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1 CHAPTER 2 Hopf Algebrs, Algebric, Forml, nd Quntum Groups 52

2 5. QUANTUM GROUPS Quntum Groups Deænition èdrinfel'dè A quntum group is noncommuttive noncocommuttive Hopf lgebr. Remrk We shll consider ll Hopf lgebrs s quntum groups. Observe, however, tht the commuttive Hopf lgebrs my be considered s æne lgebric groups nd tht the cocommuttive Hopf lgebrs my be considered s forml groups. Their property s quntum spce or s quntum monoid will ply some role. But often the èpossibly nonexistingè dul Hopf lgebr will hve the geometricl mening. The following exmples SL q è2è nd GL q è2è will hve geometricl mening. Exmple The smllest proper quntum group, i.e. the smllest noncommuttive noncocommuttive Hopf lgebr, is the 4-dimensionl lgebr H 4 = Khg;xi=èg 2, 1;x 2 ;xg+ gxè which ws ærst described by M. Sweedler. The colgebr structure is given by æègè =g æ g; æèxè =x æ 1+g æ x; "ègè =1; "èxè =; Sègè =g,1 è= gè; Sèxè =,gx Since it is ænite dimensionl its liner dul H æ 4 is lso noncommuttive noncocommuttive Hopf lgebr. It is isomorphic s Hopf lgebr to H 4. In fct H 4 is up to isomorphism the only noncommuttive noncocommuttive Hopf lgebr of dimension 4. Exmple The æne lgebric group SLènè K-cAlg,! Gr deæned by SLènèèAè, the group of n æ n-mtrices with coeæcients in the commuttive lgebr A nd with determinnt 1, is represented by the lgebr OèSLènèè = SLènè = Këx ij ë=èdetèx ij è, 1è i.e. SLènèèAè = K-cAlgèKëx ij ë=èdetèx ij è, 1è;Aè Since SLènèèAè hs group structure by the multipliction of mtrices, the representing commuttive lgebr hs Hopf lgebr structure with the digonl æ = 1 æ 2 hence æèx ik è= X x ij æ x jk ; the counit "èx ij è=æ ij nd the ntipode Sèx ij è=djèxè ij where djèxè is the djoint mtrix of X =èx ij è. We leve the veriæction of these fcts to the reder. We consider SLènè M n = A n2 s subspce of the n 2 -dimensionl æne spce. Exmple Let M q è2è = K =I s in with I the idel generted by c d b, q,1 b; c, q,1 c; bd, q,1 db; cd, q,1 dc; èd, q,1 bcè, èd, qcbè;bc, cb

3 54 2. HOPF ALGEBRAS, ALGEBRAIC, FORMAL, AND QUANTUM GROUPS We ærst deæne the quntum determinnt det q = d, q,1 bc = d, qcb in M q è2è. It is centrl element. To show this, it suæces to show tht det q commutes with the genertors ; b; c; d èd, q,1 bcè = èd, qbcè; èd, q,1 bcèc = cèd, q,1 bcè; èd, q,1 bcèb = bèd, q,1 bcè; èd, qbcèd = dèd, q,1 bcè We cn form the quntum determinnt of n rbitrry quntum mtrix in A by b det q = d, q,1 b c = d, qc b = 'èdet c d q è if ' M q è2è,! A is the lgebr homomorphism ssocited with the quntum mtrix b. c d ; b c d Given two commuting quntum 2æ2-mtrices b. The quntum c d determinnt preserves the product, since b det q è b è = det + b c b + b d c d c d q c + d c c b + d d è1è =è + b c èèc b + d d è, q,1 è b + b d èèc + d c è = c b + b c c b + d d + b d c d,q,1 è c b + b c d + d b c + b d d c è = b c c b + d d, q,1 b c d, q,1 d b c = b c c b + d d, q,1 b c d, q,1 d b c,q,1 b c è d, d, q,1 b c + qc b è = d d, q,1 d b c, q,1 b c è d, q,1 b c è =è d, q,1 b c èè d, q,1 b c è = det q b c d det q b c d In prticulr we hve æèdet q è = det q æ det q nd "èdet q è = 1. The quntum determinnt is group like element èsee 2.1.6è. Now we deæne n lgebr SL q è2èèaè =f b c d SL q è2è = M q è2è=èdet q, 1è The lgebr SL q è2è represents the functor 2M q è2èèaèjdet q b c d =1g There is surjective homomorphism of lgebrs M q è2è,! SL q è2è nd SL q è2è is subfunctor of M q è2è. Let X; Y be commuting quntum mtrices stisfying det q èxè =1=det q èy è. Since det q èxè det q èy è = det q èxy è for commuting quntum mtrices we get

