ContemporaryMathematics Infinitesimal Hopf algebras Marcelo Aguiar Abstract. Infinitesimal bialgebras were introduced by Joni and Rota [J-R]. An infinitesimalbialgebra isat the sametimean algebra anda coalgebra, in such a way that the comultiplication is a derivation. In this paper we de- fineinfinitesimalHopfalgebras,developtheirbasictheoryandpresentseveral examples. It turns out that many properties of ordinary Hopf algebras possess an infinitesimal version. We introduce bicrossproducts, quasitriangular infinites- imal bialgebras, the corresponding infinitesimal Yang-Baxter equation and a notionofDrinfeld’sdoubleforinfinitesimalHopfalgebras. 1. Introduction An infinitesimal bialgebra is a triple (A,m,∆) where (A,m) is an associative algebra,(A,∆) is a coassociative coalgebra and for each a,b∈A, ∆(ab)= ab1⊗b2+ a1⊗a2b . InfinitesimalbialgebraswereintroXducedbyJonXiandRota[J-R]inorderto provide analgebraicframeworkforthecalculusofdivideddifferences. Severalnewexamples are introduced in section 2. In particular, it is shown that the path algebra of an arbitrary quiver admits a canonical structure of infinitesimal bialgebra. In this paper we define the notion of antipode for infinitesimal bialgebras and develop the basic theory of infinitesimal Hopf algebras. Surprisingly, many of the usual properties of ordinary Hopf algebras possess an infinitesimal version. For instance, the antipode satisfies S(xy)=−S(x)S(y) and S(x1)⊗S(x2)=− S(x)1⊗S(x)2, among other properties (section 3).X X The existence of the antipode is closely related to the possibility of exponenti- ating a certain canonical derivation D :A →A that is carried by any ǫ-bialgebra. This and other related results are discussed in section 4. 2000 Mathematics Subject Classification. Primary: 16W30, 16W25. Keywordsandphrases. Hopfalgebras,derivations,Yang-Baxterequation,Drinfeld’sdouble. ResearchsupportedbyaPostdoctoralFellowshipattheCentredeRecherchesMath´ematiques andInstitutdesSciences Math´ematiques, Montr´eal,Qu´ebec. (cid:13)c0000 (copyright holder) 1 2 M. AGUIAR In section 6 we introduce the analog of “matched pairs” of groups or Hopf al- gebrasforassociativealgebras,andthe correspondingbicrossproductconstruction. Some interesting examples are given. Recall that a Lie bialgebra is a triple (g,[,],δ) where (g,[,]) is a Lie algebra, (g,δ)isaLiecoalgebraandδ :g→g⊗gisaderivation(intheLiesense). Therefore, infinitesimalbialgebrasmayalsobe seenasanassociativeanalogofLie bialgebras. Thisanalogyisreinforcedinsection5whereweintroducequasitriangularinfinites- imal bialgebras and the corresponding associative Yang-Baxter equation: r13r12−r12r23+r23r13 =0 for r ∈A⊗A . Again, most properties of ordinary quasitriangular bialgebras and Hopf algebras admit an analog in the infinitesimal context. For instance the antipode satisfies (S⊗S)(r)=r=(S−1⊗S−1)(r) . Butperhapsthemostimportantofthesepropertiesisthefactthatthereisanotion ofDrinfeld’s doubleforinfinitesimalbialgebras,satisfyingallthepropertiesonecan expect. Drinfeld’s double is defined and studied in section 7. It is an important example of the bicrossproduct construction of section 6. Recall