Representations of U ¯ q sℓ(2|1) at even roots of unity
aa r X i v : . [ m a t h . QA ] D ec REPRESENTATIONS OF U q s ℓ p | q AT EVEN ROOTS OF UNITY
A. M. SEMIKHATOV AND I. YU. TIPUNINA
BSTRACT . We construct all projective modules of the restricted quantum group U q s ℓ p | q at an even, 2 p th, root of unity. This 64 p -dimensional Hopf algebra is a com-mon double bosonization, B p X ˚ q b B p X q b H , of two rank-2 Nichols algebras B p X q with fermionic generator(s), with H “ Z p b Z p . The category of U q s ℓ p | q -modules isequivalent to the category of Yetter–Drinfeld B p X q -modules in C r “ HH YD , where coac-tion is defined by a universal R -matrix r . As an application of the projective moduleconstruction, we find the associative algebra structure and the dimension, 5 p ´ p `
4, ofthe U q s ℓ p | q center.
1. I
NTRODUCTION
We study the representation theory of a particular version of the s ℓ p | q quantum groupat even roots of unity. For an integer p ě
2, our 64 p -dimensional U q s ℓ p | q at q “ e i p { p is a double bosonization of any of the rank-2 Nichols algebras defined by the braidingmatrices(1.1) Q a “ ˜ ´ q ´ q ´ q ¸ and Q s “ ˜ ´ ´ q ´ q ´ ¸ , q “ e i p p . The interest in these Nichols algebras is motivated by the fact that they centralize ex-tended chiral algebras (vertex-operator algebras) of logarithmic models of two-dimen-sional conformal field theory with p s ℓ p q k symmetry, at the level k “ p ´ We recall thatNichols algebras are graded braided Hopf algebras “universally” associated with a braidedvector space X (see [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] and the (co)referencestherein). The “centralizer” relation between vertex-operator algebras and (braided ornonbraided) Hopf algebras is a key ingredient underlying the Kazhdan–Lusztig corre-spondence [20, 21] and the quest for nice relations between suitable categories defined in This paper was written as “
Logarithmic p s ℓ p q CFT models from Nichols algebras. 2. ” But because ithas no direct bearing on logarithmic models (which is left for the future) and in the hope that the algebraicstructures discussed here may be interesting in their own right, we do not mention logarithmic CFT in thetitle. We refer to [1, 2, 3, 4] for the origin of logarithmic conformal field theory (LCFT) and to [5] for theidea to define LCFT models as centralizers of screening operators. The screenings themselves turn out togenerate a Nichols algebra [6], hence the importance of Nichols algebras in this context.
A. M. SEMIKHATOV AND I. YU. TIPUNIN terms of Hopf algebras and representation categories of extended symmetry algebras ofthe corresponding CFT models.A “logarithmic” Kazhdan–Lusztig correspondence can be particularly nice (an equiv-alence of categories) [22, 23] (also see [24] and, among more recent papers, [25]). How-ever, the case studied in [22, 23] is nearly trivial in the language of Nichols algebras, thecase of rank one [26]; true, the braided world can be quite rich, and already the corre-sponding nonbraided Hopf algebra is U q s ℓ p q , a not altogether trivial quantum s ℓ p q at aneven (2 p th) root of unity. It is this U q s ℓ p q that features in the categorial equivalence withthe p , p q LCFT models [27, 22]; the correspondence between Hopf-algebraic and LCFTrealms, moreover, extends to modular group representations: those on the quantum groupcenter and on the torus amplitudes of the logarithmic model turn out to coincide. The ideas of the logarithmic Kazhdan–Lusztig correspondence need to be extended tohigher-rank Nichols algebras; this would show, among other things, how much of what weknow in the rank-1 case is “accidental,” and which features are indeed generic. Movingto higher-rank Nichols algebras was initiated in [32] and, with precisely the two Nicholsalgebras defined by braiding matrices (1.1), in [33].First and foremost, with the structural theory of finite-dimensional Nichols algebraswith diagonal braiding completed [15, 18, 19], knowledge about their appropriate repre-sentation categories (of Yetter–Drinfeld modules) is highly desirable. In this paper, weaddress the representation theory of the common double bosonization of the two chosenNichols algebras, which is the 64 p -dimensional U q s ℓ p | q . We first show that the cat-egory of U q s ℓ p | q modules is equivalent to the category B p X q B p X q YD C r of Yetter–Drinfeld B p X q modules in C r “ HH YD , where B p X q is the corresponding Nichols algebra and H “ Z p b Z p is the Cartan subalgebra of U q s ℓ p | q , with its coaction on HH YD definedby the universal R -matrix r P H b H . Our main result is then the construction of pro-jective U q s ℓ p | q modules. The ensuing picture is rather involved, and may be a goodillustration of the intricacies occurring at roots of unity.An essential part of the structure of projective modules can be conveniently expressedin terms of directed graphs whose vertices are simple subquotients and the edges areassociated with elements of Ext groups, weighted with some coefficients (finding whichis a major part of the existence proof for a given projective module). The paper thereforecontains a number of pictures showing graphs of projective modules. A feature that remarkably survives in the case where the representation category on the LCFT side isnot that nice [28, 29, 30, 31]. Once again on the subject of LCFT models, we note that their fusion has a good chance to be describedjust by the ring structure of a suitable category of finite-dimensional modules, finding which, difficultthough it may be, is “infinitely” easier than deriving the fusion algebra directly. The examples where thefusion algebra known or reasonably conjectured in other approaches coincides with the one taken over fromthe Hopf-algebra side are quite encouraging (see, e.g., [5, 27, 34, 35] and the references therein).
EPRESENTATIONS OF U q s ℓ p | q AT EVEN ROOTS OF UNITY 3
As an application of using the data presented in these graphs, we find the associativealgebra structure of the center of U q s ℓ p | q : Z “ Z at ‘ p p ´ qp p ´ q à j “ Z j t ‘ p p ´ q à j “ Z j st where Z at , Z j t , and Z j st are commutative associative unital algebras; all Z j st are 1-dimensional,each Z j t is 5-dimensional (contains 4 nilpotents in addition to the unit), and Z at is 10 p ´ p ´ U p X q ” U q s ℓ p | q and discuss some of its simpleproperties (including Casimir elements and the universal R -matrix). In Sec. 3, we provethat the category of its modules is equivalent to B p X q B p X q YD . In Sec. 4, we classify its simplemodules, describing them quite explicitly. We continue in Sec. 4 with listing the Ext spaces for the simple modules. We then construct the projective U p X q modules in Sec. 6.As an application of our treatment of projective modules, we find the center of U p X q inSec. 7. In an attempt to improve the readability of this inevitably technical paper, weisolate some computational details in the appendices.We use q -integers and factorials defined as r n s “ q n ´ q ´ n q ´ q ´ , r n s ! “ r s . . . r n s , all of which are assumed specialized to q “ q in (1.1).All (co)modules in this paper are finite-dimensional.
2. T HE H OPF ALGEBRA U p X qp X qp X q The notation U p X q for our 64 p -dimensional quantum group U q s ℓ p | q is a legacyof the Nichols-algebra setting, where X in B p X q is a two-dimensional braided vectorspace—in our case, specifically, the one with diagonal braiding defined by any of the twobraiding matrices in (1.1). The Hopf algebra U p X q was derived from each of these twoNichols algebras in [33]. p X qp X qp X q ” U q s ℓ p | q s ℓ p | q s ℓ p | q . Our U p X q is the algebra on B , F , k , K , C , E with the relations listed in (2.1)–(2.3). We first identify an important Hopf subalgebrain U p X q , the restricted quantum group U q s ℓ p q . It is generated by E , K , and F , with therelations(2.1) KF “ q ´ FK , EF ´ FE “ K ´ K ´ q ´ q ´ , KE “ q EK , F p “ , E p “ , K p “ . Next, U q s ℓ p q and k generate a larger algebra U ˚ q s ℓ p q with further relations kF “ q Fk , kE “ q ´ Ek , k p “ , kK “ Kk . (2.2) A. M. SEMIKHATOV AND I. YU. TIPUNIN
The other relations in U p X q — those involving fermions B and C — are(2.3) KB “ q BK , kB “ ´ Bk , KC “ q ´ CK , kC “ ´ Ck , B “ , BC ´ CB “ k ´ k ´ q ´ q ´ , C “ , FC ´ CF “ , BE ´ EB “ , F FB ´ r s F BF ` BFF “ , EEC ´ r s ECE ` CEE “ . The Hopf-algebra structure of U p X q is furnished by the coproduct, antipode, and counitgiven by D p F q “ F b ` K ´ b F , D p E q “ E b K ` b E , D p B q “ B b ` k ´ b B , D p C q “ C b k ` b C , S p B q “ ´ kB , S p F q “ ´ KF , S p C q “ ´ Ck ´ , S p E q “ ´ EK ´ , e p B q “ , e p F q “ , e p C q “ , e p E q “ , with k and K group-like.By the U p X q generators in what follows, we always mean B , F , C , E , k , K . [33] . The Hopf algebra U p X q admits a non-trivial “twist”—an invertible normalized 2-cocycle (see B.3 ) F “ b ` p q ´ q ´ q Bk b Ck ´ P U p X q b U p X q twisting by which gives rise to the second coalgebra structure r D p x q “ F ´ D p x q F . (Two coalgebra structures naturally come from the two underlying Nichols algebras.)For the U p X q generators chosen above, the coproducts are r D F “ F b ` K ´ b F ` p q ´ q ´ q FBk b Ck ´ ` p ´ q q BFk b Ck ´ , r D B “ B b k ´ ` k ´ b B , r D E “ b E ` E b K ` p q ´ q ´ q Bk b ECk ´ ` p ´ q q Bk b CEk ´ , r D C “ C b k ` k b C . We note that the new coproduct has the following skew-primitive elements: r D p Bk q “ b Bk ` Bk b k ´ , r D p FB ´ q BF q “ p FB ´ q BF q b ` K ´ k ´ b p FB ´ q BF q , r D p Ck ´ q “ Ck ´ b ` k b Ck ´ , r D p EC ´ q CE q “ b p EC ´ q CE q ` p EC ´ q CE q b Kk . EPRESENTATIONS OF U q s ℓ p | q AT EVEN ROOTS OF UNITY 5 p X qp X qp X q . We have a linear-space isomorphism(2.4) U ą b U ď Ñ U p X q , where U ą is the subalgebra in U p X q generated by E and C , and U ď is the subalgebragenerated by F , B , K , and k . A PBW basis in U ď (which we refer to as the PBW basis in U ď ) can be chosen as F n K i k j , F n BFK i k j , F n BFK i k j , F n BFBK i k j , where 0 ď n ď p ´ ď i , j ď p ´
1; the PBW basis in U ą can be chosen similarly, as E n , E n ` C , E n CE , E n CEC , where 0 ď n ď p ´
1. The PBW basis in U p X q is then given by the product ofthese two PBW bases. There are two elements of a particular form in the center of U p X q , which we call Casimir elements: C “ EFKk ` CBK k ´ q p EC ´ q CE qp FB ´ q ´ BF q Kk ` q p q ´ q ´ q k ` q ´ p q ´ q ´ q K k ´ q ´ p q ´ q ´ q K k , C “ EFK ´ k ´ ` CBK ´ k ´ ´ q ´ p EC ´ q ´ CE qp FB ´ q BF q K ´ k ´ ` q ´ p q ´ q ´ q k ´ ` q p q ´ q ´ q K ´ k ´ ´ q p q ´ q ´ q K ´ k ´ . That they are central is verified directly.
Each of the Casimir elements satisfies a minimal polynomial relationof degree p ´ p ` : ´ C ´ q p q ´ q ´ q ¯ p ´ ź s “ s ´ ź r “ ´ C ´ q ´ r p q r ´ q s ` q r ` s qp q ´ q ´ q ¯ p ´ ź r “ ´ C ´ q ´ r p q r ´ qp q ´ q ´ q ¯ “ , and the minimal relation for C is obtained from here by replacing q Ñ q ´ . This is an immediate corollary of our construction of projective U p X q -modules inSec. 6. We simply take the product of p C i ´ l q factors over all different eigenvalues l ofa given C i on the (linkage classes of) projective modules, with each factor raised to thepower given by the corresponding Jordan cell size (three for the atypical linkage class,two for each of the typical linkage classes, and one for each Steinberg module { class;see ).A somewhat more involved derivation shows that the two Casimir elements satisfy adegree- p polynomial relation, p ÿ i “ p´ q i i ˆ p ´ i ´ ˙ p q ´ q ´ q i ` q ´ i C i ´ q i C i ˘ “ . The full list of (“mixed”) relations satisfied by C and C is currently unknown. A. M. SEMIKHATOV AND I. YU. TIPUNIN
The algebra U p X q is quasitriangular, with the universal R-matrix givenby R “ r ¯ R , r “ p p q p ´ ÿ i “ p ´ ÿ j “ p ´ ÿ m “ p ´ ÿ n “ p´ q jn q ´ im ` jm ` in K i k j b K m k n , (2.5) ¯ R “ p ´ ÿ a “ q a p a ´ q p q ´ q ´ q a r a s ! E a b F a ´ b ´ p q ´ q ´ q C b B ¯ (2.6) ˆ ´ b ` q p q ´ q ´ q ¯ C b ¯ B ¯´ b ` p q ´ q ´ q ¯¯ C b ¯¯ B ¯ , where ¯ C “ EC ´ q ´ CE , ¯ B “ FB ´ q ´ BF , ¯¯ C “ C ¯ C “ CEC , ¯¯ B “ B ¯ B “ BFB . This must be possible to extract from [36]; we give the proof by direct calculationin
B.1 .The category of U p X q modules is therefore braided, with the braiding Y b Z Ñ Z b Y given by y b z ÞÑ R p q z b R p q y (where we standardly write R “ R p q b R p q ).We let U p X q - MOD denote the braided monoidal category of U p X q -modules. We recall the general properties of the universal R -matrix for further reference: R D p x q “ D op p x q R , i.e., R p q x b R p q x “ x R p q b x R p q for all x in the algebra, and R p q b R p q b R p q “ R p q b P p q b R p q P p q , R p q b R p q b R p q “ R p q P p q b P p q b R p q (with R “ R p q b R p q “ P p q b P p q , and so on). With the above R -matrix, we introduce the so-called M[onodromy]“matrix” M “ R R P U p X q b U p X q . It is known to give rise to the Drinfeld map from the space Ch of q -characters to the center Z of U p X q :(2.7) c : Ch Ñ Z : a ÞÑ a b id p M q “ a p M q M , where we standardly write M “ M b M . We recall that Ch is the space of elements of U p X q ˚ that are invariant under the left coadjoint action of U p X q . EPRESENTATIONS OF U q s ℓ p | q AT EVEN ROOTS OF UNITY 7
For the R-matrix in , the M-matrix can be written as (2.8) M “ ¯ M ¯ r ¯ R , where ¯ M “ p ´ ÿ a “ q ´ a ´ a p q ´ q ´ q a r a s ! F a K a b E a K ´ a ` b ` p q ´ q ´ q Bk b Ck ´ ˘ ˆ ` b ´ p q ´ q ¯ BKk b ¯ CK ´ k ´ ˘` b ` q p q ´ q ´ q ¯¯ BKk b ¯¯ CK ´ k ´ ˘ ¯ r “ p p ´ ÿ i “ p ´ ÿ j “ p ´ ÿ i “ p ´ ÿ j “ q i j ` i j ´ ii K i k j b K i k j . We show this in
B.2 . The quasitriangular structure corresponding to r D in is given by r R “ F ´ R F , as readily follows from R D p x q “ D op p x q R . Hence, the corresponding M-matrix is r M “ F ´ M F . We want to compare the two Drinfeld maps c : Ch Ñ Z and r c : Ă Ch Ñ Z , where Ă Ch is thespace of elements that are invariant under the “tilded” coadjoint action.The two Drinfeld maps turn out to coincide in the sense that the diagram(2.9) Ch c Ă Ch r c Z is commutative, where the horizontal arrow is a linear isomorphism b ÞÑ p b à x q “ b p x ? q , where x “ S p U ´ q U and U “ S p F q F . We prove this in the general case in B.3.1 .With the above F , it readily follows that(2.10) U “ ´ p q ´ q ´ q BCk ´ and x “ k ´ .
