreier extensions”. The terminology“Schreier extension” came from a paper of Patchkoria [23], who worked on a notion of a Schreier internalcategory in the category of monoids and proved that the category of Schreier internal categories in the cat-egory of monoids is equivalent to the category of crossed semimodules. A further step of generalization wasdone in a recent paper on split extensions of unitary magmas, where a suitable notion of split extension ofmagmas were introduced and shown to correspond to actions [12].By a result of [14] (see also [13]) saying that the category of cocommutative Hopf K -algebras is semi-abelian, where K is a field, it is known that there is an equivalence between the actions of cocommutativeHopf algebras and the split extensions of cocommutative Hopf algebras (see [22], [14], [4], [25]). Moreover,when we consider cocommutative (non-)associative bialgebras over a symmetric monoidal category C , theycan be seen as internal monoids (or magmas) in the category of cocommutative coalgebras over C , and thenwe can apply the results of [12].But what happens in the non-cocommutative case? This paper will give an answer to this question. Wedefine different split extensions that are equivalent to the actions of non-associative bialgebras, bialgebras,non-associative Hopf algebras and Hopf algebras in any symmetric monoidal category. This large contextprovides a wide variety of possible applications.The first part of this paper is devoted to the preliminaries, where we recall the definition of bialgebras andHopf algebras in a symmetric monoidal category. Key words and phrases. (non-associative) bialgebras, (non-associative) Hopf algebras, actions, split extensions, Split ShortFive Lemma.
In the second part, we define split extensions of non-associative bialgebras and show that they form acategory that is equivalent to the category of actions of non-associative bialgebras. In particular, in the caseof cocommutative bialgebras, this equivalence gives us the results in Section 4.6 in [12]. We end this sectionby a proof of a variation of the Split Short Five Lemma in the category of non-associative bialgebras, whenwe restrict it to the split extensions that we have introduced.The last part describes the case of Hopf algebras and provides some examples of split extensions ofassociative Hopf algebras. In particular, we investigate the case where a split epimorphism of Hopf algebras
G H αe satisfies the additional condition HKer ( α ) = LKer ( α ), which is a condition given in [2]in order to have an exact extension of Hopf algebras. Acknowledgements
The author would like to warmly thank her supervisors Marino Gran and Joost Vercruysse for all theadvice in the realization of this paper. Many thanks also to George Janelidze for the suggestion to exploresplit extensions and actions beyond the vector spaces case, in the monoidal categories context. It led to areal improvement of this paper. The author also would like to thank Manuela Sobral for the invitation to theUniversity of Coimbra and for the useful discussions. The author’s research is supported by a FRIA doctoralgrant of the
Communaut´e fran¸caise de Belgique .1.
Preliminaries
We recall [17] that a monoidal category is given by a triple ( C , ⊗ , I ) where C is a category, ⊗ : C × C → C a bifunctor and I is the identity element (we omit to explicit the three natural isomorphisms, the associator,the right unit and the left unit).A braided monoidal category is a 4-tuple ( C , ⊗ , I, σ ) where ( C , ⊗ , I ) is a monoidal category and σ is a braiding . A braiding consists of a family of natural isomorphisms σ X,Y : X ⊗ Y → Y ⊗ X satisfying σ X ⊗ Y,Z = ( σ X,Z ⊗ Y ) · (1 X ⊗ σ Y,Z ) ,σ X,Y ⊗ Z = (1 Y ⊗ σ X,Z ) · ( σ X,Y ⊗ Z ) . A braided monoidal category is called symmetric when σ − Y,X = σ X,Y . In this paper, we omit to denote the indexes of the braiding when it does not bring to any confusion.An algebra in a symmetric monoidal category ( C , ⊗ , I, σ ) is given by an object A ∈ C endowed with amorphism m : A ⊗ A → A , called the multiplication, and a morphism u : I → A called the unit. The algebrasthat we consider in this paper are unital, i.e. the following condition holds(1.1) m · ( u A ⊗ A ) = 1 A = m · (1 A ⊗ u A ) ,A ⊗ A AA A. mu A ⊗ A A ⊗ u A We do not require any associativity condition on algebras. A morphism of algebras f : A → B is a morphismin C such that the following two diagrams commute A BB ⊗ BA ⊗ A mm ff ⊗ f AI B. fu B u A PLIT EXTENSIONS AND ACTIONS OF BIALGEBRAS AND HOPF ALGEBRAS 3
A coalgebra is the dual notion of the notion of an algebra. In other words, a coalgebra over ( C , ⊗ , I, σ ) is anobject C ∈ C with a comultiplication ∆ : C → C ⊗ C and a counit ǫ : C → I. From now on, the coalgebraswill always be counital and coassociative(1.2) ( ǫ ⊗ C ) · ∆ = 1 C = (1 C ⊗ ǫ ) · ∆(1.3) (∆ ⊗ C ) · ∆ = (1 C ⊗ ∆) · ∆as expressed by the commutativity of the following diagrams C ⊗ C CC C ∆ ǫ ⊗ C C ⊗ ǫ C ⊗ C C ⊗ C ⊗ C.C ⊗ CC C ⊗ ∆∆ ∆ ⊗ C ∆ Similarly, a morphism of coalgebras g : C → D is a morphism in C such that the following two diagramscommute D CC ⊗ CD ⊗ D ∆∆ gg ⊗ g DI C. gǫǫ
We also recall that a bialgebra is a 5-tuple (
B, m, u, ∆ , ǫ ) where ( B, m, u ) is an algebra, ( B, ∆ , ǫ ) is acoalgebra and ∆ , ǫ are algebra morphisms (which is equivalent to ask that m and u are coalgebra morphisms)i.e. the following conditions hold(1.4) ∆ · m = ( m ⊗ m ) · (1 B ⊗ σ ⊗ B ) · (∆ ⊗ ∆) ,B ⊗ B ⊗ B ⊗ B B ⊗ B ⊗ B ⊗ B B ⊗ BBB ⊗ B ∆∆ ⊗ ∆ 1 B ⊗ σ ⊗ B m ⊗ mm (1.5) ∆ · u B = u B ⊗ u B ,BI B ⊗ B ∆ u B ⊗ u B u B (1.6) ǫ · m = ǫ ⊗ ǫ, BB ⊗ B I ǫǫ ⊗ ǫm (1.7) ǫ · u B = 1 I .BI I ǫu B FLORENCE STERCK
Moreover, a morphism in C is a morphism of bialgebras if it is a morphism of algebras and coalgebras.A non-associative Hopf algebra is a 7-tuple ( A, m, u, ∆ , ǫ, S L , S R ) where ( A, m, u, ∆ , ǫ ) is a bialgebra and S L and S R are antihomomorphisms of coalgebras and algebras, called the left and the right antipode, suchthat the following diagram commutes(1.8) A ⊗ A S L ⊗ A / / A ⊗ S R / / A ⊗ A m ●●●●●●●●● A ǫ / / ∆ < < ①①①①①①①①① I u / / A. A morphism of Hopf algebras is a morphism of bialgebras preserving the antipodes. Note that in the caseof associative Hopf algebras, the antipode is unique ( S L = S = S R ) and then and S is automatically anantihomomorphism of coalgebras and algebras. Moreover, a bialgebra morphism between associative Hopfalgebras necessarily preserves the antipode. Examples 1.1. (1) In the symmetric monoidal category (
Set , × , { ⋆ } ) of sets where σ is the twist morphism(where σ ( x, y ) = ( y, x ) for any element x of a set X and any element y of a set Y ), every object has acoalgebra structure with ∆ being the diagonal and ǫ the morphism sending every element to the singleton.Hence, a non-associative bialgebra (or algebra) is an unital magma, an associative bialgebra (or algebra) isa monoid, an associative Hopf algebra is a group.(2) In the symmetric monoidal category ( Vect K , ⊗ , K ) of vector spaces over a field K where σ is the twistmorphism (defined by σ ( x ⊗ y ) = y ⊗ x for any x ⊗ y ∈ X ⊗ Y ), we recover the notion of K -algebra, K -coalgebra, K -bialgebra and Hopf K -algebra.(3) In [8], a symmetric monoidal category was introduced such that Hom-algebras, Hom-coalgebras andHom-Hopf algebras (see [21]) coincide with the algebras, coalgebras and Hopf algebras in this symmetricmonoidal category.(4) In [9], the authors showed that Turaev’s Hopf group-coalgebras (see [24]) are Hopf algebras in asymmetric monoidal category which they called Turaev category.(5) Associative and non-coassociative bialgebras and Hopf algebras in any symmetric monoidal category C can be seen as non-associative bialgebras and Hopf algebras in C op , the opposite category, which is still asymmetric monoidal category.(6) The coquasi-bialgebras and quasi-bialgebras are respectively examples of non-associative bialgebrasin Vect and in
Vect op , see [19] for an introduction about these structures. The coquasi-Hopf algebras havedifferent antipode conditions, but under some specific assumptions it is possible to see them as non-associativeHopf algebras. An example which is both a non-associative Hopf algebras, as we defined, and a coquasi-Hopfalgebras is the structure of octonions see [1].These examples give us a glimpse of all the frameworks and cases in which the results of this paper canbe applied. Convention:
For the monoidal product of n copies A ⊗ · · · ⊗ A , the notation A n will be used. The sameconvention will be used for the morphisms, for example we denote α ⊗ α : A ⊗ A → B ⊗ B by α : A → B .For the sake of simplicity, “bialgebras” will mean “non-associative bialgebras” (unless the associativity isexplicitly mentioned). 2. Split extensions of non-associative bialgebras
Definition 2.1.
Let X and B be bialgebras in a symmetric monoidal category ( C , ⊗ , I, σ ). An action is amorphism in C , ⊲ : B ⊗ X → X , such that(2.1) ⊲ · ( u B ⊗ X ) = 1 X ,B ⊗ XX X ⊲u B ⊗ X PLIT EXTENSIONS AND ACTIONS OF BIALGEBRAS AND HOPF ALGEBRAS 5 (2.2) ⊲ · (1 B ⊗ u X ) = u X · ǫ B ,B ⊗ XB X ⊲u X · ǫ B B ⊗ u X (2.3) (1 B ⊗ ⊲ ) · (∆ ⊗ X ) = (1 B ⊗ ⊲ ) · ( σ ⊗ X ) · (∆ ⊗ X ) ,B ⊗ B ⊗ X B ⊗ B ⊗ X B ⊗ XB ⊗ B ⊗ XB ⊗ X B ⊗ ⊲ ∆ ⊗ X σ ⊗ X B ⊗ ⊲ ∆ ⊗ X (2.4) ǫ X · ⊲ = ǫ B ⊗ ǫ X ,XB ⊗ X I ǫ X ǫ B ⊗ ǫ X ⊲ (2.5) ∆ · ⊲ = ( ⊲ ⊗ ⊲ ) · (1 B ⊗ σ ⊗ X ) · (∆ ⊗ ∆) .B ⊗ B ⊗ X ⊗ X B ⊗ X ⊗ B ⊗ X X ⊗ XXB ⊗ X ∆∆ ⊗ ∆ 1 B ⊗ σ ⊗ X ⊲ ⊗ ⊲⊲ Let us note that the last two axioms mean that ⊲ is a morphism of coalgebras. The axiom (2.3) is inspiredby the condition (1) that Majid used in [18] to define a Hopf algebra crossed modules. It is also what weneed to define a bialgebra via the semi-direct product construction (see Theorem 3.3 in [20] and [22] forthe construction of the semi-direct product also called smash-product). In particular, we are interested indiagrams of the form(2.6) X ⋊ B BX π i π i , where i = 1 X ⊗ u B , i = u X ⊗ B , π = 1 X ⊗ ǫ , π = ǫ ⊗ B and X ⋊ B is the object X ⊗ B , where thebialgebra structure is given by the following morphisms in C , m X ⋊ B = ( m ⊗ m ) · (1 X ⊗ ⊲ ⊗ B ⊗ B ) · (1 X ⊗ B ⊗ σ ⊗ B ) · (1 X ⊗ ∆ ⊗ X ⊗ B ) u X ⋊ B = u X ⊗ u B , ∆ X ⋊ B = (1 X ⊗ σ ⊗ B ) · (∆ ⊗ ∆) ,ǫ X ⋊ B = ǫ X ⊗ ǫ B . By combining Figure 1 and Figure 2 in the appendix, we check that this definition provides a bialgebrastructure on X ⊗ B making the morphisms i , i and π bialgebra morphisms, and π a coalgebra morphism.Note that we can check that u X ⊗ u B is the neutral element for the multiplication thanks to the first twoaxioms (2.1) and (2.2) of the definition of action. Lemma 2.2.
