aa r X i v : . [ m a t h . QA ] F e b CATEGORICAL CONSTRUCTIONS FOR HOPF ALGEBRAS
A.L. AGORE
Dedicated to the memory of Professor Liliana Pavel
Abstract.
We prove that both, the embedding of the category of Hopf algebras intothat of bialgebras and the forgetful functor from the category of Hopf algebras to thecategory of algebras, have right adjoints; in other words: every bialgebra has a Hopfcoreflection and on every algebra there exists a cofree Hopf algebra. In this way we givean affirmative answer to a forty years old problem posed by Sweedler. On the routethe coequalizers and the coproducts in the category of Hopf algebras are explicitlydescribed.
Introduction
Hopf algebras appeared naturally in the study of Lie groups cohomology. The surveypaper [2] covers the beginnings of Hopf algebras and the roles played by H. Hopf, P.Cartier, A. Borel, J. Milnor, J. Moore, B. Konstant and M. Sweedler in the developmentof this theory. Hopf algebras became a fervid field of study especially after the appearanceof the monograph [11]. In the present paper we bring new contributions to the study ofthe category of Hopf algebras.We turn our attention to the fundamental book of Sweedler: in [11, p. 135] are stated,without any proofs, the following problems concerning Hopf algebras: given a coalgebra C there exists a free Hopf algebra on C (i.e. the forgetful functor from the category ofHopf algebras to the category of coalgebras has a left adjoint) and a free commutativeHopf algebra on C . The problem has turned out to be quite difficult: several years passeduntil Takeuchi, in [12, Sections § § A there exists a cofree Hopf algebra on A (that is the forgetful functor fromthe category of Hopf algebras to the category of algebras has a right adjoint) and acofree cocommutative Hopf algebra on A . Concerning this problem, recently H.-E. Porst[8, Corollary 4.1.4] proved that the existence of a cofree Hopf algebra on every algebraimplies the existence of a cofree cocommutative Hopf algebra on every algebra. In thepresent paper we prove Sweedler’s statement concerning the existence of a cofree Hopfalgebra on every algebra. Mathematics Subject Classification.
Key words and phrases. bialgebra, Hopf algebra, (co)product, (co)limit, (co)complete, (co)refective.The author acknowledges partial support from CNCSIS grant 24/28.09.07 of PN II ”Groups, quantumgroups, corings and representation theory”.
The paper is structured as follows. In Section 1 we introduce the notations and re-call, without proofs, some well known results pertaining to category theory that willbe intensively used throughout the paper. In Section 2 we give an explicit descriptionof coequalizers and coproducts in the category k -HopfAlg of Hopf algebras. Using theaforementioned constructions we prove, using the Special Adjoint Functor Theorem,Sweedler’s statement concerning the existence of a cofree Hopf algebra on every algebra(Theorem 2.5). 1. Preliminaries
Throughout this paper, k will be a field. Unless specified otherwise, all vector spaces,algebras, coalgebras, bialgebras, tensor products and homomorphisms are over k . Ournotation for the standard categories is as follows: k M ( k -vector spaces), k -Alg (associa-tive unital k -algebras), k -BiAlg (bialgebras over k ), k -HopfAlg (Hopf algebras over k ).We refer to [11] for further details concerning Hopf algebras.We use the standard notations for opposite and coopposite structures: A op denotes theopposite of the algebra A and C cop stands for the coopposite of the coalgebra C .Let us recall briefly some well known results from category theory, refering the readerto [4] for more details. A category C is called (co)complete if all diagrams in C have(co)limits in C . A category C is (co)complete if and only if C has (co)equalizers of allpairs of arrows and all (co)products [6, Theorem 