3. U p X q MODULES AND Y ETTER –D RINFELD B p X q MODULES
We show that the category of U p X q modules is equivalent to a category of Yetter–Drinfeld modules of the corresponding Nichols algebra. The exact statement is in below. We begin with briefly recalling the relation between U p X q and Nichols algebras. A. M. SEMIKHATOV AND I. YU. TIPUNIN p X qp X qp X q as a double bosonization. The Hopf algebra U p X q is a double bosonization[33] (also see [37, 17, 38]) of the Nichols algebra B p X q of a two-dimensional braidedvector space X with a basis p B , F q such that the corresponding braiding matrix is Q a in (1.1) . This B p X q is the quotient [18](3.1) B p X q “ T p X q{ ` BF ´ r s FBF ` F B , B , F p ˘ for p ě B p X q “ T p X q{ ` B , BFBF ´ FBFB , F ˘ if p “ B p X q “ p . Constructing its double bosonization requires choosing a cocommutative ordinaryHopf algebra H such that B p X q P HH YD . Specifically, we take H “ Z p b Z p , with thegenerators g ” k and g ” K acting on the B p X q generators F ” B and F ” F as g i ... F j “ q i , j F j , where q i , j are the entries of the braiding matrix; H coacts as F i ÞÑ g i b F i .The double bosonization(3.2) U p X q “ B p X ˚ q B p X q H (where the right-hand side is a tensor product in HH YD ) then contains the Hopf subalgebra B p X q H given by the standard, “single,” bosonization [39], and similarly for B p X ˚ q H , but with the H action and coaction changed by composing each with the antipode(hence the prime); the cross relations are p F i F j ´ q j , i F j p F i “ d i , j p ´ p g j q q in terms of thedual basis p F i in X ˚ . After a suitable change of basis, this yields our U p X q [33].Naturally, H is identified with the Hopf subalgebra in U p X q generated by k and K .Moreover, we refine (2.4) by introducing the linear space isomorphism(3.3) U ă b H b U ą Ñ U p X q , where U ă is the subalgebra in U p X q generated by F and B . Each of the algebras U ă , H ,and U ą is a Hopf algebra. The category C r is the category HH YD of Yetter–Drinfeld H -moduleswith the coaction(3.4) d : z ÞÑ z p´ q b z p q “ r p q b r p q ... z , where r is given by (2.5).Evidently, the braiding in C r is given by(3.5) u b v ÞÑ u p´ q ... v b u p q “ r p q ... v b r p q ... u . The category U p X q - MOD is equivalent as a braided monoidal category tothe category B p X q B p X q YD C r of Yetter–Drinfeld B p X q -modules in C r . We select the first braiding matrix in (1.1) because we mainly describe U p X q with the generators chosenas B , F , C , E , k , K , not those that are skew-primitive with respect to the second coproduct in . EPRESENTATIONS OF U q s ℓ p | q AT EVEN ROOTS OF UNITY 9
We prove this in the rest of this section, in several steps. We first construct a transmu-tation functor F T : U p X q - MOD Ñ B p X q B p X q YD C r (cf. [40, 41]). Its inverse, a double bosoniza-tion functor F DB : B p X q B p X q YD C r Ñ U p X q - MOD , is then constructed as a composition of thebosonization functor F B and the functor F DD sending Yetter–Drinfeld modules to mod-ules of the Drinfeld double. Both F T and F DB are vector-space-preserving functors. Wethen verify that their compositions are indeed equivalent to the corresponding identityfunctors. There is a vector-space-preserving braided monoidal functor F T send-ing each U p X q -module into a Yetter–Drinfeld B p X q -module in C r . We prove this in – . Whenever possible (and this is most often possible),we do not assume the “ground” Hopf algebra H to be cocommutative (nor is it assumedcommutative). This in fact clarifies the calculations, highlighting the “true” reasons whycertain identities hold. We frequently use the universal R -matrix properties in , for r P H b H in particular. U ă U ă U ă is an algebra in C rrr . We make U ă into an algebra object in HH YD by definingthe left (adjoint) action and left coaction as h Ż x “ h xS p h q , d : x ÞÑ x p´ q b x p q “ r p q b r p q Ż x . The Yetter–Drinfeld axiom p h Ż x q p´ q b p h Ż x q p q “ h x p´ q S p h q b h Ż x p q is then imme-diate to verify. That the product U ă b U ă Ñ U ă is a C r -morphism is also evident. U ă U ă U ă is a Hopf algebra in C rrr . We next define the coaction(3.6) D : U ă Ñ U ă b U ă : x ÞÑ x b x “ x S p r p q q b r p q Ż x . A priori, the right-hand side is an element not of U ă b U ă but of U ď b U ă . However, itis immediately verified for the generators A “ F , B that D p A q “ A b ` b A P U ă b U ă , and we then conclude that D p U ă q Ă U ă b U ă from the main axiom of braided Hopfalgebras,(3.7) D p xy q “ x p x p´ q Ż y q b x p q y . This last identity is a relatively standard statement [40, 41], but we prove it here forcompleteness:r.h.s. of (3.7) “ x S p r p q qp s p q Ż y S p t p q qq b p s p q r p q Ż x qp t p q Ż y q We do not indicate the embedding i : H Ñ U p X q explicitly, and write h xS p h q for what should be i p h q Ż x , etc.; accordingly, we do not use a special symbol for the antipode in H , and so on. “ x S p r p q1 qp r p q2 Ż y S p t p q qq b p r p q Ż x qp t p q Ż y q“ x y S p r p q t p q q b p r p q Ż x qp t p q Ż y q“ x y S p r p q q b p r p q1 Ż x qp r p q2 Ż y q“ x y S p r p q q b p r p q Ż x y q “ l.h.s. of (3.7) . Also, D is an HH YD morphism. For the H -coaction, we have to prove that x p´ q b x p q b x p q “ x p´ q x p´ q b x p q b x p q where we calculater.h.s. “ t p q s p q b p t p q Ż x S p r p q qq b p s p q r p q Ż x q“ t p q b p t p q1 Ż x S p r p q qq b p t p q2 r p q Ż x q“ t p q b p t p q1 x S p t p q2 r p q qq b p t p q3 r p q Ż x q“ t p q b p t p q1 x S p r p q t p q3 qq b p r p q t p q2 Ż x q , and l.h.s. “ t p q b p t p q1 xS p t p q2 qq S p r p q q b p r p q Ż p t p q1 xS p t p q2 qq q“ t p q b t p q1 x S p t p q4 q S p r p q q b p r p q Ż t p q2 x S p t p q3 qq , which is the same. For the H action, we readily see that h Ż D p x q “ D p h Ż x q using the factthat H is cocommutative. We thus conclude that U ă is a Hopf algebra object in C r . As in [12], we can characterize U ă inside U ď as U ă “ t x P U ď | x b p p x q “ x b u with a projection (Hopf-algebra map) p : U ď Ñ H . Then p p x q b x “ r p q b r p q Ż x . U ă “ U ă “ U ă “ B p X qp X qp X q . In C r , the braiding matrix for the generators B and F is exactly Q a in (1.1). Hence, U ă endowed with the coproduct D (and with the antipode x ÞÑ r p q S p r p q Ż x q , which we do not discuss in any further detail for brevity) is the Nicholsalgebra B p X q of the two-dimensional braided linear space X with the braiding matrix Q a .We write B p X q instead of U ă in what follows. p X qp X qp X q modules are objects of C r . Let Z P U p X q - MOD , with a U p X q action x b z ÞÑ x ... z . By restriction, Z is an H -module. We make Z into an object in C r by endowing itwith the H -coaction (3.4). EPRESENTATIONS OF U q s ℓ p | q AT EVEN ROOTS OF UNITY 11 p X qp X qp X q modules are objects of B p X qp X qp X q B p X qp X qp X q YD . On each U p X q -module Z , which is a B p X q -module by restriction, we also define the B p X q -coaction d : Z Ñ B p X q b Z z ÞÑ z ´ b z “ R p q S p r p q q b r p q R p q ... z “ r p q Ż ¯ R p q b r p q ¯ R p q ... z , (3.8)where ¯ R is defined in (2.6); the second form shows that d indeed maps to B p X q b Z . Thecoaction property z ´ b z ´ b z “ z ´ b z ´ b z is established by direct calculation.We must also verify that d is an HH YD morphism. We calculate, writing r “ r p q b r p q “ s p q b s p q “ t p q b t p q “ . . . (and assuming neither cocommutativity nor commutativityof H ): h p d z q “ p h Ż R p q S p r p q qq b h r p q R p q ... z “ h R p q S p h r p q q b h r p q R p q ... z “ h R p q S p r p q h q b r p q h R p q ... z “ R p q h S p r p q h q b r p q R p q h ... z “ R p q b r p q R p q h ... z “ d p h ... z q , and z ´ p´ q z p´ q b z ´ p q b z p q “ r p q s p q b p r p q Ż R p q S p t p q qq b s p q t p q R p q ... z “ r p q b p r p q1 Ż R p q S p t p q qq b r p q2 t p q R p q ... z “ r p q b r p q1 R p q S p r p q2 t p q qq b r p q3 t p q R p q ... z “ r p q b r p q1 R p q S p t p q r p q3 q b t p q r p q2 R p q ... z “ r p q b R p q r p q2 S p t p q r p q3 q b t p q R p q r p q1 ... z “ r p q b R p q S p t p q q b t p q R p q r p q ... z “ z p´ q b z p q´ b z p q . We next verify the Yetter–Drinfeld axiom relating the B p X q action and coaction, whichin the standard graphic notation (see, e.g., [42]) is expressed as B p X q Z ☛ ✟✍✎✡ ✠ “ B p X q Z ☛ ✟☛✡ ✠✡ With the above D , d , and braiding, this is equivalent to R p q S p s p q qp p p q t p q r p q Ż x q b p p q s p q R p q x S p r p q q t p q ... z “ x S p r p q qp s p q Ż R p q S p t p q qq b p s p q r p q Ż x q t p q R p q ... z for x P B p X q and z P Z . We prove the last identity:l.h.s. “ R p q S p s p q qp p p q r p q Ż x q b p p q s p q R p q x S p r p q1 q r p q2 ... z “ R p q S p s p q qp p p q Ż x q b p p q s p q R p q x ... z “ R p q S p s p q1 qp s p q2 Ż x q b s p q R p q x ... z “ R p q x S p s p q q b s p q R p q x ... z “ x R p q S p s p q q b s p q x R p q ... z , but r.h.s. “ x S p r p q1 qp r p q2 Ż R p q S p t p q qq b p r p q Ż x q t p q R p q ... z “ x R p q S p r p q t p q q b p r p q Ż x q t p q R p q ... z “ x R p q S p r p q q b p r p q1 Ż x q r p q2 R p q ... z , which is the same. The functor F T is monoidal. Let Y and Z be two U p X q modules. When viewedas B p X q modules, their tensor product carries the B p X q action that actually evaluates asthe U p X q action on a tensor product of its modules: x ... p y b z q “ p x r p q ... y q b p r p q Ż x q ... z “ p x S p t p q q r p q ... y q b p r p q t p q Ż x q ... z “ p x S p r p q1 q r p q2 ... y q b p r p q Ż x q ... z “ x ... y b x ... z . Essentially the same calculation shows that the functor is braided: braiding c Y , Z in thecategory B p X q B p X q YD C r evaluates as the braiding in U p X q - MOD : p y ´ r p q ... z q b r p q ... y “ R p q ... z b R p q ... y . FFF B . To establish a functor inverse to F T , we first construct a func-tor F B : B p X q B p X q YD C r Ñ B p X q H B p X q H YD , which essentially amounts to “a Radford formula forcomodules.”In this subsection, we do not need to assume the existence of a universal R -matrix r P H b H ; therefore, H is an arbitrary Hopf algebra (with bijective antipode). If H is an ordinary Hopf algebra and R is analgebra object in HH YD (a braided Hopf algebra), then Radford’s biproduct [39] (bosoniza-tion [43]) defines the structure of an ordinary Hopf algebra on the smash product R H .Its multiplication, comultiplication, and antipode are p r h qp s g q “ r p h Ż s q h g , (3.9) EPRESENTATIONS OF U q s ℓ p | q AT EVEN ROOTS OF UNITY 13
DDD : r h ÞÑ p r h q r s b p r h q r s “ p r r p´ q h q b p r p q h q , (3.10) SSS p r h q “ p s p r p´ q h qq ` S p r p q q ˘ (3.11)(where s is the antipode of H and S is the antipode of R ). Our aim is to extend bosonization to Yetter–Drinfeld modules. In the same settingas above, let Y be a Yetter–Drinfeld R -module in HH YD . We write r á y and h . y for the R and H actions, and y ÞÑ y ´ b y P R b Y and y ÞÑ y p´ q b y p q P H b Y for the coactions. Let R be a Hopf algebra in C “ HH YD and Y P RR YD C . Then Y is aYetter–Drinfeld R H-module with the R H action p r h qá y “ r áp h . y q , (3.12) and coaction ddd : y ÞÑ y r´ s b y r s “ p y ´ y p´ q q b y p q P R H b Y . (3.13) This defines a vector-space-preserving monoidal braided functor F B : RR YD C Ñ R H R H YD to the category of left–left Yetter–Drinfeld R H-modules.
First, the coaction property is readily verified for (3.13) using thefact that the R -coaction is an H -comodule morphism. Next, we must show the Yetter–Drinfeld condition in p R H q b Y , ` p r h q r s á y ˘ r´ s p r h q r s b ` p r h q r s á y ˘ r s (3.14) “ p r h q r s y r´ s b p r h q r s á y r s , where the right-hand side is simple: “ p r r p´ q h qp y ´ y p´ q q b p r p q h qá y p q . As regards the left-hand side, we first note that for z P Y , s P R , and g P H , z r´ s p s g q b z r s “ p z ´ z p´ q qp s g q b z p q “ p z ´ p z p´ q Ż s q z p´ q g q b z p q “ p z ´ p z p´ q Ż s q z p qp´ q g q b z p qp q . Using this, we readily see directly from the definitions thatl.h.s. of (3.14) “p ` r á p r p´ q h . y q ˘ ´ p ` r á p r p´ q h . y q ˘ p´ q Ż r p q q ` r á p r p´ q h . y q ˘ p q p´ q h qb ` r áp r p´ q h . y q ˘ p qp q , where we next use the condition that Y is a Yetter–Drinfeld R -module, which is ` r á p r p´ q . y q ˘ ´ ` r á p r p´ q . y q ˘ p´ q Ż r p q b ` r á p r p´ q . y q ˘ p q “ r p r p´ q Ż y ´ q b p r p q á y q , whencel.h.s. of (3.14) “ r p r p´ q Ż p h . y q ´ q r p q á p h . y q q p´ q h b p r p q á p h . y q q p q . We continue by using the fact that the R -action á is a morphism of H -comodules, “ r p r p´ q Ż p h . y q ´ q r p qp´ q p h . y q p´ q h b r p qp q á p h . y q p q and that the R coaction is a morphism of H -modules, “ r p r p´ q Ż p h Ż y ´ qq r p qp´ q p h . y q p´ q h b r p qp q áp h . y q p q “ r p r p´ q h Ż y ´ q r p´ q p h . y q p´ q h b r p q á p h . y q p q , after which the H -Yetter–Drinfeld condition for Y yields “ r p r p´ q h Ż y ´ q r p´ q h y p´ q b r p q áp h . y p q q“ r.h.s. of (3.14) . This shows that we have a functor.The functor is monoidal: for Y , Z P RR YD , the action of R on a tensor product Y b Z isgiven by the throughout map in r b p y b z q ÞÑ r b r b y b z ÞÑ r b r p´ q . y b r p q b z ÞÑ ` r áp r p´ q . y q ˘ b r p q á z . But with Y , Z P R H R H YD , the R H action on the tensor product is p r h q b p y b z q ÞÑ pp r h q r s á y q b pp r h q r s á z q“ p r r p´ q h qá y q b pp r p q h qá z q“ ` r á p r p´ q h . y q ˘ b ` r p q á p h . z q ˘ , and hence r r in the preceding formula.The functor is braided: for y P Y and z P Z , with Y , Z P RR YD , the braiding is c RR YDY , Z : y b z ÞÑ ` y ´ áp y p´ q . z q ˘ b y p q . On the other hand, when Y and Z are viewed as objects in R H R H YD , the braiding is c R H R H YDY , Z : y b z ÞÑ p y r´ s á z q b y r s “ p y ´ y p´ q qá z b y p q “ ` y ´ á p y p´ q . z q ˘ b y p q , which is the same.For Y b Z P RR YD , the coaction is the throughout map in y b z ÞÑ y ´ b y b z ´ b z ÞÑ y ´ bp y p´ q Ż z ´ qb y p q b z ÞÑ y ´ p y p´ q Ż z ´ qb y p q b z , EPRESENTATIONS OF U q s ℓ p | q AT EVEN ROOTS OF UNITY 15 which lies in R b Y b Z . Now for Y b Z P R H R H YD , the coaction is simply y r´ s z r´ s b y r s b z r s “ p y ´ y p´ q qp z ´ z p´ q q b y p q b z p q “ ` y ´ p y p´ q Ż z ´ q y p´ q z p´ q q ˘ b y p q b z p q , which is in R H b Y b Z . Applying id b e b id b id restores the previous formula. A particular case of is well known: for Y “ p R , Ż q with the leftadjoint action, we write (3.12) as (with r , p P R ) p r q Ż p “ r p r p´ q . p q S p r p q q“ p r r p´ q qp p qp s p r p´ q qqp S p r p q q q“ r r s p p q SSS p r r s q (see (3.11) for the antipode), which is a restriction of the (“nonbraided”) left adjoint actionof R H on itself. FFF DB . To construct a “double bosonization” functor F DB inverse to F T ,we compose F B with the standard functor establishing the equivalence [44] of Yetter–Drinfeld modules with modules of the Drinfeld double D p B p X q H q “ D p U ď q , F DD : B p X q H B p X q H YD Ñ D p U ď q - MOD .Applied to any Y P B p X q B p X q YD C r , F DB “ F DD F B then produces a U p X q -module becausethe action of the generators of H ˚ P D p U ď q is completely determined by the action of thegenerators of H P D p U ď q , which means that the module is actually a module of U p X q » D p U ď q{p H ˚ „ H q (the quotient by relations expressing H ˚ through H ). Indeed, for m P H ˚ , its action on Y viewed as a Yetter–Drinfeld U ď -module is standardly given by m ë y “ x m , SSS ´ p y r´ s qy y r s “ x e b m , SSS ´ p y ´ y p´ q qy y p q “ x m , S ´ p y p´ q qy y p q “ x m , S ´ p r p q qy r p q ... y . It follows that the generators of H ˚ (see B.1.2 ) act as L ë z “ K ... z , ℓ ë z “ k ... z , (3.15)and hence we have a functor F DB : B p X q B p X q YD C r Ñ U p X q - MOD . To verify that F DB F T „ U p X q - MOD , we first calculate F B F T . Applied to U ă Ă U p X q , F T gives a braided Hopf algebra with coproduct (3.6), x b x “ x S p r p q q b r p q Ż x ( x P U ă ); applying F B —using Radford’s formula (3.10)—we obtain the coproduct DDD : U ď Ñ U ď b U ď that evaluates as DDD p x h q “ x h b x h , which is the original coproduct on U ď . For Y P U p X q - MOD , similarly, F T producescoproduct (3.8), y ´ b y “ R p q S p r p q q b r p q R p q ... y ; to further apply F B , we substitutethe last formula in the “Radford formula for comodules,” Eq. (3.13), to obtain the U ă coaction ddd p y q “ R p q b R p q ... y P U ď b Y . The resulting U ď action, obviously, is the restriction of the U p X q action. To the Yetter–Drinfeld U ď -module Y thus obtained, we now apply F DD , making it into a module of theDrinfeld double D p U ď q . Then U ˚ď (a Hopf subalgebra in D p U ď q ) acts as m ë y “ x m , SSS ´ p y r´ s qy y r s “ x m , S ´ p R p q qy R p q ... y “ x m , R p q y S p R p q q ... y . With the R -matrix in and with the duality worked out in B.1.1 , we find, alongwith (3.15), that E ë y “ E ... y , C ë y “ C ... y . Comparing with the formulas in
B.1.2 shows that the resultant D p U ď q -module is in facta U p X q -module, naturally isomorphic to the original Y .We show similarly that F T F DB „ B p X q B p X q YD C r . Starting from Z P B p X q B p X q YD C r (where wewrite the action of both H and U ă as x b z ÞÑ x ... z for simplicity, and the coaction as z ÞÑ z ´ b z ) and applying F T F DB , we arrive at the Yetter–Drinfeld U ă -module with thecoaction z ÞÑ r p q Ż R p q b x ¯ R p q , SSS ´ p z ´ t p q qy r p q t p q ... z “ r p q2 Ż R p q b x ¯ R p q , SSS ´ p z ´ r p q1 qy r p q ... z “ r p q2 Ż R p q b x ¯ R p q , S ´ p r p q1 q Ż S ´ p z ´ qy r p q ... z “ r p q2 Ż R p q b x r p q1 Ż ¯ R p q , S ´ p z ´ qy r p q ... z “ R p q b x ¯ R p q , S ´ p z ´ qy z “ z ´ b z . Hence, we have constructed an inverse functor F DB to the functor F T in . The cat-egories U p X q - MOD and B p X q B p X q YD C r are equivalent. In the rest of the paper, we study themodules of U p X q .