that the underlying space of the double of a Lie bialgebra g and of an ordinary Hopf algebra H is respectively D(g)=g⊕g∗ and D(H)=H⊗H∗ . The underlying space of the double of an ǫ-bialgebra A turns out to be D(A)=(A⊗A∗)⊕A⊕A∗ Thisisyetanothermanifestationofthefactthatthetheoryofǫ-bialgebraspossesses aspects of both theories of Lie and ordinary bialgebras. Further connections be- tweenLie andinfinitesimal bialgebras,aswell asa deeper study ofbicrossproducts and quasitriangular infinitesimal bialgebras, will be presented in [A2]. An important motivation for studying infinitesimal Hopf algebra arises in the study of the cd-index of polytopes in combinatorics. Related examples will be presentedinthispaperbutthemainapplication(analgebraicproofoftheexistence of the cd-index of polytopes) will be presented in [A1]. One of these examples is providedbytheinfinitesimalHopfalgebraofallnon-trivialposets. Thisisdiscussed to some extent in sections 2 and 4. It is often assumed that all vector spaces and algebras are over a fixed field k. Sum symbols are often omitted from Sweedler’s notation: we write ∆(a) = a1⊗a2 when∆ is a coassociativecomultiplication. Compositionofmaps is writtensimply as fg. The symbol ◦ is reservedfor the circular product on an algebra (section 3). The author thanks Steve Chase for many fruitful conversations during the preparationof this work. 2. Infinitesimal bialgebras. Basic properties and examples Definition 2.1. Aninfinitesimalbialgebra(abbreviatedǫ-bialgebra)isatriple (A,m,∆) where (A,m) is an associative algebra (possibly without unit), (A,∆) is a coassociative coalgebra (possibly without counit) and, for each a,b∈A, (2.1) ∆(ab)=ab1⊗b2+a1⊗a2b . INFINITESIMAL HOPF ALGEBRAS 3 Condition 2.1 can be written as follows: ∆m=(m⊗idA)(idA⊗∆)+(idA⊗m)(∆⊗idA) Equivalently,∆:A→A⊗Ais aderivationofthe algebra(A,m)withvalues onthe A-bimodule A⊗A, or m : A⊗A →A is a coderivation [Doi] from the A-bicomodule A⊗A with values on the coalgebra (A,∆). Here A⊗A is viewed as A-bimodule via a·(x⊗y)= ax⊗y and (x⊗y)·b = x⊗yb. Dually, A⊗A is an A-bicomodule via A⊗A −∆−⊗−i−d→A A⊗(A⊗A) and A⊗A −i−dA−⊗−→∆ (A⊗A)⊗A. Remark 2.2. If an ǫ-bialgebra has a unit 1 ∈ A then ∆(1) = 0. In fact, any derivation D :A→M annihilates 1, since D(1)=D(1·1)=1·D(1)+D(1)·1= 2D(1), hence D(1)=0. If an ǫ-bialgebra has both a unit 1 ∈ A and a counit ε ∈ A∗ then A = 0. In fact, 1=(id⊗ε)∆(1)=0. Infinitesimal bialgebras were introduced by Joni and Rota (under the name infinitesimalcoalgebras)[J-R,sectionXII].EhrenborgandReaddyhavecalledthem newtonian coalgebras [E-R]. The present terminology emphasizes the analogy with thenotionofordinarybialgebras,anddoesnotfavoreitherthealgebraorcoalgebra structure over the other; as we will see next, the notion is self-dual. Since the notions of derivation and coderivation correspond to each other by duality, it follows immediately that if (A,m,∆) is a finite dimensional ǫ-bialgebra then the dual space A∗ is an ǫ-bialgebra with multiplication A∗⊗A∗ ∼=(A⊗A)∗ −∆−→∗ A∗ and comultiplication A∗ −m−→∗ (A⊗A)∗ ∼=A∗⊗A∗ . If (A,m,∆) is an arbitrary ǫ-bialgebra, then so are (A,−m,∆), (A,m,−∆), (A,−m,−∆) and also (A,mop,∆cop), where mop =mτ, ∆cop =τ∆ and τ(a⊗b)=b⊗a. In the context of Drinfeld’s double (section 7), these basic constructions will have to be combined. Examples 2.3. 