6.10]. The category C is called locallysmall (or well-powered ) if the subobjects of each C ∈ C can be indexed by a set. Dually,the category C is colocally small (or co-well-powered ) if its dual is locally small. At somepoint we will also use, in passing, the notion of locally presentable category. More detailsregarding this type of categories can be found in [1]. All categories considered above arelocally presentable. Thus, they are cocomplete by the definition of locally presentablecategories and complete by [1, Remark 1.56]. For a more detailed discussion concerningthe completeness and cocompleteness of the above categories we refer the reader to [8],[9] and [10]. A subcategory D of C is called (co)reflective in C when the inclusion functor U : D → C has a (right)left adjoint.The following categorical result play a key role in showing that the category of Hopfalgebras is a coreflective subcategory of the category of bialgebras:
Theorem 1.1. (The Special Adjoint Functor Theorem) If C is a complete and locallysmall category with a cogenerator, then a functor G : C → D has a left adjoint if andonly if it is limit preserving. Dually, if C is a cocomplete and colocally small categorywith a generator, then a functor G : C → D has a right adjoint if and only if it is colimitpreserving. Cofree Hopf algebras generated by algebras
Recall that the forgetful functor from the category of groups to the category of monoidshas a left adjoint, the so-called anvelopant group of a monoid, and a right adjoint, whichassigns to each monoid the group of its invertible elements. Therefore, if we think ofHopf algebras as a natural generalization of groups, we may expect the same behavior in the case of the embedding functor F : k -HopfAlg → k -BiAlg from the category of Hopfalgebras to the category of bialgebras. It is well known that the above embedding functor F has a left adjoint [5, Theorem 2.6.3]. We shall prove in this section that F has also aright adjoint. This result together with that fact that there exist a cofree bialgebra onevery algebra led us to the conclusion that the forgetful functor F : k -HopfAlg → k -Alghas a right adjoint, as stated in [11]. In order to prove our main result we need somepreparations. We start by recalling, for a further use, the constructions of coequalizersand coproducts in the category k -BiAlg of bialgebras.Let ( A, m A , η A , ∆ A , ε A ), ( B, m B , η B , ∆ B , ε B ) be two bialgebras and f , g : B → A betwo bialgebra maps. Consider I the two-sided ideal generated by { f ( b ) − g ( b ) | b ∈ B } .By a simple computation it can be seen than I is also a coideal. Then ( A/I, π ) is thecoequalizer of the morphisms ( f, g ) in k -BiAlg, where π : A → A/I is the canonicalprojection. Indeed for all bialgebras H and all bialgebra morphisms h : A → H suchthat h ◦ f = h ◦ g we obtain I ⊆ kerh , hence there exists an unique bialgebra map h ′ : A/I → H such that h ′ ◦ π = h . Remark 2.1.
Note that if A , B are two Hopf algebras and f, g : B → A are Hopfalgebra maps then the ideal I defined above is actually a Hopf ideal and ( A/I, π ) is thecoequalizer of the morphisms ( f, g ) in k -HopfAlg, where π : A → A/I is the canonicalprojection.Next, we recall from [5] the construction of the coproduct in the category k -BiAlg ofbialgebras. Let ( A l ) l ∈ I be a family of algebras, (cid:0)L l ∈ I A l , ( j l ) l ∈ I (cid:1) be the coproduct in k M and i : L l ∈ I A l → T (cid:0)L l ∈ I A l (cid:1) be the canonical inclusion where T (cid:0)L l ∈ I A l (cid:1) is thetensor algebra of the vector space L l ∈ I A l . Then (cid:0)` l ∈ I A l := T (cid:0)L l ∈ I A l (cid:1) /L, ( q l ) l ∈ I (cid:1) isthe coproduct of the above family in k -Alg, where L is the two sided ideal in T (cid:0)L l ∈ I A l (cid:1) generated by the set J := { i ◦ j l ( x l y l ) − i (cid:0) j l ( x l ) (cid:1) i (cid:0) j l ( y l ) (cid:1) , T (cid:0) L A l (cid:1) − i ◦ j l (1 A l ) | x l , y l ∈ A l , l ∈ I } , ν : T (cid:0)L l ∈ I A l (cid:1) → T (cid:0)L l ∈ I A l (cid:1) /L denotes the canonical projection and q l = ν ◦ i ◦ j l for all l ∈ I . Furthermore, ` l ∈ I A l is actually a bialgebra provided that ( A l ) l ∈ I is a family of bialgebras. The comultiplication and the counit are given by the uniquealgebra maps such that the following diagrams commute:(1) A l q l / / ( q l ⊗ q l ) ◦ ∆ l & & NNNNNNNNNNNN ` l ∈ I A l ∆ (cid:15) (cid:15) ` l ∈ I A l ⊗ ` l ∈ I A A l q l / / ε l GGGGGGGGGG ` l ∈ I A lε (cid:15) (cid:15) k and ` l ∈ I A l is the coproduct in k -BiAlg of the above family of bialgebras ([5, Corollary2.6.2]).Now let (cid:0) H l , m l , η l , ∆ l , ε l , S l (cid:1) l ∈ I be a family of Hopf algebras. Consider (cid:0) ( H := ` l ∈ I H l ,m, η, ∆ , ε ) , ( q l ) l ∈ I (cid:1) the coproduct of the above family in the category k -BiAlg of bialge-bras.The universal property of the coproduct yields an unique bialgebra map S : H → H opcop A.L. AGORE such that the following diagram commutes for all l ∈ I :(2) H l q l / / S l (cid:15) (cid:15) H S (cid:15) (cid:15) H lopcop q l / / H opcop With the notations above we have the following result which provides a completelydescription of the coproducts in the category k -HopfAlg of Hopf algebras: Theorem 2.2.
Let (cid:0) H l , m l , η l , ∆ l , ε l , S l (cid:1) l ∈ I be a family of Hopf algebras. The Hopfalgebra (cid:0) H := ` l ∈ I H l , m, η, ∆ , ε, S (cid:1) together with structure maps ( q l ) l ∈ I is the coproductin the category k -HopfAlg of the family (cid:0) H l , m l , η l , ∆ l , ε l , S l (cid:1) l ∈ I of Hopf algebras. Inparticular, the category k -HopfAlg is cocomplete.Proof. We will first prove that S is an antipode for the bialgebra H , i.e.(3) m ◦ (cid:0) Id ⊗ S (cid:1) ◦ ∆ = m ◦ (cid:0) S ⊗ Id (cid:1) ◦ ∆ = η ◦ ε Since S : H → H opcop defined in (2) is a bialgebra map we only need to prove that (3)holds only on the generators of H as an algebra. Indeed, let h, k be generators in H forwhich (3) holds. We obtain :( hk ) (1) S (( hk ) (2) ) = h (1) k (1) S ( k (2) ) S ( h (2) ) = ε ( k ) h (1) S ( h (2) ) = ε ( h ) ε ( k )1 H = ε ( hk )1 H It follows from here that (3) also holds for kh and thus it holds for all elements in H .Now having in mind that H := T (cid:0)L l ∈ I H l (cid:1) /L we only need to prove (3) for the elements b x ∈ H with x ∈ L l ∈ I H l , whereas the tensor algebra T (cid:0)L l ∈ I H l (cid:1) is the free algebra on L l ∈ I H l . Moreover, since L l ∈ I H l = { x ∈ Q l ∈ I H l | supp ( x ) < ∞} it is enough toshow that (3) holds for all x l ∈ H l , l ∈ I . We then have: m ◦ (cid:0) Id ⊗ S (cid:1) ◦ ∆( b x l ) = m ◦ (cid:0) Id ⊗ S (cid:1) ◦ ∆ ◦ q l ( x l ) ( ) = m ◦ (cid:0) Id ⊗ S (cid:1) ◦ ( q l ⊗ q l ) ◦ ∆ l ( x l )= m ◦ (cid:0) q l ⊗ ( S ◦ q l ) (cid:1) ◦ ∆ l ( x l ) ( ) = m ◦ (cid:0) q l ⊗ ( q l ◦ S l ) (cid:1) ◦ ∆ l ( x l )= m ◦ ( q l ⊗ q l ) ◦ ( Id ◦ S l ) ◦ ∆ l ( x l ) q l − algebra map = q l ◦ m l ◦ ( Id ◦ S l ) ◦ ∆ l ( x l )= q l ◦ η l ◦ ε l ( x l ) q l − algebra map = η ◦ ε l ( x l ) q l − coalgebra map = η ◦ ε ◦ q l ( x l )= η ◦ ε ( b x l )Hence m ◦ (cid:0) Id ⊗ S (cid:1) ◦ ∆ = η ◦ ε . In the same way it can be proved that m ◦ (cid:0) S ⊗ Id (cid:1) ◦ ∆ = η ◦ ε .Thus S is an antipode for H , as desired. Now since k -HopfAlg is a full subcategory of the category k -BiAlg it follows that (cid:0) ( H := ` l ∈ I H l , m, η, ∆ , ε ) , ( q l ) l ∈ I (cid:1) is also the coproduct of the family (cid:0) H l , m l , η l , ∆ l , ε l , S l (cid:1) l ∈ I of Hopf algebras in the category k -HopfAlg. (cid:3) We need the following well known result:
Theorem 2.3. ( [11, page 134] ) The forgetful functor F : k − BiAlg → k − Alg has aright adjoint, i.e. there exists a cofree bialgebra on every algebra.