4. S
IMPLE MODULES OF U p X qp X qp X q The following theorem is of course contained in [17, 38]; we spell out the details herein order to fix our notation and conventions.
The algebra U p X q has p simple modules, labeled as Z a , b s , r , a , b “ ˘ , s “ , . . . , p , r “ , . . . , p ´ , EPRESENTATIONS OF U q s ℓ p | q AT EVEN ROOTS OF UNITY 17 with the dimensions (4.1) dim Z a , b s , r “ $’’’’’&’’’’’% s ´ , ď s ď p , r “ , s ` , ď s ď p ´ , r “ s , s , ď s ď p ´ , r ‰ , s , p , s “ p , ď r ď p ´ . The modules in the first two lines are atypical, and the others typical.
The cases occurring in the theorem can be illustrated by the diagram in Fig. 4.1. Typical p ´ m db db db d f ‘ ‘ ‘ a r s p F IGURE
The p simple U p X q modules, labeled by s and r . modules are the “bulk” of the diagram and the greatest part of the right column. In whatfollows, we refer to these last modules (the fourth case in (4.1)) as Steinberg modules.The notation in the figure distinguishes more cases than just those in (4.1) for our laterpurposes. p X qp X qp X q -modules. We construct the simple U p X q modules ex-plicitly, by “gluing together” some modules of the algebra U ˚ q s ℓ p q defined in (2.1) and(2.2). We let simple U ˚ q s ℓ p q modules be denoted by X a , b s , r , where a , b “ ˘ , s “ , . . . , p ,and r “ , . . . , p ´
1. As a U q s ℓ p q -module, an X a , b s , r with any b and r is isomorphic to X a s (see A.1 ). In particular, there is a highest-weight vector | a , s , b , r y “ E ˇˇ a , s , b , r D “ K ˇˇ a , s , b , r D “ a q s ´ ˇˇ a , s , b , r D , and the second Cartan generator of U p X q acts on this vector as k ˇˇ a , s , b , r D “ b q ´ r ˇˇ a , s , b , r D . It follows that dim X a , b s , r “ s . 𠧧 đ BF 𠧧 đ𠧧 đ𠧧 đđ § IJđ §IJ İđ §IJ İđ §IJ İđ §IJ F IGURE L EFT : An atypical module Z a , b s , (with s “ U ˚ q s ℓ p q module in (4.2). The top state is | a , s , b , y Ð and the bottom, | a , s , b , y Ð s ´ . M IDDLE : An atypical module Z a , b s , s (with s “ U ˚ q s ℓ p q module in (4.3). The top state is | a , s , b , s y Ñ and thebottom, | a , s , b , s y Ñ s . R IGHT : The typical module Z a , b s , r ( s “ U ˚ q s ℓ p q -module in (4.5). The directions in which the generators map arecommon for all modules. U ˚ q s ℓ p q decompositions and bases of simple modules. For each of the four casesin (4.1), we now list the U ˚ q s ℓ p q decompositions of simple U p X q modules, specify the cor-responding choice of basis, and describe how the U ˚ q s ℓ p q constituents are glued togetherby the fermionic U p X q generators B and C . There are two sorts of basis vectors | y Ð m and | y Ñ m for atypical modules and two more, | y Ò m and | y Ó m , for typical modules; in all cases, C ˇˇ D Ð m “ B ˇˇ D Ñ m “ U p X q module. Atypical modules, ď s ď p ď s ď p ď s ď p , r “ r “ r “ : as U ˚ q s ℓ p q modules, these simple U p X q mod-ules decompose as(4.2) Z a , b s , “ X a , b s , ‘ X a , b s ´ , p ´ , ď s ď p , which is illustrated in Fig. 4.2, left, and as Z a , b , “ X a , b , (one-dimensional mod-ules). We choose a basis in Z a , b s , in accordance with this decomposition, as `ˇˇ a , s , b , D Ð n P X a , b s , ˘ ď n ď s ´ , `ˇˇ a , s , b , D Ñ m P X a , b s ´ , p ´ ˘ ď m ď s ´ . EPRESENTATIONS OF U q s ℓ p | q AT EVEN ROOTS OF UNITY 19
The arrows in the notation for basis vectors refer to the visualization of Z a , b s , as inFig. 4.2, left. The fermionic generators relate the two types of vectors as B ˇˇ a , s , b , D Ð n “ ´r n s ˇˇ a , s , b , D Ñ n ´ , C ˇˇ a , s , b , D Ñ m “ b ˇˇ a , s , b , D Ð m ` . Here and hereafter, we set | a , s , b , r y ‚ m “ m is outside the rangeindicated for vectors of a given type. Atypical modules, ď s ď p ´ ď s ď p ´ ď s ď p ´ , r “ sr “ sr “ s : the module decomposes as(4.3) Z a , b s , s “ X a , b s , s ‘ X a , ´ b s ` , s , as is illustrated in Fig. 4.2, middle. We choose a basis in Z a , b s , s accordingly: `ˇˇ a , s , b , s D Ð n P X a , b s , s ˘ ď n ď s ´ , `ˇˇ a , s , b , s D Ñ m P X a , ´ b s ` , s ˘ ď m ď s . The fermions act as B ˇˇ a , s , b , s D Ð n “ r s ´ n s ˇˇ a , s , b , s D Ñ n , C ˇˇ a , s , b , s D Ñ m “ b ˇˇ a , s , b , s D Ð m . Typical modules, ď s ď p ´ ď s ď p ´ ď s ď p ´ , r ‰ , sr ‰ , sr ‰ , s : the modules decompose as(4.4) Z a , b s , r “ X a , b s , r ‘ X a , ´ b s ` , r ‘ X a , ´ b s ´ , r ´ ‘ X a , b s , r ´ , ď s ď p ´ , which is illustrated in Fig. 4.2, right, and as Z a , b , r “ X a , b , r ‘ X a , ´ b , r ‘ X a , b , r ´ . Wechoose a basis in Z a , b s , r accordingly, such that the bases in respective U ˚ q s ℓ p q submodules are(4.5) `ˇˇ a , s , b , r D Ð j ˘ ď j ď s ´ , `ˇˇ a , s , b , r D Ò m ˘ ď m ď s , `ˇˇ a , s , b , r D Ó n ˘ ď n ď s ´ , `ˇˇ a , s , b , r D Ñ j ˘ ď j ď s ´ . The fermions glue the U ˚ q s ℓ p q modules together as B ˇˇ a , s , b , r D Ð j “ r j sr s s ˇˇ a , s , b , r D Ó j ´ ` b r r sr s ´ j sr s s ˇˇ a , s , b , r D Ò j , B ˇˇ a , s , b , r D Ò m “ r m s ˇˇ a , s , b , r D Ñ m ´ , C ˇˇ a , s , b , r D Ò m “ ˇˇ a , s , b , r D Ð m , B ˇˇ a , s , b , r D Ó n “ b r r sr n ` ´ s s ˇˇ a , s , b , r D Ñ n , C ˇˇ a , s , b , r D Ó n “ b r r ´ s s ˇˇ a , s , b , r D Ð n ` , C ˇˇ a , s , b , r D Ñ j “ r s s ˇˇ a , s , b , r D Ó j ` b r s ´ r sr s s ˇˇ a , s , b , r D Ò j ` . Steinberg modules, s “ ps “ ps “ p , ď r ď p ´ ď r ď p ´ ď r ď p ´ : the decomposition is(4.6) Z a , b p , r “ X a , b p , r ‘ P a , ´ b p ´ , r ´ ‘ X a , b p , r ´ , where P a , b p ´ , r is a projective U ˚ q s ℓ p q -module. This can be illustrated with muchthe same diagram as in Fig. 4.2, right, but with two differences in the middlecolumns: first, there are exactly p IJ and p İ vectors and, second, they are not a direct sum of two simple U ˚ q s ℓ p q modules; instead, the action of F and E gener-ators glues them together as İ F IJ İ E IJ İIJ İ F IJ E (the vectors are conventionally separated horizontally; we remind the reader thatthis is only the middle part of a diagram similar to the Fig. 4.2, right). We choosethe basis in Z a , b p , r in accordance with (4.6), as ˇˇ a , p , b , r D Ð n , ˇˇ a , p , b , r D Ò n , ˇˇ a , p , b , r D Ó n , ˇˇ a , p , b , r D Ñ n where n “ , . . . , p ´
1. The fermions act as B ˇˇ a , p , b , r D Ð n “ r n ´ s ˇˇ a , p , b , r D Ò n ` r n s ˇˇ a , p , b , r D Ó n ´ , B ˇˇ a , p , b , r D Ò n “ ´r n s ˇˇ a , p , b , r D Ñ n ´ , C ˇˇ a , p , b , r D Ò n “ ´ b r r s ˇˇ a , p , b , r D Ð n , B ˇˇ a , p , b , r D Ó n “ r n s ˇˇ a , p , b , r D Ñ n , C ˇˇ a , p , b , r D Ó n “ b r r ´ s ˇˇ a , p , b , r D Ð n ` , C ˇˇ a , p , b , r D Ñ n “ b r r ´ s ˇˇ a , p , b , r D Ò n ` ` b r r s ˇˇ a , p , b , r D Ó n . The remaining formulas for the U p X q action are collected in A.3 . We recall the Drinfeld map (2.7). If Z is a U p X q -module, then the (“quantum”) trace operation(4.7) Tr Z : A ÞÑ tr Z p AK p k q : U p X q Ñ C , where tr Z evaluates the ordinary trace in any chosen basis in Z , defines an element of Ch .We note that if g “ Tr Z , then it follows from (2.10) that Ă Ch Q r g : A ÞÑ tr Z p AK p q . Calculat-ing with (2.8), we see that traces over three-dimensional representations are mapped bythe Drinfeld map into the Casimir elements: c : Tr Z a , b , ÞÑ ´ a p q ´ p q ´ q ´ q C , c : Tr Z a , b , ÞÑ ´ a p q p q ´ q ´ q C .
5. Ext
SPACES FOR SIMPLE U p X qp X qp X q - MODULES
We now describe the Ext groups for simple U p X q -modules. We recall that for twomodules Z and Z , Ext p Z , Z q is a linear space with the basis identified with nontrivialshort exact sequences 0 Ñ Z Ñ Z i Z Ñ Z Ñ EPRESENTATIONS OF U q s ℓ p | q AT EVEN ROOTS OF UNITY 21 modulo a certain equivalence relation [45].The action of any U p X q generator A on Z h Z is given by r A “ r p q A ` x A where r p q A is the direct sum of the actions of U p X q generators on simple modules and x A “ x Z , Z A : Z Ñ Z are linear maps (also linear in A ). We list the x A for each extensionin what follows. spaces for typical simple modules. The nontrivial Ext p Z , Z q spaces forthe typical simple U p X q -modules are one-dimensional. These are Ext p Z a , b s , r , Z ´ a , ´ b p ´ s , p ` r ´ s q and Ext p Z a , b s , r , Z ´ a , b p ´ s , p ` r ´ s q for each pair p s , r q such that1 ď s ď p ´ , ď r ď p ´ , s ‰ r . To avoid notational complications, we here adopt the convention that Z a , b s , p ` r “ Z a , ´ b s , r for1 ď r ď p ´ spaces, writingExt p Z a , b s , r , Z ´ a , ´ b p ´ s , p ` r ´ s q “ t f p ´ s , p ` r ´ s u , (5.1)and Ext p Z a , b s , r , Z ´ a , b p ´ s , p ` r ´ s q “ t e p ´ s , p ` r ´ s u . (5.2)Here and hereafter, a and b are omitted in the right-hands sides in order not to overburdenthe notation; they are in all cases easily reconstructed from the context. The maps Z a , b s , r Ñ Z ´ a , ´ b p ´ s , p ` r ´ s by the U p X q generators associated with (5.1) are x F : ˇˇ a , s , b , r D Ò m ÞÑ d m , s ˇˇ ´ a , p ´ s , ´ b , p ` r ´ s D Ó , x C : ˇˇ a , s , b , r D Ò m ÞÑ ´ b r r s d m , s ˇˇ ´ a , p ´ s , ´ b , p ` r ´ s D Ð , x F : ˇˇ a , s , b , r D Ó m ÞÑ r s ´ r sr r s d m , s ´ ˇˇ ´ a , p ´ s , ´ b , p ` r ´ s D Ò , (5.3) x F : ˇˇ a , s , b , r D Ð m ÞÑ ´ bd m , s ´ r r s ˇˇ ´ a , p ´ s , ´ b , p ` r ´ s D Ð , x F : ˇˇ a , s , b , r D Ñ m ÞÑ b r s ´ r s d m , s ´ ˇˇ ´ a , p ´ s , ´ b , p ` r ´ s D Ñ , x C : ˇˇ a , s , b , r D Ñ m ÞÑ d m , s ´ r r sr s ´ r sr s s ˇˇ ´ a , p ´ s , ´ b , p ` r ´ s D Ò (and zero otherwise). The maps Z a , b s , r Ñ Z ´ a , b p ´ s , p ` r ´ s by the U p X q generators associatedwith (5.2) are x E : ˇˇ a , s , b , r D Ò m ÞÑ r s ` sr r ´ s sr r s d m , ˇˇ ´ a , p ´ s , b , p ` r ´ s D Ó p ´ s ´ , x E : ˇˇ a , s , b , r D Ð m ÞÑ b r s sr r ´ s s d m , ˇˇ ´ a , p ´ s , b , p ` r ´ s D Ð p ´ s ´ , x E : ˇˇ a , s , b , r D Ó m ÞÑ r s ´ s d m , ˇˇ ´ a , p ´ s , b , p ` r ´ s D Ò p ´ s , (5.4) x B : ˇˇ a , s , b , r D Ð m ÞÑ a r s s d m , ˇˇ ´ a , p ´ s , b , p ` r ´ s D Ò p ´ s , x B : ˇˇ a , s , b , r D Ò m ÞÑ ´ ab r r s d m , ˇˇ ´ a , p ´ s , b , p ` r ´ s D Ñ p ´ s ´ , x E : ˇˇ a , s , b , r D Ñ m ÞÑ ´ b r s sr r s d m , ˇˇ ´ a , p ´ s , b , p ` r ´ s D Ñ p ´ s ´ . That these formulas make Z a , b s , r h Z ´ a , ´ b p ´ s , p ` r ´ s and Z a , b s , r h Z ´ a , b p ´ s , p ` r ´ s into U p X q modulesis verified directly. spaces for atypical simple U p X qp X qp X q -modules. The Ext p Z , Z q groups for atyp-ical simple U p X q -modules are either 1- or 0-dimensional. The nonzero Ext spaces canbe arranged into several series, in addition to which there are a few “exceptional” cases,where one of the modules involved is Z a , b , . These cases can be conveniently absorbedinto the series by adopting the convention that Z a , b , “ Z a , ´ b , and setting | a , , b , y Ñ “| a , , ´ b , y Ð and | a , , b , y Ð m “ m ‰ U p X q generators map from Z to Z (againomitting the a and b indices from the notation)Ext p Z a , b s , , Z a , ´ b s ` , q “ t b s ` u , p ď s ď p ´ q x B : ˇˇ a , s , b , D Ð m ÞÑ ´r s ´ m s ˇˇ a , s ` , ´ b , D Ð m , x B : ˇˇ a , s , b , D Ñ m ÞÑ r s ´ m ´ s ˇˇ a , s ` , ´ b , D Ñ m , Ext p Z a , b s , , Z a , ´ b s ´ , q “ t c s ´ u , p ď s ď p q x C : ˇˇ a , s , b , D Ð m ÞÑ ˇˇ a , s ´ , ´ b , D Ð m , x C : ˇˇ a , s , b , D Ñ m ÞÑ ˇˇ a , s ´ , ´ b , D Ñ m , Ext p Z a , b s , s , Z ´ a , b p ´ s , q “ t f p ´ s u , p ď s ď p ´ q x F : ˇˇ a , s , b , s D Ð s ´ ÞÑ ˇˇ ´ a , p ´ s , b , D Ð , x F : ˇˇ a , s , b , s D Ñ s ÞÑ ˇˇ ´ a , p ´ s , b , D Ñ , x C : ˇˇ a , s , b , D Ñ s ÞÑ b ˇˇ ´ a , p ´ s , b , D Ð , Ext p Z a , b s , s , Z ´ a , ´ b p ´ s , q “ t e p ´ s u , p ď s ď p ´ q x E : ˇˇ a , s , b , s D Ð ÞÑ r s s ˇˇ ´ a , p ´ s , ´ b , D Ð p ´ s ´ , x E : ˇˇ a , s , b , s D Ñ ÞÑ ´r s ` s ˇˇ ´ a , p ´ s , ´ b , D Ñ p ´ s ´ , Ext p Z a , b s , s , Z a , ´ b s ´ , s ´ q “ t ¯ b s ´ u , p ď s ď p ´ q x B : ˇˇ a , s , b , s D Ð m ÞÑ ´r m s ˇˇ a , s ´ , ´ b , s ´ D Ð m ´ , x B : ˇˇ a , s , b , s D Ñ m ÞÑ r m s ˇˇ a , s ´ , ´ b , s ´ D Ñ m ´ , Ext p Z a , b s , s , Z a , ´ b s ` , s ` q “ t ¯ c s ` u , p ď s ď p ´ q x C : ˇˇ a , s , b , s D Ð m ÞÑ ˇˇ a , s ` , ´ b , s ` D Ð m ` , x C : ˇˇ a , s , b , s D Ñ m ÞÑ ˇˇ a , s ` , ´ b , s ` D Ñ m ` , Ext p Z a , b s , , Z ´ a , ´ b p ´ s , p ´ s q “ t ¯ f p ´ s u , p ď s ď p q x F : ˇˇ a , s , b , D Ð s ´ ÞÑ ˇˇ ´ a , p ´ s , ´ b , p ´ s D Ð , x F : ˇˇ a , s , b , D Ñ s ´ ÞÑ ´ ˇˇ ´ a , p ´ s , ´ b , D Ñ , EPRESENTATIONS OF U q s ℓ p | q AT EVEN ROOTS OF UNITY 23
Ext p Z a , b s , , Z ´ a , b p ´ s , p ´ s q “ t ¯ e p ´ s u , p ď s ď p q x E : ˇˇ a , s , b , D Ð ÞÑ r s s ˇˇ ´ a , p ´ s , b , p ´ s D Ð p ´ s ´ , x E : ˇˇ a , s , b , D Ñ ÞÑ r s ´ s ˇˇ ´ a , p ´ s , b , p ´ s D Ñ p ´ s , x B : ˇˇ a , s , b , D Ð ÞÑ ´ a ˇˇ ´ a , p ´ s , b , p ´ s D Ñ p ´ s , with all other generators mapping by zero in each case. It follows that the simple U p X q modules are divided into linkageclasses as follows.(1) There are 4 p p ´ q Steinberg linkage classes, labeled by a “ ˘ , b “ ˘ , and1 ď r ď p ´
1; each class contains a single module Z a , b p , r .(2) There are p p ´ qp p ´ q typical linkage classes, labeled by p a , s , r q with a “˘ and 1 ď r ă s ď p ´
1; each such class contains four simple modules Z a , ˘ s , r , Z ´ a , ˘ p ´ s , p ` r ´ s .(3) There is one atypical linkage class containing 4 p p ´ q simple modules: Z ˘ , ˘ s , s with 1 ď s ď p ´ Z ˘ , ˘ s , with 1 ď s ď p (with uncorrelated signs in eithercase).