1. Any algebra (A,m) becomes a ǫ-bialgebra by setting ∆ = 0. Dually, any coalgebra (A,∆) becomes an ǫ-bialgebra with m=0. 2. Let Q be an arbitrary quiver. Then the path algebra kQ carries a canonical ǫ-bialgebra structure. Recall that kQ = ⊕∞ kQ where Q is the set of n=0 n n paths γ in Q of length n: γ :e −a→1 e −a→2 e −a→3 ...e −a→n e . 0 1 2 n−1 n In particular, Q is the set of vertices and Q is the set of arrows. The 0 1 multiplicationisconcatenationofpathswheneverpossible;otherwiseiszero. The comultiplication is defined on a path γ =a a ...a as above by 1 2 n ∆(γ)=e0⊗a2a3...an+a1⊗a3...an+...+a1...an−1⊗en . In particular, ∆(e) = 0 for every vertex e ∈ Q0 and ∆(a) = s(a)⊗t(a) for every arrow a∈Q . 1 4 M. AGUIAR 3. The polynomial algebra k[x] is an ǫ-bialgebra with ∆(1)=0, ∆(xn)=xn−1⊗1+xn−2⊗x+...+x⊗xn−2+1⊗xn−1 for n≥1 . This is the path ǫ-bialgebra corresponding to the quiver x (cid:5)(cid:5) 1 as in example 2. Notice that the comultiplication can also be described as the map f(x)−f(y) ∆:k[x]→k[x,y], ∆(f(x))= ; x−y in other words, ∆(f(x)) is the Newton divided difference of f(x). For this reason,this structure was called the Newtonian coalgebra in [J-R]. Joni and Rota proposed the general notion of ǫ-bialgebra in order to axiomatize the situationofthisexample. Foralongtimethisremainedtheonlyexampleof ǫ-bialgebra appearing in the literature. The only work in the area seems to havebeenthatofHirschhornandRaphael[H-R],wheretheǫ-bialgebrak[x] was studied in detail in connection with the calculus of divided differences. 4. It was only recently that another natural example of ǫ-bialgebras arose, again in combinatorics, but in a different context (that of the cd-index of polytopes). The ǫ-bialgebra P of all non-trivial posets is defined as follows. As a vectorspace, P has a basisconsisting ofthe isomorphismclassesofallfinite posets P with top element 1 and bottom element 0 , except for the one- P P element poset {•}. Thus 0 6= 1 always. The multiplication of two such P P posets P and Q is P ∗Q= P −{1 } ∪ Q−{0 } P Q (cid:16) (cid:17) (cid:16) (cid:17) x,y ∈P and x≤y in P, where x≤y iff x,y ∈Q and x≤y in Q, or  x∈P and y ∈Q. This algebra possesses a unit element, namely the poset B ={0<1}. Moreover,P is an ǫ-bialgebra with comultiplication 1 ∆(P)= [0P,x]⊗[x,1P] . 0P<Xx<1P Here if x and y are two elements of a poset P, then [x,y] denotes the iso- morphism class of the poset {z ∈P / x≤z ≤y}. This ǫ-bialgebra was first considered by Ehrenborg and Hetyei [E-H], and further studied by Billera, Ehrenborg and Readdy in connection with the cd-index ofpolytopes [E-R,B-E-R].This study is continuedin example 4.7.3 and more deeply in [A1], where simple coalgebraic ideas are used to provide a proof of the existence of the cd-index of polytopes. 5. The free algebra A=khx ,x ,x ,...i is an ǫ-bialgebra with 1 2 3 n−1 ∆(xn)= xi⊗xn−1−i =1⊗xn−1+x1⊗xn−2+...+xn−1⊗1 , i=0 X INFINITESIMAL HOPF ALGEBRAS 5 where we set x =1. 