Multiplier Hopf and bi-algebras

MUL TIPLIER HOPF AND BI-ALGEBR AS K. JANSSEN AND J. VERCR UYSSE Abstract. W e prop ose a categoric al interpretation of multiplier Hopf algebras, in analogy to usual Hopf a lgebras and bialgebra s. Since the introductio n of multip lier Hopf alge br as by V an Daele in [10] such a categoric al in terpretatio n has bee n missing. W e show that a multiplier Hopf algebra can be understo o d as a coalgebr a with antipo de in a cer tain mo noidal category of alg ebras. W e show that a (p ossibly non-unital, idempo tent, no n-degenerate, k -pr o jective) algebra ov er a commutativ e ring k is a multiplier bia lgebra if and only if the category of its algebra extensions and bo th the categ ories of its left and r ight mo dules a re monoidal and fit, together with the categ ory of k -mo dules, into a diagram of strict monoida l fo rgetful functors. Intr oduction A Hopf algebra ov er a comm utativ e ring k is defined as a k -bialgebra, eq uipp ed with a n an tip o de map. A k -bialgebra can b e understo o d as a coalgebra (or comonoid) in the monoidal category of k -a lgebras; a n a n tip o de is the in v erse of t he identit y map in the conv o lutio n algebra of k -endomorphisms of the k -bialg ebra. F rom the mo dule theoretic p oint of view, a k - bialgebra can also b e understo o d a s a k -alg ebra that turns its category of left (or, equiv alen tly , righ t) mo dules in to a monoidal catego ry , suc h that the forgetf ul functor to the categor y of k -mo dules is a strict monoidal functor. During the la st decades, many generalizations of and v ariat ions on the definition of a Hopf algebra ha v e emerged in the literature, suc h as quasi-Hopf algebras [5], w eak Hopf algebras [1], Hopf a lgebroids [2] and Hopf group (co)algebras [3]. F or most of these notions, the ab ov e categorical and mo dule theoretic in terpretatio ns remain v alid in a certain form, a nd in some cases, this was exactly t he motiv ation to in tro duce suc h a new Hopf- t yp e algebraic structure. Multiplier Hopf algebras w ere in tro duced by V an Daele in [10], mo t iv ated by the theory of (discrete) quan tum gro ups. The initial data of a multiplier Hopf algebra are a non-unita l algebra A , a so-called comu ltiplication map ∆ : A → M ( A ⊗ A ), where M ( A ⊗ A ) is the multiplier algebr a of A ⊗ A , whic h is the “largest” unital algebra con ta ining A ⊗ A a s a t w o- sided ideal, and certain bijectiv e endomorphisms on A ⊗ A . It is w ell-known that the dual of a Hopf algebra is itself a Hopf alg ebra only if the o riginal Hopf algebra is finitely g enerated and pro jectiv e ov er its base ring. A very nice feature of the t heory of multiplier Hopf algebras is that it lifts this dualit y to the infinite dimensional case. In particular, the (reduced) dual of a co-F rob enius Hopf algebra is a m ultiplier Hopf algebra, rather t han a usual Hopf algebra. F r o m the module theoretic p oint of view, some asp ects in the definition of a m ultiplier Hopf algebra remain ho w ev er not completely clear. F o r example, a multiplier Hopf algebra is intro duced without defining first an appropriate notion of a “m ultiplier bialgebra”. In particular, from the definition of a m ultiplier Hopf algebra a counit can b e constructed, ra t her than b eing given as par t of the initial data. F urthermore, a categorical c har a cterization of Date : Nov ember 1 8, 2021 . 1991 Mathematics S ubje ct Classific ation. 16W30 . 1 2 K. JANSSEN A ND J. VERCRUYSSE m ultiplier Hopf algebras as in the classical and more general cases as men tioned b efore seems to b e missing or at least unclear at the momen t. In this pap er w e try to shed some ligh t o n this situation. Our pap er is o r g anized as follows. In the first Section, w e recall some notions related to non-unital algebras and non-unital mo dules. W e rep eat the construction of the m ultiplier algebra of a non-degenerate (idempotent) algebra, and sho w how this notion is related to extensions of non-unital alg ebras. W e sho w ho w non-degenerate idemp otent k -pro jectiv e algebras constitute a monoida l category . In the second Section w e then introduce the notion of a multiplier bialgebr a as a coalgebra in this monoidal category of non-degenerate idemp o t en t k -pro jectiv e algebras. W e sho w that a non-degenerate idemp otent k -pro jectiv e algebra is a multiplie r bialgebra if and only if the category of its extensions, a s well as the categories of its left and right mo dules are monoidal and fit, together with the category of mo dules o v er the commutativ e base ring, into a diagram of strict monoidal forgetful functors (see Theorem 2.9). The main difference from the unital case is that monoidality of the category of left mo dules is not equiv a lent to monoidality of the category of right mo dules . In the last Section, we recall the definition of a m ultiplier Hopf alg ebra b y V an D aele. W e sho w that a mu ltiplier Hopf alg ebra is alw a ys a m ultiplier bialgebra a nd w e giv e an in terpretat io n of the a n tip o de as a t yp e of conv olution inv erse. W e conclude the pap er by pro viding a categorical w ay to in tro duce the notions of mo dule algebra and como dule algebra o v er a m ultiplier bialgebra, whic h in t he m ultiplier Hopf algebra case were studied in [12]. Notation. Throughout, let k b e a comm utative ring. All mo dules are o ver k and linear means k -linear. Unadorned tensor pro ducts are supp osed to b e ov er k . M k denotes the category of k - mo dules. By an algebr a w e mean a k -mo dule A equipp ed with an asso ciativ e k - linear map µ A : A ⊗ A → A ; it is not assumed to p ossess a unit. An algebr a map b etw een tw o algebras is a k -linear map that preserv es the multiplication. A unital algebra is an algebra A with unit elemen t 1 A . An algebra map b etw een t wo unital algebras tha t maps the one unit elemen t to the other, will b e called a unital algebra map. A right A -mo dule M is a k - mo dule equipped with a k -linear map µ M ,A : M ⊗ A → M suc h that the asso ciativity condition µ M ,A ◦ ( µ M ,A ⊗ A ) = µ M ,A ◦ ( M ⊗ µ A ) holds. The k -mo dule o f rig ht A -linear (resp. left B -linear, ( B , A )-bilinear) maps b et w een t w o righ t A -mo dules (resp. left B -mo dules, ( B , A )- bimo dules) M and N is denoted b y Hom A ( M , N ) (resp. B Hom( M , N ), B Hom A ( M , N )). W e will shortly denote Hom k ( M , N ) = Hom( M , N ). Note that for any three k -mo dules M , N and P , where M is k - pro jectiv e, the follow ing k -linear maps are injectiv e: α M ,N ,P : M ⊗ Hom( N , P ) → Hom( N , M ⊗ P ) , α M ,N ,P ( m ⊗ f )( n ) = m ⊗ f ( n ) , (1) β N ,P ,M : Hom( N , P ) ⊗ M → Hom( N , P ⊗ M ) , β N ,P ,M ( f ⊗ m )( n ) = f ( n ) ⊗ m. (2) In fact, this is already the case if M is a lo cally pro jectiv e k -mo dule. F or an ob ject X in a category , w e denote the iden tit y morphism on X also by X . 1. Non-unit al alge bras and extensions 1.1. Non-degenerate idemp otent algebras. Let A b e an algebra, w e sa y that A is idem- p otent if A = A 2 := { P i a i a ′ i | a i , a ′ i ∈ A } . W e will use the follo wing Sw eedler-t yp e notation for idemp oten t algebras: for an elemen t a ∈ A w e denote by P a 1 a 2 a (non-unique) elemen t MUL TIPLIER HOPF AN D BI- ALGEBRAS 3 of A 2 suc h that P a 1 a 2 = a . The algebra A is said to b e non-de ge n er ate if, for all a ∈ A , w e ha v e tha t a = 0 if ab = 0 for a ll b ∈ A or ba = 0 for all b ∈ A . Example 1.1. Clearly , if A is a unital algebra, then A is a non-degenerate idempotent algebra. More g enerally , A is called an algebr a with right (r esp. left) lo c al units if, for all a ∈ A , there exists an elemen t e ∈ A such that ae = a (resp. ea = a ). If A has righ t ( o r left) lo cal units, then A is non- degenerate and idemp oten t. A c omp l e te set of rig ht (r esp. left) lo c al units for A is a subset E ⊂ A suc h that, for ev ery a ∈ A , w e can find at least one righ t (resp. left) lo cal unit e ∈ E . Let A b e a n algebra. The category of all righ t A -mo dules and righ t A -linear maps is denoted b y f M A . A righ t A -mo dule M is called idemp otent if M = M A := { P i m i a i | m i ∈ M , a i ∈ A } . A r igh t A -mo dule M is said t o b e no n-de gener ate if for a ll m ∈ M the equalities ma = 0 for all a ∈ A imply that m = 0. The full sub catego ry of f M A consisting of all non-degenerate idemp oten t k -pro jectiv e right A -mo dules, is denoted b y M A . It is ob vious that every non-degenerate idemp oten t k - pro jectiv e algebra A is in M A , taking µ A,A = µ A . Similarly , w e can introduce the category of non-degenerate idemp oten t k -pro