6. P
ROJECTIVE U p X qp X qp X q MODULES
Projective modules of U p X q are exhausted by the following list ( with a , b “ ˘ in all cases ) . ‚ Steinberg modules Q a , b p , r with ď r ď p ´ , which are the simple p-dimensionalmodules described . ‚ Typical modules Q a , b s , r with ď s ď p ´ , ď r ď p ´ and r ‰ s, describedin ; each has four simple subquotients and dimension p. ‚ Atypical modules: – The p-dimensional Q a , b p , modules described in , with simple sub-quotients each. – The p-dimensional modules Q a , b p ´ , p ´ described in , with simplesubquotients. – The p-dimensional Q a , b s , modules with ď s ď p ´ described in ,with simple subquotients. – The p-dimensional modules Q a , b s , s with ď s ď p ´ described in ,with simple subquotients. – The p-dimensional modules Q a , b , described in , with simple sub-quotients. We construct each projective module Q explicitly by choosing a basis and defining the U p X q action in that basis. The set of basis vectors is the union of the bases of all simple subquotients. This means that for each projective Q , we choose and fix a linear spaceisomorphism(6.1) m : Q Ñ ¯ Q “ à i Z i , with the direct sum of all simple subquotients of Q . In accordance with the direct sumdecomposition, we also write m “ ř i m i . p X q action and graphs. We explain how we construct the action of U p X q generators on projective modules. For a generator A , its action on v P Q is given by(6.2) r A p v q “ r p q A p v q ` r p q A p v q ` r p q A p v q , with the following ingredients.(0) r p q A “ m ´ ˝ ¯ r A ˝ m , where ¯ r is the direct sum of U p X q actions on simple mod-ules.(1) To define r p q A , we recall that in and , we chose a collection of maps x A be-tween simple U p X q modules, which we now write using a more detailed notation,as x i , jA : Z i Ñ Z j (for some simple modules Z i and Z j ). Then(6.3) r p q A “ m ´ ˝ ÿ i , k c k , i x k , i A ˝ m k , where c k , i P C are some coefficients, depending on a pair of simple subquotientsin the projective module in question. For a projective module Q , the r p q A maps can be represented as a directed graphwith the set of vertices given by the Z i in (6.1) and the set of edges correspondingto the nonzero products c k , i x k , i A (nonzero for at least one A ). The edge is directedfrom Z k to Z i . We construct such graphs in what follows, decorating the edgeswith c k , i .(2) Finally,(6.4) r p q A “ m ´ ˝ ÿ i , k h k , i A ˝ m k , where h k , i A : Z k Ñ Z i are linear maps (also linear in A ) and Z i is a descendant butnot a child of Z k in the graph defined by the c m , n x m , n A . These maps are neededfor (6.2) to be a U p X q action.Proving the existence of the projective cover Q of a simple module Z ˚ amounts to find-ing the coefficients c m , n and the maps h k , i A such that the graph has a root vertex givenby Z ˚ and Eq. (6.2) is a U p X q action (and the resultant module is maximal indecompos-able). We solve for such c m , n and h k , i A in what follows. The solution is not unique due tothe freedom of taking linear combinations of (the respective basis vectors in) isomorphic Not on pairs of isomorphism classes of simple U p X q modules. EPRESENTATIONS OF U q s ℓ p | q AT EVEN ROOTS OF UNITY 25 subquotients, but the existence of a solution proves the existence of the correspondingmodule; that it is maximal indecomposable is then verified by inspection in each particu-lar case.
We proceed with projective modules start-ing with the simple ones in and ending with those having 24 simple subquotientsin . We need the obvious notion of level in a directed graph with a root vertex ˚ . Thatvertex is assigned level 1, every child of ˚ has level 2, and so on (in the graphs we aredealing with, this defines the level of each vertex uniquely). s “ ps “ ps “ p , ď r ď p ´ ď r ď p ´ ď r ď p ´ . These are the simple (“Stein-berg”) modules discussed in and after Eq. (4.6). Each of them is also projective. ď s ď p ´ ď s ď p ´ ď s ď p ´ , r ‰ , sr ‰ , sr ‰ , s . The“bulk” of the diagram in Fig. 4.1 yields 4 p p ´ qp p ´ q modules Q a , b s , r , each of dimension8 p , labeled by 1 ď s ď p ´ , ď r ď p ´ , r ‰ s , a , b “ ˘ . Their graphs (see ) are very simple:(6.5) ` a b s r ˘ X ´ ´ a b p ´ s p ` r ´ s ¯ ´ ´ a ´ b p ´ s p ` r ´ s ¯ ` a b s r ˘ Here, ` a b s r ˘ “ Z a , b s , r are the simple subquotients, and the arrows are the (basis elementsin) the corresponding Ext and the units on the arrows are the coefficients c k , i standing infront of these elements (see (6.3)).We consider the arrow with a checkmark as an example. From the subquotients that itconnects, we see that the relevant extension is f p ´ s , p ` r ´ s in (5.1); the corresponding x A piece of the action of U p X q generators is therefore given by (5.3) times the coefficient 1.For the opposite arrow in the diagram, the same maps (5.3) should be used with a Ñ ´ a , s Ñ p ´ s , and r Ñ p ` r ´ s , and similarly for the other arrows. We omit the uninformative Z for conformity with similar, but much more complicated graphs in whatfollows, where extra symbols would complicate the picture even more. It remains to specify the h A piece of the action such that Eqs. (6.2)–(6.4) define a U p X q action. The choice of the h A maps in a given basis is not unique (the different solutionsbeing mapped into one another by basis changes), and we choose h A to be nonzero onlyfor A “ B , E , and only acting on the top subquotient. Its basis vectors (4.5), now denotedas ˇˇ a , s , b , r D Ð j △ , ˇˇ a , s , b , r D Ò m △ , ˇˇ a , s , b , r D Ó n △ , ˇˇ a , s , b , r D Ñ j △ , are than mapped into the respective vectors in the bottom subquotient, denoted as ˇˇ a , s , b , r D Ð j ▽ , ˇˇ a , s , b , r D Ò m ▽ , ˇˇ a , s , b , r D Ó n ▽ , ˇˇ a , s , b , r D Ñ j ▽ . The nonzero maps h B and h E are given by h B ˇˇ a , s , b , r D Ð j △ “ a G s , r , j ˇˇ a , s , b , r D Ó j ´ ▽ ´ ab r r ´ s s G s , r , j ˇˇ a , s , b , r D Ò j ▽ , h B ˇˇ a , s , b , r D Ò m △ “ a r s s G s , r , m ˇˇ a , s , b , r D Ñ m ´ ▽ , h B ˇˇ a , s , b , r D Ó n △ “ ab r s sr r ´ s s G s , r , n ` ˇˇ a , s , b , r D Ñ n ▽ , h E ˇˇ a , s , b , r D Ò m △ “ r s ` s ˇˇ a , s , b , r D Ò m ´ ▽ , h E ˇˇ a , s , b , r D Ð j △ “ r s s ˇˇ a , s , b , r D Ð j ´ ▽ , h E ˇˇ a , s , b , r D Ó n △ “ r s ´ s ˇˇ a , s , b , r D Ó n ´ ▽ , h E ˇˇ a , s , b , r D Ñ j △ “ r s s ˇˇ a , s , b , r D Ñ j ´ ▽ , where G s , r , m “ ´ r m ´ sr s s ` ´ ´p q r ` q ´ r q r s ´ sr s s ` r r s r s ´ s ´ r s s ¯ r m sr r ´ s s . Direct calculation shows that with the coefficients 1 in (6.5), all relations for the U p X q generators are satisfied. a : Projective cover of Z a , ba , ba , b p , p , p , . The projective cover Q a , b p , of Z a , b p , (denoted by a in Fig. 4.1) has 12 simple subquotients and is 12 p -dimensional. The correspondinggraph is shown in Fig. 6.1. Each simple subquotient Z a , b s , r is identified by its parameters ` a b s r ˘ and is in addition labeled by ℓ n , where ℓ is the level and n consecutively labelssubquotients within each level.As previously, each link Z Ñ Z corresponds to the basis element in Ext p Z , Z q (which, we recall, is one-dimensional) times the coefficient placed on the link. For ex-ample, consider the level-2-to-level-3 link ´ a ´ b p ´ ¯ ´ a ÝÝÑ ´ ´ a ´ b ¯ , which of coursestands for Z a , ´ b p ´ , ´ a ÝÝÑ Z ´ a , ´ b , . It corresponds to the element ¯ e in (the last in thelist) in accordance with the somewhat truncated notation used there, which ignores theupper indices. The coefficient ´ a on the link means that E and B map from Z a , ´ b p ´ , to Z ´ a , ´ b , as x E : ˇˇ a , p ´ , ´ b , D Ð ÞÑ ´ a ¨ ˇˇ ´ a , , ´ b , D Ð , EPRESENTATIONS OF U q s ℓ p | q AT EVEN ROOTS OF UNITY 27 ´ a b p ¯ ´ ´ a b ¯ ´ a ´ b p ´ ¯ ´ ´ a ´ b ¯ ´ a b p ¯ ´ ´ a b ¯ ´ a b p ´ p ´ ¯ ´ ´ a ´ b ¯ ´ ´ a b ¯ ´ a ´ b p ´ ¯ ´ ´ a ´ b ¯ ´ a b p ¯ ´ b abab ´ b ´ b ´ ab ´ a ´ a ´ a ´ b ´ ab ´ ´ ab ´ ´
11 1 F IGURE
Graph of the projective module Q a , b p , x E : ˇˇ a , p ´ , ´ b , D Ñ ÞÑ ´ a ¨ r´ s ˇˇ ´ a , , ´ b , D Ñ , x B : ˇˇ a , p ´ , ´ b , D Ð ÞÑ ´ a ¨ p´ a q ˇˇ ´ a , , ´ b , D Ñ , and these are the only maps by the U p X q generators between the simple subquotients Z a , ´ b p ´ , and Z ´ a , ´ b , of Q a , b p , .The same Z a , ´ b p ´ , is also linked in the graph to Z ´ a , b , , with a link decorated by ´ a ; thisis ¯ f from the list in , and the nonzero maps by U p X q generators are therefore given by x F : ˇˇ a , p ´ , ´ b , D Ð p ´ ÞÑ ´ a ¨ ˇˇ ´ a , , b , D Ð , x F : ˇˇ a , p ´ , ´ b , D Ñ p ´ ÞÑ ´ a ¨ p´q ˇˇ ´ a , , b , D Ñ . In addition to the graph, we have to specify the h A maps such that Eqs. (6.2)–(6.4)define a U p X q action. This part of the action of U p X q generators can be chosen as follows: h E ˇˇ a , p , b , D Ñ n , “ ´ a ˇˇ a , p , b , D Ñ n ´ , ´ b ˇˇ a , p ´ , b , p ´ D Ð n ´ , , h B ˇˇ a , p , b , D Ð n , “ ´ b r n s ˇˇ a , p , b , D Ñ n ´ , ` r n ` s ˇˇ a , p , b , D Ñ n ´ , ` ab r n ` s ˇˇ a , p ´ , b , p ´ D Ð n ´ , , h B ˇˇ a , p , b , D Ñ n , “ ab r n ` s ˇˇ a , p ´ , b , p ´ D Ñ n , , h C ˇˇ a , p , b , D Ð n , “ ´ ad n , p ´ ˇˇ ´ a , , b , D Ð , , h C ˇˇ a , p , b , D Ñ n , “ ad n , p ´ ˇˇ ´ a , , b , D Ñ , , h E ˇˇ a , p ´ , ´ b , D Ð n , “ ´ a ˇˇ a , p ´ , ´ b , D Ð n ´ , , h E ˇˇ a , p ´ , ´ b , D Ñ n , “ ´ a r s ˇˇ a , p ´ , ´ b , D Ñ n ´ , h B ˇˇ a , p ´ , ´ b , D Ð n , “ r n ` s ˇˇ a , p ´ , ´ b , D Ñ n ´ , ´ r n s ˇˇ a , p , b , D Ð n , , h B ˇˇ a , p ´ , ´ b , D Ñ n , “ r n ` s ˇˇ a , p , b , D Ñ n , , h E ˇˇ a , p , b , D Ñ n , “ ´ ab ˇˇ a , p , b , D Ñ n ´ , , h B ˇˇ a , p , b , D Ð n , “ b r n ` s ˇˇ a , p , b , D Ñ n ´ , , h B ˇˇ ´ a , , b , D Ð n , “ ad , n ˇˇ a , p , b , D Ð p ´ , , h B ˇˇ ´ a , , b , D Ñ n , “ ad , n ˇˇ a , p , b , D Ñ p ´ , , h E ˇˇ a , p ´ , b , p ´ D Ð n , “ ˇˇ a , p , b , D Ñ n ´ , . With these h A and with the c i , k read off from the graph, Eqs. (6.2)–(6.4) define an U p X q action, as can be verified directly. Inspection shows that the resulting module is indecom-posable and maximal. m : Projective cover of Z a , ba , ba , b p ´ , p ´ p ´ , p ´ p ´ , p ´ . The projective cover Q a , b p ´ , p ´ of Z a , b p ´ , p ´ (denoted by m in Fig. 4.1) also has 12 subquotients and is 12 p -dimensional. Its graph isshown in Fig. 6.2, with the same notation and conventions as for the preceding projectivemodule.The h piece of the action by the U p X q generators needed in (6.2) is as follows. On thebasis vectors of the top subquotient, we have h F ˇˇ a , p ´ , b , p ´ D Ð n , “ ´r s d n , p ´ ˇˇ ´ a , , b , D Ð , , h E ˇˇ a , p ´ , b , p ´ D Ð n , “ ´ a ˇˇ a , p , b , D Ñ n ´ , ´ r s ˇˇ a , p ´ , b , p ´ D Ð n ´ , ` ˇˇ a , p ´ , b , p ´ D Ð n ´ , ´ r s d , n ˇˇ ´ a , , ´ b , D Ð , , h B ˇˇ a , p ´ , b , p ´ D Ð n , “ ´ ab r n ` s ˇˇ a , p ´ , ´ b , p ´ D Ð n ´ , ` a r n s ˇˇ a , p ´ , b , p ´ D Ñ n , ´ abd , n ˇˇ ´ a , , b , D Ð , , h B ˇˇ a , p ´ , b , p ´ D Ñ n , “ ab r n ` s ˇˇ a , p ´ , ´ b , p ´ D Ñ n ´ , ´ abd , n ˇˇ ´ a , , b , D Ñ , , h C ˇˇ a , p ´ , b , p ´ D Ñ n , “ ab ˇˇ a , p ´ , b , p ´ D Ð n , ´ b r s d n , p ´ ˇˇ ´ a , , b , D Ð , . The h A maps from level-2 vectors are h E ˇˇ a , p ´ , ´ b , p ´ D Ð n , “ r s ˇˇ a , p ´ , ´ b , p ´ D Ð n ´ , , h E ˇˇ a , p ´ , ´ b , p ´ D Ñ n , “ ˇˇ a , p ´ , ´ b , p ´ D Ñ n ´ , , h B ˇˇ a , p ´ , ´ b , p ´ D Ð n , “ a r n ` s ˇˇ a , p ´ , ´ b , p ´ D Ñ n , , EPRESENTATIONS OF U q s ℓ p | q AT EVEN ROOTS OF UNITY 29 ´ a b p ´ p ´ ¯ ´ a ´ b p ´ p ´ ¯ ´ ´ a b ¯ ´ ´ a ´ b ¯ ´ ´ a ´ b ¯ ´ a b p ´ p ´ ¯ ´ ´ a b ¯ ´ a b p ¯ ´ a ´ b p ´ p ´ ¯ ´ ´ a b ¯ ´ ´ a ´ b ¯ ´ a b p ´ p ´ ¯ b ´ a ´ a b a ´ a ab b a a
11 1 1 F IGURE
