Factorizable ribbon quantum groups in logarithmic conformal field theories
aa r X i v : . [ h e p - t h ] J un FACTORIZABLE RIBBON QUANTUM GROUPS IN LOGARITHMICCONFORMAL FIELD THEORIES
A.M. SEMIKHATOVA
BSTRACT . We review the properties of quantum groups occurring as Kazhdan–Lusztigdual to logarithmic conformal field theory models. These quantum groups at even rootsof unity are not quasitriangular but are factorizable and have a ribbon structure; the mod-ular group representation on their center coincides with the representation on generalizedcharacters of the chiral algebra in logarithmic conformal field models. I NTRODUCTION
The relation of quantum groups to conformal field theory, discussed since [1, 2, 3, 4],has been formulated in the context of vertex-operator algebras as the Kazhdan–Lusztigcorrespondence [5]. In a very broad sense (and very roughly), it states that whenever“something occurs” in the representation category of a vertex-operator algebra, “some-thing similar occurs” in the representation category of an appropriate quantum group; inother words, there is a functor relating these two categories, although this functor does nothave to be either left- or right-exact. In this broad sense, the Kazhdan–Lusztig correspon-dence is therefore a principle rather than a precise statement; the details of the functorhave to be worked out in each particular case. For rational conformal field theories, acertain complication follows from the fact that the chosen vertex-operator-algebra rep-resentation category is semisimple, while the quantum-group one is not, and additional“semisimplification” (taking the quotient over tilting modules) is needed to ensure theequivalence [6]. But in logarithmic conformal field theories [7, 8], the representationcategory is already nonsemisimple on the conformal field theory side, and therefore noquotients need to be a priori taken on the quantum group side. The Kazhdan–Lusztigcorrespondence extended to the logarithmic realm shows remarkable properties and, inparticular, extends to modular-group representations [9, 10, 11].A more “physical” point of view on the Kazhdan–Lusztig correspondence originatesfrom the observation that screening operators that commute with a given vertex-operatoralgebra generate a quantum group and, moreover, the vertex-operator algebra and thequantum group are characterized by being each other’s commutant, [ vertex-operator algebra , quantum group ] = 0 , with each of the objects in this relation allowing reconstruction of the other. But thispicture also applies more as a principle than as a precise statement, and therefore needs SEMIKHATOV a clarification as well. First, the screenings proper generate only the upper-triangularsubalgebra of the quantum group in question; the entire quantum group has to be re-constructed either by introducing contour-removal operators (see [12] and the referencestherein) or, somewhat more formally, by taking Drinfeld’s double [9, 13]. Second, inseeking the commutant of a quantum group, it must be specified where it is sought, i.e.,what free-field operators are considered (in particular, what are the allowed momenta ofvertex operators or whether vertex operators are allowed at all; cf., e.g., [14, 15] in thenonlogarithmic case).For several logarithmic conformal field theories, the Kazhdan–Lusztig correspondencehas been shown to have very nice properties [9, 10, 13, 11], being “improved” comparedto the rational case. Somewhat heuristically, such an “improvement” may relate to the factthat the field content in a logarithmic model is determined not by the cohomology but bythe kernel of the screening(s) (more precisely, by the kernel of a differential constructedfrom the screenings; we recall that the rational models are just the cohomology of such adifferential, cf. [16, 17]). Most remarkably, the Kazhdan–Lusztig correspondence extendsto modular group representations. We recall that a modular group representation in alogarithmic conformal field model is generated from the characters ( χ a ( τ )) of the modelby T - and S -transformations, the latter being expressed as(1.1) χ a ( − τ ) = X b S ab χ b ( τ ) + X b ′ S ′ ab ′ ψ b ′ ( τ ) , which involves certain functions ψ b ′ , which are not characters [18, 19, 20, 13], with(1.2) ψ a ′ ( − τ ) = X b S ′ a ′ b χ b ( τ ) + X b ′ S ′ a ′ b ′ ψ b ′ ( τ ) (the χ and ψ together can be called generalized, or extended characters, for the lack ofa better name). On the other hand, in quantum-group terms, the general theory in [21](also see [22, 23]), which has been developed in an entirely different context, can beadapted to the quantum groups that are dual to logarithmic conformal field theories, withthe result that a modular group representation is indeed defined on the quantum groupcenter. This representation turns out to be equivalent to the representation generated fromthe characters.Another instance where logarithmic conformal field theories and the corresponding(“dual”) quantum groups show similarity is the fusion (Verlinde) algebra / Grothendieckring. The existing data suggest that the Grothendieck ring of the Kazhdan–Lusztig-dualquantum group coincides with or “is closely related to” the fusion of the chiral algebrarepresentations on the conformal field theory side. Two remarks are in order here: first,comparing a Grothendieck ring with a fusion algebra implies that the latter is understood“in a K -version,” when all indecomposable representations are perforce replaced with UANTUM GROUPS IN LOGARITHMIC CFT 3 direct sums (cf. a discussion of this point in [19]); second, when the logarithmic confor-mal field theory has a rational subtheory, the representations of this rational theory are tobe excluded from the comparison (this is not unnatural though, cf. [26]).The quantum groups that have so far occurred as dual to logarithmic conformal fieldtheories are a quantum sℓ (2) and a somewhat more complicated quantum group, a quo-tient of the product of two quantum sℓ (2) . They are dual to logarithmic conformal fieldtheories in the respective classes of ( p, and ( p, p ′ ) models. In either case, the Kazhdan–Lusztig-dual quantum group is at an even root of unity. In either case, the quantum grouphas a set of crucial properties, which may therefore be conjectured to be common to thequantum groups that are dual to logarithmic conformal models. These properties and theunderlying structures are reviewed here. At present, their derivation is only available ina rather down-to-earth manner, by direct calculation, which somewhat obscures the gen-eral picture. In what follows, we skip the calculation details and concentrate on the finalresults and on the interplay of different structures associated with the quantum group.We thus continue the story as seen from the quantum-group side, following the ide-ology and results in [9, 10, 13, 11]. The necessary excursions to logarithmic conformalfield theory (see [8, 27, 25, 28, 19, 9, 13] and the references therein) are basically limitedto what is needed to appreciate the similarities with quantum-group structures. When weneed to be specific (which is almost always the case, because we do not claim any gener-ality here), we choose the simplest of the two basic examples, the U q sℓ (2) quantum groupdual to the ( p, logarithmic conformal field theory models, but we indicate the proper-ties shared by the quantum group U p,p ′ dual to the ( p, p ′ ) logarithmic models whereverpossible.The quantum group dual to the logarithmic ( p, model is U q sℓ (2) at an even root ofunity(1.3) q = e iπp The three generators E , F , and K satisfy the relations(1.4) KEK − = q E, KF K − = q − F, [ E, F ] = K − K − q − q − and the “constraints”(1.5) E p = F p = 0 , K p = 1 . Whenever indecomposable representations are involved, it is of course possible (and more interesting)to consider fusion algebras where indecomposable representations are treated honestly, i.e., are not replacedby the direct sum of their irreducible subquotients [24, 25]. The correspondence with quantum groups mayalso extend from the “ K / Grothendieck-style” fusion to this case (also see below).