jectiv e left A - mo dules A M and the catego r y of non-degenerate idemp ot ent k -pro jectiv e ( A, B )-bimo dules A M B , with B another algebra. Example 1.2. If A is an algebra with righ t (resp. left) lo cal units, then the idemp oten t righ t (resp. left) A -mo dules a r e exactly those rig ht (resp. left) A -mo dules M suc h that for all m ∈ M , there exists an elemen t e ∈ A suc h that me = m (resp. em = m ). W e say that A acts with righ t (r esp. left) lo c al units on M . Remark that an idemp o ten t right (resp. left) mo dule ov er an algebra with rig ht (resp. left) lo cal units is automatically non-degenerate. The con ve rse is not true: consider an algebra with right lo cal units A , and let End A ( A ) b e the righ t A - mo dule b y putting ( f · a )( b ) = f ( ab ) for all a, b ∈ A and f ∈ End A ( A ). Then End A ( A ) is non-degenerate as right A -mo dule, but A can only act with right lo cal units o n A ∈ End A ( A ) if A has a (global) unit elemen t. If A is a unital algebra, then ev ery idemp o t ent righ t (resp. left) A -mo dule M is also unital, in the sense that m 1 A = m (resp. 1 A m = m ), for a ll m ∈ M . No w consider t wo algebras A and B . W e sa y that A is a n (algeb r a ) extension of B (or shortly a B -extensio n ) if A is a B -bimo dule a nd the multiplication of A is B -bilinear and B -balanced (the latter meaning that ( a · b ) a ′ = a ( b · a ′ ), for all a, a ′ ∈ A and b ∈ B ). W e call A a no n-de gener ate (resp. idemp otent , k -pr oje ctive ) extension of B if A is an extension of B su c h that A is non-degenerate (resp. idempo t en t, k -pro jectiv e) as a B -bimo dule. Note that an idemp oten t ( r esp. non-degenerate, k -pro jectiv e) algebra is in a canonical wa y an idemp oten t (resp. non-degenerate, k -pro jectiv e) extension of itself. It is also ob vious that a k -pro jectiv e algebra that is a B -extension is a k -pro jectiv e B -extension. Lemma 1.3. L et A b e a no n-de gener ate algebr a and an idemp otent extension of B , then A is a non-de gener ate ide mp otent extension of B . Pr o of. T ak e any a ∈ A suc h that a · b = 0 for all b ∈ B . W e hav e to sho w that a = 0. T a ke an y other a ′ ∈ A . Since A is an idemp o ten t left B -mo dule w e can write a ′ = P i b i · a ′ i ∈ B A . Hence we find that aa ′ = a ( P i b i · a ′ i ) = P i ( a · b i ) a ′ i = 0, where we used in the second equation the B -balancedness of the pr o duct of A . Since A is a no n-degenerate algebra w e conclude that a = 0.  4 K. JANSSEN A ND J. VERCRUYSSE Let now A and A ′ b e t w o (no n- degenerate) idemp o t en t extensions of B . A tr ansforma- tion from A to A ′ is a B -bilinear algebra map t : A → A ′ . The category with as ob jects non-degenerate idemp ot en t k -pro jectiv e extensions of an algebra B and a s morphisms trans- formations b et wee n the extensions is denoted b y B - Ext . Note tha t the canonical extension B of B is only an ob ject in B - Ext provided that B is a non-degenerate idemp otent k -pro jectiv e algebra. Example 1.4. Supp o se that A and B are unital algebras. Then A is a (non-degenerate) idemp oten t extension of B if and only if there is a unital a lgebra ma p f : B → A . Indeed, if A is a n idemp oten t extension of B , then define f ( b ) = 1 A · b = b · 1 A . Let no w A a nd B b e algebras with right and left lo cal units. An algebra map f : B → A is called a morphism o f alg e b r a s with right (r esp. left) lo c al units if there exists a complete set o f right (resp. left) lo cal units E ⊂ B for B suc h that f ( E ) ⊂ A is a complete set of righ t (resp. left) lo cal units for A . The algebra map f induces a natural B -bimo dule structure on A b y putting b · a · b ′ = f ( b ) af ( b ′ ) for a ll b, b ′ ∈ B and a ∈ A . One can easily see that if f is b oth a morphism of algebras with righ t lo cal units and a morphism of alg ebras with left lo cal units, then this bimo dule structure is idemp oten t (in f act, B acts with left and right lo cal units on A ), hence A is an idemp oten t extension of B . R emark 1.5 . Not ev ery idemp otent extension of algebras with rig h t (or left) lo cal units, o r, more generally , o f non-degenerate idemp oten t alg ebras, is induced by an algebra map as in Example 1.4. How eve r, in Section 1.3 w e will show tha t they are induced by a more general t yp e of morphism. 1.2. Multiplier algebras. The notion o f a multiplier a lgebra of a (p ossibly non-unital) algebra go es back to G. Hochs c hild [6] in his w ork on cohomology a nd extens ions, and to B. E. Johnson [7] in his study o f cen tra lizers in top ological algebra. A first pure a lgebraic in ves tigation of this not ion w as initiated b y J. D auns in [4]. In this Section w e will r ecall the construction of a mu ltiplier algebra for sake of complete ness and in order t o introduce the necessary notation. Giv en a n algebra A , w e consider the k -mo dules L ( A ) = End A ( A ) , R ( A ) = A End( A ) and H ( A ) = A Hom A ( A ⊗ A, A ). W e ha v e natural linear maps L : A → L ( A ) , L ( b )( a ) = λ b ( a ) = ba ; (3) R : A → R ( A ) , R ( b )( a ) = ρ b ( a ) = ab. No w consider t he linear maps ( − ) : R ( A ) → H ( A ) , ¯ ρ ( a ⊗ b ) = ρ ( a ) b, for ρ ∈ R ( A ); (4) ( − ) : L ( A ) → H ( A ) , λ ( a ⊗ b ) = aλ ( b ) , for λ ∈ L ( A ) . (5) W e define M ( A ), the multiplier algebr a of A , as the pullbac k of ( − ) and ( − ) in M k , i.e. the pullbac k of the follo wing diagram. M ( A ) / /   R ( A ) ( − )   L ( A ) ( − ) / / H ( A ) (6) MUL TIPLIER HOPF AN D BI- ALGEBRAS 5 Remark that if A is unital then A ∼ = L ( A ) ∼ = R ( A ) ∼ = H ( A ) in a canonical wa y , henc e also M ( A ) ∼ = A . W e can understand M ( A ) as the set of pairs ( λ, ρ ), where λ ∈ L ( A ) and ρ ∈ R ( A ), suc h that (7) aλ ( b ) = ρ ( a ) b, for all a, b ∈ A . Elemen ts of M ( A ) are called multipliers ; those of L ( A ) and R ( A ) are called left , resp. right multipliers . The follow ing Propo sition collects some elemen tary prop erties of M ( A ) and its relation to A . Prop osition 1.6. With notation as ab ove, the fol lowing statements hold: (i) M ( A ) is a unital alge b r a , with multiplic ation (8) xy = ( λ x ◦ λ y , ρ y ◦ ρ x ) for al l x = ( λ x , ρ x ) and y = ( λ y , ρ y ) in M ( A ) , and unity 1 := 1 M ( A ) = ( A, A ) ; (ii) ther e ar e natur al algebr a m aps ι A : A → M ( A ) , ι A ( a ) = ( λ a , ρ a ) , (9) π ℓ A : M ( A ) → L ( A ) , π ℓ A ( x ) = λ, (10) π r A : M ( A ) → R ( A ) op , π r A ( x ) = ρ, (11) wher e x = ( λ, ρ ) ∈ M ( A ) , a ∈ A and the multiplic ation on L ( A ) and R ( A ) is given by c omp osition; (iii) A is no n -de gener ate if an d only if L = π ℓ A ◦ ι A and R = π r A ◦ ι A ar e inje ctive; in this c ase ι A is inje ctive as wel l; (iv) M ( A ) is an A -bimo dule w i th actions give n by (12) x↼a := xι A ( a ) = ι A ( λ ( a )) , a⇀x = ι A ( a ) x = ι A ( ρ ( a )) , for al l x = ( λ, ρ ) ∈ M ( A ) and a ∈ A ; (v) A is a n on-de gener ate idemp otent M ( A ) -e xtension, wher e the left and right actions of x = ( λ, ρ ) ∈ M ( A ) o n a ∈ A ar e giv e n by x ✄ a = λ ( a ) , a ✁ x = ρ ( a ); (vi) if ι A is in j e ctive (e.g. if A is non- d e gener ate) then A is a two-side d ide a l in M ( A ) ; (vii) F or any M ∈ M A , we have that the formula m ✁ x = X i m i ( a i ✁ x ) , wher e m = P i m i a i ∈ M A and x ∈ M ( A ) , defines a unital rig ht M ( A ) -action on M . Henc e M A is a ful l sub c ate gory of M M ( A ) , the c ate gory of unital right M ( A ) -mo dules. Mor e over, for al l M ∈ M A , m ∈ M , a ∈ A and x ∈ M ( A ) , we have ma = m ✁ ι A ( a ) , (13) ( m ✁ x ) a = m ( x ✄ a ) , i.e. the action of A on M is M ( A ) -b ala nc e d, and ( ma ) ✁ x = m ( a ✁ x ) . (14) 6 K. JANSSEN A ND J. VERCRUYSSE Pr o of. (i). By a double application of (7 ) , we find that a ( λ x ◦ λ y )( b ) = ρ x ( a ) λ y ( b ) = ( ρ y ◦ ρ x )( a ) b, hence xy defined by fo rm ula (8) is indeed a multiplie r. (iii). This follows from the definition of the maps L and R , see (3). (iv) . This bimo dule action is induced by the algebra map ι A . Explicitly , from the righ t A -linearit y of λ and the left A -linearit y of ρ , w e obtain that ↼ and ⇀ are indeed actions. Let us verify tha t this defines indeed an A -bimo dule structure: a⇀ ( x↼b ) = a⇀ ( ι A ( λ ( b ))) = ι A ( aλ ( b )) = ι A ( ρ ( a ) b ) = ( ι A ( ρ ( a ))) ↼b = ( a⇀x ) ↼b, where w e used (12) in the third equalit y . (ii), (v), and (vi) are ob vious. (vii) . W e ha ve to che c k that the action of M ( A ) on M ∈ M A is well-define d. Supp ose that m = P i m i a i = 0 and take x ∈ M ( A ). F or all a ∈ A , w e then ha v e ( P i m i ( a i ✁ x )) a = P i m i (( a i ✁ x ) a ) = P i m i ( a i ( x ✄ a )) = m ( x ✄ a ) = 0, henc e P i m i ( a i ✁ x ) = 0 b y the non- degeneracy of M as righ t A -mo dule. T o c heck the ba la ncedness , w e compute ( m ✁ x ) a = ( X i m i ( a i ✁ x )) a = X i m i ( a i ( x ✄ a )) = m ( x ✄ a ) . The remaining assertions f ollo w immediately .  