Graph of the projective module Q a , b p ´ , p ´ . h B ˇˇ ´ a , , b , D Ð n , “ a r s d , n ˇˇ a , p ´ , b , p ´ D Ñ p ´ , , and those from level 3, h E ˇˇ a , p , b , D Ñ n , “ ˇˇ a , p ´ , b , p ´ D Ð n ´ , , h B ˇˇ a , p , b , D Ð n , “ ´ a r n ` s ˇˇ a , p ´ , b , p ´ D Ð n ´ , , h B ˇˇ a , p , b , D Ñ n , “ ´ a r n ` s ˇˇ a , p ´ , b , p ´ D Ñ n , , h E ˇˇ a , p ´ , b , p ´ D Ð n , “ a ˇˇ a , p ´ , b , p ´ D Ð n ´ , , h B ˇˇ a , p ´ , b , p ´ D Ð n , “ r n s ˇˇ a , p ´ , b , p ´ D Ñ n , , h C ˇˇ ´ a , , b , D Ð n , “ d , n ˇˇ a , p ´ , b , p ´ D Ð , , h C ˇˇ ´ a , , b , D Ñ n , “ ´ d , n ˇˇ a , p ´ , b , p ´ D Ñ , . Direct calculation shows that the above h A and the c i , k read off from the graph ensurethat Eqs. (6.2)–(6.4) define a projective U p X q module. ‘ : Projective cover of Z a , ba , ba , b s , s , s , for ď s ď p ´ ď s ď p ´ ď s ď p ´ . The projective cover Q a , b s , of Z a , b s , (any of the ‘ in Fig. 4.1) with 2 ď s ď p ´ p -dimensional. Its graph is shown in Fig. 6.3. A feature not encountered in the pre-vious graphs is the occurrence of links from a given subquotient leading to isomorphic A . M . S E M I KHA T OVAND I . YU . T I P UN I N ´ a b s ¯ ´ ´ a ´ b p ´ s p ´ s ¯ ´ ´ a b p ´ s p ´ s ¯ ´ a ´ b s ´ ¯ ´ a ´ b s ` ¯ ´ a b s ¯ ´ ´ a b p ´ s ` p ´ s ` ¯ ´ ´ a b p ´ s ´ p ´ s ´ ¯ ´ ´ a ´ b p ´ s ` p ´ s ` ¯ ´ ´ a ´ b p ´ s ´ p ´ s ´ ¯ ´ a b s ¯ ´ a ´ b s ´ ¯ ´ a ´ b s ` ¯ ´ ´ a ´ b p ´ s p ´ s ¯ ´ ´ a b p ´ s p ´ s ¯ ´ a b s ¯ r s ´ sr s s ´r s s r s s r s ´ sr s s ´ r s ´ s ab ´r s s ´r s ´ s´r s ´ s r s ´ sr s s ´r s s ´r s ´ s r s ´ sr s s r s s´r s s ´ r s ´ s ab ´r s ´ s r s ´ sr s ´ s r s ´ s F IGURE
Subquotients of the projective module Q a , b s , for 2 ď s ď p ´ EPRESENTATIONS OF U q s ℓ p | q AT EVEN ROOTS OF UNITY 31 subquotients on the next level. Two (or more) such links ` a b s r ˘ ℓ n c c ´ a b s r ¯ p ℓ ` q n ´ a b s r ¯ p ℓ ` q n mean that the U p X q generators map via c x A ` c x A , where x iA : ` a b s r ˘ ℓ n Ñ ´ a b s r ¯ ℓ ` ni are the maps in associated with each link. In other words, the corresponding basisvectors in the two isomorphic subquotients occur in linear combinations with c and c as the coefficients. The h A maps from the top-level subquotient are h E ˇˇ a , s , b , D Ð n , “ ´r s ´ sr s s ˇˇ a , s , b , D Ð n ´ , , h E ˇˇ a , s , b , D Ñ n , “ ´r s ´ s r s s ˇˇ a , s , b , D Ñ n ´ , , h B ˇˇ a , s , b , D Ð n , “ b r n sr s s ˇˇ a , s , b , D Ñ n ´ , ` a r s ´ sr s s r s ´ n s ˇˇ a , s , b , D Ñ n ´ , ` ab r s ´ sr s s d , n ˇˇ ´ a , p ´ s , b , p ´ s D Ñ p ´ s , ´ a r n sr s ´ s r s s ˇˇ a , s ` , ´ b , D Ð n , ` ab r s s pr s ´ sr s ´ n s ´ r n sr s ´ sq ˇˇ a , s , b , D Ñ n ´ , , h B ˇˇ a , s , b , D Ñ n , “ a r n ` sr s ´ s r s s ˇˇ a , s ` , ´ b , D Ñ n , , h C ˇˇ a , s , b , D Ð n , “ r ´ s sr s s d n , s ´ ˇˇ ´ a , p ´ s ` , b , p ´ s ` D Ð , ´ a r s s ˇˇ a , s ´ , ´ b , D Ð n , , h C ˇˇ a , s , b , D Ñ n , “ r s ´ sr s s d n , s ´ ˇˇ ´ a , p ´ s ` , b , p ´ s ` D Ñ , ´ a r s s ˇˇ a , s ´ , ´ b , D Ñ n , ` a r s s ˇˇ a , s , b , D Ð n ` , and those from level 2, h B ˇˇ ´ a , p ´ s , ´ b , p ´ s D Ð n , “ ´ a r s ´ s d , n ˇˇ a , s ` , ´ b , D Ð s , ´ b r n ` s s ˇˇ ´ a , p ´ s , ´ b , p ´ s D Ñ n , , h B ˇˇ ´ a , p ´ s , ´ b , p ´ s D Ñ n , “ ´ a r s ´ s d , n ˇˇ a , s ` , ´ b , D Ñ s ´ , , h B ˇˇ ´ a , p ´ s , b , p ´ s D Ð n , “ b r s ´ sr n ` s s ˇˇ ´ a , p ´ s , b , p ´ s D Ñ n , , h E ˇˇ a , s ´ , ´ b , D Ð n , “ r s ´ s ˇˇ a , s ´ , ´ b , D Ð n ´ , , The isomorphic targets could of course be identified differently, so as to “split” any given triple, butthis cannot be done for all such triples simultaneously. We did not attempt to “optimize” the possible linearcombinations, in particular because there seems to be no well-defined optimum. h E ˇˇ a , s ´ , ´ b , D Ñ n , “ r s ´ sr s ´ s ˇˇ a , s ´ , ´ b , D Ñ n ´ , , h B ˇˇ a , s ´ , ´ b , D Ð n , “ a r n ` ´ s s ˇˇ a , s ´ , ´ b , D Ñ n ´ , ´ a r n s ˇˇ a , s , b , D Ð n , , h B ˇˇ a , s ´ , ´ b , D Ñ n , “ a r n ` s ˇˇ a , s , b , D Ñ n , , h E ˇˇ a , s ` , ´ b , D Ð n , “ r s ´ sr s sr s ` s ˇˇ a , s ` , ´ b , D Ð n ´ , , h E ˇˇ a , s ` , ´ b , D Ñ n , “ r s ´ sr s s ˇˇ a , s ` , ´ b , D Ñ n ´ , , h B ˇˇ a , s ` , ´ b , D Ð n , “ ´ a r s ´ sr s ´ n s ˇˇ a , s ` , ´ b , D Ñ n ´ , , h C ˇˇ a , s ` , ´ b , D Ð n , “ r s ´ sr s s d n , s ˇˇ ´ a , p ´ s , ´ b , p ´ s D Ð , , h C ˇˇ a , s ` , ´ b , D Ñ n , “ ´r s ´ sr s s d n , s ´ ˇˇ ´ a , p ´ s , ´ b , p ´ s D Ñ , and from level 3, h B ˇˇ a , s , b , D Ð n , “ ´ b r n s ˇˇ a , s , b , D Ñ n ´ , , h B ˇˇ ´ a , p ´ s ` , b , p ´ s ` D Ð n , “ ad , n ˇˇ a , s , b , D Ð s ´ , , h B ˇˇ ´ a , p ´ s ` , b , p ´ s ` D Ñ n , “ ad , n ˇˇ a , s , b , D Ñ s ´ , , h E ˇˇ a , s , b , D Ð n , “ r s ´ sr s s ˇˇ a , s , b , D Ð n ´ , , h E ˇˇ a , s , b , D Ñ n , “ r s ´ s ˇˇ a , s , b , D Ñ n ´ , , h B ˇˇ a , s , b , D Ð n , “ ´ a r s ´ ´ n s ˇˇ a , s , b , D Ñ n ´ , . b : Projective cover of Z a , ba , ba , b s , ss , ss , s for ď s ď p ´ ď s ď p ´ ď s ď p ´ . For each 1 ď s ď p ´
2, the pro-jective module Q a , b s , s (any of the b in Fig. 4.1) is 16 p -dimensional and has 16 subquo-tients. Its graph is shown in Fig. 6.4. with all the previous conventions in force.The h A piece of the action of U p X q generators on Q a , b s , s is as follows. On the basisvectors of the top-level subquotient, we have h E ˇˇ a , s , b , s D Ð n , “ ´r s s r s ` s ˇˇ a , s , b , s D Ð n ´ , ` b r s s r s ` s ˇˇ a , s , b , s D Ð n ´ , , h E ˇˇ a , s , b , s D Ñ n , “ ´r s sr s ` s ˇˇ a , s , b , s D Ñ n ´ , ` b r s sr s ` s ˇˇ a , s , b , s D Ñ n ´ , , h B ˇˇ a , s , b , s D Ð n , “ ´ a r n sr s ` s ˇˇ a , s , b , s D Ñ n , ´ a r s sr s ` s d , n ˇˇ ´ a , p ´ s ` , b , D Ð p ´ s , ´ b r s sr s ´ n s ˇˇ a , s , b , s D Ñ n , ´ a r n ` sr s sr s ` s ˇˇ a , s ´ , ´ b , s ´ D Ð n ´ , ` ab r s ` spr n s ´ r s sr s ´ ´ n sq ˇˇ a , s , b , s D Ñ n , , h B ˇˇ a , s , b , s D Ñ n , “ a r n ` sr s sr s ` s ˇˇ a , s ´ , ´ b , s ´ D Ñ n ´ , ´ a r s sr s ` s d , n ˇˇ ´ a , p ´ s ` , b , D Ñ p ´ s ´ , . On basis vectors of the level-2 modules, the h maps are h E ˇˇ ´ a , p ´ s , b , D Ð n , “ b r s s d , n ˇˇ a , s , b , s D Ð s ´ , , h E ˇˇ ´ a , p ´ s , b , D Ñ n , “ b r s ` s d , n ˇˇ a , s , b , s D Ñ s , , E P R E S E N T A T I ON S O F U q s ℓ p | q A TE V E N R OO T S O F UN I T Y ` a b s s ˘ ´ ´ a b p ´ s ¯ ´ ´ a ´ b p ´ s ¯ ´ a ´ b s ´ s ´ ¯ ´ a ´ b s ` s ` ¯ ` a b s s ˘ ´ ´ a ´ b p ´ s ` ¯ ´ ´ a ´ b p ´ s ´ ¯ ´ ´ a b p ´ s ` ¯ ´ ´ a b p ´ s ´ ¯ ` a b s s ˘ ´ a ´ b s ´ s ´ ¯ ´ a ´ b s ` s ` ¯ ´ ´ a b p ´ s ¯ ´ ´ a ´ b p ´ s ¯ ` a b s s ˘ r s sr s ` s r s ` s´r s sr s ` s r s s ´ ´ ab r s s r s ` s´r s s r s ` sr s ` s r s sr s ` s r s s r s sr s ` sr s ` s´r s s ´ ab ´r s ` s r s ` s ´ r s ` s
111 1 1 F IGURE
Graph of the projective module Q a , b s , s for 1 ď s ď p ´ h B ˇˇ ´ a , p ´ s , b , D Ð n , “ b r n s ˇˇ ´ a , p ´ s , b , D Ñ n ´ , ` abd , n ˇˇ a , s , b , s D Ñ s , , h E ˇˇ a , s ´ , ´ b , s ´ D Ð n , “ r s ´ sr s sr s ` s ˇˇ a , s ´ , ´ b , s ´ D Ð n ´ , , h E ˇˇ a , s ´ , ´ b , s ´ D Ñ n , “ r s s r s ` s ˇˇ a , s ´ , ´ b , s ´ D Ñ n ´ , , h B ˇˇ a , s ´ , ´ b , s ´ D Ð n , “ a r n ` sr s ` s ˇˇ a , s ´ , ´ b , s ´ D Ñ n , , h C ˇˇ a , s ´ , ´ b , s ´ D Ñ n , “ a ˇˇ a , s , b , s D Ñ n ` , , h F ˇˇ ´ a , p ´ s , ´ b , D Ð n , “ ´ bd n , p ´ s ´ ˇˇ a , s , b , s D Ð , , h F ˇˇ ´ a , p ´ s , ´ b , D Ñ n , “ bd n , p ´ s ´ ˇˇ a , s , b , s D Ñ , , h B ˇˇ ´ a , p ´ s , ´ b , D Ð n , “ ´ b r n s ˇˇ ´ a , p ´ s , ´ b , D Ñ n ´ , , h C ˇˇ ´ a , p ´ s , ´ b , D Ð n , “ ´r s s d n , p ´ s ´ ˇˇ a , s ` , ´ b , s ` D Ð , , h C ˇˇ ´ a , p ´ s , ´ b , D Ñ n , “ r s s d n , p ´ s ´ ˇˇ a , s ` , ´ b , s ` D Ñ , , h E ˇˇ a , s ` , ´ b , s ` D Ð n , “ r s ` s ˇˇ a , s ` , ´ b , s ` D Ð n ´ , , h E ˇˇ a , s ` , ´ b , s ` D Ñ n , “ r s ` sr s ` s ˇˇ a , s ` , ´ b , s ` D Ñ n ´ , , h B ˇˇ a , s ` , ´ b , s ` D Ð n , “ a r n s ˇˇ a , s ` , ´ b , s ` D Ñ n , ´ a r n ` sr s ` s ˇˇ a , s , b , s D Ð n ´ , , ´ a r s ` s d , n ˇˇ ´ a , p ´ s , ´ b , D Ð p ´ s ´ , , h B ˇˇ a , s ` , ´ b , s ` D Ñ n , “ a r n ` sr s ` s ˇˇ a , s , b , s D Ñ n ´ , ´ a r s ` s d , n ˇˇ ´ a , p ´ s , ´ b , D Ñ p ´ s ´ , and on level-3 vectors, h C ˇˇ a , s , b , s D Ñ n , “ ˇˇ a , s , b , s D Ð n , , h E ˇˇ a , s , b , s D Ð n , “ r s sr s ` s ˇˇ a , s , b , s D Ð n ´ , , h E ˇˇ a , s , b , s D Ñ n , “ r s ` s ˇˇ a , s , b , s D Ñ n ´ , , h B ˇˇ a , s , b , s D Ð n , “ a r n ` s ˇˇ a , s , b , s D Ñ n , , h C ˇˇ ´ a , p ´ s ` , b , D Ð n , “ d n , p ´ s ˇˇ a , s , b , s D Ð , , h C ˇˇ ´ a , p ´ s ` , b , D Ñ n , “ ´ d n , p ´ s ´ ˇˇ a , s , b , s D Ñ , . f : Projective cover of Z a , ba , ba , b , , , . We finally describe projective covers of the one-dimensional representations ( f in Fig. 4.1). The projective module Q a , b , has dimen-sion 24 p and is built from 24 simple subquotients. Its rather involved graph is shownin Fig. 6.5. Maps into linear combinations of isomorphic subquotients, which alreadyoccurred in Figs. 6.3 and 6.4, here involve up to four modules.The h piece of the action of U p X q generators is as follows. The maps from the top are h F ˇˇ a , , b , D Ð , “ a r s ˇˇ ´ a , p ´ , ´ b , p ´ D Ð , E P R E S E N T A T I ON S O F U q s ℓ p | q A TE V E N R OO T S O F UN I T Y ´ a b ¯ ´ a b ¯ ´ ´ a ´ b p ¯ ´ ´ a b p ¯ ´ ´ a ´ b p ´ p ´ ¯ ´ ´ a b p ´ p ´ ¯ ´ a ´ b ¯ ´ ´ a b p ´ ¯ ´ ´ a ´ b p ´ ¯ ´ a b ¯ ´ a ´ b ¯ ´ a b ¯ ´ a b ¯ ´ a ´ b ¯ ´ a b ¯ ´ ´ a b p ´ p ´ ¯ ´ ´ a ´ b p ´ p ´ ¯ ´ ´ a ´ b p ¯ ´ a b ¯ ´ ´ a b p ¯ ´ ´ a ´ b p ´ p ´ ¯ ´ ´ a b p ´ p ´ ¯ ´ a ´ b ¯ ´ a b ¯ ab ´ ab ´ ´ ab ab ab ab ´ ´ ´ ´
11 1 ´
11 1 ´ ´
11 1 12 ´ ab ab ab ´ ab ´ ´ ´
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11 1 ´ ´
11 11 11 1 F IGURE
Graph of the projective module Q a , b , h E ˇˇ a , , b , D Ð , “ a r s ˇˇ ´ a , p ´ , b , p ´ D Ð p ´ , , and those from level 2 are h E ˇˇ a , , b , D Ñ n , “ r s ˇˇ a , , b , D Ñ n ´ , , h B ˇˇ a , , b , D Ð n , “ ad , n ˇˇ ´ a , p , b , D Ð p ´ , , h B ˇˇ a , , b , D Ñ n , “ ad , n ˇˇ ´ a , p , b , D Ñ p ´ , , h E ˇˇ a , , ´ b , D Ð n , “ r s ˇˇ a , , ´ b , D Ð n ´ , , h C ˇˇ a , , ´ b , D Ð n , “ d , n ˇˇ ´ a , p ´ , ´ b , p ´ D Ð , , h C ˇˇ a , , ´ b , D Ñ n , “ ´ d , n ˇˇ ´ a , p ´ , ´ b , p ´ D Ñ , , h E ˇˇ ´ a , p , ´ b , D Ñ n , “ ˇˇ ´ a , p , ´ b , D Ñ n ´ , ´ ˇˇ ´ a , p ´ , ´ b , p ´ D Ð n ´ , , h B ˇˇ ´ a , p , ´ b , D Ð n , “ a r n ` s ˇˇ ´ a , p , ´ b , D Ñ n ´ , ´ a r n ` s ˇˇ ´ a , p ´ , ´ b , p ´ D Ð n ´ , , h B ˇˇ ´ a , p , ´ b , D Ñ n , “ ´ a r n ` s ˇˇ ´ a , p ´ , ´ b , p ´ D Ñ n , , h E ˇˇ ´ a , p ´ , ´ b , p ´ D Ð n , “ ˇˇ ´ a , p ´ , ´ b , p ´ D Ð n ´ , ´ ˇˇ ´ a , p , ´ b , D Ñ n ´ , , h B ˇˇ ´ a , p ´ , ´ b , p ´ D Ð n , “ bd , n ˇˇ a , , ´ b , D Ð , ´ a r n s ˇˇ ´ a , p ´ , ´ b , p ´ D Ñ n , , h B ˇˇ ´ a , p ´ , ´ b , p ´ D Ñ n , “ bd , n ˇˇ a , , ´ b , D Ñ , , h E ˇˇ ´ a , p , b , D Ñ n , “ ˇˇ ´ a , p , b , D Ñ n ´ , ´ ˇˇ ´ a , p ´ , b , p ´ D Ð n ´ , , h B ˇˇ ´ a , p , b , D Ð n , “ a r n ` s ˇˇ ´ a , p , b , D Ñ n ´ , ´ a r n ` s ˇˇ ´ a , p ´ , b , p ´ D Ð n ´ , , h B ˇˇ ´ a , p , b , D Ñ n , “ ´ a r n ` s ˇˇ ´ a , p ´ , b , p ´ D Ñ n , , h C ˇˇ ´ a , p , b , D Ð n , “ ´ abd n , p ´ ˇˇ a , , b , D Ð , , h C ˇˇ ´ a , p , b , D Ñ n , “ abd n , p ´ ˇˇ a , , b , D Ñ , , h E ˇˇ ´ a , p ´ , b , p ´ D Ð n , “ ˇˇ ´ a , p , b , D Ñ n ´ , ´ ˇˇ ´ a , p ´ , b , p ´ D Ð n ´ , , h B ˇˇ ´ a , p ´ , b , p ´ D Ð n , “ a r n s ˇˇ ´ a , p ´ , b , p ´ D Ñ n , . Together with the c k , i specified on the links, this defines a U p X q module, which is thenseen to be maximal and indecomposable. We have constructed projective covers Q i Ñ Z i for all simple U p X q modules. That these are all projective modules can also be verified by calculating the sum ř i dim Q i ¨ dim Z i :(6.6) 4 ˆ p p ´ q p ¨ p loooooomoooooon ` p p ´ q p ´ ÿ s “ p ¨ s looooooooomooooooooon ` p p q ¨ p p ´ q looooooomooooooon ` p p q ¨ p p ´ q looooooomooooooon EPRESENTATIONS OF U q s ℓ p | q AT EVEN ROOTS OF UNITY 37 ` p ´ ÿ s “ p ¨ p s ´ q loooooooomoooooooon ` p ´ ÿ s “ p ¨ p s ` q loooooooomoooooooon ` p p q ¨ looomooon ˙ “ p “ dim U p X q (the overall 4 is for the values taken by a and b ). As an immediate corollary of the structure of projective U p X q -modules, we obtainthe minimal polynomials for the Casimir elements in , simply by finding the eigen-values on modules from each linkage class and taking the Jordan-cell size into account(which is 3 for the atypical linkage class, 2 for each of the typical linkage classes, and1 for each of the Steinberg classes; hence the multiplicities of the minimal-polynomialroots in ).As a less trivial (although generally straightforward) corollary, with find the U p X q cen-ter.
