CHAPTER 3. Hopf Algebras, Algebraic, Formal, and Quantum Groups

CHAPTER 3 Hopf Algebras Algebraic Formal and Quantum Groups 83

84 3. HOPF ALGEBRAS ALGEBRAIC FORMAL AND QUANTUM GROUPS 3. Dual Objects At the end of the rst section in Corollary 3.1.15 we saw that the dual of an H module can be constructed. We did not show the corresponding result for comodules. In fact such a construction for comodules needs some niteness conditions. With this restriction the notion of a dual object can be introduced in an arbitrary monoidal category. Denition 3.3.1. Let èc; è be a monoidal category M 2Cbe an object. An object M 2Ctogether with a morphism ev : M M! I is called a left dual for M if there exists a morphism db : I! M M in C such that èm èm 1db db 1! M M M 1ev! M ev 1 M M! Mè =1 M! M è=1 M : A monoidal category is called left rigid if each object M 2Chas a left dual. Symmetricallywe dene: an object M 2Ctogether with a morphism ev : M M! I is called a right dual for M if there exists a morphism db : I! M M in C such that èm 1db! M ev 1 M M! Mè =1 M è db 1 M! M M M! 1ev Mè=1 M: A monoidal category is called right rigid if each object M 2Chas a left dual. The morphisms ev and db are called the evaluation respectively the dual basis. Remark 3.3.2. If èm ; evè is a left dual for M then obviously èm;evè is a right dual for M and conversely. One uses the same morphism db : I! M M. Lemma 3.3.3. Let èm ; evè be a left dual for M. Then there is a natural isomorphism Mor C è M;è = MorC è; M è; i. e. the functor M : C!Cis left adjoint to the functor M : C!C. Proof. We give the unit and the counit of the pair of adjoint functors. We dene èaè :=1 A db : A! A M M and èbè :=1 B ev : B M M! B. These are obviously natural transformations. We have commutative diagrams and èa M èb M FèAè= 1Adb 1M GèBè= 1B1M db A M M M B M M M thus the Lemma has been proved by Corollary A.9.11. FèAè= 1A1M ev GèBè= 1Bev 1M The converse holds as well. If M is left adjoint to M A Mè =1 AM B M è=1 BM then the unit gives a morphism db := èiè :I! M M and the counit gives a morphism ev := èiè :M M! I satisfying the required properties. Thus we have

3. DUAL OBJECTS 85 Corollary 3.3.4. If M : C!Cis left adjoint to M : C!Cthen M is a left dual for M. Corollary 3.3.5. èm ; evè is a left dual for M if and only if there is a natural isomorphism Mor C èm ; è = MorC è;m è; i. e. the functor M : C!Cis left adjoint to the functor M : C!C. The natural isomorphism if given by and èf : M N! P è 7! èè1 M fèèdb 1 N è:n! M M N! M P è èg : N! M P è 7! èèev 1 P èè1 M gè :M N! M M P! P è: Proof. We have a natural isomorphism Mor C èm ; è = MorC è;m è; i èm;evè is a right dual for M èas a symmetric statement to Lemma 3.3.3è i èm ; evè is a left dual for M. Corollary 3.3.6. If M has a left dual then this is unique up to isomorphism. Proof. Let èm ; evè and èm! ; ev! è be left duals for M. Then the functors M and M! are isomorphic by Lemma A.9.7. In particular we have M = I M = I M! = M!. If we consider the construction of the isomorphism then we get in particular that èev! 1èè1 dbè : M!! M! M M! M is the given isomorphism. Problem 3.3.1. Let èm ; evè be a left dual for M. Then there is a unique morphism db : I! M M satisfying the conditions of Denition 3.3.1. Denition 3.3.7. Let èm ; ev M è and èn ; ev N è be left duals for M resp. N. For each morphism f : M! N we dene the transposed morphism èf : N! M è:=èn 1db M! N 1f 1 M M! N N M ev N 1! M è: With this denition we get that the left dual is a contravariant functor since we have Lemma 3.3.8. Let èm ; ev M è èn ; ev N è and èp ; ev P è be left duals for M N and P respectively. 1. We have è1 M è =1 M. 2. If f : M! N and g : N! P are given then ègfè = f g holds. Proof. 1. è1 M è = èev 1èè1 1 1èè1 dbè = 1 M.