1.3. Non-degenerate idemp otent extensions. Lemma 1.7. L et A and B b e algeb r a s. (i) Ther e is a bije ctive c orr esp onde n c e b etwe en a l g e br a ma ps ℓ : B → L ( A ) and lef t B - mo dule structur es on A such that µ A is left B - l i n e ar (i.e. µ A ( ba ⊗ a ′ ) = bµ A ( a ⊗ a ′ ) , for al l a, a ′ ∈ A and b ∈ B ); (ii) ther e is a bije ctive c orr esp ondenc e b etwe en alg e br a maps r : B → R ( A ) op and right B - mo dule structur es on A such that µ A is righ t B -line ar (i.e. µ A ( a ⊗ a ′ b ) = µ A ( a ⊗ a ′ ) b , for al l a, a ′ ∈ A and b ∈ B ); (iii) ther e is a bije ctive c orr esp ond e n c e b etwe en algebr a map s f : B → M ( A ) and B -extension structur es on A . Pr o of. ( i ) . Supp ose tha t the map ℓ exists, then w e define f or all a ∈ A and b ∈ B , b · a := ℓ ( b )( a ) . If w e no w tak e another elemen t b ′ ∈ B , then we find b · ( b ′ · a ) = ℓ ( b )( ℓ ( b ′ )( a )) = ℓ ( bb ′ )( a ) = ( bb ′ ) · a, whic h shows that the action is asso ciative. Since for an y b ∈ B , ℓ ( b ) ∈ L ( A ) is rig h t A - linear, w e obta in tha t µ A is left B -linear. Con ve rsely , if A is a left B -mo dule, then define ℓ ( b )( a ) := b · a, for all b ∈ B and a ∈ A . Since µ A is left B -linear this map is w ell-defined. Similar computa- tions as ab ov e sho w t ha t the asso ciativity of the action implies that ℓ is an a lgebra map from B to L ( A ). ( ii ) . This follo ws symme trically . ( iii ). Suppose that the map f exists. Th en we obtain t wo algebra maps π ℓ A ◦ f : B → L ( A ) MUL TIPLIER HOPF AN D BI- ALGEBRAS 7 and π r A ◦ f : B → R ( A ) op . Hence b y the first t w o parts, A will be a left and righ t B -mo dule with B - actions given by b · a = π ℓ A ( f ( b ))( a ) = f ( b ) ✄ a a nd a · b = π r A ( f ( b ))( a ) = a ✁ f ( b ) f o r all a ∈ A and b ∈ B , where ✄ and ✁ are the left and righ t M ( A )-a ctions on A (see Prop osi- tion 1 .6 (v)). Since A is an M ( A )-extension it follows immediately that A is a B -extension. Con vers ely , if A is an extension of B , then A is in particular a B -bimo dule and µ A is B - bilinear, so b y the first t w o parts, w e hav e algebra maps ℓ : B → L ( A ) a nd r : B → R ( A ) op . No w define for all b ∈ B , f ( b ) := ( ℓ ( b ) , r ( b )). Then the B - balancedness of the m ultiplication of A implies that the image of f lies in M ( A ). f is an algebra map since ℓ and r are algebra maps.  R emark 1.8 . In [1 0, App endix ], a n algebra map f : B → M ( A ) suc h tha t f ( B ) ✄ A = A = A ✁ f ( B ) was called non - d e gener ate . This coincides with our notion of A b eing an idemp o ten t B -extension, which is b y Lemma 1.3 the same as a non-degenerate idemp oten t B -extension if A is a non-degenerate algebra. The latter is alw ays asumed in [10]. Lemma 1.9. L et B b e an algebr a and A a non-de ge ner ate idemp otent k -pr o je ctive algebr a. (i) If ther e exists an algebr a map f : B → M ( A ) such that A is a non - d e gener ate id emp otent ( k -pr oje ctive ) right B -mo dule with action induc e d by f , then t her e is a functor F r : M A → M B such that the fol lowing dia g r a m c ommutes. M A F r / / " " E E E E E E E E M B | | x x x x x x x x M k Her e the unlab ele d arr ows ar e for getful functors. (ii) If ther e exists a functor F r : M A → M B r e ndering c om mutative the ab ove diagr am, then ther e is an algebr a map r : B → R ( A ) op such that A is a n on-de gener ate idemp o tent k -pr oje ctive right B -mo dule with action induc e d by r . Pr o of. ( i ) . Recall f rom Lemma 1.7 (iii) that A is a B -bimo dule, with B - actions giv en b y b · a = f ( b ) ✄ a and a · b = a ✁ f ( b ) for all a ∈ A and b ∈ B , and µ A is B -bala nced. F or an y M ∈ M A , consider m ∈ M and b ∈ B and define m · b := m ✁ f ( b ), where w e use the action of M ( A ) on M defined in Prop osition 1.6 (vii). Since the action of A on M is M ( A )-balanced, it will be B -balanced as w ell, i.e. f or all m ∈ M , a ∈ A and b ∈ B ( m · b ) a = m ( b · a ) . (15) Let us v erify that this actio n of B on M is non-degenerate and idempotent. Using the idemp otency of M as righ t A -mo dule and of A as right B -mo dule, w e find tha t M = M A = M ( AB ) = ( M A ) B = M B , where t he third equalit y , meaning ( ma ) · b = m ( a · b ) for all m ∈ M , a ∈ A and b ∈ B , follo ws directly from the definition o f the righ t B -action on M and (13). Indeed, w e hav e that ( ma ) · b = ( ma ) ✁ f ( b ) = m ( a ✁ f ( b )) = m ( a · b ). T o prov e the non-degeneracy of the right B -action on M , take a n y m ∈ M suc h that m · b = 0 for all b ∈ B . Since A is idemp ot ent as left B -mo dule, we can write any a ∈ A as a = P i b i · a i , and obtain ma = X i m ( b i · a i ) (15) = X i ( m · b i ) a i = 0 . 8 K. JANSSEN A ND J. VERCRUYSSE Since this holds f o r all a ∈ A , w e obtain by the non-degeneracy o f M as right A -mo dule that m = 0. ( ii ) . Since A is a non- degenerate idemp oten t k - pr o jectiv e algebra, we hav e in particular that A ∈ M A . Hence F r ( A ) = A is a non-degenerate idempotent ( k -pro jectiv e) right B - mo dule. By Lemma 1.7 , w e know that an y righ t B -mo dule structure on A is induced by a map r : B → R ( A ) op , provided that the multiplication map µ A of A is righ t B -linear. That the latter is the case follow s from the functorialit y of F r . Indeed, fix a ∈ A and consider the righ t A -linear map λ a ∈ L ( A ), defin ed by λ a ( a ′ ) = aa ′ , for all a ′ ∈ A . W e then ha v e that F r ( λ a ) = λ a is a morphism in M B , i.e. λ a ( a ′ · b ) = λ a ( a ′ ) · b , for all a ′ ∈ A . Th us, for all a, a ′ ∈ A w e hav e that a ( a ′ · b ) = ( aa ′ ) · b , that is, µ A is righ t B - linear.  Theorem 1.10. L et A b e a no n-de gener ate idemp otent k - p r o je ctive algebr a and B an y alge- br a. Then ther e is a bije ctive c orr esp o n denc e b etwe en the fol lowing sets o f data: (i) non-de gener ate idemp otent ( k -pr oje c tive ) B -extension structur es on A ; (ii) algebr a maps f : B → M ( A ) such that A is a non-d e gener ate idemp otent ( k -pr oje ctive ) B -bimo dule with B -actions induc e d by f ; (iii) functors F , F r and F ℓ which r e nder c om mutative the fol lowing diagr am of functors (wher e the unlab ele d arr ows ar e for getful functors); A - Ext { { v v v v v v v v v v v v v v v v v v v v v v # # H H H H H H H H H H H H H H H H H H H H H H F   B - Ext z z v v v v v v v v v $ $ H H H H H H H H H A M F ℓ / / ) ) S S S S S S S S S S S S S S S S S S B M $ $ H H H H H H H H H M B { { v v v v v v v v v M A F r o o u u k k k k k k k k k k k k k k k k k k M k (iv) functors F which r ender c om mutative the fol lowing diagr am of functors (wher e the un- lab ele d arr ows ar e for getful functors). A - Ext F / / # # H H H H H H H H H B - Ext z z v v v v v v v v v M k In an y of the ab ove e quiva lent situations, the map f : B → M ( A ) fr om p art ( ii ) c a n b e extende d uniquely to a unital algeb r a morphism ¯ f : M ( B ) → M ( A ) such that ¯ f ◦ ι B = f . Pr o of. ( i ) ⇔ ( ii ) . This equiv alence follows directly from Lemma 1.7 (iii). ( ii ) ⇒ ( iii ) . The functor F r is constructed in L emma 1.9; the functor F ℓ is constructed in a symmetrical w a y . T o construct the functor F , tak e a non-degenerate idemp oten t extension R of A . The left and right B -action on R are induced b y the functors F ℓ and F r . W e only ha v e to v erify that these a ctions imp ose that R is a B -bimo dule and that the mu ltiplication map µ R of R is B -bilinear and B -balanced. W e will pro v e the B -bala ncedness of µ R , and lea ve the other v erifications to the reader. L et r, r ′ ∈ R and b ∈ B . Since R is an idemp o t ent MUL TIPLIER HOPF AN D BI- ALGEBRAS 9 A -bimo dule, w e can write r = P i r i a i ∈ RA and r ′ = P j a ′ j r ′ j ∈ AR . W e then ha ve ( r · b ) r ′ = X i ( r i ( a i · b )) r ′ = X i r i (( a i · b ) r ′ ) = X i,j r i (( a i · b )( a ′ j r ′ j )) = X i,j r i ((( a i · b ) a ′ j ) r ′ j ) = X i,j r i (( a i ( b · a ′ j )) r ′ j ) = X i,j r i ( a i (( b · a ′ j ) r ′ j )) = X i r i ( a i ( b · r ′ )) = X i ( r i a i )( b · r ′ ) = r ( b · r ′ ) , as w an ted. Here w e used in equalities t w o and eigh t that µ R is A -balanced, and in the fifth equalit y that µ A is B - balanced. ( iii ) ⇒ ( iv ) . T rivial. ( iv ) ⇒ ( i ) . Obvious ly A is an ob ject of A - Ext , hence A = F ( A ) ∈ B - Ext . The last statemen t w as prov en in [10, Prop o sition A.5]. The map ¯ f : M ( B ) → M ( A ) is defined b y the following formulas: ¯ f ( x ) ✄ a = X i f ( x ✄ b i ) ✄ a i = X i f ( λ x ( b i )) ✄ a i , and (16) a ✁ ¯ f ( x ) = X j a ′ j ✁ f ( b ′ j ✁ x ) = X j a ′ j ✁ f ( ρ x ( b ′ j )) , (17) where x = ( λ x , ρ x ) ∈ M ( B ), a ∈ A a nd a = P i b i a i ∈ B A , a = P j a ′ j b ′ j ∈ AB .  