7. T HE U p X qp X qp X q CENTER
The algebra U p X q has a p p ´ p ` q -dimensional center. The center Z decomposes into the direct sum Z “ p ` p ´ à j “ C ¨ e j ‘ p ´ p ` à j “ C ¨ w j of linear subspaces generated by primitive central idempotents e j and central nilpotents w j . The block decomposition of the center as an associative algebra is Z “ Z at ‘ p p ´ qp p ´ q à j “ Z j t ‘ p p ´ q à j “ Z j st where Z at corresponds to atypical Z j t to typical and Z j st to Steinberg linkage class. This theorem is an application of the construction of projective U p X q modules. Wecalculate the U p X q center as the center of the basic algebra (an approach also taken for U q s ℓ p q in [46]). p X q . The basic algebra is the algebra of endomorphisms ofthe direct sum of projective modules taken with multiplicity 1 each. Basic algebra genera-tors can be chosen as primitive idempotents and nilpotents: (i) each primitive idempotent e Q is the projector on a single projective module Q , and (ii) each nilpotent w Q , n is a mor-phism Q Ñ Q defined uniquely by the condition that it sends the top subquotient of Q into an isomorphic subquotient on level two in a projective module Q , is an isomorphismof these subquotients, and acts by zero on all projective modules other than Q . Hence, n “ , . . . , N Q , where N Q is the number of level-2 subquotients in the linkage class that are isomorphic to the top subquotient of Q ; it is in fact equal to the number of level-2subquotients in Q .We describe this in more detail, invoking the structure of various projective modules:(1) Each of the 4 p p ´ q Steinberg modules (simple projective modules) contributesonly an idempotent to the basic algebra generators. These idempotents are central.(2) “Typical” projective modules contribute 4 p p ´ qp p ´ q idempotents (projectorson each of typical projective modules) and 8 p p ´ qp p ´ q nilpotents: for eachtypical projective module, these are two maps (distinguished by ˘ in front of b ) Q a , b s , r Ñ Q ´ a , ˘ b p ´ s , p ` r ´ s , r ‰ , s sending the top subquotient into a level-2 subquotient.(3) For the atypical projective modules, there are 4 p p ´ q idempotents. As regardsmaps to level two in projective modules, we see from the graphs in Figs. 6.1–6.5that the relevant number N “ N Q of such maps is as follows for the five species ofatypical projective modules:(7.1) Q a , b p , p a q : N “ , Q a , b p ´ , p ´ p m q : N “ , Q a , b s , p ‘ q : N “ , Q a , b s , s p b q : N “ , Q a , b , p f q : N “ . There are respectively 4, 4, 4 p p ´ q , 4 p p ´ q , and 4 projective modules of eachspecies, which gives the total of 4 ¨ ` ¨ ` p p ´ q ¨ ` p p ´ q ¨ ` ¨ “ p ´
16 “atypical” generators W j of the basic algebra. We now list the generators of the center. These areprimitive central idempotents (which are enumerated immediately) and central nilpotents(finding which requires more work). Their total number (see below) gives the di-mension of the center of U p X q . Each linkage class (see the list in )yields a primitive central idempotent, the projection onto that linkage class. In addition,the linkage classes except the Steinberg ones yield several central nilpotents each. Theitems describing them are listed below in the order of (rapidly) increasing complexity.(1) Each of the 4 p p ´ q Steinberg linkage classes produces a single central idempo-tent e a , b r ( r “ , . . . , p ´ p p ´ qp p ´ q “typical” linkage classes yields one central idempotent e a s , r and four central nilpotents w a , b s , r p g q , where 1 ď r ă s ď p ´ a , b , g “˘ (note that a , s , and r enumerate a linkage class, while b and g range overnilpotents inside a linkage class). These w a , b s , r p g q are the maps w a , b s , r p`q : Q a , b s , r Ñ Q a , b s , r , r ‰ , s EPRESENTATIONS OF U q s ℓ p | q AT EVEN ROOTS OF UNITY 39 and w a , b s , r p´q : Q ´ a , b p ´ s , p ` r ´ s Ñ Q ´ a , b p ´ s , p ` r ´ s , r ‰ , s , sending the top subquotient to the bottom subquotient in the same projective mod-ule.(3) The “atypical” linkage class yields one central idempotent and central nilpotentsthat are of two groups: some follow immediately (item a) and the derivation ofothers is somewhat more involved (item b).(a) There are 4 p p ´ q central nilpotents w a , b a , a “ , . . . p ´ a , b “ ˘ , onefor each atypical projective module ( Q a , b s , with 1 ď s ď p and Q a , b s , s with1 ď s ď p ´ w a , b a maps the top subquotient into the isomorphicbottom subquotient in the same projective module (and is zero on all otherprojective modules). It then follows that w a , b a w a , b a “ w b given by linear combinations ofnoncentral idempotents W B defined as follows. Each W B is a map from thetop subquotient in one of the atypical projective modules into an isomorphicsubquotient at level three in the same projective module. We can thereforewrite W B “ W W , m , where Q is a projective module and m labels its level-3subquotients that are isomorphic to its top subquotient.The graphs in Figs. 6.1–6.5 readily show that the number M of such sub-quotients and hence the number of W B yielded by each species of projectivemodules is as follows: Q a , b p , p a q : M “ , p Q a , b s , q ď s ď p ´ p ‘ q : M “ ¨ p p ´ q , Q a , b p ´ , p ´ p m q : M “ , p Q a , b s , s q ď s ď p ´ p b q : M “ ¨ p p ´ q , Q a , b , p f q : M “ ¨ p a , b q ).This gives a p p ´ q -dimensional space of noncentral “level-3” nilpotents.Taking their linear combinations(7.2) w “ ÿ x W , m W W , m “ p ´ ÿ B “ x B W B , we require the commutativity with the basic algebra generators. This w already commutes with all idempotents (which act either as identity or byzero) and, evidently, with the all “typical” nilpotents. It remains to requirethat it commute with the basic algebra generators described in item 3 in (page 38):(7.3) w W j ´ W j w “ , j “ , . . . , p ´ , Solving Eqs. (7.3) for the x W , m , we find a p p ` q -dimensional subspace ofcentral “level-3” nilpotents. There are exactly p ` linearly independent solutions w b of Eqs. (7.3) . We prove this in
B.4 by deriving the explicit form of the equations and solving them.
As a corollary, we calculate the dimension of the center of U p X q as follows, with“idem.” and “nilp.” referring to the idempotents and nilpotents described in items 1–3above: dim Z “ p p ´ q looomooon idem., item 1 ` p p ´ qp p ´ q looooooomooooooon idem., item 2 ` loomoon idem., item 3 ` p p ´ qp p ´ q looooooomooooooon nilp., item 2 ` p ´ loomoon nilp., item 3 “ p ´ p ` . It follows immediately that the algebra of the w b (solutions of (7.3)) is w b w c “ ÿ j , a , b f a , b , jb , c w a , b j where all nonzero constants f a , b , jb , c can be chosen equal to 1 by rescaling the w a , b j .
8. C
ONCLUSIONS
The full structure of projective modules is a powerful tool in studying an associativealgebra. Various consequences of the construction in this paper are to be worked out withregard to the properties relevant for the LCFT counterpart of our U q s ℓ p | q . The resultscan have a bearing on various facets of LCFT models, in the range from “spin chains,”potentially allowing physical applications (see [47, 48, 49] and the references therein), tono less exciting “categorial studies” (see [50, 51, 52] and the references therein). A spinchain that suggests itself in relation to our U q s ℓ p | q is the one composed of alternating“fundamental” p s “ , r “ q and “antifundamental” p s “ , r “ q H that defines the corresponding category of Yetter–Drinfeld modules [33]. As we see, H comes with its universal R -matrix, which also is to play a role in CFT. Of primaryinterest is the modular group action on (a subalgebra in) the U q s ℓ p | q center, which isalso linked to the study of the corresponding LCFT torus amplitudes and their modularand other properties. Tensor products of U q s ℓ p | q modules conjecturally correspond tofusion on the LCFT side.We thank B. Feigin, A. Gainutdinov, and A. Kiselev for the useful discussions. Thispaper was supported in part by the RFBR grant 13-01-00386. EPRESENTATIONS OF U q s ℓ p | q AT EVEN ROOTS OF UNITY 41 A PPENDIX
A.A.1. Simple U q s ℓ p q modules [27] . The U q s ℓ p q Hopf algebra is defined in Eqs. (2.1).Its simple modules can be labeled as X a s , where a “ ˘ and s “ , . . . , p (with dim X a s “ s ).A basis in X a s is denoted by | a , s y n , n “ , . . . s ´
1, with | a , s y being a highest-weightvector: E ˇˇ a , s D “ , K ˇˇ a , s D “ a q s ´ ˇˇ a , s D . The U q s ℓ p q action on all the | a , s y n , in a “naturally isomorphic” notation, is given in [27]. A.2. Projective U ˚ q s ℓ p q modules. The projective U ˚ q s ℓ p q -module P a , b s , r that covers X a , b s , r has a basis | a , s , b , r y b n , | a , s , b , r y a n with n “ , . . . , s ´ | a , s , b , r y x m , | a , s , b , r y y m with m “ , . . . , p ´ s ´
1. The action of U q s ℓ p q in this basis is given in [27], and the action of k is k ˇˇ a , s , b , r D a n “ b q ´ r ` n ˇˇ a , s , b , r D a n , k ˇˇ a , s , b , r D b n “ b q ´ r ` n ˇˇ a , s , b , r D b n , k ˇˇ a , s , b , r D x n “ b q s ´ r ` n ´ p ˇˇ a , s , b , r D x n , k ˇˇ a , s , b , r D y n “ b q s ´ r ` n ˇˇ a , s , b , r D y n . A.3. The rest of the U p X q action on simple modules. For completeness, we give theformulas that fully define the U p X q action on simple modules in (the action offermionic generators on basis vectors was already described there).For the atypical Z a , b s , modules in (4.2), we have K ˇˇ a , s , b , D Ð n “ a q s ´ n ´ ˇˇ a , s , b , D Ð n , k ˇˇ a , s , b , D Ð n “ b q n ˇˇ a , s , b , D Ð n , F ˇˇ a , s , b , D Ð n “ ˇˇ a , s , b , D Ð n ` , E ˇˇ a , s , b , D Ð n “ a r n sr s ´ n s ˇˇ a , s , b , D Ð n ´ , K ˇˇ a , s , b , D Ñ m “ a q s ´ m ´ ˇˇ a , s , b , D Ñ m , k ˇˇ a , s , b , D Ñ m “ ´ b q m ` ˇˇ a , s , b , D Ñ m , F ˇˇ a , s , b , D Ñ m “ ˇˇ a , s , b , D Ñ m ` , E ˇˇ a , s , b , D Ñ m “ a r m sr s ´ m ´ s ˇˇ a , s , b , D Ñ m ´ , where we set | a , s , b , y Ð n “ n ă n ą s ´
1, and | a , s , b , y Ñ m “ m ă m ą s ´
2. For s “
1, each module Z a , b , with a , b “ ˘ is 1-dimensional. The basisconsists of a single vector | a , , b , y Ð such that K ˇˇ a , , b , D Ð “ a ˇˇ a , , b , D Ð , k ˇˇ a , , b , D Ð “ b ˇˇ a , , b , D Ð , with the other generators acting trivially. For convenience in what follows (in the caseswhere Z a , b , occur along with higher-dimensional modules), we set | a , , b , y Ñ m “ m .For the atypical Z a , b s , s modules in (4.3), we have K ˇˇ a , s , b , s D Ð n “ a q s ´ n ´ ˇˇ a , s , b , s D Ð n , k ˇˇ a , s , b , s D Ð n “ b q ´ s ` n ˇˇ a , s , b , s D Ð n , F ˇˇ a , s , b , s D Ð n “ ˇˇ a , s , b , s D Ð n ` , E ˇˇ a , s , b , s D Ð n “ a r n sr s ´ n s ˇˇ a , s , b , s D Ð n ´ , K ˇˇ a , s , b , s D Ñ m “ a q s ´ m ˇˇ a , s , b , s D Ñ m , k ˇˇ a , s , b , s D Ñ m “ ´ b q ´ s ` m ˇˇ a , s , b , s D Ñ m , F ˇˇ a , s , b , s D Ñ m “ ˇˇ a , s , b , s D Ñ m ` , E ˇˇ a , s , b , s D Ñ m “ a r m sr s ´ m ` s ˇˇ a , s , b , s D Ñ m ´ , where we set | a , s , b , s y Ð n “ n ă n ą s ´
1, and | a , s , b , s y Ñ m “ m ă m ą s .For the typical modules Z a , b s , r in (4.4), K ˇˇ a , s , b , r D Ð j “ a q s ´ j ´ ˇˇ a , s , b , r D Ð j , k ˇˇ a , s , b , r D Ð j “ b q ´ r ` j ˇˇ a , s , b , r D Ð j , F ˇˇ a , s , b , r D Ð j “ ˇˇ a , s , b , r D Ð j ` , E ˇˇ a , s , b , r D Ð j “ a r j sr s ´ j s ˇˇ a , s , b , r D Ð j ´ , K ˇˇ a , s , b , r D Ò m “ a q s ´ m ˇˇ a , s , b , r D Ò m , k ˇˇ a , s , b , r D Ò m “ ´ b q ´ r ` m ˇˇ a , s , b , r D Ò m , F ˇˇ a , s , b , r D Ò m “ ˇˇ a , s , b , r D Ò m ` , E ˇˇ a , s , b , r D Ò m “ a r m sr s ´ m ` s ˇˇ a , s , b , r D Ò m ´ , K ˇˇ a , s , b , r D Ó n “ a q s ´ n ´ ˇˇ a , s , b , r D Ó n , k ˇˇ a , s , b , r D Ó n “ ´ b q ´ r ` n ` ˇˇ a , s , b , r D Ó n , F ˇˇ a , s , b , r D Ó n “ ˇˇ a , s , b , r D Ó n ` , E ˇˇ a , s , b , r D Ó n “ a r n sr s ´ n ´ s ˇˇ a , s , b , r D Ó n ´ , K ˇˇ a , s , b , r D Ñ j “ a q s ´ j ´ ˇˇ a , s , b , r D Ñ j , k ˇˇ a , s , b , r D Ñ j “ b q ´ r ` j ` ˇˇ a , s , b , r D Ñ j , F ˇˇ a , s , b , r D Ñ j “ ˇˇ a , s , b , r D Ñ j ` , E ˇˇ a , s , b , r D Ñ j “ a r j sr s ´ j s ˇˇ a , s , b , r D Ñ j ´ , where we as usual assume that the vectors outside the ranges specified in (4.5) are equalto zero. If s “
1, the decomposition degenerates to Z a , b , r “ X a , b , r ‘ X a , ´ b , r ‘ X a , b , r ´ , witha basis | a , , b , r y Ð , ` | a , , b , r y Ò m ˘ m “ , , | a , , b , r y Ñ , and with the U p X q action easilydeducible from the general case above.For the Steinberg modules Z a , b p , r in (4.6), K ˇˇ a , p , b , r D Ð n “ a q ´ n ´ ˇˇ a , p , b , r D Ð n , k ˇˇ a , p , b , r D Ð n “ b q n ´ r ˇˇ a , p , b , r D Ð n , F ˇˇ a , p , b , r D Ð n “ ˇˇ a , p , b , r D Ð n ` , E ˇˇ a , p , b , r D Ð n “ ´ a r n s ˇˇ a , p , b , r D Ð n ´ , K ˇˇ a , p , b , r D Ò n “ a q ´ n ˇˇ a , p , b , r D Ò n , k ˇˇ a , p , b , r D Ò n “ ´ b q n ´ r ˇˇ a , p , b , r D Ò n , F ˇˇ a , p , b , r D Ò n “ ˇˇ a , p , b , r D Ò n ` , E ˇˇ a , p , b , r D Ò n “ ´ a r n sr n ´ s ˇˇ a , p , b , r D Ò n ´ , K ˇˇ a , p , b , r D Ó n “ a q ´ n ´ ˇˇ a , p , b , r D Ó n , k ˇˇ a , p , b , r D Ó n “ ´ b q n ` ´ r ˇˇ a , p , b , r D Ó n , F ˇˇ a , p , b , r D Ó n “ ˇˇ a , p , b , r D Ó n ` , E ˇˇ a , p , b , r D Ó n “ a ˇˇ a , p , b , r D Ò n ´ a r n sr n ` s ˇˇ a , p , b , r D Ó n ´ , K ˇˇ a , p , b , r D Ñ n “ a q ´ n ´ ˇˇ a , p , b , r D Ñ n , k ˇˇ a , p , b , r D Ñ n “ b q n ` ´ r ˇˇ a , p , b , r D Ñ n , F ˇˇ a , p , b , r D Ñ n “ ˇˇ a , p , b , r D Ñ n ` , E ˇˇ a , p , b , r D Ñ n “ ´ a r n s ˇˇ a , p , b , r D Ñ n ´ . A PPENDIX B. P ROOFS AND CALCULATION DETAILS
B.1. The universal R -matrix of U p X q . We calculate the universal R -matrix for U p X q byrelating U p X q to the Drinfeld double of U ď (see decomposition (2.4)). EPRESENTATIONS OF U q s ℓ p | q AT EVEN ROOTS OF UNITY 43
B.1.1.