The multiplication m X ⋊ B : ( X ⊗ B ) ⊗ ( X ⊗ B ) → X ⊗ B , defined above is not associative, ingeneral. If X and B are associative, then m X ⋊ B is associative as well if and only if (2.7) ⊲ · ( m ⊗ X ) = ⊲ · (1 B ⊗ ⊲ ) , (2.8) ⊲ · (1 B ⊗ m ) = m · ( ⊲ ⊗ ⊲ ) · (1 ⊗ σ ⊗ · (∆ ⊗ X ⊗ X ) . FLORENCE STERCK
Proof.
Via the following diagram we show that if m X ⋊ B is associative then (2.7) is immediately satisfied.With similar computations we can show that the associativity also gives the condition (2.8). The other im-plication is given by the commutativity of the diagram in Figure 3 in the appendix (it is a direct computationusing the conditions (2.7) and (2.8)). B ⊗ X X ⊗ B ⊗ X X X.B ⊗ X X ⊗ B ⊗ X ( X ⊗ B ) X ⊗ B X B ⊗ X ⊗ B ⊗ XB ⊗ X B ⊗ X ( X ⊗ B ) ( X ⊗ B ) X ⊗ B ⊗ XB ⊗ X X ⊗ B ⊗ X X ⊗ B ⊗ X ⊗ BB ⊗ X B ⊗ X (2.2) associativity (1.3) (1.2)(1.7)(1.1) (1.1) ( A ) ( A ) ⊲ · (1 B ⊗ ⊲ ) ⊲ · ( m ⊗ X ) u X ⊗ B ⊗ u X ⊗ B ⊗ X ⊗ u B u X ⊗ m ⊗ X X ⊗ ⊲ mu X ⊗ (1 B ) ⊗ X X ⊗ m ⊗ X ⊗ u B m X ⋊ B X ⊗ ǫǫ ⊗ (1 B ) ⊗ X X ⊗ B ⊗ m X ⋊ B X ⊗ B ⊗ X ⊗ ǫu X ⊗ B ⊗ ∆ ⊗ X X ⊗ (1 B ) ⊗ σ ∆ ⊗ B ⊗ X B ⊗ u X ⊗ (1 B ) ⊗ X ⊲ ⊗ (1 B ) ⊗ X m X ⋊ B ⊗ X ⊗ B m X ⋊ B X ⊗ B ⊗ X ⊗ ǫ X ⊗ ⊲ X ⊗ m ⊗ X mu X ⊗ B ⊗ ⊲ X ⊗ B ⊗ ⊲ ⊗ B The trapezes (A) commute thanks to (1.2) and (1.6), as it is shown in the following diagram X ⊗ B ⊗ X X ⊗ B ⊗ X X X.X ⊗ B ⊗ X ⊗ B X ⊗ B ⊗ X ⊗ B X ⊗ B ( X ⊗ B ) X ⊗ B (1.2) (1.6) X ⊗ ⊲ m X ⊗ B ⊗ σ ⊗ B X ⊗ ⊲ ⊗ (1 B ) m X ⋊ B X ⊗ ∆ ⊗ X ⊗ B X ⊗ B ⊗ ǫ ⊗ X ⊗ ǫ m ⊗ ǫ ⊗ ǫ X ⊗ B ⊗ X ⊗ ǫ X ⊗ B ⊗ X ⊗ ǫ ⊗ ǫ X ⊗ ǫm ⊗ m (cid:3) Let us make some observations about the graph (2.6), which are analogous to the ones made in [12],
Lemma 2.3.
The following graph X ⋊ B BX π i π i , PLIT EXTENSIONS AND ACTIONS OF BIALGEBRAS AND HOPF ALGEBRAS 7 as defined in (2.6) , where i , i , π are morphisms of bialgebras, satisfies the following properties (1) π · i = 1 X , π · i = 1 B (2) π · i = u X · ǫ B , π · i = u B · ǫ X (3) m X ⋊ B · ( i · π ⊗ i · π )(1 ⊗ σ ⊗ · (∆ ⊗ ∆) = 1 X ⊗ B (4) π · m X ⋊ B · ( i ⊗ i ) = 1 X ⊗ ǫ (5) (1 B ⊗ π ) · (1 ⊗ m X ⋊ B ) · (1 B ⊗ i ⊗ i ) · (∆ ⊗ X ) = (1 B ⊗ π ) · (1 ⊗ m X ⋊ B ) · (1 B ⊗ i ⊗ i ) · ( σ ⊗ X ) · (∆ ⊗ X )(6) m X ⋊ B · (1 X ⊗ B ⊗ m X ⋊ B ) · ( i ⊗ i ⊗ X ⊗ B ) = m X ⋊ B · ( m X ⋊ B ⊗ X ⊗ B ) · ( i ⊗ i ⊗ X ⊗ B )(7) m X ⋊ B · (1 X ⊗ B ⊗ m X ⋊ B ) · ( i ⊗ X ⊗ B ⊗ i ) = m X ⋊ B · ( m X ⋊ B ⊗ X ⊗ B ) · ( i ⊗ X ⊗ B ⊗ i )(8) m X ⋊ B · (1 X ⊗ B ⊗ m X ⋊ B ) · (1 X ⊗ B ⊗ i ⊗ i ) = m X ⋊ B · ( m X ⋊ B ⊗ X ⊗ B ) · (1 X ⊗ B ⊗ i ⊗ i )(9) π is a morphism of coalgebras and π preserves the unit.Proof. The properties (1) and (2) are trivial. The condition (3) is proven via the commutativity of followingdiagram X ⊗ B X ⊗ B X ⊗ B X ⊗ B.X ⊗ B ⊗ X ⊗ B X ⊗ B ⊗ X ⊗ B ( X ⊗ B ) X ⊗ B ( X ⊗ B ) X ⊗ B X ⊗ BX ⊗ BX ⊗ BX ⊗ B (2.1)(1.5)(1.1)(1.2) X ⊗ ǫ ⊗ ǫ ⊗ B X ⊗ u B ⊗ u B ⊗ u X ⊗ B X ⊗ u B ⊗ u X ⊗ u B ⊗ B X ⊗ u X ⊗ u B ⊗ B X ⊗ σ ⊗ B ∆ ⊗ ∆ 1 X ⊗ ⊲ ⊗ B ⊗ B m ⊗ m X ⊗ B ⊗ σ ⊗ B X ⊗ ∆ ⊗ X ⊗ B X ⊗ u B ⊗ u X ⊗ B X ⊗ ǫ ⊗ ǫ ⊗ B ( i · π ) ⊗ ( i · π ) m X ⋊ B The equality (4) holds since this diagram commutes
FLORENCE STERCK
X X ⊗ B.X ⊗ B X ⊗ B ⊗ X ⊗ B X ⊗ B ⊗ X ⊗ B ( X ⊗ B ) X ⊗ BX ⊗ B (2.1)(1.5) (1.1) m X ⋊ B X ⊗ u B ⊗ u X ⊗ u B ⊗ B X ⊗ ǫ X ⊗ u X ⊗ u B ⊗ B X ⊗ u B ⊗ u X ⊗ B X ⊗ ∆ ⊗ X ⊗ B X ⊗ B ⊗ σ ⊗ B X ⊗ ⊲ ⊗ (1 B ) m ⊗ m X ⊗ ǫ The property (5) is due to (2.3) and the commutativity of the squares denoted (A) B ⊗ ( X ⊗ B ) B ⊗ X ⊗ BB ⊗ X B ⊗ X ⊗ BB ⊗ ( X ⊗ B ) B ⊗ B ⊗ XB ⊗ XB ⊗ B ⊗ XB ⊗ B ⊗ X (2.3) ( A )( A ) B ⊗ m X ⋊ B B ⊗ X ⊗ ǫ B ⊗ X ⊗ ǫ ∆ ⊗ X σ ⊗ X B ⊗ i ⊗ i B ⊗ i ⊗ i B ⊗ m X ⋊ B ∆ ⊗ X B ⊗ ⊲ B ⊗ ⊲ .We observe that the squares (A) commute thanks to the unitality of the multiplication and the counitalityof the comultiplication, as explained in the following diagram B ⊗ X B ⊗ X ⊗ B.B ⊗ X ⊗ B ( B ⊗ X ) ⊗ B B ⊗ X ⊗ B ⊗ X ⊗ BB ⊗ ( X ⊗ B ) B ⊗ XB ⊗ X B ⊗ XB ⊗ X ⊗ B B ⊗ X ⊗ B (1.2) (1.1) B ⊗ X ⊗ B ⊗ σ ⊗ B B ⊗ X ⊗ ⊲ ⊗ B ⊗ B B ⊗ m ⊗ m B ⊗ X ⊗ ǫ B ⊗ u X ⊗ B ⊗ X ⊗ u B B ⊗ u X ⊗ X ⊗ B ⊗ u B B ⊗ X ⊗ ∆ ⊗ X ⊗ B B ⊗ ∆ ⊗ X B ⊗ B ⊗ σ B ⊗ ⊲ ⊗ B B ⊗ ⊲ We prove the “partial associativity” condition (6) thanks to the following diagram
PLIT EXTENSIONS AND ACTIONS OF BIALGEBRAS AND HOPF ALGEBRAS 9 ( X ⊗ B ) ⊗ BX ⊗ B ⊗ X ⊗ B ( X ⊗ B ) X ⊗ B ⊗ X ⊗ B ( X ⊗ B ) ⊗ B.X ⊗ B ( X ⊗ B ) ⊗ B ⊗ X ⊗ B X ⊗ B X ⊗ BX ⊗ B ⊗ X ⊗ B ( X ⊗ B ) X ⊗ B ⊗ X ⊗ B ( X ⊗ B ) ( X ⊗ B ) ( X ⊗ B ) ( X ⊗ B ) ( X ⊗ B ) X ⊗ B ⊗ ( X ⊗ B ) ( X ⊗ B ) ⊗ B ⊗ X ⊗ BX ⊗ B ⊗ X ⊗ B (2.1)(1.1) + (1.5)(2.1) X ⊗ u B ⊗ X ⊗ u B ⊗ m X ⊗ u B ⊗ u X ⊗ u B ⊗ B ⊗ X ⊗ B X ⊗ B ⊗ X ⊗ ⊲ ⊗ B ⊗ B X ⊗ B ⊗ m ⊗ m X ⊗ ∆ ⊗ X ⊗ B X ⊗ B ⊗ σ ⊗ B X ⊗ ⊲ ⊗ B ⊗ B X ⊗ B ⊗ X ⊗ B ⊗ σ ⊗ B X ⊗ u B ⊗ u X ⊗ X ⊗ B ⊗ B m ⊗ mm ⊗ m X ⊗ ⊲ ⊗ B ⊗ B X ⊗ B ⊗ σ ⊗ B X ⊗ ∆ ⊗ X ⊗ B X ⊗ u B ⊗ u X ⊗ B ⊗ X ⊗ B X ⊗ B ⊗ X ⊗ ∆ ⊗ X ⊗ B m X ⋊ B m ⊗ m ⊗ X ⊗ B X ⊗ ⊲ ⊗ B ⊗ B ⊗ X ⊗ B X ⊗ ∆ ⊗ X ⊗ B ⊗ X ⊗ B X ⊗ B ⊗ σ ⊗ B ⊗ X ⊗ B X ⊗ u B ⊗ u X ⊗ B ⊗ X ⊗ B With similar computations we can show the “partial associativities” (7) and (8), moreover (9) is clear. (cid:3)
We define a split extension of bialgebras by taking inspiration of the above Lemma.