SEMIKHATOV
We note that Eqs. (1.3)–(1.4) already imply that E p , F p , and K p are central, whichthen allows imposing (1.5) (but K p , which is also central, is not set equal to unity, whichmakes the difference with a smaller but more popular version, the so-called small quantum sℓ (2) ). As a result, U q sℓ (2) is p -dimensional. The quantum group U p,p ′ dual to the ( p, p ′ ) logarithmic model is p p ′ -dimensional. We note that the “constraint” imposed on its Cartangenerator is K pp ′ = 1 . The Hopf algebra structure of U q sℓ (2) (comultiplication ∆ , counit ǫ , and antipode S )is described by(1.6) ∆( E ) = 1 ⊗ E + E ⊗ K, ∆( F ) = K − ⊗ F + F ⊗ , ∆( K ) = K ⊗ K,ǫ ( E ) = ǫ ( F ) = 0 , ǫ ( K ) = 1 ,S ( E ) = − EK − , S ( F ) = − KF, S ( K ) = K − . The simplicity of (1.4)–(1.6) is somewhat misleading. This quantum group (as well as U p,p ′ ) has interesting algebraic properties, the central role being played by its center. Quantum group center and structures on it.
On the quantum-group side, the mainarena of the Kazhdan–Lusztig correspondence is the quantum group center Z . Of course,it contains the (“quantum”) Casimir element(s) and the algebra that they generate, but thisdoes not exhaust the center.The center carries an SL (2 , Z ) representation, whose definition [21, 22, 23] requiresthree types of structure: Drinfeld and Radford maps χ and b φφφ , and a ribbon element v .The action of S = ( − ) ∈ SL (2 , Z ) on the center is given by(1.7) Ch χ ~ ~ }}}}} b φφφ AAAAA
Z Z S − k k where Ch is the space of q -characters (linear functionals invariant under the coadjointaction), and the action of T = ( ) ∈ SL (2 , Z ) essentially by (multiplication with) theribbon element,(1.8) Z v −−−−→ Z . (Our definition of b φφφ is swapped with its inverse compared to the standard conventions.)A possible way to look at the center is to first identify a number of central elements as-sociated with traces over irreducible representations and then introduce appropriate pseu-dotraces. The (“quantum”) trace over an irreducible representation gives an element of Ch , i.e., a functional on the quantum group that is invariant under the coadjoint representa-tion; these invariant functionals ( q -characters) can then be mapped into central elements.This does not cover the entire center. But then projective quantum-group modules yieldadditional q -characters, obtained by taking traces, informally speaking, of nondiagonal UANTUM GROUPS IN LOGARITHMIC CFT 5 components of the quantum group action, nondiagonal in terms of the filtration of projec-tive modules. This gives a basis γ A in the space Ch of q -characters and hence a basis inthe center.Also, the Drinfeld map χ : Ch → Z is an isomorphism of associative commutativealgebras. Therefore, the center contains (an isomorphic image of) the Grothendieck ringof the quantum group, which is thus embedded into a larger associative commutativealgebra. In Sec. 2, we review the construction of the Radford map b φφφ . In Sec. 3, we recallthe necessary facts about the (irreducible and projective) representations of the relevantquantum groups; their Grothendieck rings are also discussed there. In Sec. 4, we recall the M [onodromy] “matrix,” the Drinfeld map χ , and the ribbon element. Together with theRadford map, these serve to define the modular group action, which we finally considerin Sec. 5. R ADFORD MAP AND RELATED STRUCTURES
We consider the Radford map b φφφ : U ∗ → U ; the construction of b φφφ and its inverseinvolves a cointegral and an integral. For a Hopf algebra U , a right integral λ is a linear functional on U satisfying(2.1) ( λ ⊗ id)∆( x ) = λ ( x )1 ∀ x ∈ U. Such a functional exists in a finite-dimensional Hopf algebra and is unique up to multi-plication [29].
