math.QA 2009-12-08 1

Universal measuring coalgebras and R - transformation algebras

Universal measuring coalgebras provide an enrichment of the category of algebras over the category of coalgebras. By considering the special case of the tensor algebra on a vector space V, the category of linear spaces itself becomes enriched over co…

Authors: Marjorie Batchelor, Jordan Thomas

Univ ersal measuring coalgebras and R - transformatio n algebras M. Batc helor a J. Thomas b ∗ F ebruary 5, 2020 a Corresp ond ing Author Dept. of Pu re Mathematics an d Mathematica l S tatistics, Cen tre f or Mathematic al S ciences, Universit y of Cam bridge, Wilb erforce Road, Cam brid ge, CB3 0WB, United Kingdom T el: +44 (0) 1223 765896 , F ax: +44 (0) 1223 337920 , Email: m b 139@cam.ac.uk b Email: jwthomas@p ost.harv ard.edu Abstract Univ er sal measuring coa lg ebras provide an enric hment of the category of a lgebras ov er the ca tegory of co algebra s . By consider ing the sp ecial ca s e of the tensor alge br a on a vector space V , the categ ory of linear spaces itself beco mes enr iched over coalgebra s, and the univ ersal mea suring coalgebra is the dual coalg ebra of the tensor algebra T ( V ⊗ V ∗ ). Giv en a braiding R o n V the univ ersal meas uring coalg e br a P R ( V ) whic h preserves the g rading is naturally dual to the F adeev- T akhtadjhan-Reshitikin bialgebra A ( R ) and therefor e provides a representation of the qua ntized univ ersa l en veloping algebra as an algebra of transformations. The action of P R ( V ) descends to a ctions on quotients of the tensor algebr a, whenev er the k ernel of the quotien t map is pre s erved by the action of a g enerating subco algebra of P R ( V ). This allows repres ent a tio ns o f quantized env eloping algebras as transformation g roups o f suitably quantized spaces . Keywords: quantum groups, enveloping algebr a s, braidings, measuring co algebra s ∗ Supp orted by a Graduate R esearc h F ello wship from t he National Science F oun dation, USA. 1 The tec hnology of measurin g coalgebras p r o vides an enrichmen t of th e ca tegory of al gebras o v er coalgebras. F or fort y y ears ([9, 1]) this tec hnology has b een emplo y ed in the spirit of algebraic ge ometry , to extend str uctures arising in differential top ology to categories where the differentia l structure has b een replaced by algebraic s tructure. Chief among these applications hav e b een to reco v er geometric int erp r etations of sup ermanifolds and Lie s up eralgebras. More recent ly there h a v e b een attempts to interpret quantum grou p s in this framew ork [14, 2, 3]. The tec hnique cen tres on the universal m easur ing coa lgebra P ( A, B ), whic h is a coalgebra that compares the str ucture of algebras A and B . Initially w e h op ed to repr esen t qu an tized en ve loping algebras as sub -bialgebras of univ ersal measuring coalgebras P ( A, A ), thus act- ing as endomorph isms of suitable “fu nction” algebras A . This is the quantum ve rs ion of the alternativ e construction of the univ ersal en ve loping algebra from a representat ion of the Lie algebra as deriv ations of A [2, 3] In dev eloping the tec hnology to do this, a far simp ler truth emerged. The univ ersal mea- suring coalgebra provi d es a seemingly harmless enrichmen t of the category of ve ctor spaces o v er coalgebras. Y et when V is a vect or space with a br aidin g R : V ⊗ V → V ⊗ V , w e can ask that the universal measurin g coalgebra r esp ects this additional structure. The resulting “ R -transformation alge br a” P R ( V ) is a bialg ebra with the follo wing desirable prop erties: 1. P R ( V ) is defined by a unive r s al prop ert y 2. If V is fin ite-dimensional, then P R ( V ) is canonically isomorph ic to the dual of the F addeev-Reshetikhin-T akh ta jan (FR T) bialgebra A ( R ). 3. If V is a vec tor space on whic h the generators of a quan tized en v eloping algebra act appropriately , then P R ( V ) con tains the quantize d en velo pin g alg ebr a. 1 4. If A = T V /J is an algebra on whic h the generators of a qu antize d env eloping algebra act app ropriately , there is a homomorph ism of b ialgebras P R ( V ) to P ( A, A ). Describ ed in this setting, the quan tized en velo pin g alg ebr as are exac tly transformations of a vecto r space wh ic h preserv e a b raiding, just as the orthogonal or symp lectic algebras are transformations whic h pr eserve a form. The plan of the pap er is to p resen t the m inim um material on measurin g coalgebras required to d escrib e the construction, state the results concerning the r epresen tation of env eloping algebras as measur ing coa lgebras, and state and p ro ve the main theorem w hic h is the isomorphism of P R ( V ) w ith A ( R ) ◦ , the finite dual of the FR T b ialgebra. The pro ofs of the results on repr esen tations are con tained in sections 5 and 6. Th e necessary results ab out measuring coalgebras are included in the app endix A. The sections are as follo ws: 1. Basic definitions of measur ing coalgebras. 2. App licatio n s to classical and quan tized env eloping alg ebr as. 3. Measuring coalg ebr as and tensor algebras an d their quotien ts. 4. Definition and prop erties of R -transformation algebras. 5. Th e representa tions of U g in P ( A, A ). 6. Th e representa tions of U q g in P ( A, A ). 