Definition 2.4.
A split extension of bialgebras is given by a diagram(2.9)
A BX λκ αe , where X , A , B are bialgebras, κ , α , e are morphisms of bialgebras, such that(1) λ · κ = 1 X , α · e = 1 B (2) λ · e = u X · ǫ B , α · κ = u B · ǫ X (3) m · (( κ · λ ) ⊗ ( e · α )) · ∆ = 1 A (4) λ · m · ( κ ⊗ e ) = 1 X ⊗ ǫ (5) (1 B ⊗ λ ) · (1 B ⊗ m ) · (1 B ⊗ e ⊗ κ ) · (∆ ⊗ X ) = (1 B ⊗ λ ) · (1 B ⊗ m ) · (1 B ⊗ e ⊗ κ ) · ( σ ⊗ X ) · (∆ ⊗ X )(6) m · ( m ⊗ A ) · ( κ ⊗ e ⊗ A ) = m · (1 A ⊗ m ) · ( κ ⊗ e ⊗ A )(7) m · ( m ⊗ A ) · ( κ ⊗ A ⊗ e ) = m · (1 A ⊗ m ) · ( κ ⊗ A ⊗ e )(8) m · ( m ⊗ A ) · (1 A ⊗ κ ⊗ e ) = m · (1 A ⊗ m ) · (1 A ⊗ κ ⊗ e )(9) λ is a morphism of coalgebras preserving the unit. Remark 2.5.
We notice that the conditions λ · κ = 1 X , λ · e = u X · ǫ B and λ preserving the unit areconsequences of the axiom (4). The condition (7) follows from (3), (6) and (8), as we show in the followingdiagrams, where we use that e and κ are (bi)algebra morphisms. A A X ⊗ B A A A X ⊗ B A A AX ⊗ B A A AX ⊗ B A A A A A (6)(8) (8)(6) κ ⊗ e A ⊗ m ⊗ A κ ⊗ e A ⊗ A ⊗ m A ⊗ mκ ⊗ e m ⊗ A mmκ ⊗ e m ⊗ A ⊗ A m ⊗ A A ⊗ m ⊗ A κ ⊗ e A ⊗ mm ⊗ B ⊗ B κ ⊗ e m ⊗ A mm A ⊗ m A ⊗ A ⊗ m κ ⊗ e κ ⊗ e .Hence, by pre-composing with (1 X ⊗ λ ⊗ α ⊗ B ) · (1 X ⊗ ∆ ⊗ B ) and using the condition (3), we obtain thefollowing diagram X ⊗ A ⊗ B A A AX ⊗ A ⊗ B A A X ⊗ A ⊗ BX ⊗ A ⊗ B X ⊗ A ⊗ B X ⊗ A ⊗ BX ⊗ B (3) (3) κ ⊗ A ⊗ e A ⊗ m m X ⊗ κ ⊗ e ⊗ B X ⊗ m ⊗ B X ⊗ ∆ ⊗ B X ⊗ λ ⊗ α ⊗ B X ⊗ κ ⊗ e ⊗ B X ⊗ m ⊗ B κ ⊗ A ⊗ em ⊗ A m .Then, we conclude that condition (7) holds.Moreover, if there exists such a λ it has to be unique. Let us suppose that there exists another λ ′ satisfyingthe conditions of Definition 2.4. Then the commutativity of the following diagram shows that λ ′ has to beequal to λ . PLIT EXTENSIONS AND ACTIONS OF BIALGEBRAS AND HOPF ALGEBRAS 11
A XAA X ⊗ BA A (4)(3)(1.2) X ⊗ ǫ λλ ′ mκ ⊗ eλ ′ ⊗ α ∆ .Another important property of the split extensions of bialgebras is given by the following proposition: Proposition 2.6.
Let
A BX λκ αe be a split extension of bialgebras, then κ and e are jointly epimorphic in the category of bialgebras (and in the category of algebras)Proof. Let v, w : A → Y be 2 morphisms of bialgebras such that v · κ = w · κ and v · e = w · e . A YAA A Y A A (3) (3) ww ⊗ w ( κ · λ ) ⊗ ( e · α ) vv ⊗ v ( κ · λ ) ⊗ ( e · α )∆ m mm .The above diagram allows us to conclude that v and w are equal. Hence, κ and e are jointly epimorphic. Notethat we only use that v and w are algebra morphisms (we do not need them to be coalgebra morphisms). (cid:3) Definition 2.7.
A morphism of split extensions from
A BX λκ αe to A ′ B ′ X ′ λ ′ κ ′ α ′ e ′ is given by 3 morphisms of bialgebras g : B → B ′ , v : X → X ′ and p : A → A ′ such that the following diagramcommutes(2.10) A BX A ′ B ′ X ′ λλ ′ gv pκ ′ α ′ e ′ κ αe . We do not need to ask the commutativity of all the squares thanks to this corollary:
Corollary 2.8.
Let ( g, v, p ) be a morphism of split extensions of bialgebras A BX A ′ B ′ X ′ λλ ′ gv pκ ′ α ′ e ′ κ αe . If p · κ = κ ′ · v and p · e = e ′ · g hold, then the identities λ ′ · p = v · λ and α ′ · p = g · α follow, and conversely.Proof. Let us suppose that p · κ = κ ′ · v and p · e = e ′ · g , then by Proposition 2.6, we can prove that λ ′ · p = v · λ by checking that ( λ ′ · p ) · κ = ( v · λ ) · κ and ( λ ′ · p ) · e = ( v · λ ) · e , which is done in the following identities:( λ ′ · p ) · κ = λ ′ · κ ′ · v = v = ( v · λ ) · κ, ( λ ′ · p ) · e = λ ′ · e ′ · g = ǫ = ( v · λ ) · e. Similarly, one can check that α ′ · p = g · α . (cid:3) Proposition 2.9.
Let
A BX λκ αe be a split extension of bialgebras, then the fol-lowing diagram commutes (2.11)
A XX A A A A m λ ⊗ λmλ A ⊗ m A ⊗ ( e · α ) ⊗ ( κ · λ )∆ ⊗ A . Proof.
In order to prove the proposition, we make the three diagrams (A),(B) and (C) commute. Then wecompose them to obtain the conclusion. The diagram (A) is commutative since e and α are morphisms of(bi)algebras.( A ) A A A A.A A A A A A A A A A A A A (8)(7) A ⊗ m ⊗ A A ⊗ m m A ⊗ A ⊗ m A ⊗ m ⊗ m A ⊗ ( e · α ) ⊗ ( κ · λ ) ⊗ m A ⊗ A ⊗ ( e · α )1 A ⊗ ( e · α ) ⊗ ( κ · λ ) ⊗ ( e · α ) ⊗ ( e · α ) 1 A ⊗ m ⊗ A ( κ · λ ) ⊗ ( e · α ) m ⊗ A m ( κ · λ ) ⊗ A ⊗ A ( κ · λ ) ⊗ A ( κ · λ ) ⊗ A ⊗ A A ⊗ m ⊗ A ⊗ A A ⊗ m A ⊗ m ⊗ A ⊗ A A ⊗ ( e · α ) ⊗ ( κ · λ ) ⊗ ( e · α ) The following diagram (B) commutes thanks to the counitality of the comultiplication, the unitality of themultiplication and the fact that ( e · α ) is a morphism of (bi)algebras. PLIT EXTENSIONS AND ACTIONS OF BIALGEBRAS AND HOPF ALGEBRAS 13 ( B ) A A A A A A A A A .A A A A A A A A A (1.1) + (1.2)(1) + (2) ∆ ⊗ ∆ 1 A ⊗ ∆ ⊗ A ⊗ A A ⊗ A ⊗ σ ⊗ A A ⊗ m ⊗ A ⊗ A ⊗ A A ⊗ m ⊗ m ⊗ A A ⊗ ( κ · λ ) ⊗ ( e · α ) ⊗ A (( κ · λ ) ⊗ ( e · α )) (( κ · λ ) ⊗ ( e · α )) ⊗ ( u A · ǫ ) ⊗ ( e · α ) 1 A ⊗ ( κ · λ ) ⊗ m ⊗ A A ⊗ m ⊗ ( e · α ) ⊗ A A ⊗ ( κ · λ ) ⊗ m ⊗ A (( κ · λ ) ⊗ ( e · α )) ⊗ ( e · α )1 A ⊗ ( κ · λ ) ⊗ A ⊗ A ∆ ⊗ ∆1 A ⊗ ∆ ⊗ ∆ ⊗ A A ⊗ A ⊗ σ ⊗ A ⊗ A A ⊗ m ⊗ A ⊗ A The diagram (C) is the following, where we are using the fact that we work with coalgebra morphisms.( C ) A A A.AA A A A A A A A A A A A A A A A A A A A (3)(8)(6)(3) ∆ ⊗ ∆ 1 A ⊗ ∆ ⊗ ∆ ⊗ A A ⊗ ∆ ⊗ ∆ ⊗ A A ⊗ ∆ ⊗ A A ⊗ m A ⊗ σ ⊗ A A ⊗ σ ⊗ A m ⊗ A mm (( κ · λ ) ⊗ ( e · α )) (( κ · λ ) ⊗ ( e · α )) (( κ · λ ) ⊗ ( e · α )) (( κ · λ ) ⊗ ( e · α )) A ⊗ ( κ · λ ) ⊗ ( e · α ) ⊗ A ∆ ⊗ ∆1 A ⊗ A ⊗ m A ⊗ A ⊗ m A ⊗ m ⊗ A A ⊗ m ⊗ m ⊗ A A ⊗ m ⊗ A A ⊗ mm Finally, we combine the three above diagrams A A A A,A A A A A A A A A A A A A A A A ( B ) ( A )( C ) m A ⊗ m ⊗ A A ⊗ m m A ⊗ ( e · α ) ⊗ ( κ · λ ) ⊗ m A ⊗ m ⊗ A ( κ · λ ) ⊗ ( e · α ) m ⊗ A m ( κ · λ ) ⊗ A ⊗ A A ⊗ m ⊗ A ⊗ A A ⊗ ( e · α ) ⊗ ( κ · λ ) ⊗ ( e · α ) ∆ ⊗ ∆ 1 A ⊗ ∆ ⊗ A ⊗ A A ⊗ A ⊗ σ ⊗ A A ⊗ m ⊗ m ⊗ A A ⊗ ( κ · λ ) ⊗ ( e · α ) ⊗ A (( κ · λ ) ⊗ ( e · α )) ∆ A ⊗ ∆ ⊗ A A ⊗ σ ⊗ A and we obtain the following equality m = m · ( κ ⊗ e ) · ( m ⊗ B ) · ( λ ⊗ λ ⊗ α ) · (1 A ⊗ m ⊗ A ) · (1 A ⊗ ( e · α ) ⊗ ( κ · λ ) ⊗ m ) · (1 A ⊗ A ⊗ σ ⊗ A ) · (1 A ⊗ ∆ ⊗ A ⊗ A ) · (∆ ⊗ ∆) . By composing with λ and using the condition (4), we can conclude that λ · m = m · ( λ ⊗ λ ) · (1 A ⊗ m ) · (1 A ⊗ ( e · α ) ⊗ ( κ · λ )) · (∆ ⊗ A ) . (cid:3) Proposition 2.10.
