Generalized Frobenius Algebras and the Theory of Hopf Algebras

GENERALIZED FR OBENIUS ALGEBRAS AND THE THEO R Y OF HOPF ALGEBRAS MIODRAG CRISTIAN IOV ANOV Abstra ct. ”Co-F roben ius” coalgebras were in tro duced as dualizations of F rob enius al- gebras. Recently , it w as sho wn in [I] that th ey admit left-right sy mmetric characteriza- tions analogue to t hose of F rob enius algebras: a coalgebra C is co-F rob enius if and only if it is isomorphic to its rational d ual. W e consider th e more general qu asi-co-F rob enius (QcF) co algebras; in the first mai n result w e sho w that these also admit sy mmetric char- acterizations: a coalgebra is QcF if it is weakly isomorphic t o its (left, or equiv alently righ t) rational du al R at ( C ∗ ), in the sense that certain copro duct or pro du ct p ow ers of these ob jects are isomorphic. These sh o w that QcF coalgebras can b e view ed as general- izations of b oth co-F rob enius coalgebras and F rob enius algebras. S urprisingly , these turn out to ha ve many applications to fundamental results of Hopf algebras. T he equiv alen t chara cterizations of H opf algebras with left (or righ t) nonzero integ rals as left (or right) co-F rob enius, or QcF, or semiperfect or with nonzero rational dual all follo w immediately from th ese results. Also, the celebrated uniqueness of integ rals follo ws at th e same time as j ust another eq uiv alen t statement. Moreo ver, as a b y-p rod u ct of our metho ds, we observe a short proof for th e bijectivit y of the antip od e of a Hopf algebra with nonzero integ ral. This giv es a purely represen tation theoretic approach to man y of the basic fundamental results in the theory of Hopf algebras. Introduction A K algebra A o v er a field K is called F rob enius if A is isomorph ic to A ∗ as right A - mo dules. This is equiv alen t to there b eing an isomorphism of le ft A -mo d ules b et w een A and A ∗ . This is the m o dern algebra language form ulation f or an old question p osed b y F rob enius. Giv en a fi nite dimensional algebra w ith a basis x 1 , . . . , x n , the left m ultipli- cation by an elemen t a induces a representa tion A 7→ E nd K ( A ) = M n ( K ), a 7→ ( a ij ) i,j ( a ij ∈ K ), w here a · x i = n P j =1 a ij x j . Similarly , th e right multiplicati on p r o duces a ma- trix a ′ ij b y w r iting x i · a = n P j =1 a ′ j i x j , a ′ ij ∈ K , and this induces another represent ation A ∋ a 7→ ( a ′ ij ) i,j . F rob enius’ problem came as the natur al question of when the tw o represent ations are equiv alent. F rob eniu s algebras o ccur in many differen t fields of math- ematics, su c h as top olog y (the cohomology ring of a compact manifold with coefficients in a fi eld is a F rob enius algebra b y Poincar ´ e dualit y), top ological quantum field theory (there is a one-to-o ne corresp ondence b et w een 2- dimensional quantum field theories and Key wor ds and phr ases. coalgebra, Hopf algebra, integral, F robenius, QcF, co-F rob enius. 2000 Mathematics Subje ct Classific ation . Primary 16W30; Secondary 16S90, 16Lxx, 16Nxx, 18E40. ∗ The author was partially supp orted by the contra ct nr. 24/28.0 9.07 with U EFISCU ”Groups, quantum groups, corings and representati on theory” of CNCIS, PN II (ID 1002) THIS P APER W AS COMPILED FROM A TEX-SU BMIS SION TO arXiv. 1 2 MIODRAG CRISTIAN IOV ANOV comm utativ e F r ob enius algebras; see [Ab]), Hopf algebras (a finite dimensional Hopf al- gebra is a F rob enius algebra), and F rob enius alge bras h a v e sub s equen tly dev elop ed into a researc h subfield of algebra. Co-F rob enius coalgebras were fir st int ro d uced b y Lin in [L] as a d ualization of F r ob e- nius algebras. A coalgebra is left (right) co-F rob en ius if th ere is a monomorp hism of left (righ t) C ∗ -mo dules C ⊆ C ∗ . How ever, u nlik e the algebra case, this concept is not left- righ t sy m metric, a s an example in [L] sho ws. Nev ertheless, in the ca se of Hopf alg ebras, it w as observ ed th at left co-F rob eniu s implies right co-F rob enius. Also, a left (or righ t) co-F rob enius coalg ebra can b e infin ite dimensional, while a F rob enius algebra is necessar- ily fin ite dimensional. Co-F rob enius coalgebras are coalgebras that are b oth left and r igh t co-F rob enius . It recen tly tur ned out that this notion of co-F rob enius has a nice c haracter- ization that is analogue to the c haracterizatio ns of F r ob enius algebras and is also left-right symmetric: a coalge bra C is co-F rob enius if it is isomorphic to its left (or equiv alentl y to its righ t) rational d ual R at ( ∗ C C ∗ ) (equiv alen tly C ≃ R at ( C ∗ C ∗ ); see [I]). T his also allo w ed for a categorical c haracterizati on whic