86 3. HOPF ALGEBRAS ALGEBRAIC FORMAL AND QUANTUM GROUPS 2. The following diagram commutes M N f dbn 1 dbn 1 N N M 11f N N N g11 P N N 1evN N 1evN P g Hence we have gf =è1 ev N èèg 1 fèèdb N 1è. Thus the following diagram commutes P 1db P N N 1g1 P P N Z ZZ 1db P M M 1gf1 P P M M ev 1 1db 1 1f 1ev 1 1ev 1db P N N M M 1g1f 1 XXXX Xz P P N N M ev 1 N N M 1g1 1db P P N M M 9 1 1f 1 ev 1 1db = N M M Z ZZ ev 1 Z ZZZ~ N Problem 3.3.2. 1. In the category of Ngraded vector spaces determine all objects M that have a left dual. 2. In the category of chain complexes KComp determine all objects M that have a left dual. 3. In the category of cochain complexes KCocomp determine all objects M that have a left dual. 4. Let èm ; evè be a left dual for M. Show that db : I! M M is uniquely determined by M M and ev. èuniqueness of the dual basis.è 5. Let èm ; evè be a left dual for M. Show that ev : M M! I is uniquely determined by M M and db.

3. DUAL OBJECTS 87 Corollary 3.3.9. Let M; N have the left duals èm ; ev M è and èn ; ev N è and let f : M! N be a morphism in C. Then the following diagram commutes I dbm M M dbn f 1 N N N M : 1f Proof. The following diagram commutes f M db 1 db 1 N H HHHj 1 N N M 11f N N N 1ev N This implies èf 1 M èdb M = èè1 N ev N èè1 N 1 N fèèdb N 1 M è 1 M èdb M = è1 N ev N 1 M èè1 N 1 N f 1 M èèdb N 1 M 1 M èdb M =è1 N ev N 1 M èè1 N 1 N f 1 M èè1 N 1 N db M èdb N =è1 N èev N 1 M èè1 N f 1 M èè1 N db M èè db N =è1 N f èdb N. Corollary 3.3.10. Let M; N have the left duals èm ; ev M è and èn ; ev N è and let f : M! N be a morphism in C. Then the following diagram commutes N M f 1 M M 1f evm N N I: evn Proof. This statement follows immediately from the symmetry of the denition of a left dual. Example 3.3.11. Let M 2 R M R be an RRbimodule. Then Hom R èm:; R:è is an RRbimodule by èrfsèèxè = rfèsxè. Furthermore we have the morphism ev : Hom R èm:; R:è R M! R dened by evèf R mè=fèmè. èdual Basis Lemma:è A module M 2M R is called nitely generated and projective if there are elements m 1 ;::: ;m n 2 M und m 1 ;::: ;m n 2 Hom R èm:; R:è such that 8m 2 M : nx i=1 m i m i èmè =m: Actually this is a consequence of the dual basis lemma. But this denition is equivalent to the usual denition.