Given a split extension
A BX λκ αe of bialgebras, we can con-struct an action of bialgebras, ⊲ : B ⊗ X → X defined by ⊲ = λ · m · ( e ⊗ κ ) . Proof.
We check all the axioms of the definition of actions of bialgebras (Definition 2.1).
X XAA B ⊗ XX ⊲ u B ⊗ X λmκ e ⊗ κ B XAA B ⊗ XB ⊲ u X · ǫ B ⊗ u X λme e ⊗ κ B ⊗ X IXAA B ⊗ X ⊲ ǫ ⊗ ǫe ⊗ κ m λǫ ǫ PLIT EXTENSIONS AND ACTIONS OF BIALGEBRAS AND HOPF ALGEBRAS 15 ( B ⊗ X ) A A X .XAA B ⊗ XB ⊗ X A (1.4) ⊲⊲ ⊗ ⊲ e ⊗ k ⊗ e ⊗ k m ⊗ m λ ⊗ λe ⊗ k m λe ⊗ e ⊗ k ⊗ k ∆∆∆ ⊗ ∆ ∆ ⊗ ∆1 A ⊗ σ ⊗ A B ⊗ σ ⊗ X Moreover, the condition (2.3) for the particular action ⊲ = λ · m · ( e ⊗ κ ) is exactly the condition (5) in thedefinition of split extension of bialgebras (Definition 2.4). (cid:3) Consider the diagram(2.12)
A BX X ⋊ B BX π λ ψ ϕi π i κ αe , where the two rows are split extensions, the top row by definition and the bottom one by Lemma 2.3, wherethe action of bialgebras is given by the Proposition 2.10. The maps ϕ and ψ are defined by ψ = ( λ ⊗ α ) · ∆ , and ϕ = m · ( κ ⊗ e ) . In the following lemmas we prove step by step that ψ and ϕ are isomorphisms of split extensions. First, weprove that they are inverse to each other. Lemma 2.11.
The maps ϕ and ψ are inverse to each other.Proof. We prove in the following two diagrams that ψ · ϕ = 1 X ⊗ B and ϕ · ψ = 1 A by using the propertiesof the split extensions. X ⊗ B X ⊗ B,X ⊗ BA AA X ⊗ B X ⊗ B A ( X ⊗ B ) A X ⊗ A X ⊗ A ( X ⊗ B ) X ⊗ B (1) + (2) (4)(1.4) ψϕ κ ⊗ e mκ ⊗ e ( κ ⊗ e ) m ⊗ mm ⊗ m X ⊗ α X ⊗ ( u X · ǫ ) ⊗ ( u B · ǫ ) ⊗ B λ ⊗ α ∆∆ ⊗ ∆1 A ⊗ σ ⊗ A X ⊗ σ ⊗ B ∆ ⊗ ∆ 1 X ⊗ ( u X · ǫ ) ⊗ κ ⊗ e (1 X ) ⊗ α m ⊗ m A A X ⊗ BA A. (3) ϕψ mκ ⊗ e ∆ λ ⊗ α (cid:3) In order to prove that ψ and ϕ of (2.12) are morphisms of bialgebras, we need the following technicalLemma. Notice that this Lemma will also be convenient to express the action ⊲ := λ · m · ( e ⊗ κ ) in theparticular case of Hopf algebras (See Remark 3.5). Lemma 2.12.
Given a split extension
A BX λκ αe of bialgebras, we have (2.13) m · ( e ⊗ κ ) = m · ( κ ⊗ e ) · ( ⊲ ⊗ B ) · (1 B ⊗ σ ) · (∆ ⊗ X ) , where ⊲ = λ · m · ( e ⊗ κ ) .Proof. The equality of the lemma is proven thanks to the commutativity of the following diagram where weuse that α , κ and e are morphisms of bialgebras PLIT EXTENSIONS AND ACTIONS OF BIALGEBRAS AND HOPF ALGEBRAS 17 A ⊗ B X ⊗ B A A.A X ⊗ BA AA B ⊗ XB ⊗ XB ⊗ XB ⊗ X ⊗ BA ⊗ B B ⊗ X ⊗ B ⊗ X A A ⊗ B ⊗ X A ⊗ B (3)(1.1) + (1.2)(1) + (2) (1.4) λ ⊗ B κ ⊗ e me ⊗ κ m ( e ⊗ κ ) m ⊗ mm ⊗ B ⊗ ( u B · ǫ ) λ ⊗ m mκ ⊗ eλ ⊗ α ∆∆ ⊗ X B ⊗ σe ⊗ κ ⊗ B m ⊗ B (1 B ⊗ σ ⊗ X ) · (∆ ⊗ ∆) (1 B ⊗ σ ⊗ X ) · (∆ ⊗ ∆) e ⊗ κ ⊗ B ⊗ X m ⊗ α ⊗ α (cid:3) Lemma 2.13. ϕ := m · ( κ ⊗ e ) : X ⊗ B → A is a morphism of bialgebras.Proof. First we establish two diagrams denoted by (A) and (B). We will need them to conclude that ϕ is amorphism of algebras.(A) We use the partial associativity of the split extensions to check that the following diagram commutes X ⊗ B ⊗ X ⊗ B A A A AX ⊗ B ⊗ X ⊗ B A A A AX ⊗ B ⊗ X ⊗ B A A A (6)(8) ( κ ⊗ e ) m ⊗ A ⊗ A A ⊗ m m ( κ ⊗ e ) A ⊗ m ⊗ A m ⊗ A ( κ ⊗ e ) m ⊗ A ⊗ A m ⊗ A m m .(B) The diagram commutes since e and κ are (bi)algebra morphisms X ⊗ B A AA A A A X ⊗ B X ⊗ B X ⊗ B X ⊗ B A A A A (6)(7) m ⊗ m κ ⊗ e mκ ⊗ e κ ⊗ e κ ⊗ e κ ⊗ e A ⊗ m ⊗ A m ⊗ A m ⊗ A m ⊗ A ⊗ A m X ⊗ mm ⊗ B ⊗ B m ⊗ A ⊗ A A ⊗ m m ⊗ A .Thanks to these two diagrams and Lemma 2.12 we have the following diagram showing that ϕ := m · ( κ ⊗ e )is a morphism of algebras A A A.A X ⊗ BX ⊗ B X ⊗ B ⊗ X ⊗ B X ⊗ B ⊗ X ⊗ B ( X ⊗ B ) A A A A (2.13)( A ) ( B ) m ⊗ m m X ⊗ ∆ ⊗ X ⊗ B X ⊗ B ⊗ σ ⊗ B X ⊗ ⊲ ⊗ B ⊗ B m ⊗ A κ ⊗ em ⊗ mm ( κ ⊗ e ) ( κ ⊗ e ) A ⊗ m ⊗ A mκ ⊗ e A ⊗ m ⊗ A Finally, ϕ is a morphism of bialgebras since it is also a morphism of coalgebras: X ⊗ B ⊗ X ⊗ B A A AA X ⊗ BX ⊗ X ⊗ B ⊗ B A (1.4) ( κ ⊗ e ) m ⊗ mκ ⊗ e mκ ⊗ e ∆ ⊗ ∆1 X ⊗ σ ⊗ B ∆ ⊗ ∆1 A ⊗ σ ⊗ A ∆ .The above diagram commutes since e and κ are coalgebras morphisms. (cid:3) PLIT EXTENSIONS AND ACTIONS OF BIALGEBRAS AND HOPF ALGEBRAS 19
From this Lemma and Lemma 2.11, it is straightforward that ψ is also a morphism of bialgebras, and itbrings us to the following useful proposition. Proposition 2.14.
The diagram (2.12)
A BX X ⋊ B BX π λ ψ ∼ = ϕ ∼ = i π i κ αe commutes. Accordingly, (1 B , X , ψ ) and (1 B , X , ϕ ) are isomorphisms of split extensions of bialgebras.Proof. Thanks to Proposition 2.8, it is enough to check that the four following diagrams of (2.12) commuteto conclude that ϕ and ψ are morphisms of split extensions of bialgebras. We recall that ψ = ( λ ⊗ α ) · ∆ and ϕ = m · ( κ ⊗ e ). B AA X ⊗ BB eu X ⊗ B κ ⊗ emu A ⊗ e X AA X ⊗ BX κ X ⊗ u B κ ⊗ emκ ⊗ u A X X ⊗ BA AX X X ⊗ u B ∆ κ ⊗ κκ λ ⊗ α ∆ 1 X ⊗ ǫ B X ⊗ BA AB B u X ⊗ B ∆ e ⊗ ee λ ⊗ α ∆ ǫ ⊗ B . (cid:3) Proposition 2.15.
Given a split extension
A BX λκ αe of bialgebras, the followingproperties hold: a) κ is the kernel of α in the category of bialgebras; b) α is the cokernel of κ in the category of bialgebras.Proof. a) Let ω : D → A be a morphism of bialgebras such that α · ω = u B · ǫ . We build the morphismˆ ω : D → X by setting ˆ ω = λ · ω. First, we verify that κ · ˆ ω = w thanks to the commutativity of the following diagram A A.A A D D D (3)(1.1) + (1.2) ( κ · λ ) ⊗ ( e · α )( κ · λ · ω ) ⊗ ( u A · ǫ )∆ m ∆ κ · λ · ωω ω ⊗ ω Moreover, ˆ ω is a coalgebra morphism by construction, and an algebra morphism since the following diagramcommutes D A X X.D A A D D A A AD D (1)(1.1) + (1.2) (2.9) ω λ mm m ∆ ⊗ D ω ω ⊗ ( κ · λ · ω ) λ ω ⊗ ( ǫ · u A ) ⊗ ( κ · λ · ω ) 1 A ⊗ ( e · α ) ⊗ ( κ · λ ) ω ∆ ⊗ A A ⊗ m λω Finally, if there exists another morphism ω ′ such that κ · ω ′ = w , then by (1) ω ′ = λ · κ · ω ′ = λ · ω = λ · κ · ˆ ω = ˆ ω. Hence, κ is the kernel of α .b) Let β : A → C be a bialgebra morphism such that β · κ = u C · ǫ . We define ˜ β : B → C by˜ β = β · e. This morphism is a bialgebra morphism, and thanks to Proposition 2.6, it is enough to remark that ˜ β · α · κ = u C · ǫ = β · κ and ˜ β · α · e = β · e , to conclude that ˜ β · α = β . Moreover, if there exists another β ′ such that β ′ · α = β then, thanks to (1) we have β ′ = β ′ · α · e = β · e = ˜ β · α · e = ˜ β and α is the cokernel of κ (cid:3) Lemma 2.16.
Let ( g, v, p ) be a morphism of split extensions as in Definition 2.7, then v · ⊲ = ⊲ · ( g ⊗ v ) , where the actions are induced by the split extensions.Proof. This follows from the fact that ( g, v, p ) is a morphism of split extensions and p is a morphism ofbialgebras, as we can see in the following diagram B ⊗ X A A XB ′ ⊗ X ′ A ′ A ′ X ′ .⊲⊲ e ⊗ κ m λe ′ ⊗ κ ′ m ′ λ ′ g ⊗ v p ⊗ p p v (cid:3) Lemma 2.17.