The name integral for such a λ ∈ U ∗ is related to the fact that (2.1) is alsothe property of a right-invariant integral on functions on a group. Indeed, for a function f on a group G , ∆( f ) is the function on G × G such that ∆( f )( x, y ) = f ( xy ) , x, y ∈ G .Then the invariance property R f (? y ) = R f (?) can be written as ( R ⊗ id)∆( f ) = R f . The dual object to λ , an integral for U ∗ , is sometimes called a coin-tegral for U . We give it in the form needed below, when it is a two-sided cointegral. That the center contains the image of the Grothendieck ring but is larger than it has a counterpart inlogarithmic conformal field theory, where the set of chiral algebra characters χ a is to be extended by otherfunctions ψ a ′ in order to define a modular group action [18, 19, 13] and thus, presumably, to construct thespace of torus amplitudes (also see [20]). U ∗ is therefore assumed unimodular, which turns out to be the case for the quantum groups consideredbelow. SEMIKHATOV
A two-sided cointegral Λ is an element in U such that x Λ = Λ x = ǫ ( x ) Λ ∀ x ∈ U. Clearly, the cointegral defines an embedding of the trivial representation of U into theregular representation. The normalization λ ( Λ ) = 1 is typically understood. Let U be a Hopf algebra with a right integral λ and a two-sidedcointegral Λ . The Radford map b φφφ : U ∗ → U and its inverse b φφφ − : U → U ∗ are given by (2.2) b φφφ ( β ) = β ( Λ ′ ) Λ ′′ , b φφφ − ( x ) = λ ( S ( x )?) . ([29, 31]) . b φφφ and b φφφ − are inverse to each other and intertwine the leftactions of U on U and U ∗ , and similarly for the right actions. Here, the left- U -module structure on U ∗ is given by a⇁β = β ( S ( a )?) (and on U , bythe regular action). In particular, restricting to the space of q -characters (see A.1 ) gives b φφφ : Ch → Z . Proof.
We first establish an invariance property of the integral,(2.3) λ ( xy ′ ) y ′′ = λ ( x ′ y ) S − ( x ′′ ) . Indeed, λ ( xy ′ ) y ′′ = λ ( x ′ y ′ ) S − ( x ′′′ ) x ′′ y ′′ = λ (( x ′ y ) ′ ) S − ( x ′′ )( x ′ y ) ′′ = λ ( x ′ y ) S − ( x ′′ ) . Itthen follows that b φφφ ( b φφφ − ( x )) = b φφφ ( λ ( S ( x )?)) = λ ( S ( x ) Λ ′ ) Λ ′′ by (2.3) = λ ( S ( x ) ′ Λ ) S − ( S ( x ) ′′ ) = λ ( ǫ ( S ( x ) ′ ) Λ ) S − ( S ( x ) ′′ ) = S − ( ǫ ( S ( x ) ′ ) S ( x ) ′′ ) = x . Similarly, we calculate b φφφ − ( b φφφ ( β )) = b φφφ − ( β ( Λ ′ ) Λ ′′ ) β ( Λ ′ ) λ ( S ( Λ ′′ )?) = β ( λ ( S ( Λ ′′ )?) Λ ′ ) = β ( λ ( S ( Λ ) ′ ?) S − ( S ( Λ ) ′′ )) by (2.3) = β ( λ ( S ( Λ )? ′ )? ′′ ) = β ( λ ( Λ ? ′ )? ′′ ) = β ( λ ( ǫ (? ′ ) Λ )? ′′ ) = β ( ǫ (? ′ )? ′′ ) = β .We next show that b φφφ intertwines the left- U -module structures on U ∗ and U . With the left- U -module structure on U ∗ given by x⇁β = β ( S ( x )?) , we must prove that β ( S ( x ) Λ ′ ) Λ ′′ = xβ ( Λ ′ ) Λ ′′ , or β ( x Λ ′ ) Λ ′′ = S − ( x ) β ( Λ ′ ) Λ ′′ . But we have β ( x Λ ′ ) Λ ′′ = β ( ǫ ( x ′ ) x ′′ Λ ′ ) Λ ′′ = β (( x ′ Λ ) ′ ) S − ( x ′′ )( x ′ Λ ) ′′ = β ( ǫ ( x ′ ) Λ ′ ) S − ( x ′′′ ) ǫ ( x ′′ ) Λ ′′ = S − ( x ) β ( Λ ′ ) Λ ′′ . (cid:3) For any irreducible representation X of a quantumgroup U , the (“quantum”) trace in (A.10) is an invariant functional on U , i.e., an elementof Ch ( U ) (see A.1 ). (But the space of q -characters Ch is not spanned by q -traces overirreducible modules, as we have noted.) The Radford map sends each of the Tr X ( g − ?) functionals into the center Z of U :(2.4) b φφφ : Tr X ( g − ?) → b φφφ ( X ) ∈ Z . We use Sweedler’s notation ∆( x ) = P ( x ) x ′ x ′′ (see, e.g., [30]) with the summation symbols omittedin most cases; the defining property of the integral, for example, is then written as λ ( x ′ ) x ′′ = λ ( x ) . Here and in what follows, we use the definitions of the antipode and counit written in the form (see, e.g.,[30]) x ′ S ( x ′′ ) = S ( x ′ ) x ′′ = ǫ ( x )1 and x ′ ǫ ( x ′′ ) = ǫ ( x ′ ) x ′′ = x . Then, in particular, x ′ S − ( x ′′′ ) x ′′ = x . The reader not inclined to follow the details of the definition of g in (A.10) may think of it as just theelement that makes the trace “quantum,” i.e., invariant under the coadjoint action of the quantum group. UANTUM GROUPS IN LOGARITHMIC CFT 7
It suffices to have X range the irreducible representations of U , because traces “see” onlyirreducible subquotients in indecomposable representations. As long as the linear span of q -traces over irreducible modules is not all of the space of q -characters, the Radford-mapimage of irreducible representations does not cover the center.For any central element a ∈ Z , its action on an irreducible representation X is given bymultiplication with a scalar, to be denoted by a X ∈ C . By the Radford map properties,we have the relation a b φφφ ( X ) = a X b φφφ ( X ) in the center. In particular, the Radford-map image of (traces over) all irreducible repre-sentations is the annihilator of the radical in the center. For U q sℓ (2) , it is not difficult to verify that the right integral and the two-sidedcointegral are given by λ ( F j E m K n ) = ζ δ j,p − δ m,p − δ n,p +1 , (2.5) Λ = ζ F p − E p − p − X j =0 K j , (2.6)where we choose the normalization factor ζ = q p p − [9]. For U p,p ′ , the expres-sions for λ and Λ in [11] also hinge on the fact that p − is the highest nonzero power of theoff-diagonal quantum group generators. Another general notion that we need is that of a comodulus. For a rightintegral λ , the comodulus “measures” how much λ differs from a left integral (see [32]):it is an element a ∈ U such that (id ⊗ λ )∆( x ) = λ ( x ) a ∀ x ∈ U. A simple calculation then shows that the U q sℓ (2) comodulus is a = K . For U p,p ′ , thecomodulus is a = K p − p ′ . Q UANTUM GROUP MODULES : FROM IRREDUCIBLE TO PROJECTIVE