h is again a nalogue to a c haracterizatio n of F rob eniu s algebras: an algebra A is F rob enius iff the functors Hom A ( − , A ) (”the A -dual functor”) and Hom K ( − , K ) (”the K -dual fun ctor”) are n aturally isomorph ic. Similarly , a coalgebra is co-F rob enius if the C ∗ -dual Hom C ∗ ( − , C ∗ ) and the K -dual Hom K ( − , K ) fun ctors are isomorphic on comodu les. If a coalgebra C is finite dimensional th en it is co-F rob eniu s if and only if C is F rob enius, showing that the co-F rob enius coalgebras (or rather their dual) can b e seen as the in finite d imensional generalization of F rob enius algebras. Quasi-co-F rob eniu s (QcF) coalgebras we re introdu ced in [GTN ] (further inv estiga ted in [GMN]), as a natural dualization of qu asi-F rob en iu s algebras (QF algebras), whic h are algebras th at are self-injectiv e, cogenerators and artinian to the left, equiv alen tly , all these conditions to the right. Ho w ev er, in ord er to allo w for infi n ite dimensional QcF coalgebras (and th us obtain more a general notion), the definition w as w eak en to the follo wing: a coalge bra is said to b e left (righ t) QcF if it em b eds in ` I C ∗ (a direct copro duct of copies of C ∗ ) as left (r ight) C ∗ -mo dules. These coalgebras we re sh o wn to b ear many prop erties that w ere the d u al analogue of the p rop erties of QF algebras. Again, this turned out not to b e a left-right symmetric concept, and QcF coalgebras we re in tro duced to be the coalge bras which are b oth left and righ t QcF. Ou r firs t goal is to note that the results and tec hniques of [I] can b e extended and applied to obtain a symmetric c haracterizati on of these coalgebras. In th e fi rst m ain result w e s h o w that a coalgebra is QcF if and only if C is ”w eakly” isomorphic to Rat ( C ∗ C ∗ ) as left C ∗ -mo dules, in the sense that some (co)pro duct p o wers of these ob jects are isomorphic, and this is equiv ale nt to asking that C ∗ is ”w eakly” isomorphic to Rat ( C ∗ C ∗ ) (its righ t rational dual) as righ t C ∗ -mo dules. In fact, it is enough to hav e an isomorphism of coun table p o w ers of these ob jects. This also a llo ws for a nice catego rical c haracterizat ion, whic h sta tes that C is Q cF if and only if the ab o v e C ∗ -dual and K -dual functors are (again) ”w eakly” isomorphic. Besides realizing QcF co algebras as a left-righ t symmetric concept whic h is a generalizat ion of b oth F rob enius algebras, co-F rob enius co-algebras and co-F rob enius Hopf algebras, we note that this also provi des this c haracteriza tion of fin ite dimensional quasi-F rob enius algebras: A is Q F iff A and A ∗ are we akly isomorphic in the ab o v e sense, equiv ale ntl y , ` N A ≃ ` N A ∗ . REPRESENT A TION THEOR Y AND HOPF ALGEBRAS 3 Th us these resu lts giv e a non trivial generalizatio n of F rob enius algebras and of quasi- F rob enius algebras, and the algebras arising as dual of QcF coalgebras are en titled to b e called Generalized F r ob enius Algebras, or rather Generalized Q F Algebras. These turn out to ha v e a wide range of ap p lications to Hopf algebras. In the theory of Hopf algebras, some o f the first fundamen tal results w ere concerned with the c haracteri- zation of Hopf algebras having a n onzero in tegral. T hese are in fact generalizations of w ell kno wn resu lts from the theory of compact groups. Recall that if G is a (locally) co mpact group, then there is a uniqu e left inv arian t (Haar) mea sur e and an asso ciated in teg ral R . Considering the algebra R c ( G ) of con tin uous repr esen tativ e fun ctions on G , i.e. fu nctions f : G → R ) suc h that there are f i , g i : G → K for i = 1 , n with f ( xy ) = n P i =1 f i ( x ) g i ( y ), then this b ecomes a Hopf algebra w ith m ultiplication giv en by the usu al m ultiplication of functions, com ultiplicatio n giv en by f 7→ n P i =1 f i ⊗ g i and an tip o de S giv en b y the comp osi- tion w ith the taking of inv erses S ( f )( x ) = f ( x − 1 ). Then, the in tegral R of G r estricted to R c ( G ) b ecomes an element of R c ( G ) ∗ that has the f ollo wing p rop erty: α · R = α ( 1 ) R , with 1 b eing the constan t 1 fun ction. Such an element in a general Hopf algebra is called a left in tegral, and Hopf algebras (quant um groups) havi ng a non zero left in tegral can b e viewed as (” quantum”) generaliza tions of compact group s (the Hopf Algebra can b e though t of as the algebra of con tin uous representat iv e functions on some abstract qu an tum space). Among th e fir st the f undamenta l results in Hopf algebras w as (were) the fac t(s) that the existence of a left integ ral is equiv alent to the existence o f a righ t integral , and these