Let
A BX λκ αe and A ′ B ′ X ′ λ ′ κ ′ α ′ e ′ be two splitextensions of bialgebras, and g : B → B ′ and v : X → X ′ be two morphisms of bialgebras. Then the followingconditions are equivalent: PLIT EXTENSIONS AND ACTIONS OF BIALGEBRAS AND HOPF ALGEBRAS 21 there exists p : A → A ′ such that ( g, v, p ) is a morphism of split extensions; there exists a unique p : A → A ′ such that ( g, v, p ) is a morphism of split extensions; v · ⊲ = ⊲ · ( g ⊗ v ) .Proof. Thanks to Proposition 2.6 and Lemma 2.16, we just need to check that 3) ⇒ p : X ⋊ B → X ′ ⋊ B ′ as ˜ p = v ⊗ g. It is clear that this morphism is a morphism of coalgebras. Moreover, ˜ p is a morphism of algebras as we cansee in the following diagram( X ⊗ B ) X ⊗ B ⊗ X ⊗ B ( X ⊗ B ) ⊗ B X ⊗ B X ⊗ B ( X ′ ⊗ B ′ ) X ′ ⊗ B ′ ⊗ X ′ ⊗ B ′ ( X ′ ⊗ B ′ ) ⊗ B ′ X ′ ⊗ B ′ X ′ ⊗ B ′ . m X ⋊ B m X ′ ⋊ B ′ X ⊗ ∆ ⊗ X ⊗ B X ⊗ B ⊗ σ ⊗ B X ⊗ ⊲ ⊗ (1 B ) m ⊗ m X ′ ⊗ ∆ ⊗ X ′ ⊗ B ′ X ′ ⊗ B ′ ⊗ σ ⊗ B ′ X ′ ⊗ ⊲ ⊗ (1 B ′ ) m ⊗ m ( v ⊗ g ) v ⊗ g ⊗ v ⊗ g ( v ⊗ g ) ⊗ g v ⊗ g v ⊗ g Then we can define a morphism p : A → A ′ as ϕ A ′ · ˜ p · ψ A where ϕ and ψ are the isomorphisms in Proposition2.14. In particular, that gives us p = m · ( κ ′ ⊗ e ′ ) · ( v ⊗ g ) · ( λ ⊗ α ) · ∆ . Finally, ( g, v, p ) is a morphism of split extensions of bialgebras, X ′ A ′ A ′ X ′ ⊗ B ′ X ⊗ BA AX X (2) p κ ′ κ κ ⊗ κ ∆ v ⊗ gκ ′ ⊗ e ′ mv ∆1 X ⊗ ( u B · ǫ ) λ ⊗ α B ′ A ′ .A ′ X ′ ⊗ B ′ X ⊗ BA AB B (2) p e ′ e e ⊗ e ∆ v ⊗ gκ ′ ⊗ e ′ mg ∆( u X · ǫ ) ⊗ B λ ⊗ α Thanks to Corollary 2.8 the commutativity of these two diagrams suffices to conclude that the diagram (2.10)commutes. (cid:3)
Definition 2.18.
Let ⊲ : B ⊗ X → X and ⊲ ′ : B ′ ⊗ X ′ → X ′ be two actions of bialgebras. A morphismbetween them is defined as a pair of morphisms of bialgebras g : B → B ′ and v : X → X ′ such that v · ⊲ = ⊲ ′ · ( g ⊗ v ) . The split extensions of bialgebras (Definition 2.4) endowed with the morphisms of split extensions ofbialgebras (Definition 2.7) form the category of split extensions of bialgebras denoted by
SplitExt ( BiAlg ).The actions of bialgebras (Definition 2.1) with the morphisms of actions (Definition 2.18) form the categoryof actions of bialgebras, denoted by
Act ( BiAlg ). Theorem 2.19.
The category
SplitExt ( BiAlg ) of split extensions of bialgebras and the category Act ( BiAlg ) ofactions of bialgebras are equivalent.Proof. The functors F : SplitExt ( BiAlg ) → Act ( BiAlg ) is defined as F A BX A ′ B ′ X ′ λ ′ λv gpκ ′ α ′ e ′ κ αe = B ⊗ XB ′ ⊗ X ′ XX ′ , ⊲⊲ ′ g ⊗ v v , where ⊲ = λ · m · ( e ⊗ κ ) as in Proposition 2.10, and we have a morphism of actions thanks to Lemma 2.16.The functor G : Act ( BiAlg ) → SplitExt ( BiAlg ) is defined as G B ⊗ XB ′ ⊗ X ′ XX ′ ⊲⊲ ′ g ⊗ v v = X ⋊ B BX X ′ ⋊ B ′ B ′ ,X ′ π ′ π v gpi ′ π ′ i ′ i π i where p := v ⊗ g is given by Lemma 2.17 and the bialgebra structure of the semi-direct product X ⋊ B and X ′ ⋊ B ′ are defined as in (2.6).We observe that(2.14) F · G ( ⊲ ) = π · m X ⋊ B · ( i ⊗ i ) = ⊲, where the last equality holds thanks to the commutativity of the following diagram B ⊗ X X.X ⊗ BX ⊗ B X ⊗ B ⊗ X ⊗ B X ⊗ B ⊗ X ⊗ B ( X ⊗ B ) B ⊗ XB ⊗ X B ⊗ X B ⊗ X ⊗ B X ⊗ B (1.1)(1.2) X ⊗ B ⊗ σ ⊗ B X ⊗ ⊲ ⊗ B ⊗ B m ⊗ m X ⊗ ǫ⊲u X ⊗ B ⊗ X ⊗ u B X ⊗ ∆ ⊗ X ⊗ B ∆ ⊗ X B ⊗ σ ⊲ ⊗ B u X ⊗ X ⊗ B ⊗ u B Thanks to this observation, and the isomorphisms ϕ and ψ of (2.12), the functors F and G give rise to anequivalence of categories. (cid:3) PLIT EXTENSIONS AND ACTIONS OF BIALGEBRAS AND HOPF ALGEBRAS 23
Remark 2.20.
If we consider cocommutative bialgebras, then the category
BiAlg coc of cocommutative bial-gebras can be seen as the category of internal magmas in the category of cocommutative coalgebras. Indeed,the categorical product of cocommutative coalgebras is given by the tensor product.In this particular case, the condition (2.3) in Definition 2.1 becomes trivial and the definition can bereformulated explicitly as:
Definition 2.21.
Let X and B be cocommutative bialgebras. An action of B on X is a morphism ofcoalgebras ⊲ : B ⊗ X → X such that ⊲ · ( u B ⊗ X ) = 1 X ,⊲ · (1 B ⊗ u X ) = ǫ. Similarly, thanks to the cocommutativity we can drop the condition (5) in the definition of split extensionsof bialgebras (Definition 2.4), and this turns out to be exactly the internal version, in the category ofcoalgebras, of Definition 1.4 in [12]. Then, in the case of cocommutative bialgebras, the above theoremreduces to the results in Section 4.6 in [12]. Indeed, if C is a category with finite limits, it is in particulara cartesian monoidal category. Hence, any magma in such a category can be seen as a non-associativecocommutative bialgebra in this category (where the unique comultiplication is the diagonal). Accordingly,the results in [12] become a particular case of our theorem.Let us consider associative bialgebras. We define the categories Act ( AssBiAlg ) and
SplitExt ( AssBiAlg ).An object in
Act ( AssBiAlg ), is an action of associative bialgebras (Definition 2.1) satisfying (2.7) and (2.8),the morphisms are the morphisms of
Act ( BiAlg ). The category
SplitExt ( AssBiAlg ) is a full subcategory of
SplitExt ( BiAlg ) since the conditions (6) , (7) and (8) become redundant. Corollary 2.22.
There is an equivalence between the category
SplitExt ( AssBiAlg ) of split extensions of asso-ciative bialgebras and the category Act ( AssBiAlg ) of actions of associative bialgebras.Proof. It is clear by applying Lemma 2.2 and Theorem 2.19. (cid:3)
To end this section we prove that a variation of the Split Short Five Lemma holds in
BiAlg . Theorem 2.23.
Let ( g, v, p ) be a morphism of split extensions of bialgebras (2.15) A BX A ′ B ′ X ′ λ ′ λ gv pκ ′ α ′ e ′ κ αe then p is an isomorphism whenever v and g are.Proof. Thanks to Theorem 2.19, the diagram (2.15) is canonically isomorphic to X ⋊ B BX X ′ ⋊ B ′ B ′ .X ′ π ′ π gv v ⊗ gi ′ π ′ i ′ i π i It follows that v ⊗ g is an isomorphism whenever v and g are. (cid:3) Split extensions of non-associative Hopf algebras
In this section we consider a similar result for non-associative Hopf algebras. We prove an equivalencebetween the category of split extensions of non-associative Hopf algebras and the category of actions ofnon-associative Hopf algebras.
Convention : for the sake of simplicity, in this section “Hopf algebras” will mean “non-associative Hopfalgebras” (unless the associativity is explicitly mentioned).
Definition 3.1.
A split extension of Hopf algebras is a split extension of bialgebras(3.1)
A BX λκ αe , such that X , A , B are Hopf algebras and κ, α, e are morphisms of Hopf algebras, with an additional conditionof associativity (condition (9 ′ )) and an additional condition about the left and right antipodes (conditions(10’) and (11’)). More precisely, the split extension (3.1) satisfies(1’) λ · κ = 1 X , α · e = 1 B (2’) λ · e = u X · ǫ B , α · κ = u B · ǫ X (3’) m · (( κ · λ ) ⊗ ( e · α )) · ∆ = 1 A (4’) λ · m · ( κ ⊗ e ) = 1 X ⊗ ǫ B (5’) (1 B ⊗ λ ) · (1 B ⊗ m ) · (1 B ⊗ e ⊗ κ ) · (∆ ⊗ X ) = (1 B ⊗ λ ) · (1 B ⊗ m ) · (1 B ⊗ e ⊗ κ ) · ( σ ⊗ X ) · (∆ ⊗ X )(6’) m · ( m ⊗ A ) · ( κ ⊗ e ⊗ A ) = m · (1 A ⊗ m ) · ( κ ⊗ e ⊗ A )(7’) m · ( m ⊗ A ) · ( κ ⊗ A ⊗ e ) = m · (1 A ⊗ m ) · ( κ ⊗ A ⊗ e )(8’) m · ( m ⊗ A ) · (1 A ⊗ κ ⊗ e ) = m · (1 A ⊗ m ) · (1 A ⊗ κ ⊗ e )(9’) m · ( m ⊗ A ) · ( e ⊗ A ⊗ κ ) = m · (1 A ⊗ m ) · ( e ⊗ A ⊗ κ )(10’) S L · λ · m · ( e ⊗ κ ) = λ · m · ( e ⊗ κ ) · (1 B ⊗ S L ),(11’) ǫ ⊗ S R = λ · m · ( e ⊗ κ ) · ( S R ⊗ S R ) · (1 B ⊗ λ ) · (1 B ⊗ m ) · (1 B ⊗ e ⊗ κ ) · (∆ ⊗ X ),(12’) λ is a morphism of coalgebras preserving the unit.The following definition is inspired by the definition given in [18] in the case of associative Hopf algebras. Definition 3.2.