0 6. The algebra of dual numbers k[ε]/(ε2) is an ǫ-bialgebra with ∆(1)=0, ∆(ε)=ε⊗ε . 7. The algebra of matrices A = M (k) admits many ǫ-bialgebra structures. 2 One such is a b 0 a 0 1 0 1 c d ∆ = ⊗ − ⊗ . c d 0 c 0 0 0 0 0 0 (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) Other structures on M (k) will be discussed later (examples 5.4). 2 3. Antipodes and infinitesimal Hopf algebras Recall that if an ǫ-bialgebra A possesses both a unit and a counit then A = 0 (remark2.2). Thissimpleobservationshowsthatonecannothopetodefineanotion ofantipodeforǫ-bialgebrasasonedoesforordinarybialgebrasH,sinceforthisone mustrefertoboththeunitandcounitofH. Recallthattheantipodeofanordinary bialgebra H is defined as the inverse of id in the space Hom (H,H), which is an H k algebra under the convolution product, with unit u ε (where u :k →H is the H H H unit map u (1)=1). H If A is an ǫ-bialgebra, then the space Hom (A,A) is still an algebra under k convolution, but it does not have a unit element in general. However, one may formally adjoin a unit to this algebra and then consider invertible elements. It turnsoutthatthis simplealgebraicdevicewillprovidetherightnotionofantipode for ǫ-bialgebras, as will become clear from the examples to be discussed in this work. We recall this concept next. Let R be any k-algebra, not necessarily unital. The circular product on R is a◦b=ab+a+b ItiseasytocheckdirectlythatthisturnsRintoanassociativeunitalmonoid,with unit 0 ∈ R. This can also be seen as follows: if we adjoin a unit to R to form R+ =R⊕k, with associative multiplication (a,λ)·(b,µ)=(ab+µa+λb,λµ) and unit element (0,1), then the subset {(a,1) ∈ R+ / a ∈ R} is closed under the multiplication of R+ and contains its unit. This monoid is isomorphic to R equipped with the circular product. Now let A be an ǫ-bialgebra. The space Hom (A,A) is an algebra under con- k volution f ∗g =m(f⊗g)∆ (recall that concatenation denotes composition of maps). The circular product on this (in general, nonunital) algebra will be called the circular convolution and denoted by the symbol ⊛. Explicitly, f ⊛g =f ∗g+f +g or (f ⊛g)(a)=f(a )g(a )+f(a)+g(a) . 1 2 Definition 3.1. An infinitesimal bialgebra A is called an infinitesimal Hopf algebra if the identity map id ∈ Hom (A,A) is invertible with respect to circular k convolution. In this case, the inverse S ∈ Hom (A,A) of id is called the antipode k of A. It is characterizedby the equations (A) S(a )a +S(a)+a=0=a S(a )+a+S(a) ∀ a∈A . 1 2 1 2 6 M. AGUIAR Examples 3.2. 1. The algebra of polynomials k[x] is an ǫ-Hopf algebra. The antipode is S(xn)=−(x−1)n, that is S(p(x))=−p(x−1) . In fact, since ∆(xn)= xi⊗xj, equations (A) become i+j=n−1 −Pxi(x−1)j +xn−(x−1)n =0 i+j=n−1 X which follows from the basic identity an−bn =(a−b) aibj . i+j=n−1 X Notice that S is bijective with S−1(p(x))=−p(x+1). More generally, for any m∈Z, Sm(p(x))=(−1)mp(x−m) . In particular, S has infinite order. 2. More generally, for any quiver Q the path algebra kQ is an ǫ-Hopf algebra with antipode e−a if s(a)=t(a)=e, S(e)=−e ∀ e∈Q and S(a)= 0 (−a if s(a)6=t(a). These assertions follow from a general result on the existence of antipodes (corollary 4.3, example 4.7.2). The antipode is uniquely determined by the formulas above according to proposition 3.7. 