7. App endix Prop erties of measuring coalge b r as. W e would lik e to thank Ma rtin Hyland for man y helpful discussions. W e would also lik e to recognize the contribution of Ben F ai rb airn. His extensiv e calculatio ns in the pr eliminary stages of this work pro vided us with the exp erience on whic h to base our understand- ing. 2 1 Measuring coalgebras 1.1 Definitions Let C b e a coalgebra. W e will use S w eedler’s notation [17], so th e the com ultiplication △ : C → C ⊗ C is denoted △ c = X ( c ) c (1) ⊗ c (2) . Let A , B b e alg ebras (o v er a field k whic h w ill b e either R or C th roughout the pap er). A map σ : C → Hom k ( A, B ) is cal led a me asuring map if σ ( c )( aa ′ ) = X ( c ) σ ( c (1) )( a ) σ ( c (2) )( a ′ ) , σ ( c )(1 A ) = ǫ ( c )1 B and σ ( c )(1 A ) = ǫ ( c )1 B for a , a ′ in A and ǫ the counit. This is equiv alent to r equiring that the map σ : A → Hom( C , B ) is an algebra homomorph ism, where the algebra stru cture on Hom( C , B ) is the con v olution pro du ct determined by the com ultiplication on C . 3 C together with σ is called a me asuring c o algebr a . If ( C ′ , σ ′ ) is another measuring coalgebra, a coalgebra map ρ : C → C ′ is a morph ism of measuring coalgebras if σ = σ ′ ◦ ρ . The category of m easuring coalgebras f or a pair of algebras (A,B) has a fin al ob ject called the universal me a suring c o algebr a . Prop osition 1.1 Give n a p air of algebr as (A,B), ther e exists a me asuring c o a lgebr a P (A,B) π : P ( A, B ) → Hom( A, B ) such that if σ : C → Hom( A, B ) is any me asuring c o algebr a, ther e exists a unique map ρ : P ( A, B ) → C such that π ◦ ρ = σ . The pro ofs are deferred to App end ix A. The fun ctorial prop erties are s u mmarized in the follo wing theorem. Theorem 1.2 1. If A, B , C ar e algebr as, ther e is a map of c o algebr as P ( A, B ) ⊗ P ( B , C ) → P ( A, C ) . 2. The uni v ersal me a suring c o algebr a P ( ∗ , ∗ ) i s functorial in b oth variables. 3. In p articular P ( A, A ) is a bialgebr a and the me asuring map is an algebr a homo mor- phism. 4. If A is a bialgebr a, then Hom( A, B ) is an algebr a with the c onvolution pr o duct, P ( A, B ) is a bialgebr a, and the me a suring map is an algebr a homo morphism. 5. P ( A, k ) = A ◦ ⊂ H om ( A, k ) , the dual c o algebr a of A . 4 2 Applications of the univ ersal measuring coalgebra 2.1 Motiv ation: the em b eddings of U g and U q g Giv en that our am bition is to suggest a defin ition of R -transformation algebra wh ic h en- compasses Lie groups, Lie algebras and quan tized en v eloping algebras, in this secti on and the next, we p resen t the ma jor results describing h o w u niv ersal en ve loping algebras and their quan tized equiv alen ts can b e represent ed as sub-b ialgebras of universal measuring coalge br as. The classical mo d el that motiv ated the pro ject is as f ollo ws. Let g b e a Lie algebra, and supp ose that A is alg ebra on whic h g acts faithfully as deriv ati ons, so that th er e is a linear map φ : g → End( A ) whic h is a h omomorphism of Lie algebras. This can b e r estated in terms of m easuring coalgebras. Let C = C 1 ⊕ g b e giv en the structure of a coalgebra with 1 group -lik e and elemen ts of g primitive. The statemen t that elemen ts of g act as deriv ations is equiv alen t to sa ying th at the extension of φ to all of C sendin g 1 to the iden tit y in E nd ( A ) measures. Let π : P ( A, A ) → End( A ) be the un iv ersal m easuring m ap . Th e observ ation is that the su bbialgebra of P ( A, A ) generated b y C is exact ly the univ ersal env eloping alg ebr a of g . Theorem 2.1 1. The universal enveloping algebr a U g includes in P ( A, A ) as a me a- suring bialgebr a. 2. L et e : A → k b e an algebr a homomo rphism, and define r = e ◦ φ : g → Hom( A, k ) . Supp ose additional ly that r is inje ctive on g . If A is a bialgebr a, then U g includes in P ( A, k ) as a bialgebr a. 5 3. Wi th A , e as ab ove, the map P (1 , e ) : P ( A, A ) → P ( A, k ) , gener ate d by e ◦ π : P ( A, A ) → Hom( A, k ) , sends U g c onsid er e d as a sub algebr a of P ( A, A ) isomorphic al ly onto its i mage in P ( A, k ) . The pr o of is d eferred to sectio n 5. There are man y examples of suitable algebras A - the co ordinate ring of g , p olynomials on a v ector s pace on whic h g acts faithfully , and the exterior algebra on such a space all pro vide faithful repr esentati ons of U g as measur ing coalgebras. If G is an algebraic group with Lie algebra g and k [ G ] is the co ordinate ring of G , then k [ G ] is a bialgebra, and so con v olution in Hom( k [ G ] , k ) d etermines a bialgebra structure on P ( k [ G ] , k ), pro viding a represent ation of U g in P ( k [ G ] , k ). A stateme nt analogous to 2.1 can b e made rep lacing U g by U q g . T he construction ab o ve dep end s only on the fact that C , the generating set for U g , is a measuring coa lgebra. The generating set for U q g is also a coalgebra and th e same tec hnique will pro vide r epresen ta- tions of U q g as sub-bialgebras of su itable P ( A, A ) or P ( A, k ). Details are giv en in section 6 . Suitable algebras on whic h this coalgebra measures are constructed as quotien ts of tensor algebras of U q g -mo dules. It turns out that the tensor algebra and the univ ersal measuring coalge br a pla y complemen tary r oles. Describing this r elationship is th e sub ject of the next section. 