Irreducible (simple) and projective quantum group representations are considered be-low. By general philosophy of the Kazhdan–Lusztig duality, the irreducible quantum-group representations somehow “correspond” to irreducible chiral algebra representationsin logarithmic conformal models. In particular, the Grothendieck ring is generally relatedto fusion in conformal field theory. While direct calculation of the fusion of chiral algebra In general, the (co)integral is defined up to a nonzero factor, but factorizable ribbon quantum groups of-fer a “canonical” normalization, derived from the condition S = id on the center; in accordance with (1.7),the normalization of S is inherited from the normalization of b φφφ , and hence from that of the cointegral. SEMIKHATOV representations is typically quite difficult, this Grothendieck-ring structure may be consid-ered a poor man’s fusion (there is evidence that it is not totally meaningless). Apart fromirreducible representations, their projective covers play an important role. To be specific,we now describe some aspects of the representation theory in the example of U q sℓ (2) . There are p irreducible U q sℓ (2) -representations X ± r , which can be conveniently labeled by the ± and r p .The highest-weight vector | r i ± of X ± r is annihilated by E , and its weight is determinedby K | r i ± = ± q r − | r i ± . The representation dimensions are dim X ± r = r . Some readersmight find it suggestive to visualize the representations X ± r arranged into a “Kac table,” asingle row of boxes labeled by r = 1 , . . . , p , each carrying a “ + ” and a “ − ” representa-tion: . . . | {z } p .We next recall that the Grothendieck ring is the free Abelian group generated by sym-bols [ M ] , where M ranges over all representations subject to relations [ M ] = [ M ′ ]+[ M ′′ ] for all exact sequences → M ′ → M → M ′′ → . Multiplication in the ring is inducedby the tensor product of representations, with any indecomposable module occurring inthe tensor product replaced by a sum of its simple subquotients. U q sℓ (2) . The Grothendieck ring of U q sℓ (2) is (rather straightforwardly [9]) found to begiven by(3.1) X αr X α ′ s = r + s − X t = | r − s | +1step=2 e X αα ′ t where e X αr = ( X αr , r p, X α p − r + 2 X − αr − p , p + 1 r p − . It can also be described in terms of Chebyshev polynomials, as the quotient of the poly-nomial ring C [ x ] over the ideal generated by the polynomial b Ψ p ( x ) = U p +1 ( x ) − U p − ( x ) − , where U s ( x ) are Chebyshev polynomials of the second kind: U s (2 cos t ) = sin st sin t , s > . They satisfy the recurrence relations xU s ( x ) = U s − ( x ) + U s +1 ( x ) , s > , with the initialdata U ( x ) = 1 , U ( x ) = x . Moreover, let(3.2) P s ( x ) = ( U s ( x ) , s p, U s ( x ) − U p − s ( x ) , p + 1 s p. Under the quotient map, the image of each P s coincides with X + s for s p and with X − s − p for p + 1 s p . UANTUM GROUPS IN LOGARITHMIC CFT 9
The algebra in (3.1) is a nonsemisimple Verlinde algebra (commutative associativealgebra with nonnegative integer structure coefficients, see [33]), with a unit given by X +1 .The algebra contains the ideal V p +1 generated by X + p − r + X − r with r p − , X + p ,and X − p . The quotient over V p +1 is a semisimple Verlinde algebra and in fact coincideswith the fusion of the unitary b sℓ (2) representations of level p − . The same algebra was derived in [19] from modular transformations of the triplet W -algebra characters in logarithmic ( p, -models within a nonsemisimple generalization ofthe Verlinde formula (also see [34] for comparison with other derivations). U p,p ′ . The quantum group U p,p ′ dual to the ( p, p ′ ) logarithmic model has pp ′ irreducible rep-resentations X ± r,r ′ , r p , r ′ p ′ , with dim X ± r,r ′ = rr ′ . They can be considered arrangedinto a “Kac table” p ′ . . . . . . . . . . . . . . . . . . .. . . | {z } p , with each box carrying a “ + ” and a “ − ” representation.The Grothendieck ring structure is given by [11](3.3) X αr,r ′ X βs,s ′ = r + s − X u = | r − s | +1step=2 r ′ + s ′ − X u ′ = | r ′ − s ′ | +1step=2 e X αβu,u ′ , where e X αr,r ′ = X αr,r ′ , r p, r ′ p ′ , X α p − r,r ′ + 2 X − αr − p,r ′ , p +1 r p − , r ′ p ′ , X αr, p ′ − r ′ + 2 X − αr,r ′ − p ′ , r p, p ′ +1 r ′ p ′ − , X α p − r, p ′ − r ′ + 2 X − α p − r,r ′ − p ′ + 2 X − αr − p, p ′ − r ′ + 4 X αr − p,r ′ − p ′ , p +1 r p − , p ′ +1 r ′ p ′ − . This algebra is a quotient of C [ x, y ] as described in [11]. The radical in this nonsemisimple