1.4. The monoidal category of non-d egenerate idem p otent k -pro jective algebras. W e can now intro duce the category NdI k as the categor y whose ob jects ar e non-degenerate idemp oten t k - pro jectiv e a lgebras and whose morphisms are non- degenerate idemp o ten t ( k - pro jectiv e) extensions (or simply idempo ten t extensions, in view of Lemma 1.3). The r eason for the extra assumption tha t the alg ebras are pro jectiv e as k -mo dules is explained b y the follo wing Lemma. Let A and B b e t w o algebras. Then A ⊗ B is again an alg ebra with m ultiplicatio n (18) ( a ⊗ b )( a ′ ⊗ b ′ ) = aa ′ ⊗ bb ′ , for all a, a ′ ∈ A and b, b ′ ∈ B . Similarly , giv en an y right A -mo dule M a nd righ t B -mo dule N , M ⊗ N is a right A ⊗ B -mo dule with action ( m ⊗ n )( a ⊗ b ) = ma ⊗ nb, for all m ∈ M , n ∈ N , a ∈ A and b ∈ B . Lemma 1.11. L et A and B b e two algebr as. Supp ose that M is a non-de gen e r a te idemp otent k -pr oje ctive right A -mo dule and N a non-de gener ate ide m p otent k -p r o je ctive right B -mo dule. Then M ⊗ N is a non-de gener ate idemp otent k -pr o j e ctive right A ⊗ B -mo dule. Pr o of. The k -pro jectivit y and idemp otency of M ⊗ N are obvious. T o pro v e the non- degeneracy , remark that M is non-degenerate as righ t A -mo dule if and o nly if the map φ M : M → Hom( A, M ) , m 7→ ( a 7→ ma ) is injectiv e. Since w e assumed t ha t N and M are pro jectiv e a nd thus flat as k -mo dules, w e then obtain that the maps φ M ⊗ N : M ⊗ N → Hom( A, M ) ⊗ N and M ⊗ φ N : M ⊗ N → M ⊗ Hom( A, N ) 10 K. JANSSEN A ND J. VERCRUYSSE are b oth injectiv e. If w e comp ose these maps resp ectiv ely with the injectiv e maps β A,M ,N : Hom( A, M ) ⊗ N → Hom ( A, M ⊗ N ) and α M ,A,N : M ⊗ Ho m( A, N ) → Hom( A, M ⊗ N ) of (1), w e get the follo wing injectiv e maps ϕ M ,N : M ⊗ N → Hom( A, M ⊗ N ) , ϕ M ,N ( m ⊗ n )( a ) = ma ⊗ n and ψ M ,N : M ⊗ N → Hom( A, M ⊗ N ) , ψ M ,N ( m ⊗ n )( a ) = m ⊗ na. No w consider any P i m i ⊗ n i ∈ M ⊗ N and suppo se that ( P i m i ⊗ n i )( a ⊗ b ) = P i m i a ⊗ n i b = 0 for all a ⊗ b ∈ A ⊗ B . If w e kee p a fixed and let b v ary o ver B , then w e obtain by the injectivit y of ψ M ,N that P i m i a ⊗ n i = 0. Since this holds for all a ∈ A , w e can now apply the injectivit y of ϕ M ,N , and w e find P i m i ⊗ n i = 0. Hence M ⊗ N is no n- degenerate as righ t A ⊗ B - mo dule.  Prop osition 1.12. With notation as ab ove, NdI k is a monoidal c ate gory. Pr o of. Consider tw o ob jects A and B in NdI k . A morphism f fro m B to A in NdI k , that is a (non-degenerate) idemp otent ( k -pro jectiv e) extension A of B , will b e denoted b y f : B | / / A . The v ertical bar in the middle of the arrow reminds us to the fact that f is not an actual map. Consider t w o morphisms f : B | / / A and g : A | / / R . Sinc e w e ha v e a functor F : A - Ext → B - Ext as in Theorem 1.10 (iv), and w e hav e that R ∈ A - Ext , it follo ws that R = F ( R ) ∈ B - Ext . W e define the comp osition g ◦ f : B | / / R as this (non-degenerate) idemp oten t ( k -pro jectiv e) extens ion R of B . The iden tity morphism of A is the trivial non- degenerate idemp o t ent ( k -pro jectiv e) extension A of A . It will b e denoted by A : A | / / A . Giv en t w o ob jects A, B ∈ NdI k , a similar computation as in the pro of of Lemma 1.11 shows that A ⊗ B with m ultiplication as defined in (18) is again an ob ject in NdI k . Consider morphisms f : B | / / A and f ′ : B ′ | / / A ′ in NdI k . F r o m Lemma 1.11 it f o llo ws tha t A ⊗ A ′ is a (non-degenerate) idemp o ten t ( k -pro jectiv e) extens ion of B ⊗ B ′ . W e define f ⊗ f ′ : B ⊗ B ′ | / / A ⊗ A ′ as this extension. W e lea ve the other v erifications for NdI k to b e a monoidal category (asso- ciativit y a nd unit constraints and coherence conditions) to the reader.  Notation 1 .1 3 . F or a morphism f : B | / / A in NdI k , i.e. a non-degenerate idemp otent ( k -pro jectiv e) B -extension A , w e denote by e f : B → M ( A ) the unique algebra map w e can asso ciate to it suc h that A is a non-degenerate idemp oten t ( k - pro jectiv e) B -bimo dule with B -actions induced b y e f , and b y f : M ( B ) → M ( A ) the unique extension of e f to M ( B ) (see Theorem 1.10). F or the identit y morphism A : A | / / A on an ob ject A ∈ NdI k w e then ha v e that e A = ι A and A = M ( A ). Note that b ecause of the fact tha t A ∈ NdI k , the algebra map ι A indeed induces a non-degenerate idemp oten t ( k -pro jectiv e) A -bimo dule structure on A . The composition g ◦ f : B | / / R of a nother morphism g : A | / / R in NdI k with f is c haracterized b y the algebra map ] g ◦ f = g ◦ e f : B → M ( R ) and its unique extension g ◦ f = g ◦ f : M ( B ) → M ( R ) to M ( B ). Given another morphism f ′ : B ′ | / / A ′ in NdI k , w e ha v e for the tensor pro duct f ⊗ f ′ : B ⊗ B ′ | / / A ⊗ A ′ in NdI k that ^ f ⊗ f ′ = Ψ A,A ′ ◦ ( e f ⊗ e f ′ ) : B ⊗ B ′ → M ( A ⊗ A ′ ), where Ψ A,A ′ : M ( A ) ⊗ M ( A ′ ) → M ( A ⊗ A ′ ) is the canonical inclusion. MUL TIPLIER HOPF AN D BI- ALGEBRAS 11 R emark 1.1 4 . The category NdI k can now b e though t of a s the category whose o b jects a re non-degenerate idemp otent k - pro jectiv e algebras, and whose morphisms b et wee n tw o o b jects B and A are algebra maps e f : B → M ( A ) that induce a no n-degenerate idemp otent ( k - pro jectiv e) B - bimo dule structure on A . Theorem 1.15. We h a v e the fol lowing diagr am o f monoidal functors. NdI k M / / Alg u k U / / M k Her e Alg u k denotes the c ate gory of unital k -alg e br as, and U is a for getful functor. Mor e over, if k is a fi e ld, then we have an adjoint p air ( M , U M ) of functors, wher e U M : Alg u k → NdI k is a for g e tful functor that is ful l y faithful. Pr o of. The functor M is defined as follo ws. F or a non-degenerate idempotent k -pro jectiv e algebra A , M ( A ) is the m ultiplier algebra. If f : B | / / A is a non-degenerate idemp oten t ( k -pro jectiv e) B -extension, then we kno w b y Theorem 1.10 that this is equiv alent to the (unique) existence of an algebra map e f : B → M ( A ) that induces a non-degenerate idempo- ten t ( k - pro jectiv e) B -bimo dule structure on A , which can moreo ver b e extended to a unital algebra morphism f : M ( B ) → M ( A ). W e define M ( f ) = f . Clearly M is a functor. If k is a field, then an y k -mo dule is k -pro jectiv e and one can consider the forgetful functor U M : Alg u k → NdI k . T o see tha t ( M , U M ) is an adjo int pa ir , w e define the unit η and counit ǫ of the adjunction. F or an y A ∈ NdI k and R ∈ Alg u k , w e define η A = ι A : A → M ( A ) , ǫ R = R : M ( R ) = R → R . Since t he m ultiplier algebra of a unital alg ebra is the original algebra itself, the counit is the iden tity natural transformation. Hence U M is a fully fa it hful functor. Moreo v er, there is a natural transformation Ψ A,B : M ( A ) ⊗ M ( B ) → M ( A ⊗ B ) , whic h implies that M is indeed a monoidal functor.  2. Mul tiplier bialgebras 2.1. Multiplier bialgebras. Definition 2.1. A multiplier k -bi a lgebr a is a comonoid in the monoidal category NdI k . Let us sp end some time on restating this definition in a more explicit w ay . By definition a m ultiplier bialgebra is a triple A = ( A, ∆ , ε ) consisting of a non-degenerate idemp otent k -pro jectiv e algebra A a nd morphisms ∆ : A | / / A ⊗ A and ε : A | / / k in NdI k , suc h that (∆ ⊗ A ) ◦ ∆ = ( A ⊗ ∆) ◦ ∆ as morphisms from A to A ⊗ A ⊗ A in NdI k and ( ε ⊗ A ) ◦ ∆ = A = ( A ⊗ ε ) ◦ ∆ as morphisms from A to A in NdI k . Making use of the alg ebra maps e ∆ : A → M ( A ⊗ A ) and e ε : A → M ( k ) = k these t wo conditions can b e expressed as (19) ∆ ⊗ A ◦ e ∆ = A ⊗ ∆ ◦ e ∆ 12 K. JANSSEN A ND J. VERCRUYSSE and (20) ε ⊗ A ◦ e ∆ = ι A = A ⊗ ε ◦ e ∆ . Using, for all b, b ′ ∈ A , the no t ation b = P b 1 b 2 and b ⊗ b ′ = P e ∆( b ) ✄ ( b ⊗ b ′ ) = P ( b ⊗ b ′ ) ✁ e ∆( b ) to express the idemp otency of B and of the A -extension ∆, w e can rephrase the coasso ciativity condition (19) as (21) X ^ ∆ ⊗ A  e ∆( a ) ✄ ( b ⊗ b ′′ 1 )  ✄ ( b ⊗ b ′ ⊗ b ′′ 2 ) = X ^ A ⊗ ∆  e ∆( a ) ✄ ( b 1 ⊗ b ′ )  ✄ ( b 2 ⊗ b ′ ⊗ b ′′ ) and (22) X ( b ⊗ b ′ ⊗ b ′′ 1 ) ✁ ^ ∆ ⊗ A  ( b ⊗ b ′′ 2 ) ✁ e ∆( a )  = X ( b 1 ⊗ b ′ ⊗ b ′′ ) ✁ ^ A ⊗ ∆  ( b 2 ⊗ b ′ ) ✁ e ∆( a )  , where b ⊗ b ′ ⊗ b ′′ ∈ A ⊗ A ⊗ A and a ∈ A . The left counit condition, i.e. the first equality of (20), can also b e read as (23) ε ⊗ A  e ∆( a )  ✄ b = ab, b ✁ ε ⊗ A  e ∆( a )  = ba, for all a, b ∈ A . This formula can b e made eve n more explicit. Remark first that the fact that the A -extension ε : A | / / k is idemp oten t means exactly that e ε : A → M ( k ) = k is surjectiv e, i.e. there exists an elemen t g ∈ A such that e ε ( g ) = 1 k . F o r any b ∈ A , w e can write b = P e ε ( g ) b 1 b 2 = P ^ ε ⊗ A ( g ⊗ b 1 ) ✄ b 2 = P b 1 ✁ ^ A ⊗ ε ( b 2 ⊗ g ), the last t w o sums b eing t w o w ay s to express b due to the idempo t ency of the extension ε ⊗ A . So (23) b ecomes (24) ^ ε ⊗ A  e ∆( a ) ✄ ( g ⊗ b 1 )  ✄ b 2 = ab, b 1 ✁ ^ A ⊗ ε  ( b 2 ⊗ g ) ✁ e ∆( a )  = ba. Our definition of m ultiplier bialgebra is closely related to the one in tro duced by V an Daele in [10]. W e will mak e this relatio nship more explicit in Section 3, but w e a lready sho w in the next Prop ositions that our notions of coa sso ciativit y and counitality coincide with those of [10]. Before w e state a nd prov e these, w e first introduce some notation and pro ve a Lemma. Giv en algebras A 1 , . . . , A n , consider the m ultiplier a lgebras M ( A 1 ) , . . . , M ( A n ) and M ( A 1 ⊗ · · · ⊗ A n ). Denote b y Ψ A 1 ,...,A n : M ( A 1 ) ⊗ · · · ⊗ M ( A n ) → M ( A 1 ⊗ · · · ⊗ A n ) the linear map defined b y Ψ A 1 ,...,A n ( x 1 ⊗ · · · ⊗ x n ) ✄ ( a 1 ⊗ · · · ⊗ a n ) = ( x 1 ✄ a 1 ) ⊗ · · · ⊗ ( x n ✄ a n ) , ( a 1 ⊗ · · · ⊗ a n ) ✁ Ψ A 1 ,...,A n ( x 1 ⊗ · · · ⊗ x n ) = ( a 1 ✁ x 1 ) ⊗ · · · ⊗ ( a n ✁ x n ) , where a i ∈ A i and x i ∈ M ( A i ), for all i = 1 , . . . , n . A m ultiplier of the form Ψ A 1 ,...,A n (1 ⊗ · · · ⊗ 1 ⊗ ι A i ( a i ) ⊗ 1 ⊗ · · · ⊗ 1) ∈ M ( A 1 ⊗ · · · ⊗ A n ) will b e denoted shortly b y 1 ⊗ · · · ⊗ 1 ⊗ a i ⊗ 1 ⊗ · · · ⊗ 1, where ι A i ( a i ) and a i app ear in the i th tensorand. Lemma 2.2. L et f : A | / / B b e a morphism in NdI k , C ∈ NdI k and c ∈ C . F or any x ∈ M ( A ⊗ C ) we have the fol lowing e quality in M ( B ⊗ C ) . f ⊗ C ( x (1 ⊗ c )) = f ⊗ C ( x )(1 ⊗ c ) (25) F or any x ∈ M ( C ⊗ A ) we have the f o l lowing e quality in M ( C ⊗ B ) . C ⊗ f (( c ⊗ 1) x ) = ( c ⊗ 1) C ⊗ f ( x ) (26) MUL TIPLIER HOPF AN D BI- ALGEBRAS 13 Pr o of. W e pro ve the first equalit y , b y pro ving the equalit y of t he left m ultipliers of b oth sides; the pro of for the rig ht multipliers is completely analogo us. F or all b ∈ B and c, d ∈ C we ha v e tha t f ⊗ C ( x (1 ⊗ c )) ✄ ( b ⊗ d ) = X i ^ f ⊗ C (( x (1 ⊗ c )) ✄ ( a i ⊗ d 1 )) ✄ ( b i ⊗ d 2 ) = X i ^ f ⊗ C ( x ✄ ( a i ⊗ cd 1 )) ✄ ( b i ⊗ d 2 ) = f ⊗ C ( x ) ✄ ( b ⊗ cd ) =  f ⊗ C ( x )(1 ⊗ c )  ✄ ( b ⊗ d ) , where b = P i e f ( a i ) ✄ b i ∈ f ( A ) ✄ B and d = P d 1 d 2 ∈ C 2 , and thus a lso cd = P ( cd 1 ) d 2 .  Prop osition 2.3. A morphism ∆ : A | / / A ⊗ A in NdI k is c o asso ciative in the sense o f (19) if and only if e ∆ : A → M ( A ⊗ A ) is c o asso ciative in the sen se of [10 ] , i.e. ( a ⊗ 1 ⊗ 1)∆ ⊗ A ( e ∆( b )(1 ⊗ c )) = A ⊗ ∆(( a ⊗ 1) e ∆( b ))(1 ⊗ 1 ⊗ c ) , (27) for al l a, b, c ∈ A . Pr o of. Suppose that (19) holds, then ( a ⊗ 1 ⊗ 1) ∆ ⊗ A ( e ∆( b )(1 ⊗ c )) (25) = ( a ⊗ 1 ⊗ 1)  ∆ ⊗ A ( e ∆( b ))(1 ⊗ 1 ⊗ c )  =  ( a ⊗ 1 ⊗ 1) A ⊗ ∆( e ∆( b ))  (1 ⊗ 1 ⊗ c ) (26) = A ⊗ ∆(( a ⊗ 1) e ∆( b ))(1 ⊗ 1 ⊗ c ) . Con vers ely , supp ose that (27) ho lds. It is easy to c hec k (with metho ds similar to the ones in the pro of of Lemma 1.11) that, b y the non- degeneracy of A ⊗ A ⊗ A , for an y x ∈ A ⊗ A ⊗ A w e hav e that x = 0 if ( a ⊗ 1 ⊗ 1) ✄ x = 0, for all a ∈ A . So, to pro ve tha t the left m ultipliers of ∆ ⊗ A ( e ∆( a )) and A ⊗ ∆( e ∆( a )) are equal for all a ∈ A , it suffices to prov e that, for all a ′ , b, c, d ∈ A , w e hav e that  ( a ′ ⊗ 1 ⊗ 1 ) ∆ ⊗ A ( e ∆( a ))  ✄ ( b ⊗ c ⊗ d ) = X  ( a ′ ⊗ 1 ⊗ 1 ) ∆ ⊗ A ( e ∆( a ))(1 ⊗ 1 ⊗ d 1 )  ✄ ( b ⊗ c ⊗ d 2 ) (25) = X  ( a ′ ⊗ 1 ⊗ 1 ) ∆ ⊗ A ( e ∆( a )(1 ⊗ d 1 ))  ✄ ( b ⊗ c ⊗ d 2 ) equals  ( a ′ ⊗ 1 ⊗ 1 ) A ⊗ ∆( e ∆( a ))  ✄ ( b ⊗ c ⊗ d ) (26) = A ⊗ ∆ (( a ′ ⊗ 1) e ∆( a )) ✄ ( b ⊗ c ⊗ d ) = X  A ⊗ ∆(( a ′ ⊗ 1) e ∆( a ))(1 ⊗ 1 ⊗ d 1 )  ✄ ( b ⊗ c ⊗ d 2 ) , and this is the case b y ( 27). Here w e used again the notation d = P d 1 d 2 ∈ A 2 .  Prop osition 2.4. A morphism ∆ : A | / / A ⊗ A in NdI k is c ounital in the sense of (2 0 ) if an d only if e ∆ : A → M ( A ⊗ A ) is c ounital i n the sense of [10 ] , i.e. ε ⊗ A ( e ∆( a )(1 ⊗ b )) = ι A ( ab ) = A ⊗ ε (( a ⊗ 1) e ∆( b )) , (28) for al l a, b ∈ A . Pr o of. Suppose that (20) holds. W e pro ve the first equalit y of (28): ε ⊗ A ( e ∆( a )(1 ⊗ b )) (25) = ε ⊗ A ( e ∆( a )) ι A ( b ) (20) = ι A ( a ) ι A ( b ) = ι A ( ab ) , 14 K. JANSSEN A ND J. VERCRUYSSE where w e used that M ( k ⊗ A ) ∼ = M ( A ) in the first equalit y . Con vers ely , supp ose that w e are giv en ( 2 8). Then for a ll a, a ′ , b ∈ A w e hav e  ε ⊗ A ( e ∆( a )) ✄ b  a ′ = ε ⊗ A ( e ∆( a )) ✄ ( ba ′ ) =  ε ⊗ A ( e ∆( a )) ι A ( b )  ✄ a ′ (25) = ε ⊗ A ( e ∆( a )(1 ⊗ b )) ✄ a ′ (28) = ι A ( ab ) ✄ a ′ = ( ab ) a ′ = ( ι A ( a ) ✄ b ) a ′ , suc h that b y the non-degeneracy o f A it follow s that ε ⊗ A ( e ∆( a )) ✄ b = ι A ( a ) ✄ b , for all a, b ∈ A . By a similar computation, one sho ws that the rig h t multipliers of ε ⊗ A ( e ∆( a )) and ι A ( a ) a re equal.  2.2. Monoidal structures on mo dule categories. In this Section w e will sho w that a non-degenerate idemp o t ent k -pro jectiv e algebra A is a multiplier bialgebra if a nd only if the category of its non-degenerate idemp oten t k -pro jectiv e algebra extensions and b o t h the categories of its non-degenerate idemp otent k -pro jectiv e left a nd rig ht mo dules are monoidal and fit, together with the category o f k - mo dules, into a diagra m of strict monoidal forgetful functors (in the sense of [8]). Lemma 2.5. L et A b e a non-de g e ner ate idemp otent ( k -p r o je ctive) algebr a and ∆ : A | / / A ⊗ A a non-de gener ate idem p otent ( k -pr oje c tive) extens i o n. F or any two M , N ∈ M A , we have M ⊗ N ∈ M A with right A -action define d b y (29) ( m ⊗ n ) · a := X i,j ( m i ⊗ n j )(( a i ⊗ b j ) ✁ e ∆( a )) , for al l a ∈ A, m ∈ M , n ∈ N and wher e m = P i m i a i ∈ M A and n = P j n j b j ∈ N A . Pr o of. By Lemma 1.11 M ⊗ N is a non- degenerate idemp oten t k -pro jectiv e righ t A ⊗ A - mo dule. Proposition 1.6 (vii) then implies that M ⊗ N is a unita l right M ( A ⊗ A )-mo dule. (2 9) means nothing else than ( m ⊗ n ) · a = ( m ⊗ n ) ✁ e ∆( a ), where ✁ is exactly the aforemen tioned righ t M ( A ⊗ A )-action on M ⊗ N . Hence the rig h t A -a ction on M ⊗ N is w ell-defined and asso ciativ e. L et us pro v e tha t this action is also non-degenerate. Recall that since ∆ is an idemp oten t extension, giv en a ⊗ a ′ ∈ A ⊗ A , we can find elemen ts a i ∈ A and b i ⊗ b ′ i ∈ A ⊗ A suc h that a ⊗ a ′ = P i e ∆( a i ) ✄ ( b i ⊗ b ′ i ). Now take any m ⊗ n ∈ M ⊗ N , then w e find ( m ⊗ n )( a ⊗ a ′ ) (13) = ( m ⊗ n ) ✁ ι A ⊗ A ( a ⊗ a ′ ) = X i ( m ⊗ n ) ✁ ι A ⊗ A ( e ∆( a i ) ✄ ( b i ⊗ b ′ i )) (12) = X i ( m ⊗ n ) ✁ ( e ∆( a i ) ι A ⊗ A ( b i ⊗ b ′ i )) = X i (( m ⊗ n ) ✁ e ∆( a i )) ✁ ι A ⊗ A ( b i ⊗ b ′ i ) = X i (( m ⊗ n ) · a i ) ✁ ι A ⊗ A ( b i ⊗ b ′ i ) (13) = X i (( m ⊗ n ) · a i )( b i ⊗ b ′ i ) . Supp ose no w that ( m ⊗ n ) · a = 0, for all a ∈ A . Then w e kno w, using the ab o v e computation, that for all a ⊗ a ′ ∈ A ⊗ A , ( m ⊗ n )( a ⊗ a ′ ) = P i (( m ⊗ n ) · a i )( b i ⊗ b ′ i ) = 0. Since w e know already that M ⊗ N is non- degenerate as righ t A ⊗ A -mo dule, w e find that m ⊗ n = 0, hence M ⊗ N is also non-degenerate as right A -mo dule. Finally , let us v erify that the action of A on M ⊗ N is idemp otent. This follo ws from M ⊗ N = ( M ⊗ N )( A ⊗ A ) = ( M ⊗ N )(( A ⊗ A ) ✁ e ∆( A )) (14) = (( M ⊗ N )( A ⊗ A )) ✁ e ∆( A ) = ( M ⊗ N ) ✁ e ∆( A ) = ( M ⊗ N ) · A, MUL TIPLIER HOPF AN D BI- ALGEBRAS 15 where w e used in the first and the fourth equality the idemp otency of M ⊗ N as righ t A ⊗ A - mo dule and in the second one the idemp otency of A ⊗ A a s r ig h t A -mo dule (with A -action induced b y e ∆, the a lgebra ma p asso ciated to the idemp oten t extension ∆).  