6 3 Measuring coalgebras and tensor algebras and their quo- tien ts. The success of the pro j ect of reco v ering the classical transformation algebras and their quan tized v ersions encouraged us to consid er replacing the v ector space of linear trans- formations Hom( V , W ) by the universal measur ing coalgebra P ( T V , T W ) where T V (resp T W ) is the tensor algebra on V (resp W ). T his pr o vides an enrichmen t of the category of v ector spaces o ve r coalgebras. Prop osition 3.1 1. L et V , W b e ve ctor sp ac es, and let H a c o al geb r a to gether with a map σ : H → Hom( V , T W ) Then σ extends uniquely to a me asuring map ˆ σ : H → Hom( T V , T W ) . 2. If H is in fact a bialgebr a then ˆ σ i s also an algebr a hom omorphism. Pro of 1. The map σ can b e though t of as a map σ : V → Hom( H , T W ) . But since T ( V ) is an algebra and H is a coalg ebra, Hom( H , T W ) has the structure of an alge br a, where αβ ( b ) = X ( b ) α ( b (1) ) β ( b (2) ) . 7 The un iv ersal prop ert y of tensor algebras th en extends σ to an algebra map σ : T V → Hom( H , T W ) or equiv alen tly ˆ σ : H → Hom( T V , T W ) measures as required. 2. Th e sec ond p art follo ws from the third part of 1.2, whic h is prov ed in the App endix. The ob ject of interest h ere is sub coalgebra of P ( T V , T W ) whose elemen ts determine maps from V to W . Definition Defin e P ( V , W ) = { p ∈ P ( T V , T W ) : π ( p )( V ) ⊂ W } . (Here π is the measuring map π : P ( T V , T W ) → Hom( T V , T W ).) It is easy to c hec k that P ( V , W ) is a su b coalgebra of P ( T V , T W ). Replacing Hom( V , W ) by P ( V , W ) gives an enric hment of th e catego ry of linear spaces o ver the categ ory of coalge br as. In the case of particular in terest wh ere V = W , P ( V , V ) will b e denoted simply b y P ( V ). By the third part of 3.1, P ( V ) is a bialgebra. Comp osition giv es Hom( T V , T V ) and h ence P ( T V , T V ) (resp P ( V )) has th e structure of a bialgebra. The other common multiplicativ e structure on Hom( D, A ) is con volutio n when D is a coalge br a and A is an algebra, F or a ny vect or sp ace W , the concept of a dual sp ace can b e ”enric hed” by considering th e unive rs al measuring coalgebra P ( T W, k ). When the v ector 8 space W is replaced b y a coalg ebra D this coalg ebra b ecomes a bialgebra. Prop osition 3.2 1. L et C b e any c o algebr a, W any ve ctor sp ac e and let µ : C → Hom( W, k ) b e any line ar map. Then µ extends unique ly to a me asuring map ˆ µ : C → Hom( T W, k ) 2. If D is a c o algebr a, c omultiplic ation in D extends uniquely to a bialgebr a structur e on T D . 3. Supp ose D is a c o algebr a so that H om ( D , k ) is an algebr a under c onvolution. If C is a bialgebr a and µ is an algebr a homomorp hism, then ˆ µ i s also a bialgebr a map fr o m C to ( T D ) ◦ . Pro of 1. As b efore, Hom( C , k ) has the structure of an algebra (since C is a coalgebra), so that µ : W → Hom ( C, k ) extends to ˆ µ : T W → Hom( C, k ) 2. Th e inclusion D ⊗ D ⊂ T D ⊗ T D , pro vides a linear map △ : D → T D ⊗ T D . By the universal prop erty of the tensor algebra, this extends uniquely to an alge br a 9 homomorphism T △ : T D → T D ⊗ T D . Routine verificatio n sho ws th at T △ is coassociativ e. S imilarly , the counit ǫ : D → k extends to an algebra homomorphism T : T D → k . Agai n , th e requ ired iden tities can b e ve rifi ed by direct calculation. 3. Th is is a direct app lication of the fourth part of 1.2 w hic h will b e prov ed in the app end ix. In the eve nt that W = V ⊗ V ∗ , observe that we can iden tify V ⊗ V ∗ in T ( V ⊗ V ∗ ) with V ⊗ V ∗ in T V ⊗ T V ∗ , and h ence make the inclusion T ( V ⊗ V ∗ ) ⊂ T V ⊗ T V ∗ . The case of in terest is when D = V ⊗ V ∗ , wh ere there are isomorphisms Hom( V , V ) = V ⊗ V ∗ = ( V ⊗ V ∗ ) ∗ The Killing form id en tifies the space V ⊗ V ∗ with its dual. Thus the v ector space V ⊗ V ∗ carries b oth an algebra structure and a coalg ebr a structure, but these t wo stru ctures are as incompatible as p ossible. Th e function of P ( V ) is to r epair the incompatibilit y . Moreo v er, the ab o ve identi fi cations are as algebras (with the algebra structure on the last b eing giv en b y con v olution). Th us the tw o apparent ly different cases of σ and µ ab o v e in fact describ e the same s ituation. 10 Notation . It quic kly b ecomes confusing wh ether V ⊗ V ∗ (or T W ) is b eing r egarded as an algebra or a coalgebra. When there is danger of confusion, we will wr ite V ⊗ a V ∗ (or T a W ) when considering these spaces as an algebra, and V ⊗ c V ∗ (or T c W ) when they are considered as coalg ebras. Theorem 3.3 P ( V ) = ( T a ( V ⊗ c V ∗ )) ◦ Pro of T his is nearly tautologica l: b y the definition of P ( V ), th e measuring map π : P ( V ) → Hom ( T V , T V ) restricts to a linear map π : P ( V ) → Hom( V , V ). But Hom( V , V ) = V ⊗ a V ∗ = Hom( V ⊗ c V ∗ , k ). Th us by the universal prop ert y of P ( T ( V ⊗ c V ∗ ) , k ), th e follo wing diagram comm utes P ( T ( V ⊗ a V ∗ , k ) P ( V ) Hom( V , V ) Hom( V ⊗ c V ∗ , k , ) Hom( T a ( V ⊗ c V ∗ ) , k ) ❄ ✲ ❄ ❄ ✲ π ρ π ∼ = and the m ap ρ : P ( V ) → P ( T ( V ⊗ V ∗ ) , k ) = ( T ( V ⊗ V ∗ )) ◦ . is unique. C on ve rsely , the m easur ing map