are equiv alent to the (co)represen tation th eoretic p r op erties of the u nderlying coalgebra of H of b eing left co-F rob enius, righ t co-F rob enius, left (or righ t) Q cF, or h a ving n onzero rational dual. These w ere results obtained in several initiating researc h pap ers on Hopf algebras [LS, MTW, R, Su, Sw1]. Then the natural question of wh ether the inte gral in a Hopf algebra is u nique arose (i.e. the space of left inte grals R l or that of right in tegrals R r is one dimensional), whic h w ould generalize the results f rom compact groups. The ans w er to this qu estion turn ed out p ositive , as it w as p ro v ed by Sulliv an in [Su]; alternate pro ofs follo w ed afterwards (see [Ra, S t]). An other v ery imp ortan t result is th at of Radford, wh o sho w ed that the ant ip ode of a Hopf algebra with nonzero in tegral is alwa ys bijectiv e. W e re-obtain all these results as a b ypro d uct of our co-represent ation theoretic results and generalizations of F rob enius algebras; they will turn out to b e an easy app lication of these general resu lts. W e also note a v ery short pro of of the bijectivit y of the an tip o de by constructing a certain deriv ed como du le structure on H , obtai ned b y using the antipo d e and the so called distinguished grou p lik e element of H , and the prop erties of the como dule H H . The only wa y we need to use the Hopf algebra structure of H is through the classical F un damen tal theorem of Hopf mo dules which give s an isomorphism of H -Hopf mo dules R l ⊗ H ≃ R at ( H ∗ H ∗ ); ho w ev er, w e will only n eed to use that this is a isomorphism of comod ules. W e th us find almost purely representa tion th eoretic p ro ofs of all th ese classical fundamental results from the theory of Hopf algebras, which b ecome immediate easy applications of the more general results on the ”generaliz ed F rob enius algebras”. T h us, the metho d s and results in this pap er are also inte nded to emphasize the p oten tial of these represent ation theoretic app roac hes. 4 MIODRAG CRISTIAN IOV ANOV 1. Quasi-co-Frobenius Coalgebras Let C b e a co algebra ov er a field K . W e den ote b y M C (resp ectiv ely C M ) the category of righ t (left) C -comodules and by C ∗ M (resp ectiv ely M C ∗ ) the cate gory of left (righ t) C ∗ -mo dules. W e use the s implified Sweedler’s σ -n otation f or the com ultiplicati on ρ : M → M ⊗ C of a C -como dule M , ρ ( m ) = m 0 ⊗ m 1 . W e will alwa ys use the inclusion of categories M C ֒ → C ∗ M , where the left C ∗ -mo dule s tructure on M is giv en by c ∗ · m = c ∗ ( m 1 ) m 0 . Let S b e a set of representati ve s for the types of isomorph ism of simp le left C -comod ules and T b e a set of represen tativ es for the simple right como dules. It is well kno wn that we ha v e an isomorphism of left C -como dules (equiv alently right C ∗ -mo dules) C ≃ L S ∈S E ( S ) n ( S ) , where E ( S ) is an injectiv e en v elop e of the left C -comodu le S and n ( S ) are p ositiv e in tegers. Similarly , C ≃ L T ∈T E ( T ) p ( T ) in M C , with p ( T ) ∈ N (w e use the same notation for en v elop es of left mo dules and for those of r igh t mo dules, as it will alwa ys b e understo o d from the co nt ext what t yp e of mo du les we refer to). Also C ∗ ≃ Q S ∈S E ( S ) ∗ in C ∗ M and C ∗ ≃ Q T ∈T E ( T ) ∗ in M C ∗ . W e refer the reader to [A], [DNR ] or [Sw] for these results and other basic facts of coalgebras. W e w ill use the finite top ology on du als of v ector spaces: giv en a v ector space V , this is the linear top ology on V ∗ that h as a basis of neigh b our h o o ds of 0 formed by the sets F ⊥ = { f ∈ V ∗ | f | F = 0 } for finite dimensional subspaces F of V . W e also write W ⊥ = { x ∈ V | f ( x ) = 0 , ∀ f ∈ W } for subsets W of V ∗ . An y top ologica l reference will b e with resp ect to this top ology . F or a mo dule M , w e conv ey to write M ( I ) for the copro duct (d irect sum) of I co pies of M and M I for the pro du ct of I copies of M . W e r ecall the follo wing definition from [GTN] Definition 1.1. A c o algebr a C is c al le d right (left) quasi-c o-F r ob enius, or shortly right QcF c o algebr a, if ther e is a monomor phism C ֒ → ( C ∗ ) ( I ) of right (left) C ∗ -mo dules. C is c al le d QcF c o algebr a if it is b oth a left and right QcF c o algebr a. Recall that a coalge bra C is called right semip erfect if the catego ry M C of right C - comod ules is semip erfect, that is, every righ t C -comod u le has a p ro jectiv e co v er. Th is is equiv alent to the fact that E ( S ) is finite d imensional for all S ∈ S (see [L]). In fact, this is the definition w e will n eed to use. F or con v enience, w e also r ecall the follo w in g very