Consider aga in a non-degenerate idemp oten t k -pro jectiv e algebra A . Suppose that the follo wing is a diagram of monoidal categories and strict monoidal fo r g etful functors. (30) A - Ext z z v v v v v v v v v $ $ H H H H H H H H H A M # # H H H H H H H H H M A { { v v v v v v v v v M k Since A is an o b ject in A - Ex t , A ⊗ A is also an ob ject in A - Ext , i.e. there is a non-degenerate idemp oten t ( k -pr o jectiv e) extension ∆ : A | / / A ⊗ A . R ecall from Theorem 1.10 that ∆ induces an algebra map e ∆ : A → M ( A ⊗ A ) b y putting (31) e ∆( b ) ✄ ( a ⊗ a ′ ) = b · ( a ⊗ a ′ ) , ( a ⊗ a ′ ) ✁ e ∆( b ) = ( a ⊗ a ′ ) · b, for all a ⊗ a ′ ∈ A ⊗ A and b ∈ A . Then by Lemma 2.5 we kno w that f or any tw o M , N ∈ M A , w e hav e an A -mo dule structure on M ⊗ N . The follow ing Lemma asserts that this A -mo dule structure on M ⊗ N coincides with the A -mo dule structure giv en b y the monoidal structure on M A . Lemma 2.6. L et A b e a non-d e gener ate idemp otent k -pr oje c tive algebr a such that (30) is a diag r a m of monoidal c ate gories and strict monoidal for getful functors. Then for any two M , N ∈ M A , we have w i th notation as ab ove (32) ( m ⊗ n ) · a = X i,j ( m i ⊗ n j )(( a i ⊗ b j ) ✁ e ∆( a )) , for al l a ∈ A, m ∈ M and n ∈ N , wher e m = P i m i a i ∈ M A and n = P j n j b j ∈ N A . Pr o of. Becaus e of the idemp otency as A ⊗ A -mo dule, it suffices to pro ve that f or all m ⊗ n ∈ M ⊗ N , a ⊗ a ′ ∈ A ⊗ A and b ∈ A ,  ( m ⊗ n )( a ⊗ a ′ )  · b = ( m ⊗ n )  ( a ⊗ a ′ ) ✁ e ∆( b )  . Consider the morphisms ψ : A → M , ψ ( a ) = ma and φ : A → N , φ ( a ) = na in M A . Since M A is a monoidal category , w e kno w that ψ ⊗ φ : A ⊗ A → M ⊗ N , ( ψ ⊗ φ )( a ⊗ a ′ ) = ma ⊗ na ′ = ( m ⊗ n )( a ⊗ a ′ ) is a right A -linear map. Hence, (( m ⊗ n )( a ⊗ a ′ )) · b = (( ψ ⊗ φ ) ( a ⊗ a ′ )) · b = ( ψ ⊗ φ )( ( a ⊗ a ′ ) ✁ e ∆( b )) = ( m ⊗ n )  ( a ⊗ a ′ ) ✁ e ∆( b )  , where w e used the righ t A -linearity of ψ ⊗ φ together with (31) in the second equalit y .  Lemma 2.7. L et A b e a n o n-de gener ate idem p otent k -pr oje ctive algebr a such that (30) is a di- agr am of monoidal c ate gories an d strict monoidal for getful functors. Then the non -de gener ate idemp otent ( k -pr oje ctive ) extension ∆ as defin e d ab ove is c o asso ciative in the sens e of (19) . 16 K. JANSSEN A ND J. VERCRUYSSE Pr o of. Since A ∈ M A , M A is a monoidal category and the forg etful functor to M k is strict monoidal, w e know tha t the ma p f : A ⊗ ( A ⊗ A ) → ( A ⊗ A ) ⊗ A, f ( a ⊗ ( a ′ ⊗ a ′′ )) = ( a ⊗ a ′ ) ⊗ a ′′ is an isomorphism of righ t A -mo dules. F urthermore, by Lemma 2.6 the right A -mo dule structure on these isomorphic right A - mo dules can b e computed using the fo rm ula (32). Pe rforming the computation of f (( b ⊗ ( b ′ ⊗ b ′′ )) · a ) = ( ( b ⊗ b ′ ) ⊗ b ′′ ) · a explicitly results exactly in the form ula (22). By a left-righ t symm etric argumen t, using the monoidal structure on A M , w e find that also formula (2 1) holds, and therefore ∆ is coasso ciativ e.  Supp ose, as ab o ve , that A is a non-degenerate idemp o ten t k -pro jectiv e algebra suc h that (30) is a diagram of monoidal categories and strict monoidal forgetful functors. Then, in particular, k ∈ A - Ext , and therefore, b y Theorem 1.10, there is a non-degenerate idemp oten t extension ε : A | / / k , determined b y e ε : A → M ( k ) = k : a 7→ 1 k · a = a · 1 k . Recall also that b ecause of the idemp o tency of ε , e ε is surjectiv e. W e denote by g ∈ A a fixed elemen t suc h that e ε ( g ) = 1 k . Lemma 2.8. L et A b e a n o n-de gener ate idem p otent k -pr oje ctive algebr a such that (30) is a di- agr am of monoidal c ate gories an d strict monoidal for getful functors. Then the non -de gener ate idemp otent ( k -pr oje ctive ) extension ∆ as defin e d ab ove is c ounital in the sense of (20) . Pr o of. By assump tion, k is the monoidal unit of the monoidal category M A , and for an y M ∈ M A the map r M : M ⊗ k → M , r M ( m ⊗ t ) = mt f or m ∈ M and t ∈ k , is an isomorphism of righ t A -mo dules. In particular w e hav e t ha t, f o r a ll a, b ∈ A r A (( b ⊗ 1 k ) · a ) = r A ( b ⊗ 1 k ) a = ba. By (32) w e find ( b ⊗ 1 k ) · a = ( b 1 ⊗ 1 k )  ( b 2 ⊗ g ) ✁ e ∆( a )  = b 1  ( A ⊗ e ε )(( b 2 ⊗ g ) ✁ e ∆( a ))  . In the second equalit y w e used the form ula for e ε giv en ab ov e this Lemma. If w e apply r A to the last expression w e obtain exactly t he right hand side of (24). Similarly , the left hand side of (24) is obtained, using the isomorphism k ⊗ A ∼ = A in the monoidal category A M o f left A -mo dules.  W e no w arriv e at the main r esult of t his Section. Theorem 2.9. L et A b e a non-de gen er ate idem p otent k -pr oje ctive algebr a, then ther e is a bije ctive c orr esp ondenc e b etwe en structur es of a multiplier bialgebr a on A and stru ctur es of monoidal c ate g ories on M A , A M a n d A - Ex t such that a l l for getful functors in diagr am (30) ar e strict monoidal. Pr o of. Suppose first t hat (30) is a diagram of monoidal categories and strict mo no idal forgetful functors. Then b y Lemma 2.7 a nd Lemma 2.8, there are (non-degenerate) idemp otent algebra ( k -pro jectiv e) extensions ∆ : A | / / A ⊗ A and ε : A | / / k which turn A in to a m ultiplier bialgebra. Con vers ely , if A is a multiplier bialgebra, with comultiplication ∆ and counit ε , then w e kno w b y Lemma 2 .5 that there is a w ell-defined functor ⊗ : M A × M A → M A . Us ing the coasso ciativity of ∆, w e can pro v e b y sim ilar argumen ts as in Lemma 2.7 that, for a ll M , N , P ∈ M A (resp. in A M ), the isomorphism M ⊗ ( N ⊗ P ) ∼ = ( M ⊗ N ) ⊗ P holds. Since ε : A | / / k is a non-degenerate idemp oten t k -pro jectiv e extens ion, we kno w that k ∈ A - Ext , so in par ticular k is a non-degenerate idemp oten t k - pro jectiv e left and right A - mo dule. Let us c hec k that k is a monoidal unit in M A . T o prov e that the map r M : M ⊗ k → MUL TIPLIER HOPF AN D BI- ALGEBRAS 17 M , r M ( m ⊗ t ) = mt for m ∈ M and t ∈ k , is an isomorphism of righ t A -mo dules, w e can use the same (conv erse) reasoning as in Lemma 2.8 . Next, we will sho w that tha t the k -linear isomorphism ℓ M : k ⊗ M → M , ℓ M ( t ⊗ m ) = tm is also right A -linear. T a ke any m ∈ M and a ∈ A . Using the idemp otency of M as rig h t A -mo dule, w e can write m = P i m i a i ∈ M A . As b efore, ta k e g ∈ A such that e ε ( g ) = 1 k . Then w e hav e m = P i m i ℓ A  ( e ε ⊗ A )( g ⊗ a i )  . F urt hermore, using form ula (29), w e find ℓ M ((1 k ⊗ m ) · a ) = X i ℓ M  (1 k ⊗ m i )(( g ⊗ a i ) ✁ e ∆( a ))  = X i m i ℓ A  ( e ε ⊗ A )(( g ⊗ a i ) ✁ e ∆( a ))  . If w e m ultiply the last expression with an arbitrary b = P b 1 b 2 ∈ A = A 2 , w e further obtain X i m i ℓ A  ( e ε ⊗ A )  ( g ⊗ a i ) ✁ e ∆( a )   b = X i m i ℓ A  ( e ε ⊗ A )  ( g ⊗ a i ) ✁ e ∆( a )   ℓ A  ( e ε ⊗ A )( g ⊗ b 1 )  b 2  = X i m i  ℓ A  ( e ε ⊗ A )  ( g ⊗ a i ) ✁ e ∆( a )  ℓ A  ( e ε ⊗ A )( g ⊗ b 1 )   b 2 = X i m i  ℓ A ◦ ( e ε ⊗ A )    ( g ⊗ a i ) ✁ e ∆( a )  ( g ⊗ b 1 )  b 2 = X i m i  ℓ A ◦ ( e ε ⊗ A )   ( g ⊗ a i )  e ∆( a ) ✄ ( g ⊗ b 1 )   b 2 = X i m i ℓ A  ( e ε ⊗ A )( g ⊗ a i )  ℓ A  ( e ε ⊗ A )  e ∆( a ) ✄ ( g ⊗ b 1 )   b 2 = mab = ℓ M (1 k ⊗ m ) ab. Here we used the idemp otency of A and the extension ε in the first equality , the asso ciativit y of the A -action on M in t he second equalit y , the multiplic ativit y of the morphisms ℓ A and e ε ⊗ A in the third a nd p enultimate equalit y , the multiplier prop erty (7) in the fourth equalit y , and (24) in the la st equalit y . F rom t he fact that M is non- degenerate a s r ig h t A -mo dule it follo ws then that ℓ M ((1 k ⊗ m ) · a ) = ℓ M (1 k ⊗ m ) a , as desired. The v erification of the coherence conditions of the asso ciat ivity and unit constraints is left to the reader. This completes t he pro of that M A is a monoidal category and the forgetful functor to M k is a strict monoidal f unctor. Similarly , o ne pro v es that A M is a monoidal catego r y . Finally , ha ving t w o ob jects P , Q ∈ A - Ext we can construct P ⊗ Q ∈ A - Ext , by using the left A -mo dule structure as computed in A M , the righ t A -mo dule structure as in M A and the k -algebra structure as in (1 8). It is then easy to v erify that A - Ext b ecomes a monoidal catego ry so that all forgetful functors in diagra m (30) are strict mono ida l. T o end the pro of, w e hav e to sho w that b oth constructions are m ut ua lly in vers e. Starting with a m ultiplier bialgebra, it is an immediate consequence of our constructions that the reconstructed com ultiplication and counit coincide with the origina l o nes. Con vers ely , if w e start with the monoidal structures, cons truct the multiplier bialgebra and reconstruct 18 K. JANSSEN A ND J. VERCRUYSSE the mono idal structure s, it follows fr o m Lemma 2.6 t hat we reco v er the origina l monoidal structures.  