π : P ( T ( V ⊗ c V ∗ ) , k ) → Hom( T ( V ⊗ c V ∗ ) , k ) restricts to a linear map π : P ( T ( V ⊗ c V ∗ ) , k ) → Hom( V ⊗ c V ∗ , k ) = Hom( V , V ) . Th e resulting map of measur ing coalgebras P ( T ( V ⊗ V ∗ ) , k ) → P ( V ) pr o vides an in v erse to ρ . Since the iden tification 11 Hom( V ⊗ c V ∗ , k ) ∼ = ( V ⊗ a V ∗ ∼ = Hom( V , V ) preserve s the m ultiplicativ e structur e of these sp aces, the maps ρ and its in v erse are bial- gebra maps. The maps indu ced b y such σ , µ of 3.1 , 3.2 descend to quotien ts of T V pro vid ed the ideal in question is preserve d. T his is the c hief to ol for constru cting env eloping algebras, b oth classical and quan tized, as sub-bialgebras of univ ersal measuring coalebras. Prop osition 3.4 1. Supp ose that C is a c o algebr a and that σ : C → End( T V ) me a- sur es. L et J b e an ide al in T V . Supp ose additional ly that σ ( C )( J ) ⊂ J . Then σ induc es a me asuring map ˜ σ : C → End( T V /J ) 2. If ˆ µ : C → Hom( T V , k ) me asur es, and J is an ide al as ab ove su c h that c ( J ) = 0 for al l c ∈ C , then ˆ µ desc ends to a me a suring map ˆ µ : C → Hom( T V /J , k ) . Pro of F or b oth parts of the prop osition the argument is the same. The statemen t that σ ( C )( J ) ⊂ J (resp ˆ µ ( C )( J ) = 0) sa ys th at σ indu ces a m ap σ : C → E nd ( T V /J, T V /J ) (resp ˆ µ descends to a map ˆ µ : C → H om ( T V /J, k )). These maps retain the measurin g prop erty . 4 R -transformation alg ebras The resu lts of the previous sections ha ve encouraged us to consider u niv ersal measuring bialgebras P ( V ) as candidates for the alge br a of transformations of a v ector sp ace V . Where th e v ector space b ecomes equipp ed w ith a sp ecified b raiding R : V ⊗ V → V ⊗ V , we can describ e the R -tr ansforma tion algebr a P R ( V ) as the sub-b ialgebra of P ( V ) consisting 12 of elemen ts that pr eserv e the braiding. T h e R -transformation algebra in corp orates the unive rsal env eloping algebra and its qu an tized version as sp ecial cases. Moreo v er, d escrib ed in this setting, P R ( V ) is easily r ecognized as the du al coalge br a of the FR T bialgebra A ( R ), when V is finite-dimensional. 4.1 The braiding on T V . The b ialgebra P R ( V ) will b e constructed as a sub -bialgebra of P ( T V , T V ) whic h preserv es a br aiding on T V w hic h exte n d s the one giv en on V . Recall that a br aide d algebr a A is an algebra together with an inv ertible linear op erator Ψ : A ⊗ A → A ⊗ A suc h that 1. Ψ satisfies the braid equation on A ⊗ A ⊗ A , that is: (id ⊗ Ψ )(Ψ ⊗ id)(id ⊗ Ψ) = (Ψ ⊗ id)(id ⊗ Ψ)(Ψ ⊗ id) . 2. Th e m ultiplication m : A ⊗ A → A and u nit ν : 1 → A satisfy th e follo wing consistency conditions: Ψ( m ⊗ id) = (id ⊗ m )(Ψ ⊗ id)(id ⊗ Ψ) : A ⊗ A ⊗ A → A ⊗ A, Ψ(id ⊗ m ) = ( m ⊗ id)(id ⊗ Ψ)(Ψ ⊗ id) : A ⊗ A ⊗ A → A ⊗ A, (1) Ψ( m ⊗ ν ) = ( ν ⊗ m ) : A → A ⊗ A, Ψ( ν ⊗ m ) = ( m ⊗ ν ) : A → A ⊗ A. F or fur ther details on braided algebras and related stru ctures, refer to [13]. No w let ( V , R ) b e a br aide d ve ctor sp ac e , that is, V is a vec tor space and R : V ⊗ V → V ⊗ V is an inv ertible linear op erator satisfying the braid equation. T hen T V can b e given th e 13 structure of a braided algebra. Prop osition 4.1 If R is a br aid op er ator for V , then 1. R extends to a map Ψ m,n Ψ m,n : ⊗ m V ⊗ ⊗ n V → ⊗ n V ⊗ ⊗ m V . wher e Ψ 1 , 1 = R : V ⊗ V → V ⊗ V . 2. Writing Ψ for P m,n Ψ m,n , then Ψ gives T V the structur e of a br aide d algebr a. 3. If J is an ide al of T V such that Ψ J = J , then Ψ gives T V /J the structur e of a br aide d algebr a. Pro of . T he map Ψ m,n is obtained by usin g R on adj acen t factors one pair at a time, and observing that the braid iden tit y en s ures that this is w ell-defined. See Ma jid [12] for details. 4.2 Definition of P R ( V ) Definition L et V b e a v ector space with braiding R . Let C b e a coalge br a and supp ose the map σ : C → End( T V ) is a measuring map. S ay that C pr ese rve s ( V , R ) if: σ ( c )( V ) ⊂ V ∀ c ∈ C, R ( σ ⊗ σ )(∆( c ))( v ⊗ w ) = ( σ ⊗ σ )(∆( c )) R ( v ⊗ w ) ∀ c ∈ C ; ∀ v , w ∈ V . The next lemma follo ws from a sim p le applicatio n of equations 1 and the braiding condi- tion: 14 Lemma 4.2 T a ke C a c o algebr a and supp ose the map σ : C → End( T V ) is a me asuring that pr e serves ( V , R ) . Then C pr eserves the br aiding Ψ on al l of T V , that is, Ψ( σ ⊗ σ )(∆( c ))( v ⊗ w ) = ( σ ⊗ σ )(∆( c ))Ψ( v ⊗ w ) ∀ c ∈ C ; ∀ v , w ∈ T V . If C is a measuring coalgebra wh ic h preserv es R , th en the sub-bialgebra generated by C in P ( V ) w ill a lso preserve R . I f su b-bialgebras C and B of P ( T V , T V ) preserv e R , then so do es the b ialgebra generated b y C and B . So, there is a largest su b-bialgebra of P ( T V , T V ) whic h p reserv es R . This is our preferred candidate for the role of R -transformation alge- bra. Definition T he R -tr ansformat ion algebr a , denoted P R ( V ), of th e braided ve ctor space ( V , R ), is the u nique maximal sub-bialgebra of P ( V ) whic h presev es R . F rom the remarks ab ov e, P R ( V ) is also the maximal sub coalgebra of P ( V ) whic h preserv es R - the requirement that it b e a bialgebra imp oses no restriction. The coalgebras measuring T V to T V and preserving R form a full sub category of th e coalge br as measur ing T V to T V . Th e definition of P R ( V ) sho ws that it is the final ob ject in this category . W e ha v e inciden tally shown that P R ( V ) is a final ob ject in another catego ry . Prop osition 4.3 L et H R b e the c ate gory whose obje cts ar e bialgebr as H to gether with actions of H on V such that the action on V ⊗ V (induc e d by c omultiplic ation in H ) pr e se rve s R . Then P R ( V ) is also a final obje ct in the c a te gory H R . This will b e useful in p ro ving the d u alit y b etw een P R ( V ) and A ( R ) in th e follo wing su b- section. 