The generators E and C can be viewed as functionals on the subalgebra U ď suchthat x E , F K i k j y : “ ´ q ´ q ´ , x C , BK i k j y : “ q ´ q ´ , and all other evaluations of E and C on the PBW basis elements in U ď vanish. We let ˜ U ą temporarily denote the algebra generated by the functionals E and C with the product ofany two functionals b and g defined standardly as x bg , x y “ x b , x y x g , x y , where D x “ x b x is the coproduct on U ď . It then readily follows that the relations EEC ´ r s ECE ` CEE “ CC “ U ą . Hence, polynomials in E and C canbe expressed linearly in terms of E i , E i ` C , E i CE , E i CEC , i ě
0. By induction on thepower of the relevant generator, we then find that these tentative PBW basis elementshave the following nonzero evaluations on the PBW basis elements in U ď : x E n , F n K i k j y “ p´ q n p q ´ q ´ q ´ n q n p n ´ q r n s ! , x E n ` C , F n BFK i k j y “ p´ q n ` p q ´ q ´ q ´ n ´ q p n ` qp n ´ q r n ` s ! , x E n C , F n BK i k j y “ p´ q n p q ´ q ´ q ´ n ´ q n p n ´ q r n s ! , x E n CE , F n BFK i k j y “ p´ q n ` p q ´ q ´ q ´ n ´ q n p n ´ q ` ` q n ´ r n s ˘ r n s ! , x E n CE , F n ` BK i k j y “ p´ q n ` p q ´ q ´ q ´ n ´ q p n ` qp n ´ q r n ` s ! , x E n CEC , F n BFBK i k j y “ p´ q n ` p q ´ q ´ q ´ n ´ q p n ` qp n ´ q r n s !(and all other evaluations vanish). It then follows that E p “ U ą , and ˜ U ą , with itsPBW basis modeled on that in U ą , is isomorphic to U ą as an algebra.To diagonalize the “nondiagonal” part of the pairing U ą b U ď Ñ C , we define X n “ r n s E n ´ CE ´ ` q ´ n ` r n ´ s ˘ E n C , Y n “ E n ´ CE ´ q ´ E n C . Then x X n , F n BK i k j y “ p´ q n ` q p n ´ qp n ´ q r n s ! p q ´ q ´ q n , x X n , F n ´ BFK i k j y “ , x Y n , F n BK i k j y “ , x Y n , F n ´ BFK i k j y “ p´ q n q n p n ´ q r n ´ s ! p q ´ q ´ q n . B.1.2.
We next extend ˜ U ą by generators L , ℓ P H ˚ , also functionals on U ď , that werequire to commute with the generators of U ă exactly as K and k do: FL “ q LF , BL “ q ´ LB , F ℓ “ q ´ ℓ F , B ℓ “ ´ ℓ B , where for a P U ď and a functional b , we evaluate the product using the Drinfeld-doubleformula(B.1) a b “ p a á b à S ´ p a qq a (where D a “ a b a is the coproduct on U ď ). It then follows that x L , K m k n y “ q n ´ m , x ℓ, K m k n y “ p´ q n q m (with the other evaluations vanishing). Using (B.1), we then establish further “cross-commutator” relations: F E ´ EF “ ´ L ´ K ´ q ´ q ´ , BC ´ CB “ ´ ℓ ´ k ´ q ´ q ´ , as well as FC ´ CF “ BE ´ EB “ KL “ LK , etc.).This essentially (modulo easily reconstructible details) shows that U p X q is the quotientof the Drinfeld double of U ď by the Hopf ideal generated by the relations L “ K and ℓ “ k .The (inverse) universal R -matrix is then inherited from the Drinfeld double in the form R ´ “ p ´ ÿ a “ p´ q a q ´ a p a ´ q p q ´ q ´ q a r a s ! ´ E a b F a ´ q a ´ X a b F a B ´ q p q ´ q ´ q Y a ` b F a BF ´ q p q ´ q ´ q E a CEC b F a BFB ¯ r ´ , where r ´ “ p p q p ´ ÿ i “ p ´ ÿ j “ p ´ ÿ m “ p ´ ÿ n “ p´ q jn q im ´ in ´ jm K i k j b K m k n . From here, the universal R -matrix R “ p S ´ b id q R ´ (where S is the antipode of U p X q )follows in the form R “ r ¯ R , with r in (2.5) and¯ R “ p ´ ÿ a “ q a p a ´ q p q ´ q ´ q a r a s ! ´ E a b F a ´ q ´ ` q ´ a r a s E a ´ CE ´ p ` q ´ a r a ´ sq E a C ˘ b F a B ` q ´ p q ´ q ` E a CE ´ q E a ` C ˘ b F a BF ´ q ´ p q ´ q E a CEC b F a BFB ¯ , which can be readily rewritten in the factored form as in (2.6). B.2. Proof of 2.4.4.
The M-matrix (see ) can obviously be written as M “ p r ¯ R r q ¯ R ,where we calculate r ¯ R r from the definition of r and ¯ R : r ¯ R r “ p p q p ´ ÿ i “ p ´ ÿ j “ p ´ ÿ m “ p ´ ÿ n “ p ´ ÿ i “ p ´ ÿ j “ p ´ ÿ m “ p ´ ÿ n “ p´ q jn ` j n ˆ q ´ im ` jm ` in q ´ i m ` j m ` i n p K m k n b K i k j q ¯ R p K i k j b K m k n q . EPRESENTATIONS OF U q s ℓ p | q AT EVEN ROOTS OF UNITY 45
Here, we next move all K and k factors to the right of ¯ R ; after simple changes of sum-mation variables, i Ñ i ´ m and j Ñ j ´ n , we then see that the summations over m and n can give a nonzero result only for terms with particular values of i and j , e.g., i “ m ´ a and j “ n or i “ m ´ ´ a and j “ n ´
1, etc. (depending on the term taken in theexpression for ¯ R ). This reduces the m and n summations to the form p ´ ÿ m “ p ´ ÿ n “ q xm ` yn K m k n , where x and y are integer linear combinations of other summation indices. Next, splittingthe range r , p ´ s of m into r , p ´ s Y r p , p ´ s and shifting m by p in the secondhalf of the range, we obtain that the entire sum is proportional to 1 ` p´ q x , which actsas a selection rule for the parity of one of the remaining summation indices involved in x .The same is repeated for the n sum, yielding the factor 1 ` p´ q y and another selectionrule. The result of these straightforward manipulations is r ¯ R r “ p ´ ÿ a “ q ´ a ´ a p q ´ q ´ q a r a s ! ˆ ˆ F a b E a ` q a ´ F a B b ` q ´ a r a s E a ´ CE ´ p ` q ´ a r a ´ sq E a C ˘ p k b k ´ q´ q ´ a ´ p q ´ q F a BF b ` E a CE ´ q E a ` C ˘ p Kk b K ´ k ´ q´ q ´ p q ´ q p F a BFB b E a CEC qp Kk b K ´ k ´ q ˙ ˆ p p ´ ÿ i “ p ´ ÿ j “ p ´ ÿ i “ p ´ ÿ j “ q i j ` i j ´ ii ´ K i ` a k j b K i ´ a k j ¯ . This is immediately verified to be equal to ¯ M ¯ r in . B.3. Coincidence of two Drinfeld maps.
Let A be a Hopf algebra and F an invertiblenormalized two-cocycle, i.e., an invertible element F P A b A such that F F b F F b F “ F b F F b F F and e p F q F “ F e p F q “ , where F “ F b F “ F b F (and e is the counit). This standardly defines a new Hopfalgebra structure—the one with the same product and counit, and with the coproduct andantipode given by r D p x q “ F ´ D p x q F , r S p x q “ U ´ S p x q U @ x P A , where(B.2) U “ S p F q F . We also note that r S p x q “ x ´ S p x q x , where(B.3) x “ S p U ´ q U . B.3.1. Theorem.
Let A be a quasitriangular Hopf algebra and F an invertible normalized2-cocycle. Then diagram (2.9) is commutative, i.e., r b p r M q r M “ b p M q M for any b P Ch . Proof.
First, we have a linear space isomorphism Ch Ñ Ă Ch given by b ÞÑ p b à x q . In-deed, if b P Ch , which amounts to the condition that b p xy q “ b p S p y q x q for all x , y P A ,then the functional r b : x ÞÑ b p x x q is invariant under the “tilded” coadjoint action, i.e., r b p xy q “ r b p r S p y q x q for all x , y P A .We next note two simple consequences of the cocycle condition: j b F j b F “ j F b j F b j F , (B.4) F b F j b j “ j F b j F b j F , (B.5)where F ´ “ j b j . From (B.4), applying the antipode and multiplying, we find theidentity j S p F j q F “
1, whence, for U in (B.2), it follows that U ´ “ j S p j q . Hence, x “ S p U ´ q U “ S p f q S p f q S p F q F , and we calculate r b p r M q r M “ b ` xj M F ˘ j M F “ b ` S p f q S p F q F j M F f ˘ j M F where we next note that S p F q F j b j “ S p F q F b F as a simple consequence of (B.5),and we can therefore continue “ b ` S p f q S p F q F M F f ˘ F M F “ b ` S p f q S p F q M F F f ˘ M F F (by the property M D p x q “ D p x q M of the M-matrix), which after directly applying thecocycle condition becomes “ b ` S p f q S p F F q M F F f ˘ M F “ b ` S p F q M F ˘ M F “ b ` M F S ´ p F q ˘ M F . This is the same as b ` M ˘ M . EPRESENTATIONS OF U q s ℓ p | q AT EVEN ROOTS OF UNITY 47
B.4. Proof of 7.3.2.
We solve Eqs. (7.3). We recall that W j are the 32 p ´
16 nilpotentbasic algebra generators defined in item 3 in , and the unknowns x W , m (see (7.2)) areassociated with the 16 p ´ W W , m defined initem 3b in .For a projective module Q “ Q a , b r , s , we let W Q a , b r , s , m be denoted as W a , b r , s p m q . The corre-sponding unknowns, accordingly, are then written as x a , b r , s p m q , and we moreover drop theuninformative “ x ” and distinguish the variables pertaining to the five species of projectivemodules in – by an individual letter each:(B.6) x a , b r , s p m q “ $’’’’’’’&’’’’’’’% v a , b p p m q , s “ p , r “ p a q , p a , b p ´ p m q , s “ r “ p ´ p m q , w a , b s p m q , ď s ď p ´ , r “ p ‘ q , s a , b s p m q , ď s “ r ď p ´ p b q , z a , b p m q , s “ , r “ p f q . We recall that the argument m labels level-3 subquotients in a given projective modulethat are isomorphic to the top subquotient. It is convenient, for uniformity, to let m takenot consecutive values (e.g., from 1 to 4 for the projective module Q a , b , in Fig. 6.5) butthe values that the relevant subquotients are already assigned in the graphs (which are 2,4, 5, and 7 in Fig. 6.5). The argument in W a , b r , s p m q is then of course understood in thesame way.Each W a , b r , s p m q , as well as each W j in (7.3), can be regarded as a linear operator onthe vector space with basis consisting of all simple subquotients of all projective mod-ules, and is therefore completely determined by the coefficients with which it sends eachsubquotient into (linear combinations of) others. By definition, the top subquotient ismapped (into a level-3 subquotient by each W and a level-2 subquotient by each W j ) withthe coefficient 1, while all other coefficients are obtained from the graphs in Figs. 6.1–6.5,simply from the condition that the basic algebra elements be U p X q intertwiners. This isillustrated in Fig. B.1.The commutativity in (7.3) then becomes the commutativity condition for the corre-sponding matrices. We illustrate this with equations involving the six basic algebra gen-erators W Q a , b , , n , 1 ď n ď
6, sending the top one-dimensional subquotient of Q a , b , intoisomorphic level-2 subquotients (see , item 3). The isomorphic subquotients occur inthe projective modules Q ´ a , b p , , Q ´ a , ´ b p , , Q ´ a , b p ´ , p ´ , Q ´ a , ´ b p ´ , p ´ , Q a , ´ b , , and Q a , b , ; we selectthe first one in this (arbitrary) ordering. Because the corresponding basic algebra genera-tor W Q a , b , , acts by zero on all projective modules except Q a , b , , commutator equation (7.3)takes the form ` a b s r ˘ c c ´ a b s r ¯ . . . . . . ´ a b s r ¯ c { c ´ a b s r ¯ c { c ` a b s r ˘ c c . . . . . . ´ a b s r ¯ ´ a b s r ¯ . . . . . . F IGURE
B.1.