R emark 2.10 . If A is an algebra with unity , then (30) will b e a diagram of monoidal categories and strict mono idal functors if and only if M A is a mo no idal category and the forgetful functor M A → M k is a strict mono ida l functor if a nd only if A M is a monoida l category and the forgetful functor A M → M k is a strict monoidal functor. Ind eed, since there are algebra isomorphisms A ∼ = L ( A ) ∼ = R ( A ) op ∼ = M ( A ), the data of (i), (ii) and (iii) of Lemma 1.7 are in bijectiv e correspondence in the unital case, fro m whic h this statemen t can b e easily deriv ed. 3. Mul tiplier Hopf algebras In this Section w e recall the definition of a m ultiplier Hopf alg ebra b y A. V an Da ele. W e sho w that the an tip o de of a multiplier Hopf algebra A can b e in t erpreted as a con v olutio n in vers e in a certain “m ultiplicativ e structure” asso ciated to the underlying multiplier bialgebra of A . 3.1. V an Daele’s definition. Let us first in tro duce the following Sw eedler-type notation. Let A b e a non-degenerate alg ebra and e ∆ : A → M ( A ⊗ A ) an algebra map. Supp ose that e ∆ satisfies the fo llo wing conditions for a ll a, b ∈ A : e ∆( a )(1 ⊗ b ) ⊆ ι A ⊗ A ( A ⊗ A ) , (33) ( a ⊗ 1) e ∆( b ) ⊆ ι A ⊗ A ( A ⊗ A ) . (34) Giv en elemen ts a, b ∈ A , we denote by a (1 ,b ) ⊗ a (2 ,b ) ∈ A ⊗ A (summation implicitly understo o d) the elemen t suc h tha t e ∆( a )(1 ⊗ b ) = ι A ⊗ A ( a (1 ,b ) ⊗ a (2 ,b ) ). Similarly , w e denote by a ( b, 1) ⊗ a ( b, 2) the elemen t in A ⊗ A suc h that ( b ⊗ 1) e ∆( a ) = ι A ⊗ A ( a ( b, 1) ⊗ a ( b, 2) ) . Recall from [10] the fo llo wing definitions. The algebra map e ∆ : A → M ( A ⊗ A ) is called c o asso ciative if the follo wing prop ert y holds: ( a ⊗ 1 ⊗ 1)( e ∆ ⊗ A )  e ∆( b )(1 ⊗ c )  = ( A ⊗ e ∆)  ( a ⊗ 1) e ∆( b )  (1 ⊗ 1 ⊗ c ) , i . e . (35) b (1 ,c )( a, 1) ⊗ b (1 ,c )( a, 2) ⊗ b (2 ,c ) = b ( a, 1) ⊗ b ( a, 2)(1 ,c ) ⊗ b ( a, 2)(2 ,c ) , for all a, b, c ∈ A (to b e able to let e ∆ ⊗ A (resp. A ⊗ e ∆) act on e ∆( b )(1 ⊗ c ) (resp. ( a ⊗ 1) e ∆( b )), A ⊗ A is identifie d with its image in M ( A ⊗ A ) under ι A ⊗ A ). A multiplier Hopf algebr a is a non-degenerate k -pro jectiv e a lg ebra A , equipped with a coasso ciativ e algebra map e ∆ : A → M ( A ⊗ A ) that satisfie s (33) and (34), and suc h that the follo wing maps are bijective : T 1 : A ⊗ A → A ⊗ A, T 1 ( a ⊗ b ) = a (1 ,b ) ⊗ a (2 ,b ) ; T 2 : A ⊗ A → A ⊗ A, T 2 ( a ⊗ b ) = b ( a, 1) ⊗ b ( a, 2) . V an Daele prov es that the com ultiplication of a m ultiplier Hopf algebra A is a no n- degenerate algebra map in the sense of Remark 1 .8 . F urthermore, A can b e endo w ed with an algebra map e ε : A → k that satisfies the fo llo wing counit conditio ns: ( A ⊗ e ε )  ( a ⊗ 1) e ∆( b )  = b ( a, 1) e ε ( b ( a, 2) ) = ab, (36) ( e ε ⊗ A )  e ∆( a )(1 ⊗ b )  = e ε ( a (1 ,b ) ) a (2 ,b ) = ab, (37) MUL TIPLIER HOPF AN D BI- ALGEBRAS 19 for all a, b ∈ A (see [10, Theorem 3.6]). Moreov er, there exists an “an tip o de” map S : A → M ( A ) that satisfies the follow ing prop erties: m 1 ( S ⊗ A )  e ∆( a )(1 ⊗ b )  = S ( a (1 ,b ) ) ✄ a (2 ,b ) = e ε ( a ) b, , (38) m 2 ( A ⊗ S )  ( a ⊗ 1) e ∆( b )  = b ( a, 1) ✁ S ( b ( a, 2) ) = a e ε ( b ) , (39) for all a, b ∈ A (see [10, Theorem 4.6]), where m 1 : M ( A ) ⊗ A → A and m 2 : A ⊗ M ( A ) → A are the natur a l ev a lua tion maps. Finally , b y [11, Prop osition 1.2] it follows that A is a n algebra with lo cal units, so in par t icular A is idempo t en t. Remark that in [1 0] k is supposed to b e a field. Therefore, e ε is automa t ically surjectiv e (i.e. non-degenerate in the sense of Remark 1.8). W e no w easily arriv e at the follo wing. Prop osition 3.1. A multiplier Hopf algebr a is a multiplier bialgebr a. Pr o of. By the observ ations made ab ov e, we know that A is a non-degenerate idemp otent k -pro jectiv e algebra, equipped with non-degenerate algebra maps e ∆ : A → M ( A ⊗ A ) and e ε : A → k = M ( k ). So b y Remark 1.8 A ⊗ A and k are non- degenerate idempo ten t ( k - pro jectiv e) extensions of A , i.e. e ∆ and e ε giv e rise to morphisms ∆ : A | / / A ⊗ A and ε : A | / / k in NdI k . The equalities (3 5), (36) and (37) hold if and only if they hold after applying the injectiv e algebra maps ι A ⊗ A ⊗ A and ι A . Then ∆ is a coasso ciativ e and counital com ultiplication b y Prop osition 2.3 and Prop osition 2.4. This show s that A is a m ultiplier bialgebra.  An immediate consequence of the previous Prop osition is that for a m ultiplier Hopf algebra A we ha v e a diagram of monoidal categories and strict monoidal forgetful functors as in (30) . R emark 3.2 . The definition o f a m ultiplier Hopf algebra differs from the classical definition of a Hopf algebra, not only in its range of generality , but also in its initial set-up. Classically , a Hopf algebra is defined as a bialgebra ha ving an an tip o de. As w e hav e se en ab o v e, a m ultiplier Hopf algebra is a m ultiplier bialgebra tha t p osse sses an an t ip o de, but the conv erse is not true. In fact, for a m ultiplier Hopf algebra the maps T 1 and T 2 are supp osed to b e bijectiv e. In case of an algebra A with unit, these conditions a re equiv alen t to A b eing a Hopf alg ebra, as we will see in what follows. Recall that for an y (usual) bialgebra A , one can de v elop the theory of Hopf- G alois extensions o v er A . In this theory , w e study (righ t) A -como dule algebras B , and define the coinv ariants of B as B co A = { b ∈ B | ρ ( b ) = b ⊗ 1 } , where ρ : B → B ⊗ A, ρ ( b ) = b [0] ⊗ b [1] is the A -coaction on B . Then B co A ⊂ B is called a (righ t) A -Galois extension if the canonical map can : B ⊗ B co A B → B ⊗ A, can( b ′ ⊗ b ) = b ′ b [0] ⊗ b [1] is bijectiv e. A result of Sc hauen burg [9] states that a bialgebra A is a Hopf algebra if A itself is a faithfully flat right A -Galois extens ion. If w e consider a bialgebra as righ t como dule algebra o v er itself (via its com ult iplicatio n map), then A co A ∼ = k and can = T 2 . Similarly , b y considering A as left como dule algebra ov er itself, w e obtain T 1 as the canonical map of the left A -Galois extension k ⊂ A . Therefore, w e can intuitiv ely unde rstand a m ultiplier Hopf algebra as b eing a m ultiplier bialgebra A suc h that k ⊂ A is a left a nd right “Hopf-Ga lois extension”. The construction of the antipo de can then b e understoo d a s a generalizaton of Sc hauenburg’s result to the non-unital case. Remark that in V an Daele’s o r ig inal definition k is supposed to b e a field, so the faithfully flatness condition is a ut o matically satisfied. 20 K. JANSSEN A ND J. VERCRUYSSE 3.2. The an t ip ode as con volution in v erse. Classically , the an tip o de of a Hopf algebra A is the inv erse of the iden tity ma p in the conv olutio n algebra End ( A ). If A is a m ultiplier bialgebra, then End( A ) is no longer a (con v o lution) a lgebra. In this section w e sho w that it is how eve r p ossible to put a ric her structure on the k -mo dule Hom( A, M ( A )), that mak es it p ossible to define the antipo de as a kind o f con v o lution in v erse of the map ι A . Lemma 3.3. L et A b e a n o n-de gener ate idemp otent algebr a and c onsid e r R = Hom( A, M ( A )) . Then (i) R is a non-de ge n er ate A -bimo dule with actions ( b · f · b ′ )( a ) = f ( b ′ ab ) , for al l a, b, b ′ ∈ A and f ∈ R ; (ii) R is a non-de g e n er ate A -bimo dule with actions ( b⇀f ↼b ′ )( a ) = ι A ( b ) f ( a ) ι A ( b ′ ) , for al l a, b, b ′ ∈ A and f ∈ R ; (iii) for al l b, b ′ ∈ A and f ∈ R we have, b⇀ ( b ′ · f ) = b ′ · ( b⇀f ) , ( f · b ) ↼b ′ = ( f ↼b ′ ) · b ; b⇀ ( f · b ′ ) = ( b⇀f ) · b ′ , ( b · f ) ↼b ′ = b · ( f ↼b ′ ); henc e R is an A ⊗ A op -bimo dule. Pr o of. ( i ) . W e only che c k the non-degeneracy of the left action. T ake f ∈ R and supp ose that b · f = 0 for all b ∈ A , then ( b · f )( a ) = f ( ab ) = 0 for all a, b ∈ A . Since A is idempo t en t, w e then find that f = 0. ( ii ) . Again, we only pro of that R is non-degenerate as left A -mo dule. T ak e f ∈ R and supp ose that b⇀f = 0, for all b ∈ A . Then, for any a ∈ A , we ha v e 0 = ι A ( b ) f ( a ) (12) = ι A ( b ✁ f ( a )), so by the injectivit y of ι A w e find 0 = b ✁ f ( a ) = ρ a ( b ), fo r all a, b ∈ A (where w e used the notation f ( a ) = ( λ a , ρ a ) ∈ M ( A )) . Hence ρ a = 0 for a ll a ∈ A , whic h implies that f ( a ) = 0, f o r all a ∈ A , that is f = 0. ( iii ) . Easy to c heck .  