15 4.3 R -Admissabl e coactions and R-Admissable actions. The ai m is to r elate th e R -transformation al gebra P R ( V ) to the dual bialg ebr a of the F addeev-Reshetikhin-T akh ta jan (FR T) bialgebra asso ciated with R , A ( R ). T o d o so, we will describ e in parallel a category in w h ic h A ( R ) is an initial ob j ect and a categ ory in whic h P R ( V ) is a final ob ject. The follo wing lemma stat es that act ions and coactions of bialgebras on a v ector sp ace extend to the whole tensor alge b r a. Lemma 4.4 1. L et H b e a bialgebr a with an action a : H ⊗ V → V . Comultipl ic ation in H e xtends this action to an action T a : H ⊗ T V → T V . 2. L et B b e a bialgebr a with a c o action c : V → V ⊗ B . Multiplic ation in B extends this c o action to a c o action T c : T V → T V ⊗ B which is an algebr a homom orphism. Pro of T he first statemen t is a restate ment of the fir st p art of 3.1, and is only r estated for comparison w ith the second. Th e second mak es use of the defin ing prop ert y of the tensor pro du ct. V ery elemen tary algebra establishes the equiv ale nce of an action a : H ⊗ V → V with an algebra h omomorp hism a : H → V ⊗ V ∗ . The coal gebra structure of V ∗ ⊗ V , w hile equally elemen tary feels less familiar: co multiplicat ion in V ∗ ⊗ V arises from th e un it u in the algebra E nd ( V ) = V ⊗ V ∗ : △ = 1 ⊗ u ∗ ⊗ 1 : V ∗ ⊗ V → V ∗ ⊗ V ⊗ V ∗ ⊗ V . The counit ǫ is simply the ev aluati on map. The argumen t relating actions with homor- phisms into V ∗ ⊗ V dualizes to giv e the result that coactions c : V → V ⊗ B corresp ond 16 to coalgebra m aps c : V ∗ ⊗ V → B . Using this definition of actions and coactions, the con ten t of 4.4 could b e expr essed by sa ying that the action a (resp coaction c ) extends to maps T i a : H → ⊗ i ( V ⊗ a V ∗ ) , T i c : ⊗ ( V ∗ ⊗ c V ) → B (2) for all i . W e defi ne admissable actions and co actions to b e those whic h pr eserv e R, as follo ws. Definition I f B b e a bialgebra for wh ic h V is a como dule, sa y the coaction c : V → V ⊗ B is (V,R) admissable (or simply admissable) if V ⊗ V V ⊗ V ⊗ B V ⊗ V ⊗ B V ⊗ V ❄ ✲ ❄ ✲ c R c R ⊗ 1 B If H is a bialgebra for which V is a mo dule, sa y the a : H ⊗ V → B is (V,R) admissable (or simp ly admissable) if a ◦ 1 H ⊗ R = R ◦ a : H ⊗ V ⊗ V → V ⊗ V . 17 V ⊗ V H ⊗ V ⊗ V ⊗ B H ⊗ V ⊗ V ⊗ B V ⊗ V ❄ ✲ ❄ ✲ c R c R ⊗ 1 B Using 2, the defin ition of admissable translates to a statemen t th at c factors through a co equalizer (resp . the image of a lies in an equalizer). Denote by τ the map whic h twists the m iddle t wo factors τ = 1 V ∗ ⊗ tw ist ⊗ 1 V : V ∗ ⊗ V ⊗ V ∗ ⊗ V → V ∗ ⊗ V ⊗ V ∗ ⊗ V . Define tw o maps from V ∗ ⊗ V ⊗ V ∗ ⊗ V to itself: α = τ ◦ R ∗ ⊗ 1 V ⊗ V ◦ τ , β = τ ◦ 1 V ∗ ⊗ V ∗ ⊗ R ◦ τ . Define C to b e the coequalizer of α and β , and E to b e the equalizer, th us C = V ∗ ⊗ V ⊗ V ∗ ⊗ V /im ( α − β ) (3) and E = { w ∈ V ∗ ⊗ V ⊗ V ∗ ⊗ V : αw = β w } . (4) The translation of the statemen t that a or c pr eserv es R can then b e stated as a lemma. 18 Lemma 4.5 1. A c o action c of a bialgebr a B on V is admissable if and only if T 2 c : ( V ∗ ⊗ V ) ⊗ ( V ∗ ⊗ V ) → B factors thr ough C. 2. An action a of a bialebr a H on V ∗ is admissable if and only if imT 2 a ( H ) ⊂ E ⊂ ( V ⊗ V ∗ ) ⊗ ( V ∗ ⊗ V ) . Pro of Like all pro ofs of this nature, th ere is no difficulty b ey ond that of un ra v elling the definitions and displaying the m aterial in a manner in which the claim b ecomes obvious. W e will sho w th e statemen t for an act ion to b e admissable. The ingredients here are the follo wing. First, the com ultiplication H is sim p ly V ⊗ V ∗ H H ⊗ H ( V ⊗ V ∗ ) ⊗ ( V ⊗ V ∗ ) ❄ ✲ ❄ ✲ r c r ⊗ r where c is 1 V ⊗ u V ∗ ⊗ V ⊗ 1 ∗ V . T he action of H on V is then 19 V ⊗ V ∗ ⊗ V H ⊗ V V ❄ ✲ ❩ ❩ ❩ ❩ ❩ ⑦ r a where a is con traction on the second and third factors. W riting ( V ⊗ V ∗ ) ⊗ ( V ⊗ V ∗ ) as V ⊗ V ⊗ V ∗ ⊗ V ∗ , the admissabilit y condition can set in the follo wing diagram H ⊗ H ⊗ V ⊗ V V ⊗ V ⊗ V ∗ ⊗ V ∗ ⊗ V ⊗ V V ⊗ V ⊗ V ∗ ⊗ V ∗ ⊗ V ⊗ V V ⊗ V V ⊗ V ❄ ❄ ❄ ✲ ❄ ✲ ✑ ✑ ✑ ✸ ◗ ◗ ◗ s a b c R Here a = R ⊗ 1 ⊗ 1 , b = 1 ⊗ R ∗ ⊗ 1 and c = 1 ⊗ 1 ⊗ R . T he h orizon tal maps are the action - in th is case giv en by con traction on the last four fac tors. E v id en tly the diagram co mmutes if the map a is used. Admissabilit y is the statemen t that the d iagram also comm utes for the map c . But by the definition of R ∗ , this comm utes if and only if the square with b comm utes. Th us the diagram comm utes if, on the image of H ⊗ H ⊗ , a = b . But this is exactly the statemen t th at the image of H ⊗ H in the coequalizer. By its construction, P R ( V ) is readily seen to b e the final ob ject in the category of bialgebras H equipp ed with a admissable action on V . T he initial ob ject in