useful results on inj ective (pro jectiv e) como d ules, the first one originally giv en in [D] Prop ositio n 4, p.34 and the second one b eing Lemma 15 fr om [L]: [D, Prop osition 4] L et Q b e a finite d im en sional right C -comod ule. Then Q is injec- tiv e (pro jectiv e) as a left C ∗ -mo dule if and only if it is injectiv e (pr o jectiv e) as right C -como du le. [L, Lemma 15] L et M b e a finite-dimensional righ t C -como d u le. Th en M is an injectiv e righ t C -como du le if and only if M ∗ is a pro jectiv e left C -como dule. W e note the foll o wing prop osition that will b e useful in what follo ws; (i) ⇔ (ii) w as giv en in [GTN] and our approac h al so giv es here a d ifferen t pro of, along with the new charac- terizatio ns. Prop osition 1.2. L et C b e a c o algebr a. Then the fol lowing assert ions ar e e quivalent: (i) C is a right Q cF c o algebr a. (ii) C i s a right to rsionless mo dule, i.e. ther e is a mono morphism C ֒ → ( C ∗ ) I . (iii) Ther e exist a dense morphism ψ : C ( I ) → C ∗ , that is, the image of ψ is dense in C ∗ . (iv) ∀ S ∈ S , ∃ T ∈ T such that E ( S ) ≃ E ( T ) ∗ . REPRESENT A TION THEOR Y AND HOPF ALGEBRAS 5 Pro of. (i) ⇒ (ii) ob vious. (ii) ⇔ (iii) Let ϕ : C → ( C ∗ ) I ≃ ( C ( I ) ) ∗ b e a monomorphism of right C ∗ -mo dules. Let ψ : C ( I ) → C ∗ b e defined by ψ ( x )( c ) = ϕ ( c )( x ). It is straight forward to see that the fact that ϕ is a morph ism of left C ∗ -mo dules is equiv alent to ψ b eing a morphism in M C ∗ , and that ϕ injectiv e is equiv alent to (Im ψ ) ⊥ = 0, wh ich is furth er equiv alent to Im ψ is dense in C ∗ (for example, by [DNR] Corollary 1.2.9). (ii),(iii) ⇒ (iv) As Im ψ ⊆ Rat ( C ∗ C ∗ ), Rat ( C ∗ C ∗ ) is dense in C ∗ , so C is right semip erfect b y Prop osition 3.2.1 [DNR]. Th us E ( S ) is fi nite dimensional ∀ S ∈ S . Also b y (ii) ther e is a m onomorphism ι : E ( S ) ֒ → Q j ∈ J E ( T j ) ∗ with T j ∈ T , and as dim E ( S ) < ∞ there is a monomorphism to a finite dir ect sum E ( S ) ֒ → Q j ∈ F E ( T j ) ∗ ( F finite, F ⊆ J ). Indeed, if p j are the p ro jections of Q j ∈ J E ( T j ) ∗ , then note that T j ∈ J k er p j ◦ ι = 0, so there m ust b e T j ∈ F k er p j ◦ ι = 0 for a fi n ite F ⊆ J . Then E ( S ) is injectiv e also as right C ∗ -mo dules (see for example [DNR], Corolla ry 2.4.19), and so E ( S ) ⊕ X = L j ∈ F E ( T j ) ∗ for some X . By [I, Lemma 1.4], the E ( T j ) ∗ ’s are lo cal indecomp osable, and as they are also cyclic p ro jectiv e w e ev en tually get E ( S ) ≃ E ( T j ) ∗ for some j ∈ F . Th is can b e easily seen by noting that there is at lea st o ne n onzero morphism E ( S ) ֒ → E ( S ) ⊕ X = L j ∈ F E ( T j ) ∗ → L j ∈ F T ∗ j → S k (one lo oks at Jacobson radicals) and this pro jection th en lifts to a morp hism f : E ( S ) → E ( T k ) ∗ as E ( S ) is ob viously pr o jectiv e; this h as to b e surjectiv e since E ( T k ) ∗ is cyclic lo cal, and then f splits; hence E ( S ) ≃ E ( T k ) ∗ ⊕ Y with Y = 0 as E ( S ) is indecomp osable. (iv) ⇒ (i) Any isomorphism E ( S ) ≃ E ( T ) ∗ implies E ( S ) fi n ite dimensional b ecause then E ( T ) ∗ is cycli c rational; therefore E ( T ) ≃ E ( S ) ∗ . Thus f or eac h S ∈ S there is exactly one T ∈ T s u c h th at E ( S ) ≃ E ( T ) ∗ . If T ′ is the set of th ese T ’s, then: C ≃ M S ∈S E ( S ) n ( S ) ֒ → M S ∈S E ( S ) ( N ) ≃ M T ∈T ′ ⊆T ( E ( T ) ∗ ) ( N ) ֒ → ( M T ∈T ( E ( T ) ∗ ) p ( T ) ) ( N ) ⊆ ( Y T ∈T ( E ( T ) ∗ ) p ( T ) ) ( N ) = C ∗ ( N )  F rom the ab ov e p ro of, we see that when C is right QcF, the E ( S )’s are finite dimensional pro jectiv e for S ∈ S , and w e also conclude the follo wing result already kno wn from [GTN ] (in fact these cond itions are eve n equiv alen t); see also [DNR, Th eorem 3.3.4]. Corollary 1.3. If C is right Q cF, then C is also rig ht semip erfe ct and pr oje ctive as right C ∗ -mo dule. W e also immediately conclude the follo wing Corollary 1.4. A c o algebr a C is QcF i f and only if the app lic ation { E ( S ) | S ∈ S } ∋ Q 7→ Q ∗ ∈ { E ( T ) | T ∈ T } is wel l define d and bije ctive. Definition 1.5. (i) L et C b e a c ate gory having pr o ducts. We say that M , N ∈ C ar e we akly π -isomorphic if and only if ther e ar e some sets I , J suc h that M I ≃ N J ; we write M π ∼ N . (ii) L et C b e a c ate gory having c opr o ducts. We say that M , N ∈ C ar e we akly σ - isomorphic if and only if ther e ar e some sets I , J such that M ( I ) ≃ N ( J ) ; we wr ite M σ ∼ N . 