Ver-linde algebra (with a unit given by X +1 , ) is generated by the algebra action on X + p,p ′ ; the quotientover the radical coincides with the fusion of the ( p, p ′ ) Virasoro minimal model.The above algebra is a viable candidate for the (“ K -type”) fusion of W -algebra representationsin the logarithmic ( p, p ′ ) -models (see [13, 26]). Irreducible quantum-group modules can be “glued” together to produce indecom-posable representations. Already for U q sℓ (2) , its indecomposable representations (which It may be worth emphasizing that a Verlinde algebra structure involves not only an associative commu-tative structure but also a distinguished basis (the above quotient is that of Verlinde algebras). In particular,the reconstruction of the Verlinde algebra from its block decomposition as an associative algebra (the struc-ture of primitive idempotents and elements in the radical in the algebra) requires extra information, cf. [19]. have been classified, rather directly, in [10] or can be easily deduced from a more generalanalysis in [35]) are rather numerous. Apart from the projective modules, to be consid-ered separately in , indecomposable representations are given by families of modules W ± r ( n ) , M ± r ( n ) , and O ± r ( n, z ) that can be respectively represented as X ± r • x ± (cid:29) (cid:29) X ± r • x ± (cid:1) (cid:1) x ± (cid:29) (cid:29) . . . x ± (cid:1) (cid:1) x ± (cid:29) (cid:29) X ± r • x ± (cid:1) (cid:1) X ∓ p − r • X ∓ p − r • . . . X ∓ p − r • (with r p − , and integer n > the number of X ± r modules), X ± r • x ± (cid:1) (cid:1) x ± (cid:29) (cid:29) . . . x ± (cid:1) (cid:1) x ± (cid:29) (cid:29) X ± r • x ± (cid:1) (cid:1) x ± (cid:29) (cid:29) X ∓ p − r • X ∓ p − r • . . . X ∓ p − r • X ∓ p − r • (with r p − , and integer n > the number of X ∓ p − r modules), and X ± r • x ± (cid:5) (cid:5) x ± (cid:29) (cid:29) X ± r • x ± (cid:1) (cid:1) . . . X ± r • x ± (cid:25) (cid:25) X ∓ p − r • X ∓ p − r • . . . X ∓ p − r • X ± r • z x ± z x ± h h (with r p − , z = z : z ∈ CP , and integer n > the number of the X ± r modules).The small x + i and x − i , i = 1 , , are basis elements chosen in the respective spaces C =Ext U q sℓ (2) ( X + r , X − p − r ) and C = Ext U q sℓ (2) ( X − p − r , X + r ) ; they in fact generate the algebra Ext • s (with the Yoneda product) with the relations x + i x + j = x − i x − j = x +1 x − + x +2 x − = x − x +2 + x − x +1 = 0 (see [10] for the details).Interestingly, a very similar picture (the “zigzag,” although not the “ O ” modules) alsooccurred in a different context [36, 37]. The representation category decomposes into subcategories as follows. For U q sℓ (2) ,the familiar (“quantum”) Casimir element(3.4) C = ( q − q − ) EF + q − K + q K − satisfies the minimal polynomial relation Ψ p ( C ) = 0 , where [9](3.5) Ψ p ( x ) = ( x − β ) ( x − β p ) p − Y s =1 ( x − β s ) , β s = q s + q − s . UANTUM GROUPS IN LOGARITHMIC CFT 11
This relation yields a decomposition of the representation category into the direct sum offull subcategories C ( s ) such that ( C − β s ) acts nilpotently on objects in C ( s ) . Because β s = β s ′ for s = s ′ p , there are p + 1 full subcategories C ( s ) for s p . Each C ( s ) with s p − contains precisely two irreducible modules X + s and X − p − s (becausethe Casimir element acts by multiplication with β s on precisely these two) and infinitelymany indecomposable modules. The irreducible modules X + p and X − p corresponding tothe respective eigenvalues β p and β comprise the respective categories C ( p ) and C (0) . The process of constructing the extensions stops at projectivemodules — projective covers of each irreducible representation. Taking direct sums ofprojective modules then gives projective covers of all indecomposable representation.A few irreducible representations are their own projective covers; these are X ± p for U q sℓ (2) and X ± p,p ′ for U p,p ′ . The other irreducible representations have projective coversfiltered by several irreducible subquotients.For U q sℓ (2) , the projective cover P ± r of X ± r , r = 1 , . . . , p − , can be represented as (3.6) X ± r • z z $ $ X ∓ p − r • (cid:25) (cid:25) X ∓ p − r • (cid:5) (cid:5) X ± r • It follows that dim P ± r = 2 p . For U p,p ′ , besides irreducible projective modules of dimension pp ′ , there are p − p ′ − projective modules of dimension pp ′ and p − p ′ − projectivemodules of dimension pp ′ (see [11], where a diagram with 16 subquotients is also given). Regarding this picture for projective modules (as well as more involved pictures in [11]),it is useful to keep in mind that because of the periodicity in powers of q , the top and thebottom subquotients sit in the same grade (measured by eigenvalues of the Cartan genera-tor K ), as do the two “side” subquotients. A picture that makes this transparent and whichshows the states in projective modules can be drawn as follows. Taking U q sℓ (2) with p = 5 and choosing r = 3 for example, we first represent the X − r = X − and X + p − r = X +2 In diagrams of this type, first, the arrows are directed towards submodules; second, it is