the category of bialgebras 20 B equipp ed with admisab le coactions on V is exactly the FR T algebra. T o see this, it is only necessary to giv e the standard definition of the FR T algebra A ( R ) in a b asis ind ep endent fashion. By 3.2 a coalgebra structure on D endo ws the tensor algebra T D with the structure of a bialgebra. Notice that if K is a coideal in T D , the ideal generated by K in T D is also a coideal. A rou tin e calc ulation shows that im ( α − β ) is a coideal in V ∗ ⊗ V . Definition L et J d enote the co ideal generated by im ( α − β ). Define A ( R ), the FR T algebra, to b e the b ialgebra A ( R ) = T ( V ∗ ⊗ V ) /J. The maps j : V ∗ ⊗ V → A ( R ) is evidently a coalgebra map. Moreo ver, in th e ligh t of the pr eceding lemma, we ha v e the follo wing prop osition. Prop osition 4.6 (The universal pr op erty of the FR T algebr a.) If B is a bialgebr a with an admissable c o action c on V, then ther e i s a unique map ρ : A ( R ) → B such that 21 V V ⊗ A ( R ) V ⊗ B ❄ ✲ ❩ ❩ ❩ ❩ ❩ ⑦ r a Th u s A ( R ) as constru cted as ab ov e has the pr op ert y exp ected of the FR T construction, and by uniqueness, this construction is equiv alen t to the more common basis dep endent definition of A ( R ). The main theorem is the observ ation that 3.3 and 4.6 com bine to allo w a simple identifi- cation of the R -transform ation algebra and the FR T algebra. Theorem 4.7 P R ∗ ( V ∗ ) = A ( R ) ◦ . Remark 1. It is easy to v erify that if R : V ⊗ V → V ⊗ V is a braidin g (of a fin ite dimensional v ector space) then the dual m ap R ∗ : V ∗ ⊗ V ∗ → V ∗ ⊗ V ∗ is also a braiding. 2. Coactions on V to corresp ond to ac tions on V ∗ hence the theorem is stated in terms of P R ∗ ( V ∗ ). Ho w eve r, P R ( V ) = P R ∗ ( V ∗ ) op . See lemma 4.9. Pro of F rom 3.3 w e ha ve that P ( V ∗ ) = ( T ( V ∗ ⊗ V )) ◦ . The essen tial step is to apply the con ten t of 4.6. If B is a bialgebra and c : V → V ⊗ B is an admissable coaction, then 22 T 2 c : ( V ∗ ⊗ V ) ⊗ ( V ∗ ⊗ V ) → B factors through ( V ∗ ⊗ V ) ⊗ ( V ∗ ⊗ V ) /J wher e J is k er ( α − β ) from 3. Thus c ◦ : B ◦ → ( V ∗ ⊗ V ) ∗ has its image in J ⊥ , that is, the act ion of B ◦ is admissable. Thus, in particular, the action of A ( R ) ◦ on V ∗ is adm iss able, and A ( R ) ◦ ⊂ ( T ( V ∗ ⊗ V ) ◦ ) is con tained in P R ∗ ( V ). If a : H → V ∗ ⊗ V is an admissable action of a bialgebra H on V ∗ , then T 2 a : H → ( V ∗ ⊗ V ) ⊗ ( V ∗ ⊗ V ) has its image in E of 4. Then, s in ce m ultiplication in H induces H ◦ → H ◦ ⊗ H ◦ , the map T 2 a d ualizes to give a coaction ψ = ( T 2 a ) ◦ : ( V ∗ ⊗ V ) ⊗ ( V ∗ ⊗ V ) ∗ → H ◦ . Since T 2 a had its image in E , ψ factors through C of 3. In particular, this holds for H = P R ∗ ( V ), and hence ψ factors through a map from A ( R ) to ( P R ∗ ( V )). Corollary 4.8 P R ( V ) op = A ( R ) ◦ The corollary is an immediate consequence of the follo wing lemma. Lemma 4.9 P R ( V ) ∼ = P R ∗ ( V ∗ ) op . Pro of T his is simp ly the observ ation th at n othing goes wr on g when taking d uals: if ρ : C → En d( V ) is a linear map, define ρ ′ : C → End( V ∗ ) via ρ ′ ( c ) = ( ρ ( c )) ∗ . If ρ ( c ) preserve d 23 R , then ( ρ ( c )) ∗ will p r eserv e R ∗ . Thus as measuring coalgebras, P R ( V ) ∼ = P R ∗ ( V ∗ ). Sin ce ( αβ ) ∗ = α ∗ β ∗ for α, β in En d( V ) the order of m ultiplication is the rev erse of the usu al in P R ∗ ( V ∗ ). 5 Pro of of 2.1 for the classical case Recall that φ : g → End( A ) is a faithfu l repr esen tation of g as deriv ations on A . Notice that C = C 1 ⊕ g can b e give n the structur e of a coalge br a by setting 1 to b e grouplike, and elemen ts in g to b e primitiv e. Let ˆ φ : C → End( A ) map 1 to the id entit y map on A and equal φ when r estricted to g . Then ˆ φ is an injectiv e measuring map. Pro of of Theorem 2.1 1. Let π : P ( A, A ) → End( A ) b e the unive rs al measur ing map. By the un iv ersal pr op ert y of m easuring coalgebras there is a unique coalg ebr a map ρ : C → P ( A, A ) th at satisfies π ◦ ρ = ˆ φ . Then, ρ is in jectiv e b ecause ˆ φ is in jectiv e. Let U d enote the su b-bialgebra of P ( A, A ) generated by the image of C . Consider th e ideal J in P ( A, A ) generated by the set of elemen ts of the f orm ρX ρY − ρY ρX − ρ [ X , Y ] . Lemma 5.1 The ide al J is in the kernel of the me asuring map π . Mor e over, J is also a c oide al. Pro of Notice that since g is represen ted in End( A ) and since multiplicat ion in P ( A, A ) is defi n ed via comp osition in En d( A ), π ( Z ) = 0 for all Z in { ρX ρY − ρY ρX − ρ [ X , Y ] } . Similarly , since π is an algebra homomorphism, all of the ideal J generated by elemen ts Z m ust also lie in the k ernel of π . That J is a coideal can b e v erified by c hecking directly 24 that the space spanned b y ρX ρY − ρY ρX − ρ [ X , Y ] is a coideal. Lemma 5.2 If J is any c oid e al in P ( A, A ) which lies in the kernel of π , then J = 0 . Pro of If J is a coideal in P ( A, A ) which is con tained in the ke r n el of π , then observe that P ( A, A ) /J has the unive r s al pr op erty whic h charac terizes the unive rs al measuring coalge br a, hence J = 0 b y the uniqueness of P ( A, A ). Since ρ is th us a linear map from g to an asso ciativ e algebra U wh ic h satisfies the iden tit y ρX ρY − ρY ρX − ρ [ X , Y ] = 0 , b y the universal p rop erty of universal env eloping