6 MIODRAG CRISTIAN IOV ANOV The study of ob jects of a (suitable) catego ry C up to π (resp ectiv ely σ )-isomorphism is the study of the lo calization of C with resp ect to the class of all π (or σ )-isomorphisms. Recall that in the ca tegory C M of left como dules, copro du cts are the usual d irect sums of (r igh t) C ∗ -mo dules and the p ro duct C Q is giv en, for a family of como dules ( M i ) i ∈ I , by C Q i ∈ I M i = R at ( Q i ∈ I M i ). F or easy future reference, we in tro duce the f ollo wing conditions: (C1) C σ ∼ Rat ( C ∗ C ∗ ) in C M (or equiv alen tly , in M C ∗ ). (C2) C π ∼ Rat ( C ∗ C ∗ ) in C M . (C3) R at ( C I ) ≃ R at ( C ∗ J ) for some sets I , J . (C2’) C π ∼ R at ( C ∗ C ∗ ) in M C ∗ . Lemma 1.6. Either one of the c onditions (C1), (C2), (C3), (C2’) implies that C is QcF (b oth left and right). Pro of. Ob viously (C2’) ⇒ (C2). In all of the ab o v e conditions one ca n find a mon omor- phism of right C ∗ -mo dules C ֒ → ( C ∗ ) J , and th us C is r igh t QcF. Th en eac h E ( S ) for S ∈ S is finite d imensional and p ro jectiv e by Corollary 1.3. W e first sho w th at C is also left semip erfect, along the same lines as the pro ofs of [I], Prop osition 2. 1 and [I] Prop o- sition 2.6. F or s ak e of completeness, w e includ e a short ve rsion of these argumen ts h er e. Let T 0 ∈ T an d assume, b y con tradiction, that E ( T 0 ) is infin ite dimens ional. W e first sho w that Rat ( E ( T 0 ) ∗ ) = 0. Indeed, assu me otherwise. Then, since C ∗ = Q T ∈T E ( T ) ∗ p ( T ) and C = L S ∈S E ( S ) n ( S ) as righ t C ∗ -mo dules, it is straigh tforw ard to see that either one of co nditions (C1-C3) implies that Rat ( E ( T 0 ) ∗ ) is injectiv e as left comodu le, as a direct summand in an injectiv e como dule. Thus, as Rat ( E ( T 0 ) ∗ ) 6 = 0, there is a monomorp hism E ( S ) ֒ → R at ( E ( T 0 ) ∗ ) ⊆ E ( T 0 ) ∗ for some indecomp osable in jectiv e E ( S ) ( S ∈ S ). This sho ws th at E ( S ) is a direct su mmand in E ( T 0 ) ∗ , s in ce E ( S ) is injectiv e also as right C ∗ - mo dule (b y the ab ov e cited [D , Prop ositio n 4]). Bu t th is is a con tradiction since E ( S ) is finite dimen sional and E ( T 0 ) ∗ is indecomp osable by [I, Lemma 1.4] an d dim E ( T 0 ) ∗ = ∞ . Next, use [I, Prop osition 2.3] to get an exact sequence 0 → H → E = M α ∈ A E ( S α ) ∗ → E ( T ) → 0 with S α ∈ S . Since the E ( S α ) ∗ ’s are injectiv e in C ∗ M by [L, Lemma 15], we m a y as- sume, b y [I, Prop osition 2.4] that H con tains no nonzero injectiv e righ t comod ules. F or some β ∈ A 6 = ∅ , put E ′ = L α ∈ A \{ β } E ( S α ) ∗ . Th en one sees that H + E ′ = E (otherwise, since there is an epimorph ism E ( T ) = E H → E H + E ′ , the finite dimensional rational right C ∗ -mo dule  E H + E ′  ∗ w ould b e a nonzero rational submo du le of E ( T ) ∗ ), and this pro vides an epimorph ism H → H H ∩ E ′ ≃ H + E ′ E ′ ≃ E ( S β ) ∗ . But E ( S β ) ∗ is pr o jectiv e, so this epimor- phism sp lits, and this comes in contradict ion with the assumption on H (the E ( S β ) ∗ ’s are injectiv e in C ∗ M ).  No w, we note that if a coalgebra C is QcF, then all the conditions (C1)-(C3) are fu lfi lled. Indeed, we ha v e that eac h E ( S ) ( S ∈ S ) is isomorphic to exactly one E ( T ) ∗ ( T ∈ T ) and actually all E ( T ) ∗ ’s are isomorphic to s ome E ( S ). Then: REPRESENT A TION THEOR Y AND HOPF ALGEBRAS 7 (C1) C ( N ) = ( M S ∈S E ( S ) n ( S ) ) ( N ) = M S ∈S E ( S ) ( N ) = M T ∈T E ( T ) ∗ ( N ) = M T ∈T E ( T ) ∗ ( p ( T ) × N ) = M T ∈T ( E ( T ) ∗ p ( T ) ) ( N ) = ( R atC ∗ ) ( N ) where w e use that Rat ( C ∗ ) = L T ∈T E ( T ) ∗ p ( T ) as righ t C ∗ -mo dules for left and r ight semip er- fect coalgebras (see [DNR , Corollary 3.2.17]) (C2) C Y N C = Rat ( C N ) = C Y N M S ∈S E ( S ) ( n ( S )) = C Y N C Y S ∈S E ( S ) n ( S ) ( ∗ ) = C Y S ∈S E ( S ) n ( S ) × N = C Y S ∈S E ( S ) N = C Y T ∈T E ( T ) ∗ N = C Y T ∈T E ( T ) ∗ N × p ( T ) = C Y N C Y T ∈T E ( T ) ∗ p ( T ) = C Y N Rat ( Y T ∈T ( E ( T ) p ( T ) ) ∗ ) = C Y N Rat (( M T ∈ T E ( T ) p ( T ) ) ∗ ) = C Y N Rat ( C ∗ ) where for (* ) w e hav e used [I, Lemma 2.5] and the fact that E ( T ) ∗ are all rational since E ( T ) are fin ite dim en sional in this case (the pro du ct in th e category of left como dules is understo o d w henev er C Q is written); also (C3) holds b ecause R at ( C N ) = C Q T ∈T E ( T ) ∗ N b y the computations in lines 1 an d 3 in the ab o v e equation and b ecause Rat ( C ∗ N ) = Rat ( Y N Y T ∈T E ( T ) ∗ p ( T ) ) = C Y T ∈T E ( T ) ∗ p ( T ) × N = C Y T ∈T E ( T ) ∗ N Com bining all of the ab ov e we obtain the follo wing n ice symmetric c haracteriz ation wh ic h extends the one of co-F rob eniu s coalgebras from [I] and those of co-F rob enius Hopf algebras and F r ob enius Algebras. Theorem 1.7. L et C b e a c o algebr a. Then the fol lowing assertions ar e e quivalent. (i) C is a QcF c o algebr a. (ii) C σ ∼ Rat ( C ∗ C ∗ ) or C π ∼ Rat ( C ∗ C ∗ ) in C M or Rat ( C I ) ≃ Rat ( C ∗ J ) in C M (or M C ∗ ) for some sets I , J . (iii) C ( N ) ≃ ( Rat ( C ∗ )) ( N ) or C Q N C ≃ C Q N Rat ( C ∗ ) or Rat ( C N ) ≃ Rat ( C ∗ N ) as left C - c omo dules (right C ∗ -mo dules) 8 MIODRAG CRISTIAN IOV ANOV (iv) The left hand side version of (i)- (iii). (v) The asso ciation Q 7→ Q ∗ determines a duality b etwe en the finite d imensional inje ctive left c omo dules and finite dimensional inje ctiv e right c omo dules. 