understoodthat the quantum group action on each irreducible representation is changed in agreement with the arrowsconnecting a given subquotient with others. This is of course true for the “two-floor” indecomposablemodules considered above, but is even more significant for the projective modules, where the X ± r • −→ X ∓ p − r • extensions alone do not suffice to describe the quantum group action. Constructing the quantum groupaction there requires some more work, but is not very difficult for each of the quantum groups consideredhere, as explicit formulas in [9, 11] show. irreducible modules as ••• ooooo ????? and • • ZZZZ and then construct their extension • • ••• ooooo ?????ZZZZ (cid:7) (cid:7) (cid:14)(cid:14)(cid:14)(cid:14)(cid:14) which actually gives a Verma module (the arrowis directed to the submodule). From this moduleand a contragredient one, we further construct theprojective module P +2 as the extension • • ••• • • ••• ooooo ?????ZZZZ (cid:7) (cid:7) (cid:14)(cid:14)(cid:14)(cid:14)(cid:14)oooooooo ?????????ZZZZZZZZ Q Q $$$$$$$$$ (cid:22) (cid:22) -------- L L (cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24) where pairs of nearby dots represent states that actually sit in the same grade. The results in [35] go be-yond the Grothendieck ring for the quantum group closely related to U q sℓ (2) : tensorproducts of the indecomposable representations are evaluated there. It follows from [35]that the U q sℓ (2) Grothendieck ring (3.1) is in fact the result of “forceful semisimplifi-cation” of the following tensor product algebra of irreducible representations. First, if r + s − p , then, obviously, only irreducible representations occur in the decomposi-tion: X αr ⊗ X βs = r + s − M t = | r − s | +1step=2 X αβt (the sum contains min( r, s ) terms). Next, if r + s − p > and is even, r + s − p = 2 n with n > , then X αr ⊗ X βs = p − r − s − M t = | r − s | +1step=2 X αβt ⊕ n M a =1 P αβp +1 − a . Finally, if r + s − p > and is odd, r + s − p = 2 n + 1 with n > , then X αr ⊗ X βs = p − r − s − M t = | r − s | +1step=2 X αβt ⊕ n M a =0 P αβp − a . We note that in each of the last two formulas, the first sum in the right-hand side contains p − max( r, s ) terms, and therefore disappears whenever max( r, s ) = p (see [35] for thetensor products of other modules in ). (1) It follows that the irreducible U q sℓ (2) representations produce only (themselvesand) projective modules in the tensor algebra. Because tensor products of any UANTUM GROUPS IN LOGARITHMIC CFT 13 modules with projective modules decompose into projective modules, we can con-sistently restrict ourself to only the irreducible and projective modules (in otherwords, there is a subring in the tensor algebra). This is a very special situation,however, specific to U q sℓ (2) (and the slightly larger algebra in [35]); generically,indecomposable representations other than the projective modules occur in tensorproducts of irreducible representations. In particular, the true tensor algebra behindthe Grothendieck ring in (3.3) is likely to involve various other indecomposable modulesin the product of irreducible representations.
Already for U q sℓ (2) , specifying the full tensor algebra means evaluating the products of all the representations listedin .(2) Reiterating the point in [10], we note that the previous remark fully applies to fu-sion of the chiral algebra (triplet W -algebra [38, 8, 27]) representations in ( p, logarithmic conformal field theory models, once the fusion is taken not in the K -version but with “honest” indecomposable representations [24, 25]. Whileit is possible to consider such a fusion of only irreducible and projective W -algebra modules, the full fusion algebra must include all of the “ W ,” “ M ,” and“ O ” indecomposable modules of the triplet algebra (with the last ones, somewhatintriguingly, being dependent on z ∈ CP ). Projective modules serve another, somewhat technical but usefulpurpose. It was noted in that traces over irreducible representations do not span theentire space Ch of q -characters. Projective modules provide what is missing: they allowconstructing pseudotraces Tr P ( g − ? σ ) (for certain maps and modules σ : P → P ) thattogether with the traces Tr X ( g − ?) over irreducible representations span all of Ch : abasis γ A in Ch can be constructed such that with a subset of the γ A is given by tracesover irreducible representations and the rest by pseudotraces associated with projectivemodules in each full subcategory.The strategy for constructing the pseudotraces is as follows. For any (reducible) module P and a map σ : P → P , the functional(3.7) γ : x Tr P ( g − xσ ) is a q -character if and only if (cf. (A.2))(3.8) γ ( xy ) − γ ( S ( y ) x ) ≡ Tr P ( g − x [ y, σ ]) . It is possible to find reducible indecomposable modules P and maps σ satisfying (3.8).This requires taking P to be the projective module in a chosen full subcategory (one ofthose containing more than one module). For U q sℓ (2) , this is(3.9) P r = P + r ⊕ P − p − r , r p − , I thank V. Schomerus for this remark and a discussion of this point. and for