3. ThealgebraPofnon-trivialposetsisanǫ-Hopf algebra. Anexplicitformula for the antipode is: ∞ S(P)= (−1)n [0 ,x ][x ,x ]...[x ,1 ] . P 1 1 2 n−1 P nX=1 0P<x1<.X..<xn−1<1P This will discussed in detail in example 4.7.3. 4. The ǫ-bialgebra A=khx ,x ,x ,...i of example 2.3.5 is an ǫ-Hopf algebra 1 2 3 with antipode n+1 S(x )= (−1)k x x ...x n n1−1 n2−1 nk−1 kX=1 (n1,...,nkX)∈C+(n+1,k) where C+(n+1,k)={(n ,... ,n ) / n ∈Z+, n +...+n =n+1} is the 1 k i 1 k set of strict compositions of n+1 into k parts. See example 4.7.4. 5. The algebra of dual numbers (example 2.3.6) is an ǫ-Hopf algebra. The antipode is simply S = −id. The same is true for the ǫ-bialgebra M (k) of 2 example 2.3.7. 6. Not every ǫ-bialgebra possesses an antipode. Consider the following comul- tiplication on the polynomial algebra k[x]: ∆(1)=0, ∆(xn)=xn⊗x+xn−1⊗x2+...+x2⊗xn−1+x⊗xn for n>0 . INFINITESIMAL HOPF ALGEBRAS 7 It is easy to see that this endows k[x] with the structure of an ǫ-bialgebra (different from that of example 3, but closely related to its graded dual). In particular ∆(x)=x⊗x. If there were an antipode S, then we would have −x S(x)x+S(x)+x=0⇒S(x)= ∈/ k[x] 1+x which is a contradiction. Remark 3.3. In all previous examples, S(1) = −1. More generally, for any ǫ-Hopf algebra A and u∈Ker∆, S(u)=−u. In fact, equation (A) gives 0=u S(u )+u+S(u)⇒S(u)=−u . 1 2 Theantipodeofanǫ-Hopf algebrasatisfiesmanypropertiesanalogoustothose of the antipode of an ordinary Hopf algebra, which we will present next. We need some basic general results first. Lemma 3.4. Let A,B be algebras and C,D coalgebras. (a) If φ : C → D is a morphism of coalgebras then φ∗ : Hom (D,A) → k Hom (C,A), φ∗(f)=fφ, is a morphism of (circular) convolution monoids. k (b) Ifφ:A→Bisamorphismofalgebrasthenφ :Hom (C,A)→Hom (C,B), ∗ k k φ (f)=φf, is a morphism of (circular) convolution monoids. ∗ Proof. Any morphism of algebras preserves the corresponding circular prod- ucts, so it is enough to check that ordinary convolutionis preserved in either case. This is well-known. The next lemma is meaningful for nonunital algebras (or noncounital coalgebras) only,since a unital multiplicationis alwayssurjective (anda counitalcomultiplica- tion injective) . Lemma 3.5. (a) Let C and D be coalgebras, u ∈ Ker∆ and v ∈ Ker∆ . C D Then C⊗D is a coalgebra with ∆(c⊗d)=(c1⊗v)⊗(c2⊗d)+(c⊗d1)⊗(u⊗d2) . (b) Let A and B be algebras, γ ∈(CokermA)∗ and δ ∈(CokermB)∗. Then A⊗B is an algebra with (a⊗b)·(a′⊗b′)=δ(b)aa′⊗b′+γ(a′)a⊗bb′ . Proof. To prove (a) we calculate (id⊗∆)∆(c⊗d)=(c1⊗v)⊗∆(c2⊗d)+(c⊗d1)⊗∆(u⊗d2) =(c1⊗v)⊗(c2⊗v)⊗(c3⊗d)+(c1⊗v)⊗(c2⊗d1)⊗(u⊗d2)+(c⊗d1)⊗(u⊗d2)⊗(u⊗d3) , and (∆⊗id)∆(c⊗d)=∆(c1⊗v)⊗(c2⊗d)+∆(c⊗d1)⊗(u⊗d2) =(c1⊗v)⊗(c2⊗v)⊗(c3⊗d)+(c1⊗v)⊗(c2⊗d1)⊗(u⊗d2)+(c⊗d1)⊗(u⊗d2)⊗(u⊗d3) . thus (id⊗∆)∆=(∆⊗id)∆ as needed. Case (b) is dual. Recall that if an ǫ-bialgebra has a unit 1 then ∆(1) = 0. Dually, if it has a counit ǫ then ǫ(Imm)=0, so we can view ǫ∈(Cokerm)∗. Thus, lemma 3.5 may be applied as follows. Lemma 3.6. Let (A,m,∆) be an ǫ-bialgebra. 