Supp ose no w that A is a m ultiplier bialgebra. Supp ose furt hermore tha t the com ultipli- cation of A satisfies conditions (33) and (34 ) . Using Sw eedler-t yp e not a tion as intro duced in the previous Section, we can endo w R with tw o “lo cal m ultiplication structures”, i.e. t wo linear maps a s follo ws: ∗ 1 : R ⊗ R → Hom( A, R ) , f ⊗ g 7→ ( b 7→ f ∗ b g ) , ∗ 2 : R ⊗ R → Hom( A, R ) , f ⊗ g 7→ ( b 7→ f ∗ b g ) , where f ∗ b g ( a ) = µ M ( A )  ( f ⊗ g )( e ∆( a )(1 ⊗ b ))  = f ( a (1 ,b ) ) g ( a (2 ,b ) ) a nd f ∗ b g ( a ) = µ M ( A )  ( f ⊗ g )(( b ⊗ 1 ) e ∆( a ))  = f ( a ( b, 1) ) g ( a ( b, 2) ). T hese multiplications satisfy the f ollo wing asso ciativit y condition. Lemma 3.4. With notation as ab ov e , the f o l lowing holds for al l f , g , h ∈ R and a, b ∈ A : f ∗ a ( g ∗ b h ) = ( f ∗ a g ) ∗ b h. MUL TIPLIER HOPF AN D BI- ALGEBRAS 21 Pr o of. T ak e an y c ∈ A , then w e find  f ∗ a ( g ∗ b h )  ( c ) = f ( c ( a, 1) )( g ∗ b h )( c ( a, 2) ) = f ( c ( a, 1) ) g ( c ( a, 2)(1 ,b ) ) h ( c ( a, 2)(2 ,b ) ) = f ( c (1 ,b )( a, 1) ) g ( c (1 ,b )( a, 2) ) h ( c (2 ,b ) ) = ( f ∗ a g )( c (1 ,b ) ) h ( c (2 ,b ) ) =  ( f ∗ a g ) ∗ b h  ( c ) , where w e used (35) in the third equalit y .  F urt hermore, w e ha ve t he following linear maps α ∈ Hom( A, R ) , α ( b )( a ) = α b ( a ) = e ε ( a ) ι A ( b ); β 1 : R → Hom( A, R ) , β 1 ( f )( b ) = β b ( f ) = b · f ; β 2 : R → Hom( A, R ) , β 2 ( f )( b ) = β b ( f ) = f · b. Then the following unitalit y condition holds for t he multiplic ativ e structure w e defined on R . Lemma 3.5. With notation as ab ov e , we have for al l f ∈ R and b ∈ A , α b ∗ 1 f = β 1 ( b⇀f ) , f ∗ 2 α b = β 2 ( f ↼b ) . Pr o of. W e only sho w the first equalit y , the second one follows b y a similar computation. T a k e an y a, c ∈ A , then we che c k that ( α b ∗ c f )( a ) = α b ( a (1 ,c ) ) f ( a (2 ,c ) ) = ι A ( b ) e ε ( a (1 ,c ) ) f ( a (2 ,c ) ) = ι A ( b ) f ( ac ) = ( c · ( b⇀f ))( a ) = β c ( b⇀f )( a ) . Here w e used (37) in the third equation.  It no w makes sense to define what is a con volution inv erse in R = Hom( A, M ( A )). Giv en f ∈ R , w e say that ¯ f ∈ R is a (left-right) c o nvolution inverse of f in R if ¯ f ∗ 1 f = α = f ∗ 2 ¯ f . Prop osition 3.6. L et A b e a multiplier Hopf algebr a. Then a line ar map S ∈ Hom( A, M ( A )) is the antip o de of A if and only if S is a (le f t-righ t) c onvo lution inverse of ι A in Hom( A, M ( A )) . Pr o of. A linear map S : A → M ( A ) is a (left-right) conv o lution in v erse of ι A if and only if, for all a, b ∈ A , the follo wing holds: ( S ∗ b ι A )( a ) = S ( a (1 ,b ) ) ι A ( a (2 ,b ) ) (12) = ι A ( S ( a (1 ,b ) ) ✄ a (2 ,b ) ) equals α b ( a ) = ι A ( b ) e ε ( a ) and ( ι A ∗ a S )( b ) = ι A ( b ( a, 1) ) S ( b ( a, 2) ) (12) = ι A ( b ( a, 1) ✁ S ( b ( a, 2) )) is equal to α a ( b ) = ι A ( a ) e ε ( b ). Since ι A is injectiv e, these form ulas are equiv a len t to (38) and (39), res p ectiv ely . Because of the uniqueness of the an tip o de of a multiplie r Hopf a lgebra, the statemen t follows .  3.3. (Co)mo dule algebras ov er a m ultiplier bialgebra. By definition a multiplie r bial- gebra A is a coalgebra in the monoidal category of non-degenerate idemp oten t k -pro jectiv e algebras NdI k . W e define a right A -c o m o dule algebr a as to b e a r ig h t como dule ov er t he coa lg ebra A in NdI k . Th us a righ t A -como dule alg ebra B = ( B , ρ ) consists o f a non-degenerate idempotent k -pro jectiv e algebra B and a morphism ρ : B | / / B ⊗ A suc h that ( ρ ⊗ A ) ◦ ρ = ( B ⊗ ∆) ◦ ρ, (40) 22 K. JANSSEN A ND J. VERCRUYSSE as morphisms from B to B ⊗ A ⊗ A , and ( B ⊗ ε ) ◦ ρ = B , (41) as morphisms from B to B . Making use of the asso ciated algebra map e ρ : B → M ( B ⊗ A ), (40) and (41) can b e translated in to ρ ⊗ A ◦ e ρ = B ⊗ ∆ ◦ e ρ, (42) B ⊗ ε ◦ e ρ = ι B . (43) Prop osition 3.7. A morphism ρ : B | / / B ⊗ A in NdI k is c o asso ciative in the sense of (42) if and only if ρ ⊗ A ( e ρ ( b )(1 ⊗ a )) = B ⊗ ∆( e ρ ( b ))(1 ⊗ 1 ⊗ a ) , (44) for al l a ∈ A and b ∈ B , if and only i f ( c ⊗ 1 ⊗ 1) ρ ⊗ A ( e ρ ( b )(1 ⊗ a )) = B ⊗ ∆(( c ⊗ 1) e ρ ( b ))(1 ⊗ 1 ⊗ a ) , (45) for al l a ∈ A and b, c ∈ B . Pr o of. Completely analogous to the pro of of Propo sition 2.3.  Prop osition 3.8. A morphism ρ : B | / / B ⊗ A in NdI k is c ounital in the sense of (4 3) if and only if B ⊗ ε ( e ρ ( b )(1 ⊗ a )) = e ε ( a ) ι B ( b ) , (46) for al l a ∈ A and b ∈ B . Pr o of. The pro of is quite analo gous to the one of Prop osition 2.4. W e will pro of one direction. Supp ose that (43) ho lds. F or b ′ = P e ε ( g ) b ′ 1 b ′ 2 ∈ B = B 2 w e hav e that B ⊗ ε ( e ρ ( b )(1 ⊗ a )) ✄ b ′ = X ^ B ⊗ ε  ( e ρ ( b )(1 ⊗ a )) ✄ ( b ′ 1 ⊗ g )  ✄ b ′ 2 = X ^ B ⊗ ε ( e ρ ( b ) ✄ ( b ′ 1 ⊗ ag ) ) ✄ b ′ 2 = X B ⊗ ε ( e ρ ( b )) ✄ e ε ( a ) b ′ (43) = ι B ( b ) ✄ e ε ( a ) b ′ = e ε ( a ) ι B ( b ) ✄ b ′ , hence the left m ultipliers of b oth sides of (46) are equal for all a ∈ A and b ∈ B ; the equalit y of the righ t m ultipliers follo ws in a similar w ay . Here w e used in the third equalit y that e ε ( a ) b ′ = P e ε ( a ) e ε ( g ) b ′ 1 b ′ 2 = P e ε ( ag ) b ′ 1 b ′ 2 .  By a right A -mo dule algebr a w e mean a non-unital algebra in the mo no idal category M A of non-degenerate ide mp otent k - pro jectiv e righ t A -mo dules. That is a n algebra B that is at the same time a non-degenerate idempotent k -pro jectiv e righ t A -mo dule, suc h that the m ultiplicatio n µ B : B ⊗ B → B is a righ t A -linear map. The latter means that µ B ( b ⊗ b ′ ) · a = µ B (( b ⊗ b ′ ) · a ) = µ B (( b ⊗ b ′ ) ✁ e ∆( a )) , for all b, b ′ ∈ B and a ∈ A , and where w e used the right A - action o n B ⊗ B as in Lemma 2.6. R emark 3.9 . In [12] como dule algebras and mo dule alg ebras w ere defined ov er a multiplier Hopf algebra. Just like m ultiplier Hopf algebras are in particular multiplier bialgebras, (co)mo dule algebras o v er a m ultiplier Ho pf algebra are particular instances of (co) mo dule algebras ov er the underlying multiplier bialgebra. Let us make this corresp ondence a bit MUL TIPLIER HOPF AN D BI- ALGEBRAS 23 more explicit. Note that (44) is an equalit y in M ( B ⊗ A ⊗ A ). In [12], a so-called coas- so ciativ e coaction of A on B is an algebra map e ρ : B → M ( B ⊗ A ) suc h that a similar coasso ciativit y condition as (44) holds, but under the extra assumption that e ρ ( b )(1 ⊗ a ) ⊆ ι B ⊗ A ( B ⊗ A ) , (47) (1 ⊗ a ) e ρ ( b ) ⊆ ι B ⊗ A ( B ⊗ A ) , (48) for all a ∈ A and b ∈ B . These equations (47) and (48) “ force” the equalit y ( 4 4) t o b e in ι B ⊗ A ⊗ A ( B ⊗ A ⊗ A ), hence obtaining an equalit y in B ⊗ A ⊗ A (by the injectivit y o f ι B ⊗ A ⊗ A ). It is a ctually this equalit y that expresses the coasso ciativit y in [12]. F urthermore, (4 6) states that the coaction e ρ : B → M ( B ⊗ A ) is counital in the sense of [12]. Ac kno w ledgemen t. The authors w ould like to thank A. V a n Daele for explaining them the concept of m ultiplier Hopf algebra during the join t Algebra seminars of the univ ersities of An tw erp en, Brussel, Leuv en and Hasselt in 200 8. W e thank Matthew Da ws for p ointing out references [4], [6 ] and [7]. JV thanks the F und for Scien tific Researc h–Flanders (Belgium) (F.W.O- Vlaanderen) for a P ostdo ctoral F ellow ship. Reference s [1] G. B¨ ohm, F. Nill and K. Szlach´ anyi, W ea k Hopf algebra s. I. Integral theory and C ∗ -structure, J. Alge br a , 221 (2), (199 9 ) 38 5–438 . [2] G. B¨ ohm and K. Szlac h´ anyi, Hopf a lgebroids with bijectiv e antipo des : axioms, integrals, and duals, J. Alge br a , 274 (2), (200 4) 70 8–75 0 . [3] S. Caenep eel and M. De Lombaerde, A categor ical a pproach to Tura ev’s Hopf g roup-coa lgebras, Comm. Alge br a , 34 (7), (2006 ) 26 31–26 57. [4] J. Dauns, Multiplier rings and primitive ideals, T r ans. A mer. Math. So c. , 145 , (196 9) 12 5–158 . [5] V. G. Drinfel ′ d, Quasi-Ho pf algebr as, Algebr a i A naliz , 1 (6), (1989 ) 114 –148. [6] G. Ho chsc hild, Cohomolo g y and repr esentations of as so ciative algebras , Duke Math. J. , 14 , (19 4 7) 9 21– 948. [7] B. E . Johnson, An introduction to the theory of centralizers, Pr o c. L ondon Math. So c. ( 3) , 14 , (19 64) 299–3 20. [8] S. Mac Lane, Catego ries for the working mathematician, volume 5 of Gr aduate T exts in Mathematics , Springer-V er lag, New Y or k, 1998, second edition. [9] P . Sc hauenburg, A bia lg ebra that admits a Hopf-Galois extension is a Hopf alg e bra, Pr o c. Amer. Math. So c. , 1 25 (1), (1 997) 83– 85. [10] A. V a n Daele, Multiplier Hopf a lgebras, T r ans. A mer. Math. So c. , 342 (2), (199 4 ) 917–93 2. [11] A. V a n Daele and Y. Zha ng, Corepres e n tation theory of m ultiplier Hopf a lgebras. I, Internat. J. Math. , 10 (4), (1999 ) 5 03–5 39. [12] A. V an Dae le a nd Y. H. Zha ng, Galois theory for multiplier Hopf algebra s with integrals, Algebr. R ep- r esent. The ory , 2 (1), (1 999) 83–10 6 . F a cul ty of Engineering, Vrije Universiteit Br ussel (VUB), B-1050 Brussels, Belgium E-mail addr ess : krja nsse@ vub.ac .be URL : homep ages. vub.ac.be/~krjansse E-mail addr ess : jver cruy@ vub.ac .be URL : homep ages. vub.ac.be/~jvercruy