Mapping projective modules by an element of the basic algebra.The top subquotient of the projective module on the left is sent into an isomorphiclevel-2 subquotient in the projective module on the right. Isomorphic descendantsare then mapped into one another with the coefficients given by ratios of theweights associated with edges of the graphs. Those children of ` a b s r ˘ on the leftthat have no isomorphic subquotients among the children of ` a b s r ˘ on the right aremapped to zero. The procedure continues similarly to the lower-lying levels. (B.7) W Q a , b , , ´ z a , b p q W a , b , p q ` z a , b p q W a , b , p q ` z a , b p q W a , b , p q`` z a , b p q W a , b , p q ¯ ´ v ´ a , b p p q W ´ a , b p , p q W Q a , b , , “ . This equation for the unknowns z a , b p q , z a , b p q , z a , b p q , z a , b p q , and v ´ a , b p p q isstill an “operator” equation in the sense that it is written in terms of maps. A “scalar”equation follows by applying (B.7) to the top subquotient of Q a , b , . Simple analysis as inFig. B.1 readily shows that, with W Q a , b , , sending the top subquotient of Q a , b , as ´ a b ¯ Ñ ´ a b ¯ , some other relevant subquotients of Q a , b , are mapped by W Q a , b , , as ´ a b ¯ Ñ ´ ´ a b ¯ , ´ a b ¯ Ñ ´ a b ¯ , ´ a b ¯ Ñ , ´ a b ¯ Ñ . Also, W ´ a , b p , p q maps as ´ a b ¯ Ñ a ´ a b ¯ . This gives the equation av ´ a , b p p q ` z a , b p q ´ z a , b p q “ . EPRESENTATIONS OF U q s ℓ p | q AT EVEN ROOTS OF UNITY 49
The full list of equations that follow from commuting with W Q a , ` , , n is av ´ a , ` p p q ` z a , ` p q ´ z a , ` p q “ , av ´ a , ´ p p q ´ z a , ` p q ´ z a , ` p q “ , p ´ a , ` p ´ p q ´ z a , ` p q ` z a , ` p q “ , p ´ a , ´ p ´ p q ` z a , ` p q ` z a , ` p q “ , ´ w a , ´ p qr s ´ z a , ` p q ´ a w a , ´ p qr s “ , s a , ` p q ´ a r s z a , ` p q ` r s z a , ` p q ` r s z a , ` p q “ a occurring as coefficients).Similarly, the 4 p p ´ q equations that follow from commuting with W Q ` , ` s , s , n , 1 ď s ď p ´
2, are as follows: s ` , ` p qr s ` z ` , ` p q ` z ` , ` p q ` z ` , ` p q “ , s ` , ` s p q ` w ´ , ´ p ´ s p qr s s “ , ď s ď p ´ , s ` , ` s p q ´ w ´ , ` p ´ s p qr s s “ , ď s ď p ´ , s ` , ` s ` p qr s ` s ´ s ` , ´ s p qr s s ` s ` , ´ s p qr s s “ , ď s ď p ´ , ´ s ` , ´ s ` p qr s ` s ` s ` , ` s p qr s s ` s ` , ` s p qr s s “ , ď s ď p ´ , s ` , ` p ´ p qr s ` s ` , ` p ´ p qr s ` p ` , ´ p ´ p q “ Z ` , ` s , s (the top subquo-tient of Q ` , ` s , s ) are Q ´ , ´ p ´ , , Q ` , ` , , Q ´ , ` p ´ , , and Q ` , ´ , for s “ Q ´ , ´ p ´ s , , Q ´ , ` p ´ s , , Q ` , ´ s ´ , s ´ ,and Q ` , ´ s ` , s ` for 2 ď s ď p ´ W Q a , b r , s , n of basic algebra generators. Part of thesystem can be written “uniformly,” with the equations labeled by s and having the samefunctional form for any s . But there are also “boundary effects”: the two length- p p ´ q series of projective modules ( b and ‘ ) are followed at the end by modules of a somewhatreduced structure ( m and a ), and are also “joined” by the f projective module with an“enhanced” structure. Both these effects are well seen in the above formulas ( z in thefirst and p p ´ in the last equation). But it is possible to give the full solution of the system, which can be written relativelycompactly. The comparative complexity or simplicity of the explicit solution—and in-deed of the procedure of solving—depends rather strongly on the choice of free variablesin terms of which the others are to be expressed. In choosing the free variables, we wereguided by the desire to avoid final formulas with the number of terms growing with p ; thisturned out to be possible, and was actually a factor underlying the success in solving thesystem explicitly. Numerous variations of our choice are of course possible. A drawbackof the specific choice that we make is that the formulas become slightly sensitive to theparity of p ; we therefore write the solution explicitly only for odd p . Specifically, the2 p ` z ` , ` p q , z ` , ` p q , z ´ , ´ p q , z ` , ` p q , w ´ , ´ p q , w ` , ´ i ` p q , i “ , . . . , p ´ ´ , w ` , ` i p q , i “ , . . . , p ´ , s ` , ´ i ´ p q , i “ , . . . , p ´ ´ , s ` , ` i p q , i “ , . . . , p ´ ´ , p ` , ` p ´ p q . The other 14 p ´ s relations, occurring because of the special structure of the Q a , b , projectivemodule: z ´ , ` p q “ ´ z ´ , ´ p q , z ` , ´ p q “ ´ z ` , ` p q , z ´ , b p q “ z ´ , ´ p q ´ w ` , ` p ´ p qr s , z ` , ´ p q “ z ` , ` p q , z ´ , b p q “ ´ z ´ , ´ p q ´ p ` , ` p ´ p q , z ` , ´ p q “ z ` , ` p q , z ` , ` p q “ ´ z ` , ` p q ´ z ` , ` p q ´ s ` , ´ p qr s ` z ´ , ´ p q , z ` , ´ p q “ ´ z ` , ` p q ´ z ` , ` p q ´ s ` , ´ p qr s , z ´ , b p q “ ´ bz ` , ` p q ` z ` , ` p q ` p ` , ` p ´ p q ` s ` , ´ p qr s , s ´ , b p q “ b r s z ` , ` p q ´ r s z ` , ` p q ` w ` , ` p ´ p q ´ s ` , ´ p q , w ´ , b p q “ b r s z ` , ` p q ´ b r s z ` , ` p q ´ b r s p ` , ` p ´ p q´ b r s s ` , ´ p q ´ b r s w ´ , ´ p q , w ` , b p q “ ´ b r s z ` , ` p q ´ b r s z ` , ` p q ´ b r s s ` , ´ p q ` b r s w ` , ` p q , EPRESENTATIONS OF U q s ℓ p | q AT EVEN ROOTS OF UNITY 51 w ´ , ` p q “ w ´ , ´ p q ´ r s z ` , ` p q , w ´ , ´ p q “ ´ r sr s z ` , ` p q ´ r sr s z ` , ` p q ´ r s s ` , ´ p q ` r s s ` , ` p ´ p q , w ´ , b p q “ ´ b r sr s z ` , ` p q ` b r sr s z ` , ` p q ` b r s s ` , ´ p q´ b r s s ` , ` p ´ p q ` b r s w ´ , ´ p qr s . Here and hereafter, b “ ˘ . Next, there are 14 p p ´ q ´
31 “serial” relations for “generic”values of s , which in our solution are split into even and odd ones: w ` , b i p q “ b r i s w ` , ` i p q ´ b r i s w ` , ´ i ´ p qr i ´ sr i ´ s , ď i ď p ´ , w ` , b i ` p q “ b r i ` s w ` , ´ i ` p q ´ b r i ` s w ` , ` i p qr i ´ sr i s , ď i ď p ´ ´ , w ´ , b i p q “ ´ b r i s s ` , ´ p ´ i p q ` b r i s s ` , ` p ` ´ i p qr i ´ s , ď i ď p ´ , w ´ , b i ` p q “ ´ b r i ` s s ` , ` p ´ ´ i p q ` b r i ` s s ` , ´ p ´ i p qr i ´ s , ď i ď p ´ ´ , w ` , ´ i p q “ w ` , ` i p q ´ r i ´ sr i s z ´ , ´ p q , ď i ď p ´ , w ` , ` i ` p q “ w ` , ´ i ` p q ´ r i sr i ` s z ´ , ´ p q , ď i ď p ´ ´ , w ´ , b i p q “ ´ b r i ´ sr i s z ` , ` p q ´ r i ´ sr i s z ` , ` p q´ r i ´ sr i s s ` , ´ p qr s ` r i s s ` , ´ p ´ i p q , ď i ď p ´ , w ´ , b i ` p q “ b r i sr i ` s z ` , ` p q ´ r i sr i ` s z ` , ` p q´ r i sr i ` s s ` , ´ p qr s ` r i ` s s ` , ` p ´ ´ i p q , ď i ď p ´ ´ , s ´ , b i p q “ ´ b r i sr i ` s z ` , ` p q ´ r i sr i ` s z ` , ` p q` w ` , ´ p ´ i p qr i s ´ r i sr i ` s s ` , ´ p qr s , ď i ď p ´ ´ , s ´ , b i ` p q “ b r i ` sr i ` s z ` , ` p q ´ r i ` sr i ` s z ` , ` p q` w ` , ` p ´ ´ i p qr i ` s ´ r i ` sr i ` s s ` , ´ p qr s , ď i ď p ´ ´ , s ` , ´ i p q “ s ` , ` i p q ´ r i sr i ` s z ´ , ´ p q , ď i ď p ´ ´ , s ` , ` i ` p q “ s ` , ´ i ` p q ´ r i ` sr i ` s z ´ , ´ p q , ď i ď p ´ ´ , s ´ , b i p q “ b w ` , ´ p ´ i p qr i s ´ b r i s w ` , ` p ´ ´ i p qr i ` s r i ` s , ď i ď p ´ ´ , s ´ , b i ` p q “ b w ` , ` p ´ ´ i p qr i ` s ´ b r i ` s w ` , ´ p ´ ´ i p qr i ` s r i ` s , ď i ď p ´ ´ . s ` , b i p q “ b r i s s ` , ´ i ` p qr i ` s ´ b r i s s ` , ` i p q , ď i ď p ´ ´ , s ` , b i ` p q “ b r i ` s s ` , ` i ` p qr i ` s ´ b r i ` s s ` , ´ i ` p q , ď i ď p ´ ´ , And finally, the 14 high- s relations, whose form is largely determined by the somewhatspecial structure of Q a , b p , and Q a , b p ´ , p ´ , are s ` , ` p ´ p q “ ´ r sr s z ` , ` p q ` r sr s z ` , ` p q ` r s w ´ , ´ p qr s ` r s s ` , ´ p q ´ r s s ` , ` p ´ p q , s ` , b p ´ p q “ b r s z ` , ` p q ´ b r s z ` , ` p q ´ b r s p ` , ` p ´ p q´ b r s s ` , ´ p q ´ b w ´ , ´ p qr s , s ´ , b p ´ p q “ ´ b r s z ` , ` p q ´ b r s z ` , ` p q ´ b r s s ` , ´ p q ` b w ` , ` p qr s , s ` , ` p ´ p q “ ´ r s z ` , ` p q ` r s z ` , ` p q ` w ´ , ´ p qr s ` s ` , ´ p q ´ r s z ´ , ´ p q , s ` , ´ p ´ p q “ ´ r s z ` , ` p q ` r s z ` , ` p q ` w ´ , ´ p qr s ` s ` , ´ p q , p ´ , b p ´ p q “ b z ` , ` p q ´ z ` , ` p q , p ` , ´ p ´ p q “ z ´ , ´ p q ` p ` , ` p ´ p q , v ` , ` p p q “ w ` , ` p ´ p qr s ´ z ´ , ´ p q , v ` , ´ p p q “ w ` , ` p ´ p qr s v ´ , b p p q “ z ` , ` p q ´ b z ` , ` p q . This completes the list of formulas expressing 14 p ´ p ` EFERENCES [1] V. Gurarie,
Logarithmic operators in conformal field theory,
Nucl. Phys. B410 (1993) 535 [hep-th { Polymers and percolation in two-dimensions and twisted N “ supersymmetry , Nucl. Phys.B382 (1992) 486–531 [hep-th { Extended conformal algebras generated by a multiplet of primary fields , Phys. Lett.B 259 (1991) 448.
EPRESENTATIONS OF U q s ℓ p | q AT EVEN ROOTS OF UNITY 53 [4] M.R. Gaberdiel and H.G. Kausch,
Indecomposable fusion products , Nucl. Phys. B477 (1996) 293–318 [hep-th { A rational logarithmic conformal field theory , Phys. Lett. B 386 (1996) 131–137 [hep-th { A local logarithmic conformal field theory , Nucl. Phys. B538 (1999) 631–658[hep-th { Nonsemisimple fusion algebras and theVerlinde formula , Commun. Math. Phys. 247 (2004) 713–742 [hep-th { The Nichols algebra of screenings , Commun. Contemp. Math.14 (2012) 1250029, arXiv:1101.5810.[7] W. D. Nichols,
Bialgebras of type one , Commun. Algebra 6 (1978) 1521–1552.[8] S.L. Woronowicz,
Differential calculus on compact matrix pseudogroups ( quantum groups ), Com-mun. Math. Phys. 122 (1989) 125–170.[9] G. Lusztig, IntroductiontoQuantumGroups. Birkhäuser, 1993.[10] M. Rosso, Quantum groups and quantum shuffles , Invent. math. 133 (1998) 399–416.[11] N. Andruskiewitsch and M. Graña,
Braided Hopf algebras over non-abelian finite groups , Bol. Acad.Nacional de Ciencias (Cordoba) 63 (1999) 45–78 [arXiv:math { Pointed Hopf algebras , in: NewdirectionsinHopfalgebras,MSRI Publications 43, pages 1–68. Cambridge University Press, 2002.[13] N. Andruskiewitsch and H.-J. Schneider,
On the classification of finite-dimensional pointed Hopf al-gebras , Ann. Math. 171 (2010) 375–417 [arXiv:math { The Weyl groupoid of a Nichols algebra of diagonal type , Invent. Math. 164 (2006)175–188.[15] I. Heckenberger,
Classification of arithmetic root systems , Adv. Math. 220 (2009) 59–124 [math.QA { The Nichols algebra of a semisimple Yet-ter–Drinfeld module , Amer. J. Math. 132 (2010) 1493–1547.[17] N. Andruskiewitsch, D Radford, and H.-J. Schneider,
Complete reducibility theorems for modulesover pointed Hopf algebras , J. Algebra, 324 (2010) 2932–2970 [arXiv:1001.3977].[18] I.E. Angiono,
On Nichols algebras with standard braiding , Algebra & Number Theory 3 (2009) 35–106 [arXiv:0804.0816].[19] I.E. Angiono,
A presentation by generators and relations of Nichols algebras of diagonal type andconvex orders on root systems , arXiv:1008.4144, J. Europ. Math. Soc.[20] D. Kazhdan and G. Lusztig,
Tensor structures arising from affine Lie algebras,
I, J. Amer. Math. Soc.6 (1993) 905–947; II, J. Amer. Math. Soc. 6 (1993) 949–1011; III, J. Amer. Math. Soc. 7 (1994)335–381; IV, J. Amer. Math. Soc. 7 (1994) 383–453.[21] M. Finkelberg,
An equivalence of fusion categories , Geometric and Functional Analysis (GAFA) 6(1996) 249–267.[22] B.L. Feigin, A.M. Gainutdinov, A.M. Semikhatov, and I.Yu. Tipunin,
Kazhdan–Lusztig correspon-dence for the representation category of the triplet W -algebra in logarithmic CFT , Theor. Math. Phys.148 (2006) 1210–1235 [arXiv:math { The triplet vertex operator algebra W p p q and the restricted quantumgroup at root of unity , arXiv:0902.4607 [math.QA].[24] D. Adamovi´c and A. Milas, Lattice construction of logarithmic modules for certain vertex algebras ,Selecta Math. New Ser., 15 (2009) 535–561, arXiv:0902.3417.[25] A. Tsuchiya and S. Wood,
The tensor structure on the representation category of the W p triplet alge-bra , arXiv:1201.0419. [26] A.M. Semikhatov, Fusion in the entwined category of Yetter–Drinfeld modules of a rank- Nicholsalgebra , Theor. Math. Phys. 173(1) (2012) 1329–1358.[27] B.L. Feigin, A.M. Gainutdinov, A.M. Semikhatov, and I.Yu. Tipunin,
Modular group representationsand fusion in logarithmic conformal field theories and in the quantum group center , Commun. Math.Phys. 265 (2006) 47–93 [arXiv:hep-th { Logarithmic extensions of min-imal models: characters and modular transformations , Nucl. Phys. B757 (2006) 303–343 [arXiv:hep-th { Kazhdan–Lusztig-dual quan-tum group for logarithmic extensions of Virasoro minimal models , J. Math. Phys. 48 (2007) 032303[math.QA { Fusion rules and boundary conditions in the c “ tripletmodel , arXiv:0905.0916; A modular invariant bulk theory for the c “ triplet model , J. Phys. A44(2011) 015204, arXiv:1008.0082.[31] I. Runkel, M.R. Gaberdiel, and S. Wood, Logarithmic bulk and boundary conformal field theory andthe full centre construction , arXiv:1201.6273.[32] A.M. Semikhatov,
Virasoro central charges for Nichols algebras , arXiv:1109.1767 [math.QA], in:“Conformal field theories and tensor categories,” Mathematical Lectures from Peking University Bai,C.; Fuchs, J.; Huang, Y.-Z.; Kong, L.; Runkel, I.; Schweigert, C. (Eds.) 2014, IX 67–92.[33] A.M. Semikhatov and I.Yu. Tipunin,
Logarithmic p s ℓ p q CFT models from Nichols algebras. 1 , J. Phys.A: Math. Theor. 46 (2013) 494011 [arXiv:1301.2235].[34] P.V. Bushlanov, B.L. Feigin, A.M. Gainutdinov, and I.Yu. Tipunin
Lusztig limit of quantum sl(2) atroot of unity and fusion of (1,p) Virasoro logarithmic minimal models , Nucl. Phys. B818 (2009) 179–195 [arXiv:0901.1602];[35] P.V. Bushlanov, A.M. Gainutdinov, and I.Yu. Tipunin,
Kazhdan-Lusztig equivalence and fusion ofKac modules in Virasoro logarithmic models , arXiv:1102.0271.[36] I. Angiono and H. Yamane,
The R-matrix of quantum doubles of Nichols algebras of diagonal type ,arXiv:1304.5752 [math.QA].[37] I. Heckenberger and H. Yamane,
Drinfel’d doubles and Shapovalov determinants , Revista de la UniónMatemática Argentina 51 no. 2 (2010) 107–146 [arXiv:0810.1621 [math.QA]].[38] I. Heckenberger and H.-J. Schneider,
Yetter–Drinfeld modules over bosonizations of dually pairedHopf algebras , arXiv:1111.4673.[39] D. Radford,
Hopf algebras with a projection , J. Algebra 92 (1985) 322–347.[40] S. Majid,
Braided groups and algebraic quantum field theories , Lett. Math. Phys. 22 (1991) 167–176.[41] S. Majid,
Transmutation theory and rank for quantum braided groups , Math. Proc. Camb. Phil. Soc.113 (1993) 45–70.[42] Yu.N. Bespalov,
Crossed modules, quantum braided groups, and ribbon structures , Theor. Math.Phys. 103 (1995) 621–637;
Crossed modules and quantum groups in braided categories , arXiv:q-alg/9510013.[43] S. Majid,
Crossed products by braided groups and bosonization , J. Algebra 163 (1994) 165–190.[44] S. Majid,
Doubles of quasitriangular Hopf algebras , Comm. Alg. 19 (1991) 3061–3073.[45] S. Maclane, Homology, Springer Verlag, 1963.[46] Y. Arike,
Symmetric linear functions of the restricted quantum group ¯ U q sl p C q , arXiv:0706.1113[math.QA]; A construction of symmetric linear functions of the restricted quantum group U q p sl q ,arXiv:0807.0052 [math.QA]. EPRESENTATIONS OF U q s ℓ p | q AT EVEN ROOTS OF UNITY 55 [47] A.M. Gainutdinov and R. Vasseur,
Lattice fusion rules and logarithmic operator product expansions ,Nucl. Phys. B 868, 223–270 (2013) [arXiv:1203.6289].[48] A.M. Gainutdinov, H. Saleur, and I.Yu. Tipunin,
Lattice W-algebras and logarithmic CFTs , arXiv:1212.1378.[49] A.M. Gainutdinov, J.L. Jacobsen, N. Read, H. Saleur, and R. Vasseur,
Logarithmic conformal fieldtheory: a lattice approach , J. Phys. A: Math. Theor. 46 (2013) 494012 [arXiv:1303.2082].[50] J. Fuchs, C. Schweigert, and C. Stigner,
Modular invariant Frobenius algebras from ribbon Hopfalgebra automorphisms , Journal of Algebra 363 (2012) 29–72 [arXiv:1106.0210 [math.QA]].[51] J. Fuchs, C. Schweigert, and C. Stigner,
Higher genus mapping class group invariants from factoriz-able Hopf algebras , ZMP-HH/12-13, Hamburger Beitr. zur Math. 447 [arXiv:1207.6863 [math.QA]].[52] J. Fuchs, C. Schweigert, and C. Stigner,
From non-semisimple Hopf algebras to correlation func-tions for logarithmic Conformal Field Theory , Hamburger Beitr. zur Mathematik 468, ZMP-HH/13-2[arXiv:1302.4683 [hep-th]].L
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