Let X and B be Hopf algebras, ⊲ : B ⊗ X → X is an action of Hopf algebras if it is anaction of bialgebras such that the following additional conditions are satisfied(3.2) ⊲ · (1 B ⊗ ⊲ ) = ⊲ · ( m ⊗ X ) , (3.3) ⊲ · (1 B ⊗ m ) = m · ( ⊲ ⊗ ⊲ ) · (1 B ⊗ σ ⊗ X ) · (∆ ⊗ X ⊗ X ) , (3.4) ⊲ · (1 B ⊗ S L ) = S L · ⊲, (3.5) ⊲ · ( S R ⊗ S R ) · (1 B ⊗ ⊲ ) · (∆ ⊗ X ) = ǫ ⊗ S R . These conditions can be expressed by the commutativity of the following diagrams
PLIT EXTENSIONS AND ACTIONS OF BIALGEBRAS AND HOPF ALGEBRAS 25 B ⊗ X XB ⊗ XB ⊗ B ⊗ X (3.2) ⊲ B ⊗ ⊲ ⊲m ⊗ X B ⊗ X X.X ( B ⊗ X ) B ⊗ X B ⊗ X ⊗ X (3.3) ⊲ ∆ ⊗ X ⊗ X B ⊗ σ ⊗ X ⊲ ⊗ ⊲m B ⊗ m B ⊗ X XXB ⊗ X (3.4) ⊲⊲ S L B ⊗ S L X B ⊗ XB ⊗ XB ⊗ XB ⊗ X (3.5) ⊲ ∆ ⊗ X B ⊗ ⊲ S R ⊗ S R ǫ ⊗ S R Note that whenever S L = S R , the condition (3.5) follows from (3.2) and (3.4). We notice that, whenwe consider associative Hopf algebras, the conditions (3.4) and (3.5) are trivially satisfied thanks to theuniqueness of the antipode.We define the map Θ : B ⊗ X → X ⊗ B as the compositionΘ := ( ⊲ ⊗ B ) · (1 B ⊗ σ ) · (∆ ⊗ X ) . We will use this map to obtain shorter computations. We re-formulate the conditions (3.3), (2.1), (3.2) and(2.2) in terms of Θ. These new conditions will help us to prove that the semi-direct product is a Hopf algebrawhen we construct it with an action as defined above (Definition 3.2).
Lemma 3.3.
Let ⊲ : B ⊗ X → X be an action of Hopf algebra, the morphism Θ := ( ⊲ ⊗ B ) · (1 B ⊗ σ ) · (∆ ⊗ X ) satisfies the following conditions (3.6) ( m ⊗ B ) · (1 X ⊗ Θ) · (Θ ⊗ X ) = Θ · (1 B ⊗ m )(3.7) Θ · ( u B ⊗ X ) = 1 X ⊗ u B , (3.8) (1 X ⊗ m ) · (Θ ⊗ B ) · (1 B ⊗ Θ) = Θ · ( m ⊗ X ) , (3.9) Θ · (1 B ⊗ u X ) = u X ⊗ B ,B ⊗ X X ⊗ BX ⊗ X ⊗ BX ⊗ B ⊗ XB ⊗ X ⊗ X ΘΘ ⊗ X X ⊗ Θ m ⊗ B B ⊗ m X B ⊗ XX ⊗ B u B ⊗ X Θ1 X ⊗ u B B ⊗ X X ⊗ BX ⊗ B ⊗ BB ⊗ X ⊗ BB ⊗ B ⊗ X Θ1 B ⊗ Θ Θ ⊗ B X ⊗ mm ⊗ X B B ⊗ XX ⊗ B B ⊗ u X Θ u X ⊗ B .Proof. We only show the two first equalities since the computations are similar. First we prove (3.6) via thefollowing diagram, where the key part is given by (3.3). X ⊗ B ⊗ X X ⊗ B ⊗ X ( X ⊗ B ) X ⊗ B.X ⊗ BX ⊗ BB ⊗ X ⊗ BB ⊗ B ⊗ XB ⊗ X B ⊗ X B ⊗ X B ⊗ X B ⊗ X ⊗ BB ⊗ X ⊗ B ( B ⊗ X ) B ⊗ X ⊗ B ⊗ X ( B ⊗ X ) ⊗ B (1.3) (3.3)1 X ⊗ ΘΘ · (1 B ⊗ m )Θ ⊗ X X ⊗ ∆ ⊗ X X ⊗ B ⊗ σ X ⊗ ⊲ ⊗ B B ⊗ X ⊗ ∆ ⊗ X B ⊗ X ⊗ B ⊗ σ ⊲ ⊗ ⊲ ⊗ B B ⊗ ∆ ⊗ X ⊗ X B ⊗ B ⊗ σ B,XX ∆ ⊗ X ⊗ X B ⊗ B ⊗ m B ⊗ σ ⊲ ⊗ B ∆ ⊗ X ⊗ X B ⊗ σ ⊗ X ⊲ ⊗ B ⊗ X ∆ ⊗ B ⊗ X ⊗ X B ⊗ σ BB,X ⊗ X ∆ ⊗ X ⊗ X ⊗ B B ⊗ σ ⊗ X ⊗ B ∆ ⊗ X ⊗ X ⊗ B B ⊗ σ B,XX B ⊗ m ⊗ B m ⊗ B The condition (2.1) provides directly the condition (3.7) as we can see in the following diagram
X B ⊗ XB ⊗ XB ⊗ X ⊗ BX ⊗ BX (2.1)(1.5) u B ⊗ X u B ⊗ X ⊗ u B ∆ ⊗ X B ⊗ σ⊲ ⊗ B X ⊗ u B .The other equalities follow from (3.2) and (2.2), the proofs being similar to the ones given above. (cid:3) Starting from an action of Hopf algebras, we can define a split extension of Hopf algebras(3.10) X ⋊ B BX π i π i PLIT EXTENSIONS AND ACTIONS OF BIALGEBRAS AND HOPF ALGEBRAS 27 where the structure of X ⋊ B is given by m X ⋊ B = ( m ⊗ m ) · (1 X ⊗ Θ ⊗ B ) u X ⋊ B = u X ⊗ u B , ∆ X ⋊ B = (1 X ⊗ σ ⊗ B ) · (∆ ⊗ ∆) ,ǫ X ⋊ B = ǫ X ⊗ ǫ B ,S X ⋊ B L = Θ · ( S L ⊗ S L ) · σ,S X ⋊ B R = Θ · ( S R ⊗ S R ) · σ. Thanks to Lemma 2.3, we already know that (3.10) is a split extension of bialgebras. It is also easy tocheck that i , i and π are morphisms of Hopf algebras as it is shown in the following diagrams for the leftantipode, similar computations work for the right antipode, B X ⊗ BB B ⊗ XB X ⊗ B B ⊗ X (3.9) u X ⊗ B B ⊗ u X u X ⊗ B σS L S L ⊗ S L Θ X X ⊗ BX B ⊗ XX X ⊗ B B ⊗ X (3.7) X ⊗ u B u B ⊗ X X ⊗ u B σS L S L ⊗ S L Θ B X ⊗ B.B B ⊗ X ⊗ BB B ⊗ XX ⊗ B B ⊗ X B ⊗ X (2.4) (1.2) ǫ ⊗ B ǫ ⊗ ǫ ⊗ B σ S L ⊗ S L ǫ ⊗ B S L ∆ ⊗ X B ⊗ σ⊲ ⊗ B B ⊗ ǫ Furthermore, thanks to (2.3) one can show that S X ⋊ B L and S X ⋊ B R are antihomomorphisms of coalgebrasand thanks to (2.3), (3.4), (3.5), (3.6) and (3.8) one can show that they are antihomomorphisms of algebras.Moreover, we check that the above construction satisfies the antipode conditions (1.8) thanks to the followingtwo diagrams X ⊗ B X ⊗ B X X ⊗ B X ⊗ B,X ⊗ B ( X ⊗ B ) X ⊗ B ⊗ XX ⊗ B ⊗ X ( X ⊗ B ) X ⊗ B X ⊗ B X ⊗ B ⊗ XX ⊗ BX ⊗ B X ⊗ B (1.8) (3.7) (3.8)∆ X ⋊ B X ⊗ B ⊗ S X ⋊ B R m X ⋊ B ( u X ⊗ u B ) · ( ǫ ⊗ ǫ ) ∆ ⊗ B X ⊗ S R ⊗ ǫ X ⊗ X ⊗ u B m ⊗ B ∆ ⊗ ∆ 1 X ⊗ σ ⊗ B X ⊗ B ⊗ σ X ⊗ B ⊗ S R ⊗ S R X ⊗ S R ⊗ B ⊗ S R X ⊗ X ⊗ m X ⊗ σ X ⊗ B ⊗ Θ m ⊗ m X ⊗ Θ ⊗ B X ⊗ m ⊗ X X ⊗ Θ1 X ⊗ u B ⊗ X X ⊗ B X ⊗ B B X ⊗ B X ⊗ B.X ⊗ B ( X ⊗ B ) B ⊗ X ⊗ BB ⊗ X ⊗ B ( X ⊗ B ) X ⊗ B X ⊗ B B ⊗ X ⊗ BX ⊗ B X ⊗ B X ⊗ B (1.8) (3.9) (3.6)∆ X ⋊ B S X ⋊ B L ⊗ X ⊗ B m X ⋊ B ( u X ⊗ u B ) · ( ǫ ⊗ ǫ ) X ⊗ ∆ ǫ ⊗ S L ⊗ B u X ⊗ B ⊗ B X ⊗ m ∆ ⊗ ∆ 1 X ⊗ σ ⊗ B σ ⊗ X ⊗ B S L ⊗ S L ⊗ X ⊗ B S L ⊗ X ⊗ S L ⊗ B m ⊗ B ⊗ B σ ⊗ B Θ ⊗ X ⊗ B m ⊗ m X ⊗ Θ ⊗ B B ⊗ m ⊗ B Θ ⊗ B B ⊗ u X ⊗ B Moreover, the conditions (3.8) and (3.6) imply that X ⋊ B BX π i π i satisfies the con-dition (9 ′ ) of the Definition 3.1 as it is shown in the following diagram( B ⊗ X ) ( X ⊗ B ) X ⊗ B ⊗ X ⊗ B ( X ⊗ B ) .X ⊗ B X ⊗ BX ⊗ B ( X ⊗ B ) X ⊗ B ⊗ X ⊗ B ( X ⊗ B ) ( B ⊗ X ) ( B ⊗ X ) B ⊗ X ⊗ B B ⊗ X ⊗ B X ⊗ B ( B ⊗ X ) ( X ⊗ B ) X ⊗ B ( B ⊗ X ) X ⊗ B ⊗ X X ⊗ B ⊗ X X ⊗ B ( X ⊗ B ) X ⊗ B ( X ⊗ B ) X ⊗ B (1.1)(1.1)(3.6)(3.8) u X ⊗ B ⊗ X ⊗ B ⊗ X ⊗ u B X ⊗ Θ ⊗ B X ⊗ Θ ⊗ B X ⊗ Θ ⊗ B ⊗ X ⊗ B m ⊗ (1 B ) X ⊗ B ⊗ Θ (1 X ) ⊗ m Θ ⊗ X ⊗ B m ⊗ m ⊗ X ⊗ B Θ ⊗ B ⊗ X X ⊗ m ⊗ X X ⊗ ΘΘ ⊗ Θ 1 X ⊗ Θ ⊗ B m ⊗ m B ⊗ X ⊗ Θ 1 B ⊗ m ⊗ B Θ ⊗ B u X ⊗ B ⊗ X ⊗ B ⊗ X ⊗ u B X ⊗ B ⊗ X ⊗ Θ ⊗ B X ⊗ B ⊗ m ⊗ m m ⊗ m X ⊗ Θ ⊗ B m ⊗ m X ⊗ Θ ⊗ B m ⊗ B X ⊗ m Finally, the condition (10’) and (11’) hold thanks to (3.4), (3.5) and (2.14), and we can conclude that (3.10)is a split extension of Hopf algebras as defined in Definition 3.1.On the other hand, if we have a split extension of Hopf algebras, we can define an action of Hopf algebras.Thanks to Proposition 2.9 and condition (9 ′ ), we can prove two identities which are crucial properties, forour purpose, of a split extension of Hopf algebras. Lemma 3.4.