4 5. QUANTUM GROUPS 55 det q èxy è = 1, hence SL q è2è is quntum submonoid of M q è2è nd SL q è2è is bilgebr with digonl nd æ " = = 1 1 æ To show tht SL q è2è hs n ntipode we ærst deæne homomorphism of lgebrs T M q è2è,! M q è2è op by T = d,qb,q,1 c We check tht T K,! M q è2è op vnishes on the idel I. T èb, q,1 bè =T èbèt èè, q,1 T èèt èbè=,qbd + q,1 qdb = We leve the check of the other deæning reltions to the reder. Furthermore T restricts to homomorphism of lgebrs S SL q è2è,! SL q è2è op since T èdet q è= T èd, q,1 bcè =T èdèt èè, q,1 T ècèt P èbè =d, q,1 è,q,1 cèè,qbè = det q hence T èdet q,1è = det q,1=insl q è2è. One veriæes esily tht S stisæes Sèx è1è èx è2è = "èxè for ll given genertors of SL q è2è, hence S is left ntipode by Symmetriclly S is right ntipode. Thus the bilgebr SL q è2è is Hopf lgebr or quntum group. Exmple The æne lgebric group GLènè K-cAlg,! Gr deæned by GLènèèAè, the group of invertible næn-mtrices with coeæcients in the commuttive lgebr A, is represented by the lgebr OèGLènèè = GLènè =Këx ij ;të=èdetèx ij èt,1è i.e. GLènèèAè = K-cAlgèKëx ij ;të=èdetèx ij èt, 1è;Aèè Since GLènèèAè hs group structure by the multipliction of mtrices, the representing commuttive lgebr hs Hopf lgebr structure with the digonl æ = 1 æ 2 hence æèx ik è= X x ij æ x jk ; the counit "èx ij è=æ ij nd the ntipode Sèx ij è=tædjèxè ij where djèxè is the djoint mtrix of X =èx ij è. We leve the veriæction of these fcts from liner lgebr to the reder. The digonl pplied to t gives æètè =t æ t Hence tè= detèxè,1 è is grouplike elementinglènè. This reæects the rule detèabè = detèaè detèbè hence detèabè,1 = detèaè,1 detèbè,1. ;

5 56 2. HOPF ALGEBRAS, ALGEBRAIC, FORMAL, AND QUANTUM GROUPS Exmple Let M q è2è be s in the exmple We deæne GL q è2è = M q è2èëtë=j with J generted by the elements t æ èd, q,1 bcè, 1 The lgebr GL q è2è represents the functor GL q è2èèaè =f b c d 2M q è2èèaèjdet q b c d invertible in Ag In fct there is cnonicl homomorphism of lgebrs M q è2è,! GL q è2è. A homomorphism of lgebrs ' M q è2è,! A hs unique continution to GL q è2è iæ det q è' èisinvertible M q - M q è2èëtë - G q A with t 7! b det,1 q Thus GL c d q è2èèaè is subset of M q è2èèaè. Observe tht M q è2è,! GL q è2è is not surjective. Since the quntum determinnt preserves products nd the product of invertible elements is gin invertible we get GL q è2è is quntum submonoid of M q è2è, hence ægl q è2è,! GL q è2èægl q è2è with æ = æ nd æètè =tæt. We construct the ntipode for GL q è2è. We deæne T M q è2èëtë,! M q è2èëtë op by T = t d,qb,q,1 c nd T ètè = det q = d, q,1 bc As in T deænes homomorphism of lgebrs. We obtin n induced homomorphism of lgebrs S GL q è2è,! GL q è2è op or GL q è2è op -point ingl q è2è since Sètèd, q,1 bcè, 1è=èSèdèSèè, q,1 SècèSèbèèSètè, Sè1è=èt 2 d, q,1 t 2 cbèèd, q,1 bcè, 1=t 2 èd, q,1 bcè 2, 1=. Since S stisæes P Sèx è1è èx è2è = "èxè for ll given genertors, S is left ntipode by Symmetriclly S is right ntipode. Thus the bilgebr GL q è2è is Hopf lgebr or quntum group. Exmple Let slè2è be the 3-dimensionl vector spce generted by the mtrices X = 1 ; Y = ; H = 1 1,1 Then slè2è is subspce of the lgebr Mè2è of 2æ2-mtrices over K. We esily verify ëx; Y ë = XY,Y X = H, ëh; Xë = HX,XH = 2X, nd ëh; Y ë = HY,Y H =,2Y,