algebras there is a unique algebra homo- morphism µ : U g → U whic h agrees with ρ when r estricted to g . But th e map µ is not just an alge br a h omomor- phism; it is a b ialgebra map b ecause ρ is a coalgebra map. W e n o w app eal to the follo wing basic, if initially surprisin g, fact ab out coalg ebra maps: they enj o y a rigidit y that algebra homomorph isms lac k. Prop osition 5.3 L et B b e a c o algebr a. If C is anoth er c o algebr a and ν : B → C is a map of c o algebr as, then ν is inje ctive if and only if it is inje ctive on the first c or adic al filtr ation of B. 25 Pro of [15 ] page 65. It is th us sufficien t to sh ow that µ is injectiv e on the first coradical filtration of U g , whic h is just C [17]. But µ = ρ on C , and is th us inj ectiv e. 2. The structure of this p art is p arallel to that of the fi rst p art. The same coalgebra C can b e used, and the measuring map from C to k is simply r = e ◦ φ . This map measures, pro vidin g a map ˆ ρ : C → P ( A, k ) . As b efore, let ˆ U b e the subalgebra of P ( A, k ) generatd by the image of C , and tak e ˆ J to b e the ideal generated b y ˆ ρX ˆ ρY − ˆ ρY ˆ ρX − ˆ ρ [ X, Y ] . The pr o of con tinues as ab o ve , relying on the follo wing t w o lemmas as b efore. Lemma 5.4 The ide al ˆ J is in the kernel of the me asuring map π . Mor e over, ˆ J is also a c oide al. Lemma 5.5 If ˆ J is any c oide al in P ( A, k ) which lies in the kernel of π , then ˆ J = 0 3. T ak e π : P ( A, A ) → End( A ) an d ˜ π : P ( A, k ) → Hom( A, k ) to b e the resp ectiv e unive rsal measurings. And let µ : U g → P ( A, A ) b e th e in j ection from part 1 and let ˜ µ : U g → P ( A, k ) b e the injection from part 2. Then P (1 , e ) ◦ µ and ˜ µ provide tw o bialgebra maps from U g to P ( A, k ). W e m us t sho w that they are the same map. By th e unive rsal pr op ert y of U g , they are th e same if they are the same on g . By th e u niv ersal prop erty of P ( A, k ), they are the s ame on g if they giv e the same action when comp osed 26 with ˜ π . As maps from g to Hom( A, k ), these t wo maps satisfy: ˜ π ◦ ( P (1 , e ) ◦ µ ) = ( ˜ π ◦ P (1 , e )) ◦ µ = e ◦ π ◦ µ = e ◦ φ, ˜ π ◦ ˜ µ = r = e ◦ φ, and the tw o maps are thus the same. P articular c hoices of A reco v er the un iv ersal en ve loping algebra as the sub-b ialgebra of P ( A, A ) generated by g . The follo wing resu lts follo w easily from Theorem 2.1. Corollary 5.6 1. U g includes in P ( C ∞ ( G ) , C ∞ ( G )) . 2. If V is a faithful r epr esentation of g , then U g includes in P ( T V , T V ) wher e T V is the tensor algebr a on V 3. If V is a faithful r epr e sentation of g , then U g includes in P (Sym V , Sym V ) wher e Sym V is the symmetric algebr a on V . 4. If V is a f aithful r epr esentation of g , then U g includes in P ( V V , V V ) wher e V V is the exterior algebr a on V . 5. L et e denote the homomorphism e : C ∞ ( G ) → R which evaluates a function at the identity e of G . Then P (1 , e ) : P ( C ∞ ( G ) , C ∞ ( G )) → P ( C ∞ ( G ) , R ) identifies the c opy of U g in P ( C ∞ ( G ) C ∞ ( G )) with the p ointe d subbialgebr a of P ( C ∞ ( G ) C ∞ ( G )) with the identity as the unique gr oup-like element. Remark 1. Even if S is just a linear space of deriv ations of A , the constru ction still 27 generates the unive rs al en ve loping algebra of the Lie algebra generate d by S . In par- ticular, if g is semisimple, U g will b e generated in P ( A, A ) as long as φ : g → End( A ) is inj ectiv e on the generators of g . Th is is the classica l parallel of the quantize d situation discussed b elo w. 2. Th e third example d emonstrates the fact that the u niv ersal en v eloping algebra can b e infi nite dimensional ev en w hen the alg ebra A is finite dimensional. In this w a y , the univ ersal en ve loping algebra is realized not as some external abstract con- struction app ended to a Lie algebra, but is a ve ry natural bialgebra of transformations (in the enric hed setting) of a linear space. Moreo ve r, this in terpretation of the role of the univ ersal env eloping algebra app lies w ithout fu r ther adj ustment to the case of quan- tized en ve loping algebras. It is only necessary to int ro duce appropriate analogues of the symmetric and exterio r algebras, and the fu nction ring of a Lie group G . 6 Quan tized En v eloping Algebras 6.1 Definitions and Basic F a ct s Let g b e a finite-dimensional complex semisimple Lie algebra of r ank n with Cartan matrix ( a ij ) and let d i b e the coprime p ositiv e in tegers su ch that the matrix ( d i a ij ) is symmetric. Let q b e a fi xed complex n umb er, not a ro ot-of-unit y , and set q i : = q d i . T he algebra U q g is the complex asso ciativ e algebra with 4 n generators E i , F i , K i , K − 1 i , 1 ≤ i ≤ n and relations: K i K j = K j K i , K i K − 1 i = K − 1 i K i = 1 , K i E j K − 1 i = q a ij i E j , K i F j K − 1 i = q − a ij i F j , 28 E i F j − F j E i = δ ij K i − K − 1 i q i − q − 1 i 1 − a ij X r =0 ( − 1) r "" 1 − a ij r ## q i E 1 − a ij − r i E j E r i = 0 , i 6 = j 1 − a ij X r =0 ( − 1) r "" 1 − a ij r ## q i F 1 − a ij − r i F j F r i = 0 , i 6 = j where: "" 1 − a ij r ## q = [ n ] q ! [ r ] q ![ n − r ] q ! , [ n ] q = q n − q − n q − q − 1 . There is a Hopf algebra structure on U q g giv en b y: ∆( K i ) = K i ⊗ K i , ∆( K − 1 i ) = K − 1 i ⊗ K − 1 i , ∆( E i ) = E i ⊗ K i + 1 ⊗ E i , ∆( F i ) = F i ⊗ 1 + K − 1 i ⊗ F i , ǫ ( K i ) = 1 , ǫ ( E i ) = ǫ ( F i ) = 0 , S ( K i ) = K − 1 i , S ( E i ) = − E i K − 1 i , S ( F i ) = − K i F i . A quant u m ve rsion of the PBW th eorem [4] shows th at the generating coalg ebra C = C [ E i , F i , K i , K − 1 i ] 1 ≤ i ≤ n is a s ub coalgebra of U q g . Chin and Musson [5] [6] and M ¨ uller [16] ha v e s h o wn that the coalgebra structure of U q g is particularly simp le: Prop osition 6.1 U q g is p ointe d with c or adic al U 0 = C [ K i , K − 1 i ] 1 ≤ i ≤ n and first term of the c or adic al filtr ation P i U 0 + U 0 E i + U 0 F i . 