1.1. Categorical c haracterization of QcF coalgebras. W e giv e no w a characte riza- tion similar to the fun ctorial c haracterizations of co-F r ob enius coalge bras an d of F rob enius algebras. F or a set I let ∆ I : C M − → ( C M ) I b e the diagonal functor and let F I b e the comp osition functor F I : C M ∆ I − → ( C M ) I L I − → C M that is F I ( M ) = M ( I ) for any left C -como du le M . Theorem 1.8. L et C b e a c o algebr a. Then the fol lowing assertions ar e e quivalent: (i) C is QcF. (ii) The functors Hom C ∗ ( − , C ∗ ) ◦ F I and Hom( − , K ) ◦ F J fr om C M = R at ( C ∗ M ) to C ∗ M ar e natur al ly isom orphic for some sets I , J . (iii) The functors Hom C ∗ ( − , C ∗ ) ◦ F N and Hom( − , K ) ◦ F N ar e natur al ly isom orphic. Pro of. S ince for an y left como d u le M , there is a natural isomorphism of left C ∗ -mo dules Hom C ∗ ( M , C ) ≃ Hom( M , K ), then for any sets I , J and any left C -como dule M we hav e the follo wing natural isomorphism s: Hom( M ( I ) , K ) ≃ Hom C ∗ ( M ( I ) , C ) ≃ Hom C ∗ ( M , C I ) ≃ Hom C ∗ ( M , Rat ( C I )) Hom C ∗ ( M ( J ) , C ∗ ) ≃ Hom C ∗ ( M , ( C ∗ ) J ) ≃ Hom( M , Rat ( C ∗ ) J ) Therefore, by the Y oneda Lemma, the functors of (ii) are naturally isomorph ic if and only if Rat ( C I ) ≃ R at ( C ∗ J ). Thus, by Th eorem 1.7 (ii), these functors are iso morph ic if and only if C is QcF. Moreo v er, in this case, b y the same theorem the sets I , J can b e c hosen coun table.  R emark 1.9 . T h e ab ov e theorem states that C is QcF if and only if the fun ctors C ∗ - dual Hom( − , C ∗ ) and K -dual Hom( − , K ) from C M to C ∗ M are isomorphic in a ”we ak” meaning, in the sense that they are isomorphic only on the ob jects of the form M ( N ) in a w a y that is n atur al in M , i.e. they are isomorphic on the sub categ ory of C M consisting of ob jects M ( N ) with morphisms f ( N ) induced b y an y f : M → N . If we consider the categ ory C of fu nctors fr om C M to C ∗ M with morph isms the classes (whic h are not n ecessarily sets) of natural transformations b et w een functors, then the isomorphism in (ii) can b e restated as (Hom C ∗ ( − , C ∗ )) I ≃ (Hom( − , K )) J in C , i.e. the C ∗ -dual and the K -dual functors are w eakly π -isomorphic ob jects of C . 2. Appl ica tions to Hop f Algebras Before giving th e main app lications to Hopf algebras, we start with t w o easy prop ositions that will cont ain the main ideas of the applications. First, for a QcF coalgebra C , let ϕ : S → T b e the fun ction defined by ϕ ( S ) = T if and only if E ( T ) ≃ E ( S ) ∗ as left C ∗ -mo dules; b y the ab ov e Corollary 1.4, ϕ is a bijection. Prop osition 2.1. (i) L et C b e a QcF c o algebr a and I , J sets such that C ( I ) ≃ ( R at ( C ∗ )) ( J ) as right C ∗ -mo dules. If one of I , J is finite then so i s the other. (ii) L et C b e a c o algebr a. Then C is c o-F r ob enius if and only if C ≃ Rat ( C ∗ C ∗ ) as left C ∗ -mo dules and if and only if C ≃ Rat ( C ∗ C ∗ ) as right C ∗ -mo dules. REPRESENT A TION THEOR Y AND HOPF ALGEBRAS 9 Pro of. (i) C is left and r igh t semip erfect (Corollary 1.3), so using again [DNR, Corollary 3.2.17 ] w e h a v e Rat ( C ∗ C ∗ ) = L T ∈T E ( T ) ∗ p ( T ) = L S ∈S E ( S ) p ( ϕ ( S )) and we get L S ∈S E ( S ) n ( S ) × I ≃ L S ∈S E ( S ) p ( ϕ ( S )) × J . F rom h er e, since the E ( S )’s are indecomp osable injectiv e comod ules w e get an equiv alence of sets n ( S ) × I ∼ p ( ϕ ( S )) × J (or directly , by ev aluating the so cle of these como dules). This finish es the pr o of, as n ( S ) , p ( ϕ ( S )) are fin ite. (ii) If C is co-F rob enius, C is also QcF and a monomorphism C ֒ → Rat ( C ∗ C ∗ ) of right C ∗ -mo dules implies L S ∈S E ( S ) n ( S ) ֒ → L T ∈T E ( T ) ∗ p ( T ) ≃ L S ∈S E ( S ) p ( ϕ ( S )) and w e get n ( S ) ≤ p ( ϕ ( S )) for all S ∈ S ; similarly , a s C is also left co-F rob enius w e get n ( S ) ≥ p ( ϕ ( S )) for all S ∈ S . Hence n ( S ) = p ( ϕ ( S )) for all S ∈ S and th is implies C = L S ∈S E ( S ) n ( S ) ≃ L T ∈T E ( T ) p ( T ) = Rat ( C ∗ C ∗ ). Con v ersely , if C ≃ Rat ( C ∗ C ∗ ) by the pr o of of (i), wh en I and J ha v e one ele ment we get that n ( S ) = p ( ϕ ( S )) for all S ∈ S wh ic h implies that w e also ha v e C = L T ∈T E ( T ) p ( T ) ≃ L S ∈S E ( S ) ∗ n ( S ) = R at ( C ∗ C ∗ ) so C is co-F rob enius.  