U p,p ′ this is the direct sum(3.10) P r,r ′ = P + r,r ′ ⊕ P − p − r,r ′ ⊕ P − r,p ′ − r ′ ⊕ P + p − r,p ′ − r ′ , plus the “boundary” cases where either r = p or r ′ = p ′ , with two terms in the sum (here, P ± r,r ′ is the projective cover of the irreducible representation X ± r,r ′ ). In all cases, σ is a linearmap that sends the bottom module in the filtration of each projective module into “thesame” module at a higher level in the filtration. Such maps are not defined uniquely (e.g.,they depend on the choice of the bases and, obviously, on the “admixture” of lower-lyingmodules in the filtration), but anyway, taken together with the traces over irreduciblerepresentations, they allow constructing a basis in Ch . For U q sℓ (2) , there is a singlepseudotrace for each r in (3.9) obtained by letting σ send the bottom of both diamondsof type (3.6) into the top. This gives just p − linearly independent elements of Ch . Thestructure for U p,p ′ is somewhat richer, and the counting goes as follows [11]. There are not onebut three other copies of the bottom subquotient in each of the four projective modules in (3.10).For the parameters thus emerging, constraints follow from (3.8). Of the remaining differentmaps satisfying (3.8), there is just one (the map to the very top) for each of the ( p − p ′ − modules of form (3.10), two for each of the ( p − p ′ modules, and two more for each of the p ( p ′ − modules. This gives the total of ( p − p ′ −
1) + ( p − p ′ + p ( p ′ − linearly independent pseudotraces. Together with the traces over pp ′ irreducible representations,we thus obtain (3 p − p ′ − linearly independent elements of Ch . Radford-map images of the basis γ A of traces and pseudotracesin Ch give a basis b φφφ A = b φφφ ( γ A ) in the quantum-group center Z . This basis plays an important role in what follows, beingone of the two special bases related by S ∈ SL (2 , Z ) . The other special basis is associatedwith the Drinfeld map considered in the next section. Projective modules are also a crucial ingre-dient in finding the quantum group center. Central elements are in a correspondencewith bimodule endomorphism of the regular representation. We recall that viewed as aleft module, the regular representation decomposes into projective modules, each enter-ing with the multiplicity given by the dimension of its simple quotient. Generalizing thispicture to a bimodule decomposition shows that the multiplicities are in fact tensor fac-tors with respect to the right action. A typical block of the bimodule decomposition ofthe regular representation looks as follows: with respect to the left action, it is a sumof projective modules in one full subcategory, with each projective (externally) tensored
UANTUM GROUPS IN LOGARITHMIC CFT 15 with a suitable simple module. For U q sℓ (2) , where the subquotients are few and thereforethe picture is not too complicated, it can be drawn as [9] X + r ⊠ X + r } } {{{{{{ ! ! CCCCCC X − p − r ⊠ X − p − r } } {{{{{{ ! ! CCCCCC X − p − r ⊠ X + r ! ! CCCCCC X − p − r ⊠ X + r } } {{{{{{ X + r ⊠ X − p − r ! ! CCCCCC X + r ⊠ X − p − r } } {{{{{{ X + r ⊠ X + r X − p − r ⊠ X − p − r With respect to the right action, the picture is totally symmetric, but with the subquotientsplaced as above, the structure of their extensions has to be drawn as X + r ⊠ X + r ' ' * * X − p − r ⊠ X − p − r r r s s X − p − r ⊠ X + r , , X − p − r ⊠ X + r ) ) X + r ⊠ X − p − r r r X + r ⊠ X − p − r p p X + r ⊠ X + r X − p − r ⊠ X − p − r Pictures of this type immediately yield the number of central elements and their asso-ciative algebra structure . First, each block yields a primitive idempotent e I , which is justthe projector on this block; second, there are maps sending A ⊠ B bimodules into “thesame” bimodules at lower levels, yielding nilpotent central elements. For U q sℓ (2) , thebimodule decomposition contains p − blocks of the above structure, plus two more givenby X + p ⊠ X + p and X − p ⊠ X − p ; in each of the “complicated” blocks, there are two bimoduleautomorphisms under which either the top X + r ⊠ X + r or the top X − p − r ⊠ X − p − r goes intothe corresponding bottom one, yielding two two central elements w + r and w − r with zeroproducts among themselves. Therefore, the (3 p − -dimensional center decomposes intoa direct sum of associative algebras as Z U q sℓ (2) = I (1) p ⊕ I (1)0 ⊕ p − M r =1 B (3) r , where the dimension of each algebra is shown as a superscript. For U p,p ′ , there are several intermediate levels in the filtration of projective modules, and hencethe nilpotent elements are more numerous and have a nontrivial multiplication table (see [11] forthe details); the center is (3 p − p ′ − -dimensional and decomposes into a direct sum ofassociative algebras as(3.11) Z U p,p ′ = I (1) p,p ′ ⊕ I (1)0 ,p ′ ⊕ p − M r =1 B (3) r,p ′ ⊕ p ′ − M r ′ =1 B (3) p,r ′ ⊕ M r,r ′ ∈I A (9) r,r ′ , where the dimension of each algebra is shown as a superscript (and where |I | = ( p − p ′ − ).The B (3) algebras are just as in the previous formula, and each A (9) r,r ′ is spanned by a primitive idempotent e r,r ′ (acting as