29 These authors use this result together with Prop osition 5.3 to sho w: Prop osition 6.2 E very bi-ide al of U q g c ont ains E i and F i for some i . Remark Th ese r esu lts are only explicitly stated in [5] for simple g , although it follo ws from the w ork of [16] that they extend to semisimp le g . Also note that the result th at we cite is slight ly wea ker th an the result giv en in [5] (as extended b y [16]). It is all th at we will n eed. This p ro vides a stronger v ersion of Prop osition 5.3: Corollary 6.3 L et B b e a b ialgebr a. If ν : U q g → B is a map of bialgebr as, then ν is in- je ctive if and only if it is inje ctive on the gener ating c o a lgebr a C = C [ E i , F i , K i , K − 1 i ] 1 ≤ i ≤ n . W e will also n eed the follo wing result from the represen tation theory of qu an tized en v elop- ing algebras. Refer to [4] for terminology and pro of. Prop osition 6.4 The finite-dimensional typ e 1 mo dules of U q g c arry a br aiding that c om- mutes with the action of U q g , wher e this action is define d on V ⊗ V usi ng the c omultipli- c ation in U q g . 6.2 Statemen t and Pro of of Theorem 2.1 in the Quantum Case T ak e C = C [ E i , F i , K i , K − 1 i ] 1 ≤ i ≤ n , the coalgebra of generators for U q g . Let φ : C → End( A ) b e a faithful measuring that preserves the defining relations of U q g in End( A ), that is to sa y that A can b e made into a U q g mo du le. Theorem 6.5 1. The qu antize d enveloping algebr a U q g includes in P ( A, A ) as a me a- suring bialgebr a. 2. L et e : A → k b e an algebr a homomorphism, and define r = e ◦ φ : C → Hom ( A, k ) . 30 Supp ose additiona l ly that r i s inje ctive on C . If A i s a bialgebr a, then U q g includes in P ( A, k ) as a bialgebr a. 3. Wi th A , r as ab ove, the map P (1 , e ) : P ( A, A ) → P ( A, k ) , gener ate d by e ◦ π : P ( A, A ) → Hom( A, k ) , sends U q g c onsider e d as a sub algebr a of P ( A, A ) isomorphic al ly onto its i mage in P ( A, k ) . The pro of is the same as in the classical case w ith t wo mo difications. First, instead of Prop osition 5.3, use Corollary 6.1. Secondly , instead of the universal prop ert y of U g , us e the fact that a map φ : C → E nd( A ) that p reserv es the defining relations of U q g in End( A ) giv es rise to a unique algebra m ap µ : U q g → En d( A ) that ag r ees with φ on C . 6.3 Examples of Algebras on which U q g measures Let V b e a mo d ule for U q g whic h is f aithful on the generators. T hen b y Pr op osition 3.1, U q g measures T V to T V and the measurin g map is an algebra homomorp h ism from U q g to End( T V ). By Theorem 2.1, U q g em b eds as a sub-bialgebra U q g ⊂ P ( T V , T V ) . The v ector space V has a b raiding R b y Prop osition 6.4 and U q g preserv es ( V , R ). Th us, in the ligh t of the discussion f ollo win g 4.2, U q g em b eds as a sub-bialgebra U q g ⊂ P R ( V ) . 31 Remark Th e iden tification of P R ( V ) with A ( R ) ◦ means that U q g em b eds in the du al of the FR T bialge br a of any representa tion on wh ich the generators of U q g act faithfully . There are also in teresting quotien t algebras of T V , on whic h U q g measures. W e construct these as follo ws. L et f b e a complex p olynomial, and su pp ose that V is a v ector space with br aiding R as ab o ve. F ollo wing [10] define the algebra χ f ,R to b e the quotien t of the tensor algebra T V b y th e tw o-sided id eal generated by f ( R )( V ⊗ V ). Remark Th is constru ction is p ossible for any op erator R : V ⊗ V → V ⊗ V , bu t c ho osing R to b e a braiding m ak es it less like ly that the quotien t will b e trivial (see [11] for d etails). Note that it migh t b e necessary to choose an appropriate n ormalization on R to obtain a non-trivial quotient. Note that if χ f ,R is non-trivial, then V ⊂ χ f ,R . By Prop osition 3.4, U q g measures T V to T V and the measurin g map is an algebra h omomorphism from U q g to End( χ f ,R ). Thus, as long as χ f ,R is non-trivial, U q g em b eds as a bialgebra U q g ⊂ P ( χ f ,R , χ f ,R ) . W e list a few examples of non-trivial χ f ,R . Example V ector represen tation of U q sl n . The ve ctor rep resen tation V of U q sl n is n -dimensional and is th e quantum v ersion of the defining rep resen tation of sl n . Let e i,j b e the n × n matrix with a 1 in the ( i, j ) p osition and 0 elsewhere. W e can choose a basis su c h that V is given by: K i = q − 1 e i,i + q e i +1 ,i +1 + X k 6 = i,i +1 e k ,k , 32 E i = e i +1 ,i , F i = e i,i +1 . The asso ciated braiding (appr op r iately normalized) is: R = q X i ( e ii ⊗ e ii ) + X i 6 = j ( e ii ⊗ e j j ) + ( q − q − 1 ) X i

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