8 M. AGUIAR (a) Suppose that A has a unit 1. View A⊗A as a coalgebra as in lemma 3.5 (a) with u=v =1. Then m:A⊗A→A is a morphism of coalgebras. (b) Suppose that A has a counit ǫ. View A⊗A as an algebra as in lemma 3.5 (b) with γ =δ =ǫ. Then ∆:A→A⊗A is a morphism of algebras. Proof. To prove(a)we needto show that A⊗A m //A commutes. ∆(cid:15)(cid:15) (cid:15)(cid:15)∆ // (A⊗A)⊗(A⊗A) A⊗A m⊗m We calculate (2.1) ∆m(x⊗y)=∆(xy) = x1⊗x2y+xy1⊗y2, and (m⊗m)∆(x⊗y)=(m⊗m) (x1⊗1)⊗(x2⊗y)+(x⊗y1)⊗(1⊗y2) =x1⊗x2y+xy1⊗y2 , (cid:16) (cid:17) as needed. Case (b) is dual. ThepreviousresultdoesnotsaythatAisanordinarybialgebra,sincethecoalgebra or algebra structures on A⊗A are not the usual tensor product structures. The antipode of an ordinary Hopf algebra reverses multiplications and comul- tiplications. The analogous result for ǫ-Hopf algebras is as follows. Proposition 3.7. Let A be an ǫ-Hopf algebra with antipode S. Then (a) S(xy)=−S(x)S(y), (b) S(x1)⊗S(x2)=−S(x)1⊗S(x)2. Proof. We present the proof of (a), (b) being dual. Suppose first that A has a unit 1. View A⊗A as a coalgebra as in lemma 3.6 (a). Then m : A⊗A → A is a morphism of coalgebras, so by lemma 3.4 (a), m∗ : Homk(A,A) →Homk(A⊗A,A) preserves circular convolutions. Hence m = m∗(id) is invertible with inverse m∗(S) (with respect to circular convolution). On the other hand, let ν ∈ Homk(A⊗A,A) be ν(x⊗y) = −S(x)S(y). We need to show that ν = m∗(S) (since m∗(S)(x⊗y) = S(xy)). Since m∗(S) is the inverse of m, it suffices to check that m⊛ν =0. We calculate (m⊛ν)(x⊗y)=m(x1⊗1)ν(x2⊗y)+m(x⊗y1)ν(1⊗y2)+m(x⊗y)+ν(x⊗y) =−x S(x )S(y)−xy S(1)S(y )+xy−S(x)S(y) 1 2 1 2 =(−x S(x )−S(x))S(y)+xy S(y )+xy 1 2 1 2 (A) (A) = xS(y)+xy S(y )+xy =x(S(y)+y S(y )+y) = 0. 1 2 1 2 as needed. (We used that S(1)=−1, which we know from remark 3.3.) ThiscompletestheproofwhenAhasaunit. Thegeneralcasecandereducedto thisoneasfollows: adjoinaunittoAtoformtheunitalalgebraA+ =A⊕kasinthe paragraphprecedingdefinition3.1. ItiseasytocheckthatA+ isanǫ-Hopf algebra, with comultiplication ∆(a,λ) = (a1,0)⊗(a2,0) and antipode S(a,λ) = (S(a),−λ). Since the result holds for A+, it also does for its ǫ-Hopf subalgebra A. Amorphismofǫ-bialgebrasisalinearmapφ:A→B thatisbothamorphism of algebras and coalgebras: mB(φ⊗φ)=φmA and (φ⊗φ)∆A =∆Bφ . INFINITESIMAL HOPF ALGEBRAS 9 For instance, proposition 3.7 says precisely that S :(A,m,∆)→(A,−m,−∆) is a morphism of ǫ-bialgebras. A morphismofǫ-Hopf algebrasis a morphismofǫ-bialgebrasthatfurthermore preservesthe antipodes: φS =S φ. AsforordinaryHopfalgebras,this turnsout A B to be automatic. Proposition 3.8. Let Aand B beǫ-Hopf algebras andφ:A→B a morphism of ǫ-bialgebras. Then φS =S φ, i.e. φ is a morphism of ǫ-Hopf algebras. A B Proof. By lemma 3.4, there are morphisms of monoids φ∗ :Hom (B,B)→Hom (A,B), φ∗(f)=fφ k k and φ :Hom (A,A)→Hom (A,B), φ (f)=φf . ∗ k k ∗ Since φ∗(id ) = f = φ (id ) and inverses are preserved, we must have φ∗(S ) = B ∗ A B φ (S ), i.e. S φ=φS . ∗ A B A Example 3.9. Let A = khx ,x ,x ,...i and B = k[x] be the algebras of 1 2 3 examples 2.3.5 and 2.3.3. Recall from examples 3.2 that n+1 S(xn)=−(x−1)n and S(x )= (−1)k x x ...x . n n1−1 n2−1 nk−1 Xk=1 (n1,...,nkX)∈C+(n+1,k) The