Let
A BX λκ αe be a split extension of Hopf algebras, we have (3.11) λ · m · ( e ⊗ κ ) · ( m ⊗ X ) = λ · m · ( e ⊗ ( κ · λ )) · (1 B ⊗ m ) · (1 B ⊗ e ⊗ κ ) , (3.12) λ · m · ( e ⊗ κ ) · (1 B ⊗ m ) = m · ( λ ⊗ λ ) · ( m ⊗ m ) · ( e ⊗ κ ⊗ e ⊗ κ ) · (1 B ⊗ σ ⊗ X ) · (∆ ⊗ X ⊗ X ) , PLIT EXTENSIONS AND ACTIONS OF BIALGEBRAS AND HOPF ALGEBRAS 29 B ⊗ X XB ⊗ XB ⊗ X (3.11) λ · m · ( e ⊗ κ )1 B ⊗ ( λ · m · ( e ⊗ κ )) λ · m · ( e ⊗ κ ) m ⊗ X B ⊗ X X.X ⊗ XB ⊗ X ⊗ B ⊗ XB ⊗ X B ⊗ X ⊗ X (3.12) λ · m · ( e ⊗ κ )∆ ⊗ X ⊗ X B ⊗ σ ⊗ X ( λ · m · ( e ⊗ κ )) m B ⊗ m Proof.
Thanks to Proposition 2.9, the result follows as we can check in the following diagrams, where we usethat e, α, κ are morphisms of bialgebras. B ⊗ X B ⊗ X A AAXXAA B ⊗ AB ⊗ A B ⊗ X B ⊗ A B ⊗ A B ⊗ XA B ⊗ X B ⊗ A B ⊗ A X ⊗ A X ⊗ A X ⊗ A X A A A X B ⊗ X A A A A (9 ′ ) (2.11)(1.1)(1.1) + (1.3) m ⊗ X e ⊗ κ me ⊗ e ⊗ κ A ⊗ m mm ⊗ A ∆ ⊗ e ⊗ κ (1 B ) ⊗ m ( u X · ǫ ) ⊗ e ⊗ A A ⊗ ( e · α ) ⊗ ( κ · λ ) 1 X ⊗ m X ⊗ λ m A ⊗ ( e · α ) ⊗ ( κ · λ ) 1 A ⊗ m λ ⊗ λ m B ⊗ m e ⊗ A B ⊗ e ⊗ κ B ⊗ m B ⊗ λ e ⊗ κ mλλ ∆ ⊗ A e ⊗ e ⊗ A e ⊗ e ⊗ κ B ⊗ e ⊗ κ ( e · α ) ⊗ ( κ · λ )0 FLORENCE STERCK B ⊗ X ( B ⊗ X ) A A .B ⊗ X B ⊗ X ( B ⊗ X ) ⊗ X A A A A X XB ⊗ X A A AA A AB ⊗ X B ⊗ X A A A A (1) + (2)(9 ′ ) (2.11) B ⊗ σ ⊗ X ( e ⊗ κ ) m ⊗ m ∆ ⊗ ∆ ⊗ X B ⊗ σ ⊗ X ( e ⊗ κ ) ⊗ κ A ⊗ ( e · α ) ⊗ ( κ · λ ) m ⊗ m ⊗ A A ⊗ m λ m B ⊗ m e ⊗ κ A ⊗ m me ⊗ κ m ⊗ A m λ ∆ ⊗ A A ⊗ ( e · α ) ⊗ ( κ · λ )∆ ⊗ A A ⊗ σ ⊗ A ∆ ⊗ X mm ⊗ m ⊗ A e ⊗ κ ⊗ κ A ⊗ ( e · α ) ⊗ ( κ · λ ) (cid:3) This lemma implies that the action of bialgebras defined by (2.10) ( ⊲ = λ · m · ( e ⊗ κ )) satisfies the conditions(3.2) and (3.3). Hence, this action becomes an action of Hopf algebras since the conditions (3.4) and (3.5)are given by the conditions (10’) and (11’) . Remark 3.5.
The construction of the action of Hopf algebras given by ⊲ = λ · m · ( e ⊗ κ ) can be reformulatedwithout λ , when we compose it by κ . Indeed, by pre-composing by (1 B ⊗ X ⊗ e ) · (1 B ⊗ X ⊗ S R ) · (1 B ⊗ σ ) · (∆ ⊗ X ) and post-composing by m , the two components of the equality (2.13), we obtain the followingequality(3.13) κ · ⊲ = m · ( m ⊗ A ) · ( e ⊗ κ ⊗ e ) · (1 B ⊗ X ⊗ S R ) · (1 B ⊗ σ ) · (∆ ⊗ X ) . We notice that thanks to the condition (8 ′ ), this is equivalent to κ · ⊲ = m · (1 A ⊗ m ) · ( e ⊗ κ ⊗ e ) · (1 B ⊗ X ⊗ S R ) · (1 B ⊗ σ ) · (∆ ⊗ X ) . When the symmetric monoidal category is
Vect K , κ can be viewed as an inclusion and (3.13) give us a wayto construct the action without λ .The actions of Hopf algebras (Definition 3.2) endowed with the morphisms of actions of bialgebras (Defi-nition 2.18), where the pair is a pair of morphisms of Hopf algebras, form the category Act ( Hopf ) of actionsof Hopf algebras. The split extensions of Hopf algebras with the morphisms given by Definition 2.7 , wherethe triple is a triple of morphisms of Hopf algebras, form the category
SplitExt ( Hopf ) of split extensions Hopfalgebras.
Theorem 3.6.
There is an equivalence between the category
SplitExt ( Hopf ) of split extensions of Hopf algebrasand the category Act ( Hopf ) of actions of Hopf algebras.Proof. Let ( g, v, p ) be a morphism in
SplitExt ( Hopf ), then it is clear that ( v, g ) is a morphism in
Act ( Hopf ). Onthe other hand, if ( v, g ) is a morphism of actions of Hopf algebras, the triple ( g, v, v ⊗ g ) is a morphism of splitextensions of Hopf algebras since v ⊗ g preserves the antipode. Moreover, the isomorphisms ϕ := m · ( κ ⊗ e )and ψ := ( λ ⊗ α ) · ∆ (in (2.14)) form an isomorphism in SplitExt ( Hopf ) since they are morphisms of Hopfalgebras, as we can see in the following diagram
PLIT EXTENSIONS AND ACTIONS OF BIALGEBRAS AND HOPF ALGEBRAS 31 A A A A,A A X ⊗ B B ⊗ X B ⊗ X B ⊗ X B ⊗ X ⊗ B X ⊗ B (2.13) S L mσ S L ⊗ S L ∆ ⊗ X B ⊗ σ ⊲ ⊗ B κ ⊗ e κ ⊗ ee ⊗ κ mm where the left square commutes since S L is an antihomomorphism of algebras (a similar computation holdsfor S R ). In conclusion, we obtain our statement thanks to the observations about the split extension (3.10)and Lemma 3.4. (cid:3) Remark 3.7.
Whenever, we consider Hopf algebras such that S L = S R , the semi-direct product in (3.10)also satisfies this property ( S X ⋊ B L = S X ⋊ B R ) and Theorem 3.6 can be restricted to such Hopf algebras.Moreover, the condition (11 ′ ) in Definition 3.1 is trivially satisfied thanks to (10’) and (3.11), and as we havealready noticed the condition (3.5) in Definition 3.2 always holds.In the case of associative Hopf algebras, we can define SplitExt ( AssHopf ) and
Act ( AssHopf ). The actionsof associative Hopf algebras are actions of Hopf algebras where the condition (3.4) and (3.5) always holdsthanks to the uniqueness of the antipode. A split extension of associative Hopf algebras is the same as in
SplitExt ( Hopf ) where the conditions (6 ′ ), (7 ′ ), (8 ′ ) and (9 ′ ) become trivial. Moreover, the conditions (10’)and (11’) are not required, they become properties that any split extension of associative Hopf algebras has. Corollary 3.8.
There is an equivalence between the category
SplitExt ( AssHopf ) of split extensions of asso-ciative Hopf algebras and the category Act ( AssHopf ) of actions of associative Hopf algebras. Let us notice that, since any morphism in
SplitExt ( Hopf ), SplitExt ( AssHopf ) and
SplitExt ( AssBiAlg ) is amorphism in
SplitExt ( BiAlg ), the Split Short Five Lemma also holds in these categories.In some sense, this result, in the associative Hopf algebras, is similar to a property obtained for the exactcleft sequences of associative Hopf K -algebras (with bijective antipodes) investigated by [3] (see Lemma3.2.19). We would like to emphasize the differences and shared properties between the definition of an exactcleft sequence of associative Hopf algebras and the definition of split extension of associative Hopf algebras(Definition 3.1), in the symmetric monoidal category Vect K of vector spaces. First, we recall the definitionof an exact cleft sequence of associative Hopf algebras [2]. Definition 3.9.
The sequence of morphisms of associative Hopf algebras(3.14) C ′ B ′ A ′ ι π , is exact if1) ι is injective,2) π is surjective,3) ker ( π ) = C ′ ι ( A ′ ) + ( ker ( π ) is the kernel in Vect and ι ( A ′ ) + = { x ∈ ι ( A ′ ) | ǫ ( x ) = 0 } ) ,4) ι ( A ′ ) = LKer ( π ) = { x ∈ C ′ | ( π ⊗ C ′ ) · ∆( x ) = u B ′ ⊗ x } . Definition 3.10.
Let (3.14) be an exact sequence of associative Hopf algebras, then the sequence(3.15) C ′ B ′ A ′ ι πχξ , is cleft if and only if there exist a morphism of A ′ -modules ξ : C ′ → A ′ (i.e. the equality ξ · m · ( ι ⊗ C ′ ) = m · (1 A ′ ⊗ ξ ) holds) and a morphism of B ′ -comodules χ : B ′ → C ′ (i.e. the equality ( π ⊗ C ′ ) · ∆ ⊗ χ = (1 B ′ ⊗ χ ) · ∆is satisfied) such that the following two equations hold(3.16) ξ · χ = u A ′ · ǫ, (3.17) m · (( ι · ξ ) ⊗ ( χ · π )) · ∆ = 1 C ′ . Remark that in [2, 3] the above definition is not the definition of an exact cleft sequence, but it isequivalent to it by Lemma 3.1.14 in [2], It is straightforward to observe that the conditions (3.16) and(3.17) of the sequence (3.15) are the same as the conditions (2’) and (3’) in Definition 3.1. Moreover, let
A BX λκ αe be a split extension of associative Hopf algebras, then λ is a X -module morphism(thanks to Proposition 2.9) and e is a B -comodule morphism.However, there are major differences. Indeed, a split extension of associative Hopf algebras (Definition3.1) is (in general) not exact in the sense of [2]. Conversely, the Hopf algebra morphism π in the exact cleftsequence (3.15) is not a split epimorphism of Hopf algebras, since χ is neither a morphism of algebras nora morphism of coalgebras (see [3] for such an example). Then, it is clear that one definition does not implythe other and vice versa. Nevertheless, there are sequences of associative Hopf algebras that are exact cleftsequences and split extensions of associative Hopf algebras. For example, any exact sequence of associativeHopf algebras (3.14) such that π is a split epimorphism is an example of the both definitions (see example2) in V ect K ).To end this paper, we investigate the two main symmetric monoidal categories of interest Set and
Vect K .On one hand, we specify our results in Set . Split extensions of Hopf algebras in the category of sets.