88 3. HOPF ALGEBRAS ALGEBRAIC FORMAL AND QUANTUM GROUPS Let M 2 R M R. M as a right Rmodule is nitely generated and projective im has a left dual. The left dual is isomorphic to Hom R èm:; R:è. If M R is nitely P generated projective then we use db : R! M R Hom R èm:; R:è n with dbè1è = m i=1 i R m i. In fact we have è1 R evèèdb R 1èèmè = è1 R P evèè m i R m i P R mè= m i m i èmè =m. Wehave furthermore èev R 1èè1 R dbèèfèèmè = èev P n R 1èè f i=1 R m i P P R m i èèmè = fèm i èm i èmè =fè m i m i èmèè = fèmè for all m 2 M hence èev R 1èè1 R dbèèfè =f. Conversely if M has a left dual M then ev : M R M! R denes a homo : M! Hom R èm:; R:è in R M R by èm èèmè = evèm R mè. We dene Pmorphism n m i=1 i m i := dbè1è 2 M M then m =è1evèèdb 1èèmè =è1 P evèè m i m i P mè = m i èm i èèmè so that m 1 ;::: ;m n 2 M and èm 1 è;::: ;èm n è 2 Hom R èm:; R:è form a dual basis for M i.e. M is nitely generated and projective asanrmodule. Thus M and Hom R èm:; R:è are isomorphic by the map. Analogously Hom R è:m; :Rè is a right dual for M i M is nitely generated and projective as a left Rmodule. Problem 3.3.3. Find an example of an object M in a monoidal category C that has a left dual but no right dual. Denition 3.3.12. Given objects M; N in C. An object ëm; Në is called a left inner Hom of M and N if there is a natural isomorphism Mor C è M; Nè = Mor C è; ëm; Nëè i. e. if it represents the functor Mor C è M; Nè. If there is an isomorphism Mor C èp M; Nè = MorC èp; ëm; Nëè natural in the three variable M; N; P then the category C is called monoidal and left closed. If there is an isomorphism Mor C èm P; Nè = MorC èp; ëm; Nëè natural in the three variable M; N; P then the category C is called monoidal and right closed. If M has a left dual M in C then there are inner Homs ëm; ë dened by ëm; Në :=N M. In particular left rigid monoidal categories are left closed. Example 3.3.13. 1. The category of nite dimensional vector spaces is a monoidal category where each object has a èleft and rightè dual. Hence it is èleft and rightè closed and rigid. 2. Let Ban be the category of ècomplexè Banach spaces where the morphisms satisfy k fèmè kk m k i. e. the maps are bounded by 1 or contracting. Ban is a monoidal category by the complete tensor product M bn. InBan exists an inner Hom functor ëm; Në that consists of the set of bounded linear maps from M to N made into a Banach space by an appropriate topology. Thus Ban is a monoidal closed category. 3. Let H be a Hopf algebra. The category HMod of left Hmodules is a monoidal category èsee Example 3.2.4 2.è. Then Hom K èm; Nè is an object in HMod by the multiplication èhfèèmè := X h è1è fèmsèh è2è è

3. DUAL OBJECTS 89 as in Proposition 3.1.14. Hom K èm; Nè is an inner Hom functor in the monoidal category HMod. The isomorphism : Hom K èp; Hom K èm; Nèè = HomK èp M; Nè can be restricted to an isomorphism Hom H èp; Hom K èm; Nèè = HomH èp M; Nè; Pbecause èfèèhèp mèè P P = èfèè h è1è p P h è2è mè = fèh è1è pèèh è2è mè = èhè1è èfèpèèèèh è2è mè= h è1è èfèpèèsèh P è2è èh è3è mèè = hèfèpèèmèè P = hèèfè èpmèè and conversely èhèfèpèèèèmè = h è1è èfèpèèsèh è2è èmèè = h è1è èèfè èp P Sèh è2è èmèè = èfèèh è1è p h è2è Sèh è3è èmè =èfèèhp mè =fèhpèèmè. Thus HMod is left closed. If M 2 HMod is a nite dimensional vector space then the dual vector space M := Hom K èm; Kè again is an Hmodule by èhfèèmè :=fèsèhèmè: Furthermore M is a left dual for M with the morphisms db : K 3 1 7! X i m i m i 2 M M and ev : M M 3 f m 7! fèmè 2 K where m i and m i are a dual basis of the vector space M. Clearly we have è1evèèdb 1è=1 M and èev 1èè1dbè = 1 M since M is a nite dimensional vector space. We have to show that db and ev are Hmodule homomorphisms. We have P P èh dbè1èèèmè =èhè m i P m i èèèmè =è h è1è m i h è2è m i èèmè = P P èhè1è m i èèèh è2è m i èèmèè = èh è1è P m i èèm i èsèh è2è èmèè = hè1è Sèh è2è èm = "èhèm = "èhèè m i m i èèmè ="èhè dbè1èèmè = dbè"èhè1èèmè = dbèh1èèmè; hence h dbè1è = dbèh1è. Furthermore we have h evèf mè =hfèmè = P h è1è fèsèh è2è èh è3è mè= P èh è1è fèèh è2è mè= P evèhè1è f h è2è mè = evèhèf mèè: 4. Let H be a Hopf algebra. Then the category of left Hcomodules èsee Example 3.2.4 3.è is a monoidal category. Let M 2 HComod be a nite dimensional vector space. Let m i be a basis for M and let the comultiplication of the comodule be èm i è= P h ij m j. Then we have èh ik è= P h ij h jk. M := Hom K èm; Kè becomes a left Hcomodule èm j è:= P Sèh ij èm i. One veries that M is a left dual for M. Lemma 3.3.14. Let M 2Cbe an object with left dual èm ; evè. Then 1. M M is an algebra with multiplication r := 1 M ev 1 M : M M M M! M M