Our main results now follow:
Theorem 2.4.
The embedding functor F : k − Hopf Alg → k − BiAlg has a rightadjoint, i.e. the category of Hopf algebras is a coreflective subcategory of the category ofbialgebras.Proof.
We apply the Special Adjoint Functor Theorem (Theorem 1.1): since k -HopfAlgis a locally presentable category by [8, 4.3.1 and 4.1.3] (although epimorphisms of Hopfalgebras are not necessarily surjective maps [3]), this category in particular has a gener-ator and is cocomplete and is colocally small. By Theorem 2.2 the result follows. (cid:3) Theorem 2.5.
The forgetful functor F : k − Hopf Alg → k − Alg has a right adjoint,i.e. there exists a cofree Hopf algebra on every algebra.Proof.
It follows from Theorem 2.3 and Theorem 2.4 by composing the right adjointfunctors. (cid:3)
We proved, using the Special Adjoint Functor Theorem, the existence of a cofree Hopfalgebra on every bialgebra without indicating explicitly his construction. The followingnatural problem arises:
Problem:
Give an explicit construction of the cofree Hopf algebra on an bialgebra (resp.algebra).
We expect that the right adjoint of the embedding functor from the category of Hopfalgebras to the category of bialgebras to assign to every bialgebra B his ”biggest” sub-bialgebra H that has an antipode. Acknowledgements
The author wishes to thank Professor Gigel Militaru, who suggested the problem studiedhere, for his great support and for the useful comments from which this manuscript hasbenefitted, as well as the referee for valuable suggestions and for indicating the papers[9] and [10].
A.L. AGORE
References [1] J. Ad´amek, J. Rosick´y, Locally Presentable and Accessible Categories, Cambridge University Press,1994[2] N. Andruskiewitsch, W. F. Santos, The beginnings of the theory of Hopf algebras, to appear in
ActaAppl. Math. [3] A. Chirvasitu, On epimorphisms and monomorphisms of Hopf algebras, to appear in
J. Algebra ∼ ∼ pareigis/Vorlesungen/01WS/advalg.pdf[7] B. Pareigis, M.E. Sweedler, On generators and cogenerators, Manuscripta Math. , (1970), 49-66[8] H.-E. Porst, Universal constructions for Hopf algebras, J. Pure Appl. Algebra , (2008), 2547-2554[9] H.-E. Porst, Fundamental constructions for coalgebras, corings and comodules, Appl. Categor.Struct. , (2008), 223-238[10] H.-E. Porst, On corings and comodules, Arch. Math. (Brno) , (2006), 419-425[11] M.E. Sweedler, Hopf Algebras, Benjamin New York, 1969[12] M. Takeuchi, Free Hopf algebras generated by coalgebras, J. Math. Soc. Japan (1971), 561-582 Department of Mathematics, Academy of Economic Studies, Piata Romana 6, RO-010374Bucharest 1, Romania
E-mail address ::