map φ:A→B, φ(x )=xn, is clearly a morphism of ǫ-bialgebras. Since n it must preserve the antipodes, we deduce that n+1 −(x−1)n = (−1)k xn1−1xn2−1...xnk−1 kX=1 (n1,...,nkX)∈C+(n+1,k) n+1 n = (−1)k#C(n+1,k)xn+1−k =− (−1)k#C(n+1,k+1)xn−k k=1 k=0 X X fromwhichweobtainthe basicfactthatthe numberofstrictcompositionsofn+1 into k+1 parts is n #C(n+1,k+1)= . k (cid:18) (cid:19) A finite dimensional subbialgebra of an ordinary Hopf algebra is necessarily a Hopf subalgebra. This is a consequence of the following basic fact: if R is a finite dimensional unital subalgebra of a unital algebra S and x ∈ R is invertible in S, then x is already invertible in R. To deduce the corresponding property of ǫ-Hopf algebras, first note that if R is a finite dimensional subalgebra of an arbitrary (nonunital) algebra S and x ∈ R is circular invertible in S, then x is alreadycircularinvertible in R. This follows fromthe previous fact applied to R+, S+ and the element (x,1). Proposition 3.10. IfB isafinitedimensionalǫ-subbialgebraofanǫ-Hopf algebra A, then B is an ǫ-Hopf subalgebra. 10 M. AGUIAR Proof. Let i : B → A be the inclusion. Hom (B,B) is a finite dimensional k subalgebra of Hom (B,A) via i . Considering i∗ : Hom (A,A) → Hom (B,A) we k ∗ k k see that i is circular invertible in Hom (B,A). By the preceding remark, id is k B invertible in Hom (B,B). k We turn to the study of antipodes in relation to the basic constructions of section 2. Proposition 3.11. Let (A,m,∆) be an ǫ-Hopf algebra with antipode S. 1. S is the antipode for (A,−m,−∆). 2. If S is bijective, then S−1 is the antipode for (A,−m,∆) and (A,m,−∆). 3. Conversely, if (A,−m,∆) or (A,m,−∆) admit an antipode S¯, then S¯ is the inverse of S with respect to composition. Proof. 1. Equations(A) coincidefor(A,m,∆)and(A,−m,−∆),sothis assertion is clear. 2. We first show that for any a,b∈A, (∗) S−1(a)S−1(b)=−S−1(ab) Since S is bijective, we can write a = S(x) and b = S(y). By proposition 3.7, S(xy) = −S(x)S(y). Hence xy = −S−1 S(x)S(y) , which rewrites as S−1(a)S−1(b)=−S−1(ab), as needed. (cid:16) (cid:17) Now from (A) we deduce 0=a S(a )+a+S(a) 1 2 ⇒0=S−1 a S(a ) +S−1(a)+S−1S(a) 1 2 (cid:16) (cid:17) (∗) ⇒ 0=−S−1(a )a +S−1(a)+a . 1 2 Similarly, from the other half of (A) one deduces 0 = −a S−1(a )+a+ 1 2 S−1(a). These say that S−1 is the antipode for both (A,−m,∆) and (A,m,−∆). 3. Suppose (A,−m,∆) admits an antipode S¯. Byproposition3.7(a),S :(A,−m)→(A,m)isamorphismofalgebras. Hence, by lemma 3.4 (a), S :Hom ((A,∆),(A,−m))→Hom ((A,∆),(A,m)), f 7→Sf , ∗ k k is a morphism of circular convolution monoids. Now, S (id) = S and ∗ S (S¯) = SS¯. We deduce that SS¯ is the inverse of S with respect to cir- ∗ cular convolution. Hence SS¯=id. One deduces similarly that S¯S =id, by using the morphism S∗. Proposition 3.12. Let (A,m,∆) be an ǫ-Hopf algebra with antipode S. Then so is (A,mop,∆cop), with the same antipode S. Proof. TheconvolutionproductonHom (A,A)isoppositetotheconvolution k product on Hom (Acop,Aop): k m(f⊗g)∆=mτ(g⊗f)τ∆=mop(g⊗f)∆cop . Hence the same is true for the circular products. In particular, the inverse of id is the same in both monoids.