90 3. HOPF ALGEBRAS ALGEBRAIC FORMAL AND QUANTUM GROUPS and unit u := db : I! M M ; 2. M M is a coalgebra with comultiplication and counit :=1 M db 1 M : M M! M M M M " := ev : M M! I: Proof. 1. The associativity isgiven by èr1èr =è1 M ev 1 M 1 M 1 M èè1 M ev 1 M è=1 M ev ev 1 M =è1 M 1 M 1 M ev 1 M èè1 M ev 1 M è=è1rèr. The axiom for the left unit is rèu1è = è1 M ev 1 M èèdb 1 M 1 M è=1 M 1 M. 2. is dual to the statement for algebras. Lemma 3.3.15. 1. Let A be an algebra inc and left M 2Cbe a left rigid object with left dual èm ; evè. There is a bijection between the set of morphisms f : A M! M making M a left Amodule and the set of algebra morphisms e f : A! M M. 2. Let C be acoalgebra inc and left M 2Cbe a left rigid object with left dual èm ; evè. There is a bijection between the set of morphisms f : M! M C making M a right Ccomodule and the set of coalgebra morphisms e f : M M! C. Proof. 1. By Lemma 3.3.14 the object M M is an algebra. Given f : A M! M such that M becomes an Amodule. By Lemma 3.3.3 we associate e f := èf 1èè1 dbè : A! A M M! M M. The compatibility of e f with the multiplication is given by the commutative diagram A A @ H HHHj @@ ef e f M M M M 11db A A M M 1 e f@ @@R ef 11 The unit axiom is given by 1f 1 A M M 1db 11 r r11 f 1 A M M M M u I A f 11 db 1db 1 XXXXXXXXz 11ev 1 1ev 1 M M H HHHHj u1 A M M A M M f 1 M M 11 A M M f 1 M M A 1db @ @@ 1 H @ f 1 HHHj @@R M M ef

3. DUAL OBJECTS 91 Conversely let g : A! M M be an algebra homomorphism and consider eg := è1 evèèg 1è : A M! M M M! M. Then M becomes a left Amodule since A A M 1eg A M and @ XX X XX @@R 1g1 X gg1 XXXXX Xz A M M M 11ev g111 g1 : r1 M M M M M 111ev M M M M H HHHH u1 db 1 XXXXXXX 1 X XXX Hj Xz A M M M M g1 eg 1ev 11 M M M 1ev XXXXXXXXXXX Xz 1ev M A M g1 A AA 1ev A AA eg A AU M commute. 2. èm ; evè is a left dual for M in the category C if and only if èm ; dbè is the right dual for M in the dual category C op. So if we dualize the result of part 1. we have to change sides hence 2.