6 5. QUANTUM GROUPS 57 so tht slè2è becomes Lie sublgebr of Mè2è L, which is the Lie lgebr of mtrices of trce zero. The universl enveloping lgebr U èslè2èè is Hopf lgebr generted s n lgebry the elements X; Y; H with the reltions ëx; Y ë=h; ëh; Xë =2X; ëh; Y ë=,2y As consequence of the Poincre-Birkhoæ-Witt Theorem ètht we don't proveè the Hopf lgebr Uèslè2èè hs the bsis fx i Y j H k ji; j; k 2 Ng. Furthermore one cn prove tht ll ænite dimensionl U èslè2èè-modules re semisimple. Exmple We deæne the so-clled q-deformed version U q èslè2èè of Uèslè2èè for ny q 2 K, q 6= 1;,1 nd q invertible. Let U q èslè2èè be the lgebr generted by the elements E; F; K; K with the reltions KK = K K =1; KEK = q 2 E; KFK = q,2 F; EF, FE = K, K q, q,1 Since K is the inverse of K in U q èslè2èè we write K,1 = K. The representtion theory of this lgebr is fundmentlly diæerent depending on whether q is root of unity or not. We show tht U q èslè2èè is Hopf lgebr or quntum group. We deæne. æèeè =1æ E + E æ K; æèf è=k,1 æ F + F æ 1; æèkè =K æ K; "èeè ="èf è=; "èkè =1; SèEè =,EK,1 ; SèF è=,kf; SèKè =K,1 First we show tht æ cn be expnded in unique wy to n lgebr homomorphism æ U q èslè2èè,! U q èslè2èè æ U q èslè2èè. Write U q èslè2èè s the residue clss lgebr KhE; F; K; K,1 i=i where I is generted by KK,1, 1; K,1 K, 1; KEK,1, q 2 E; KFK,1, q,2 F; EF, FE, K, K,1 q, q,1 Since K,1 must be mpped to the inverse of æèkè wemust hve æèk,1 è=k,1 æ K,1.Now æ cn be expnded in unique wy to the free lgebr æ KhE; F; K; K,1 i,! U q èslè2èè æ U q èslè2èè. We hve æèkk,1 è=æèkèæèk,1 è = 1 nd similrly æèk,1 Kè=1. Furthermore we hve æèkek,1 è=æèkèæèeèæèk,1 è=èk æ Kè è1 æ E + E æ KèèK,1 æ K,1 è=kk,1 æ KEK,1 + KEK,1 æ K 2 K,1 = q 2 è1 æ E +

7 58 2. HOPF ALGEBRAS, ALGEBRAIC, FORMAL, AND QUANTUM GROUPS E æ Kè =q 2 æèeè =æèq 2 Eè nd similrly æèkfk,1 è=æèq,2 F è. Finlly we hve æèef, FEè=è1æE + E æ KèèK æ F + F æ 1è,èK æ F + F æ 1èè1 æ E + E æ Kè = K æ EF + F æ E + EK æ KF + EF æ K,K æ FE, K E æ FK, F æ E, FEæ K = K æ èef, FEè+èEF, FEèæ K = K æ èk, K è+èk, K è æ K q, q,1 K, K =æ q, q,1 hence æ vnishes on I nd cn be fctorized through unique lgebr homomorphism æu q èslè2èè,! U q èslè2èè æ U q èslè2èè In similr wy, ctully much simpler, one gets n lgebr homomorphism " U q èslè2èè,! K To check tht æ is cossocitive it suæces to check this for the genertors of the lgebr. We hve èææ 1èæèEè =èææ 1èè1 æ E + E æ Kè =1æ 1 æ E +1æ E æ K + E æ K æ K =è1æ æèè1 æ E + E æ Kè =è1æ æèæèeè. Similrly we get èæ æ 1èæèF è=è1æ æèæèf è. For K the clim is obvious. The counit xiom is esily checked on the genertors. Now we show tht S is n ntipode for U q èslè2èè. First deæne S KhE; F; K; K,1 i,! U q èslè2èè op by the deænition of S on the genertors. We hve P Sèxè1è SèKK,1 è=1=sèk,1 Kè; SèKEK,1 è=,kek,1 K,1 =,q 2 EK,1 = Sèq 2 Eè; SèKFK,1 è=,kkfk,1 =,q,2 KF = Sèq,2 F è; SèEF, FEè=KFEK,1, EK,1 KF = KFK,1 KEK, EF = K,1, K K, K,1 = S q, q,1 q, q,1 So S deænes homomorphism of lgebrs S U q èslè2èè,! U q èslè2èè. Since S stisæes èx è2è = "èxè for ll given genertors, S is left ntipode by Symmetriclly S is right ntipode. Thus the bilgebr U q èslè2èè is Hopf lgebr or quntum group. This quntum group is of centrl interest in theoreticl physics. Its representtion theory is well understood. If q is not root of unity then the ænite dimensionl U q èslè2èè-modules re semisimple. Mny more properties cn be found in ëkssel Quntum Groupsë.