The ab ov e Prop osition 2.1 (ii) shows that the results of this pap er are a generalizati on of the results in [I ]. Prop osition 2.2. L e t C b e a (QcF) c o algebr a such that C k ≃ Rat ( C ∗ C ∗ ) in M C ∗ and C l ≃ R at ( C ∗ C ∗ ) in C ∗ M , k , l ∈ N . Then C is c o-F r ob enius and k = l = 1 . Pro of. As in the pro of of P r op osition 2.1 we get k · n ( S ) = p ( ϕ ( S )) f or all S ∈ S . Similarly , usin g C l ≃ ( Rat ( C ∗ C ∗ )) in C ∗ M w e get l · p ( T ) = n ( ϕ − 1 ( T )) for T ∈ T i.e. n ( S ) = l · p ( ϕ ( S )). These t w o equations gi ve k = l = 1 and the conclusion follo ws as in Prop osition 2.1 (ii).  Let H b e a Hopf algebra o v er a basefield k . Recall that a left in tegral for H is an element λ ∈ H ∗ suc h that α · λ = α (1) λ , for all α ∈ H ∗ . The space of left in tegrals for H is denoted b y R l . The righ t int egrals and the sp ace of righ t in tegrals R r are defined b y analogy . F or basic facts on Hopf algebras we r efer to [A], [DNR], [M] and [S w]. Th e Hopf algebra structure will come in to p la y only through a basic Th eorem of Hopf algebras, the fun damen tal theorem of Hopf mo d ules whic h yields the isomorphism of right H -Hopf mo dules R l ⊗ H ≃ R at ( H ∗ H ∗ ). Th is isomorph ism is given by t ⊗ h 7→ t ↽ h = S ( h ) ⇀ t , where for x ∈ H , α ∈ H ∗ , x ⇀ α is defin ed b y ( x ⇀ α )( y ) = α ( y x ) and α ↽ x = S ( x ) ⇀ α . Y et, we will only need that this is an isomorphism of right H -como dules (left H ∗ -mo dules). Similarly , H ⊗ R r ≃ R at ( H ∗ H ∗ ). Theorem 2.3. (Lin, L arson, Swe e d ler, Sul livan) If H is a Hopf algebr a, then the fol lowing assertions ar e e qu i valent. (i) H is a right c o-F r ob enius c o algebr a. (ii) H is a right QcF c o algebr a. (iii) H is a right semip erfe ct c o algebr a. (iv) Rat ( H ∗ H ∗ ) 6 = 0 . (v) R l 6 = 0 . (vi) dim R l = 1 . (vii) The left hand side version of the ab ove. Pro of. (i) ⇒ (ii) ⇒ (i ii) is clea r and (iii) ⇒ (iv) is a pr op erty of semip erfect coalgebras (see [DNR, Section 3.2]). (iv) ⇒ (v) follo ws by the isomorphism R l ⊗ H ≃ R at ( H ∗ H ∗ ) and (vi) ⇒ (v) is tr ivial. 10 MIODRAG CRISTIAN IOV ANOV (v) ⇒ (i), (vi) and (vi i). Since R l ⊗ H ≃ R at ( H ∗ H ∗ ) in M H , w e ha v e H ( R l ) ≃ Rat ( H ∗ H ∗ ) so by Theorem 1.7 H is QcF (b oth left and right); it then follo ws that R r 6 = 0 (by the left h and v ersion of (ii) ⇒ (v)) and H ( R r ) ≃ R at ( H ∗ H ∗ ). W e can n o w apply Prop ositions 2.1 and 2.2 to first get that dim R l < ∞ , dim R r < ∞ and then that H is co-F rob enius (both left and righ t) so (i) and (vii) h old. Again b y Prop osition 2.2 we get that, more precisely , dim R l = dim R r = 1.  The follo wing corollary w as the initial form of the result pro v ed by Sw eedler [Sw1] Corollary 2.4. The fol lowing ar e e quivalent for a Hopf algebr a H : (i) H ∗ c ontains a finite dimensio nal left ide al. (ii) H c ontains a lef t c oide al of finite c o dimension. (iii) R l 6 = 0 . (iv) Rat ( H ∗ ) 6 = 0 . Pro of. (i) ⇔ (ii) It can b e seen by a straight forward computation that there is a bijectiv e corresp onden ce b et w een fin ite dimensional left ideals I of H ∗ and coideals K of finite co dimension in H , giv en b y I 7→ K = I ⊥ . Moreo ve r, it follo ws that any suc h finite dimensional ideal I of H ∗ is of the form I = K ⊥ with dim( H/K ) < ∞ , s o I = K ⊥ ≃ ( H /K ) ∗ is then a rational left H ∗ -mo dule, thus I ⊆ R at ( H ∗ ). This shows that (ii) ⇒ (iv) also holds, while (iii) ⇒ (ii) is trivial.  The bijectivity of the an tip o de. Let t b e a nonzero left int egral for H . Then it is easy to see that the one dimensional vec tor space k t is a t wo sid ed ideal of H ∗ . Also, b y the defin ition of in tegrals, k t ⊆ R at ( H ∗ H ∗ ) = Rat ( H ∗ H ∗ ) (since H is semip erfect as a coalge bra). T h us k t also has a left comultiplica tion t 7→ a ⊗ t , a ∈ H and then b y the coassociativit y and counit prop ert y for H k t , a has to b e a grouplike elemen t. This element is cal led the distinguishe d gr oupl ike elemen t of H . In p articular t · α = α ( a ) t, ∀ α ∈ H ∗ . See [DNR, Chapter 5] for some more details. F or any righ t H -comod ule M den ote a M the left H -como dule stru cture on M with co- m ultiplication M ∋ m 7→ m a − 1 ⊗ m a 0 = aS ( m 1 ) ⊗ m 0 ( S d enotes the antip o de). It is str aigh tforw ard to see that this d efines an H -como dule structure. Prop osition 2.5. The map p : a H → R at ( H ∗ ) , p ( x ) = x ⇀ t is a surje ctive morphism of left H -c omo dules (right H ∗ -mo dules). Pro of. Since the ab o v e isomorphism H ≃ R l ⊗ H ≃ Rat ( H ∗ ) is giv en by h 7→ t ↽ h = S ( h ) ⇀ t , w e get the surjectivit y of p . W e need to sh ow that p ( x ) − 1 ⊗ p ( x ) 0 = x a − 1 ⊗ p ( x a 0 ) and for this, h a ving the left H -comod u le structure of Rat ( H ∗ ) in mind, it is enough to sho w that for all α ∈ H ∗ , p ( x ) 0 α ( p ( x ) − 1 ) = p ( x ) · α = α ( x a − 1 ) p ( x a 0 ). In deed, for g ∈ H we ha v e: (( x ⇀ t ) · α )( g ) = t ( g 1 x ) α ( g 2 ) = t ( g 1 x 1 ε ( x 2 )) α ( g 2 ) = t ( g 1 x 1 ) α ( g 2 x 2 S ( x 3 )) = t ( g 1 x 1 )( α ↽ x 3 )( g 2 x 2 ) = t (( g x 1 ) 1 )( α ↽ x 2 )(( gx 1 ) 2 ) = ( t · ( α ↽ x 2 ))( gx 1 ) = ( α ↽ x 2 )( a ) t ( gx 1 ) ( a is the distinguish ed group like of H ) = α ( aS ( x 2 ))( x 1 ⇀ t )( g ) and this ends th e pr o of.  