1) Any split extension of groups (0.1) is a split extension of associative Hopf algebras when the sym-metric monoidal category is
Set . In particular, Corollary 3.8 becomes the well-known equivalence ofcategories between split extensions of groups and group actions.2) In
Set , non-associative Hopf algebras will be structures given by a set G , with a non-associativemultiplication, a neutral element 1, left inverses and right inverses such that(3.18) g − L g = 1 = gg − R . In particular, the non-zero octonions [10, 15] are equipped with a non-associative multiplicationsatisfying (3.18). More generally, any loop satisfies (3.18). We can describe what split exten-sions of this algebraic structure should be in order to be equivalent to actions of such an alge-braic structure. Indeed, a split extension should be a split morphism of these algebraic struc-tures
A BX κ αe such that the following conditions are satisfied for any a ∈ A, b ∈ B, x ∈ X (3’) ( a ( e · α )( a − R ))( e · α )( a ) = a (6’) ( κ ( x ) e ( b )) a = κ ( x )( e ( b ) a )(7’) ( κ ( x ) a ) e ( b ) = κ ( x )( ae ( b ))(8’) a ( κ ( x ) e ( b )) = ( aκ ( x )) e ( b )(9’) ( e ( b ) a ) κ ( x ) = e ( b )( aκ ( x ))(10’) e ( b − R ) − L = e ( b )(11’) (cid:16) e ( b − R ) (cid:16) e ( b − R ) − R (cid:0) x − R e ( b − R ) (cid:1)(cid:17)(cid:17) e ( b − R ) − R = x − R .The other conditions are trivially satisfied with λ ( a ) = a ( e · α )( a − R ).On the other hand, in the symmetric monoidal category Vect K of vector spaces over a field K , we give someparticular cases of associative split extensions of Hopf algebras. Split extensions of Hopf algebras in the category of vector spaces.
1) We consider a split epimor-phism α of associative Hopf K -algebras,(3.19) A BHKer ( α ) κ α αe , PLIT EXTENSIONS AND ACTIONS OF BIALGEBRAS AND HOPF ALGEBRAS 33 where
HKer ( α ) is the kernel of α in the category Hopf K of Hopf K -algebras, κ α stands for the equalizer in Vect K of (1 A ⊗ u B ⊗ A ) · ∆ and (1 A ⊗ α ⊗ A ) · (∆ ⊗ A ) · ∆,(3.20) A A ⊗ B ⊗ AHKer ( α ) (1 A ⊗ α ⊗ A ) · (∆ ⊗ A ) · ∆ κ α (1 A ⊗ u B ⊗ A ) · ∆ . We recall that the equalizer in
Vect K of f, g : A → B is given by { a ∈ A | f ( a ) = g ( a ) } . We also define thefollowing equalizers A B ⊗ ALKer ( α ) ( α ⊗ A ) · ∆ κ α,L u B ⊗ A , A A ⊗ BRKer ( α ) (1 A ⊗ α ) · ∆ κ α,R A ⊗ u B . Proposition 3.11.
A split epimorphism (3.19) satisfying the condition
HKer ( α ) = LKer ( α ) is an extensionof associative Hopf algebras (Definition 3.1) in the symmetric monoidal category Vect K .Proof. First, we recall that for any morphism α in AssHopf K it is well-known that the following conditionsare equivalent (see [2]) • HKer ( α ) = LKer ( α ), • HKer ( α ) = RKer ( α ), • LKer ( α ) = RKer ( α ), • LKer ( α ) is an associative Hopf algebra, • RKer ( α ) is an associative Hopf algebra.Let A B αe , be a split epimorphism of associative Hopf algebras satisfying the condition HKer ( α ) = LKer ( α ), since A is an associative Hopf algebra we can define the following section of κ α : HKer ( α ) → A λ = m · (1 A ⊗ ( S · e · α )) ⊗ ∆ . First, we use the condition
HKer ( α ) = RKer ( α ) on the kernel to prove that λ factors through HKer ( α ), A A A ⊗ BA ⊗ B A A A A A A A A A A A A A (1.3)(1.2) (1.4)(1.8) A ⊗ ( S · e · α ) m ⊗ u B A ⊗ A ⊗ ∆ (1 A ⊗ ( S · e · α )) m ⊗ m ∆ 1 A ⊗ ( S · e · α ) mm ⊗ m ∆ ∆ 1 A ⊗ ∆1 A ⊗ σ A ⊗ ( S · e · α ) ⊗ ǫ ∆ ⊗ ∆1 A ⊗ A ⊗ σ A ⊗ ( S · e · α ) ⊗ α ⊗ ( S · α )1 A ⊗ σ ⊗ A ∆ ⊗ ∆1 A ⊗ σ ⊗ A (1 A ) ⊗ α ∆1 A ⊗ α . By using that λ factors through RKer ( α ) = HKer ( α ), we prove that ∆ · λ = ( λ ⊗ λ ) · ∆. Indeed, the centralrectangle commutes since RKer ( α ) = HKer ( α ), the commutativity of the part (A) is clarified in the seconddiagram A A A A A A A ⊗ B ⊗ AA A A A A A A AA A ( A ) (1.1) λ ⊗ λ ∆ 1 A ⊗ ∆ 1 A ⊗ α ⊗ A ∆ 1 A ⊗ ( S · e · α ) mλ m ⊗ A A ⊗ ( S · e ) ⊗ A ∆∆1 A ⊗ u B ⊗ A ∆1 A ⊗ ( S · e · α ) m ∆ .The diagram (A) commutes as we can see in the following diagram PLIT EXTENSIONS AND ACTIONS OF BIALGEBRAS AND HOPF ALGEBRAS 35 A A A .A A A A A A A A A A A A A ⊗ B ⊗ A A ⊗ B ⊗ AA A A A A X A A A A A A A A A A A A A A (1.4)( ass )(1.3)(1.2) (1.1)(1.8) (1 A ⊗ ( S · e · α )) m ⊗ m (1 A ) ⊗ u A ⊗ (1 A ) σ A ,A ⊗ (1 A ) m ⊗ A ⊗ mm ⊗ (1 A ) ⊗ m A ⊗ m ⊗ A A ⊗ σ ⊗ (1 A ) m ⊗ (1 A ) ⊗ m A ⊗ m ⊗ A (1 A ) ⊗ ∆ ⊗ (1 A ) (1 A ⊗ ( S · e · α )) m ⊗ m ⊗ m (1 A ) ⊗ ( S · e · α ) ⊗ A ⊗ ( S · e · α )1 A ⊗ σ (1 A ) ⊗ ∆ (1 A ⊗ ( S · e · α )) (1 A ) ⊗ σ (1 A ) ⊗ ( S · e · α ) m ⊗ mm A ⊗ ( S · e · α )∆∆∆ ⊗ ∆ ∆ ⊗ A ∆ ⊗ ∆(1 A ) ⊗ ǫ ⊗ (1 A ) (1 A ⊗ ( S · e · α )) (1 A ) ⊗ m ⊗ A A ⊗ ( S · e · α ) ⊗ ( e · α ) ⊗ ( S · e · α ) ⊗ A ⊗ ( S · e · α )∆ ⊗ (1 A ) A ⊗ σ ⊗ A A ⊗ σ ⊗ A A ⊗ σ ⊗ (1 A ) A ⊗ ( S · e · α ) ⊗ ( S · S · e · α ) ⊗ A ⊗ ( S · e · α )(1 A ) ⊗ σ ⊗ (1 A ) A ⊗ m ⊗ (1 A ) A ⊗ σ ⊗ (1 A ) A ⊗ σ ⊗ A A ⊗ ( S · e ) ⊗ A m ⊗ ( α ) ⊗ m ∆∆ ⊗ A A ⊗ α ⊗ A A ⊗ ( S · e ) ⊗ A m ⊗ A m ⊗ A ∆ ⊗ ∆ ⊗ (1 A ) ∆ ⊗ S ⊗ (1 A ) (1 A ) ⊗ S ⊗ (1 A ) ∆ ⊗ ∆ The condition (3 ′ ) is trivially respected. The condition (4 ′ ) is also satisfied by this definition of λ thanks tothe commutativity of the following diagram, where we use that HKer ( α ) = RKer ( α ), HKer ( α ) ⊗ B A A.A ⊗ B A HKer ( α ) ⊗ B A A A A A A ⊗ B A ⊗ B ⊗ A A ⊗ B ⊗ A A A ⊗ B A ⊗ B ( A ⊗ B ) A A A HKer ( α ) ⊗ B A A (1.1)(1.1)(1.8) HKer ( α ) = RKer ( α ) ( ass )(1.4) κ α ⊗ ǫκ α ⊗ ǫ ∆ 1 A ⊗ u A ⊗ ( S · e · α )1 A ⊗ ( S · e ) m ⊗ m A ⊗ σ A ⊗ ∆ ⊗ A A ⊗ e ⊗ A ∆ ⊗ ∆ 1 A ⊗ σ ⊗ A A ⊗ σ ⊗ A m ⊗ mκ α ⊗ e mκ α ⊗ B (1 A ) ⊗ ( S · e · α ) ∆ ⊗ B A ⊗ e ⊗ ( S · e ) ⊗ ( S · e · α )∆ ⊗ ∆1 A ⊗ ǫ ⊗ A A ⊗ B ⊗ σ (1 A ) ⊗ σ A ⊗ m ⊗ A A ⊗ m A ⊗ α A ⊗ u B (1 A ⊗ e ) ∆ m A ⊗ ( S · e · α ) m The last condition (5 ′ ) is left to the reader, to prove it we use the fact that λ · m · ( e ⊗ κ α ) factors through HKer ( α ).To conclude, it is a split extension of associative Hopf algebras. (cid:3) Notice that this proposition can be extended to any symmetric monoidal category with equalizers that arepreserved by all endofunctors on C of the form − ⊗ X and X ⊗ − .2) In the symmetric monoidal category ( Vect K , ⊗ , K ), an exact sequence of associative Hopf algebras(Definition 3.9)(3.21) C ′ B ′ A ′ ι πe , such that π is a split epimorphism of Hopf algebras, is an exact cleft sequence and a split extension ofassociative Hopf algebras. Indeed, since the condition 4) in Definition 3.9 is equivalent to the condition LKer ( π ) = HKer ( π ) [2], this example is a particular case of Proposition 3.11. Due to Definition 3.9 thesequence (3.21) has to be isomorphic to the following one, C ′ C ′ C ′ ( LKer ( π )) + HKer ( π ) κ π πe ,where HKer ( π ) = LKer ( π ).3) If we consider cocommutative associative Hopf K -algebras, then we can drop the condition HKer ( α ) = RKer ( α ) in the Proposition 3.11. So any split epimorphism of cocommutative associative Hopf algebrasinduces a split extension as defined in 3.1 (and an exact cleft sequence). The Corollary 3.8 becomes thewell-known equivalence between points over B and B -module Hopf algebras [25]. PLIT EXTENSIONS AND ACTIONS OF BIALGEBRAS AND HOPF ALGEBRAS 37 conclusion and perspectives To sum up, we defined the category
SplitExt ( BiAlg ) and proved that this category is equivalent to thecategory
Act ( BiAlg ). Moreover, we proved that a suitable version of the Split Short Five Lemma holds whenthe split extensions occurring in it belong to the category
SplitExt ( BiAlg ). These results were proved to holdalso in the category of Hopf algebras, and we gave some examples of split extensions of Hopf algebras in thecategories of sets and of vector spaces. It is worthwhile to observe that any isomorphism γ : A → A of Hopfalgebras determines a split extension in the sense of Definition 3.1 as indicated in the following diagram A AI uǫ γγ − .This elementary example motivates the study of internal structures in the context of non-associative bialgebrasand Hopf algebras. Indeed, thanks to this example, a discrete reflexive graph(4.1) A AI uǫ A A A , is a reflexive graph in Hopf such that his “legs” are in
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Appendix
This appendix contains three figures given below. The monoidal product is denoted by juxtaposition. Onone hand, by combining the diagrams of Figure 1 and Figure 2, we show that the structure of X ⋊ B asdefined in (2.6) gives a bialgebra structure. On the other hand, since the diagram of Figure 3 commutes, wecan conclude that whenever X and B are associative bialgebras and (2.7) and (2.8) are satisfied m X ⋊ B asdefined in (2.6) is associative, which is a part of the proof of Lemma 2.2. Institut de Recherche en Math´ematique et Physique, Universit´e catholique de Louvain, Chemin du Cyclotron2, 1348 Louvain-la-Neuve and D´epartement de Math´ematique, Universit´e Libre de Bruxelles, Campus de la PlaineCP 210 Boulevard du Triomphe 1050 Bruxelles, Belgium