REPRESENT A TION THEOR Y AND HOPF ALGEBRAS 11 Let π b e th e comp osition map a H p − → R at ( H ∗ H ∗ ) ∼ − → H ⊗ R r ≃ H , wh er e the isomorphism H ⊗ R r ≃ H ∗ H ∗ is the left analog ue of R l ⊗ H ≃ R at ( H ∗ H ∗ ). S in ce H H is pro jectiv e in H M , this su r jectiv e map splits by a morph ism of left H -comod ules ϕ : H ֒ → a H , so π ϕ = Id H . Then we can find another pro of of: Theorem 2.6. The ant ip o de of a c o-F r ob enius Hopf algebr a is bije ctive. Pro of. Since the inj ectivit y of S is immed iate from the injectivit y of the map H ∋ x 7→ t ↽ x ∈ H ∗ , as noticed b y Swe edler [Sw1], w e only observe the surjectivit y . The fact that ϕ is a m orp hism of como du les reads ϕ ( x ) a − 1 ⊗ ϕ ( x ) a 0 = x 1 ⊗ ϕ ( x 2 ), i.e. aS ( ϕ ( x ) 2 ) ⊗ ϕ ( x ) 1 = x 1 ⊗ ϕ ( x 2 ), and since a = S ( a − 1 ) = S 2 ( a ), by applyin g Id ⊗ επ w e get S ( a − 1 ) S ( ϕ ( x ) 2 ) επ ( ϕ ( x ) 1 ) = x 1 επ ϕ ( x 2 ) = x 1 ε ( x 2 ) = x , so x = S ( επ ( ϕ ( x ) 1 ) ϕ ( x ) 2 a − 1 ).  Referen ces [A] E. A b e, Hopf Algebr as , Cambridge Univ. Press, 1977. [Ab] L. Abrams, Two-dimensional top olo gic al quantum field the ories and F r ob enius algebr as , J. Knot The- ory R amifications 5 (1996), no. 5, 569587. [AF] D. And erson, K. F uller, Rings and Cate gories of Mo dules , Grad. T ex ts in Math., Springer, Berlin- Heidelb erg-New Y ork, 1974. [D] Y . Doi, H om olo gic al Co algebr a , J. Math. S oc. Japan 33 (1981), 31-50. [DNR] S. D˘ asc˘ alescu, C. N ˘ ast˘ asescu, S ¸ . R aian u, Hopf Algebr as: an intr o duction . V ol. 235. Lecture N otes in Pure and A pplied Math. V ol.235, Marcel Dekker, New Y ork, 2001. [F] C. F aith, Algebr a II : Ring The ory . V ol 191, Sp ringer-V erlag, Berlin-Heidelb erg-New Y ork, 1976. [GTN] J. Gomez-T orrecillas, C. N˘ ast˘ asescu, Q uasi-c o-F r ob enius c o algebr as , J. Algebra 174 (1995), 909-923. [GMN] J. Gomez-T orrecillas, C. Manu, C. N˘ ast˘ asescu, Quasi-c o-F r ob enius c o al gebr as II , Comm. Algebra V ol 31, No. 10, pp. 5169-5177, 2003. [I] M.C. Iov ano v, Co-F r ob enius Co algebr as , J. A lgebra 303 (2006), no. 1, 146–153; eprint arXiv:math/0604251, http://xxx.lanl.go v/abs/math.QA/0604251. [L] B.I-P eng Lin, Semip erfe ct c o al gebr as , J. Algebra 30 (1974), 559-601. [LS] R.G. Larson; M.E. Sweedler, A n asso ciative ortho gonal bil ine ar form for Hopf algebr as , Amer. J. Math. 91 1969 7594. [M] S . Montgomery , Hopf algebr as and their actions on rings , Amer. Math. So c., Pro vidence, RI, 1993. [MTW] C. Menini, B. T orreci llas Jov er, R . Wisbauer, Str ongly r ational c omo dules and se mip erfe ct Hopf algebr as over QF rings , J. Pure A ppl. A lgebra 155 (2001), no. 2-3, 237255. [R] D.E. R adford, Fini teness c onditions for a Hopf algebr a with a nonzer o inte gr al , J. Algebra 46 (1977), no. 1, 189195. [Ra] S ¸ . Raian u, An e asy pr o of f or the uniqueness of i nte gr als . Hopf algebras and quantum groups (Brussels, 1998), 237–24 0, Lecture Notes in Pure and App l. Math., 209, Dek ker, New Y ork, 2000. [Su] J.B. Sulliv an, The uniqueness of inte gr als f or Hopf algebr as and some existenc e the or ems of inte gr als for c ommutative Hopf algebr as , J. Algebra 19 1971 426440. [Sw] M.E. Sweedler, Hopf Alge br as , Benjamin, New Y ork, 1969. [Sw1] M.E. Swee dler, I nte gr als for Hopf algebr as , Ann. Math 89 (1969), 323-335. [St] S ¸ . Drago¸ s The uniqueness of inte gr als (a homolo gic al appr o ach) , Comm. Algebra 23 (1995), no. 5, 1657166 2. [vD] A. v an Daele, The Haar me asur e on finite quantum gr oups , Proc. Amer. Math. Soc. 125 (1997), no. 12, 3489-3500. [vD1] K. V an Daele, An algebr aic fr amework for gr oup duality , Adv . Math. 140 (1998), n o. 2, 323366. Miodrag Cristia n Iov anov University of B ucharest, F acul ty of Ma the ma tics, Str. A cademie i 14 RO-0 10014, Bucharest, Romania and 12 MIODRAG CRISTIAN IOV ANOV St a te Un iversity of New Y ork (Buff alo) Dep ar tment of Ma thema tics, 244 Ma thema ti cs Buildi ng Buff alo, NY 14260-2900, U SA E–mail addr ess : yovanov@gmai l.com