identity on A (9) r,r ′ ) and eight radical elements v ր r,r ′ , v ւ r,r ′ , v տ r,r ′ , v ց r,r ′ , w ↑ r,r ′ , w → r,r ′ , w ↓ r,r ′ , w ← r,r ′ that have the nonzero products v ր r,r ′ v տ r,r ′ = w ↑ r,r ′ , v ր r,r ′ v ց r,r ′ = w → r,r ′ , v ւ r,r ′ v տ r,r ′ = w ← r,r ′ , v ւ r,r ′ v ց r,r ′ = w ↓ r,r ′ . D RINFELD MAP AND FACTORIZABLE AND RIBBON STRUCTURES M -matrix and the Drinfeld map. For a quasitriangular Hopf algebra U with theuniversal R -matrix R , the M -matrix is the “square” of the R -matrix, defined as M = R R ∈ U ⊗ U. It satisfies the relations (∆ ⊗ id)( M ) = R M R ,M ∆( x ) = ∆( x ) M ∀ x ∈ U. (4.1) Indeed, using (A.5), we find (∆ ⊗ id)( R ) = R R and then using (A.4) we obtain thefirst relation in (4.1). Next, it follows from (A.3) that R R ∆( x ) = ( R ∆( x )) op R =(∆ op ( x ) R ) op R = ∆( x ) R R , that is, the second relation in (4.1) .The Drinfeld map χ : U ∗ → U is defined as χ : β ( β ⊗ id)( M ) , that is, if we write the M -matrix as(4.2) M = X I m I ⊗ n I , then χ ( β ) = P I β ( m I ) n I .Whenever χ : U ∗ → U is an isomorphism of vector spaces, the Hopf algebra U iscalled factorizable [39]. Equivalently, this means that m I and n I in (4.2) are two bases in U . ([40]) . In a factorizable Hopf algebra U , by restriction to Ch ( see A.1 ) ,the Drinfeld map defines a homomorphism Ch ( U ) → Z ( U ) of associative algebras.Proof. We first show that χ ( β ) is central for any β ∈ Ch : for any x ∈ U , we calculate χ ( β ) x = P I β ( m I ) n I x = P I β ( m I x ′′ S − ( x ′ )) n I x ′′′ . But because M ∆( x ) = ∆( x ) M and β ( xy ) = β ( S ( y ) x ) , we obtain that χ ( β ) x = P I β ( S ( x ′ ) x ′′ m I ) x ′′′ n I = x χ ( β ) .Next, to show that χ : Ch → Z is a homomorphism of associative algebras, we recall that theproduct of two functionals is defined as βγ ( x ) = ( β ⊗ γ )(∆( x )) , and therefore, using the first UANTUM GROUPS IN LOGARITHMIC CFT 17 relation in (4.1), we have χ ( βγ ) = ( β ⊗ γ ⊗ id)((∆ ⊗ id)( M )) = ( β ⊗ γ ⊗ id)( R M R ) =( γ ⊗ id)( R χ ( β ) R ) = χ ( β )( γ ⊗ id)( R R ) = χ ( β )( γ ⊗ id)( M ) = χ ( β ) χ ( γ ) . (cid:3) For the quantum groups considered here, the above homomorphism is in fact an iso-morphism (cf. [40, 41]). M -matrix, and R -matrix. The quantum groups U originating from logarithmic conformal models are notquasitriangular , but are nevertheless factorizable in the following sense: the M -matrixcan be expressed through an R that is the universal R -matrix of a somewhat larger quan-tum group ¯ D . This is true for both U q sℓ (2) and U p,p ′ , with the extension to ¯ D realizedin each case by introducing the generator k = K / . In other words, in each case, thereis a quasitriangular quantum group ¯ D with a set of generators k, . . . , with a universal R -matrix R , such that R R turns out to belong to U ⊗ U , where U is the Hopf subalgebrain ¯ D generated by K = k and the other ¯ D generators. In the respective cases, U is either U q sℓ (2) or U p,p ′ .The universal R -matrix for ¯ D , in turn, comes from constructing the Drinfeld dou-ble [42] of the quantum group B generated by screenings in the logarithmic conformalfield model [9, 13]. For ( p, models, B is the Taft Hopf algebra with generators E and k , with kEk − = q E , E p = 0 , and k p = 1 . We then take the dual space B ∗ , whichis a Hopf algebra with the multiplication, comultiplication, unit, counit, and antipodegiven by(4.3) h βγ, x i = h β, x ′ ih γ, x ′′ i , h ∆( β ) , x ⊗ y i = h β, yx i , h , x i = ǫ ( x ) , ǫ ( β ) = h β, i , h S ( β ) , x i = h β, S − ( x ) i for β, γ ∈ B ∗ and x, y ∈ B . The Drinfeld double D ( B ) is a Hopf algebra with theunderlying vector space B ∗ ⊗ B and with the multiplication, comultiplication, unit, counit,and antipode given by those in B , by Eqs. (4.3), and by(4.4) xβ = β ( S − ( x ′′′ )? x ′ ) x ′′ , x ∈ B, β ∈ B ∗ . The resulting Hopf algebra D ( B ) is canonically endowed with the universal R -matrix [42].The doubling procedure also introduces the dual κ to the Cartan element k , which isthen to be eliminated by passing to the quotient over (the Hopf ideal generated by) kκ − It was noted in [41] that “Drinfeld’s proof of [40, 3.3] shows more than what is actually stated in [40,3.3].” It actually follows that χ ( βγ ) = χ ( β ) χ ( γ ) whenever β ∈ Ch ( U ) and γ ∈ U ∗ . The standard definition of a factorizable quantum group [39] involves the universal R -matrix as well,which is the reason why we express some caution; the R -matrix in the M -matrix property (4.1) is not anelement of U ⊗ U . In particular, U is not unimodular in our case (but U ∗ is!). The screenings generate only the upper-triangular subalgebra of the Kazhdan–Lusztig-dual quantumgroup; to these upper-triangular subalgebra, we add Cartan generator(s) constructed from zero modes ofthe free fields involved in the chosen free-field realization. This gives the B quantum group. (it follows that kκ is central in the double). The quotient ¯ D is still quasitriangular, butevaluating the M -matrix and the ribbon element for it shows that they are turn out tobe those for the (Hopf) subalgebra generated by K ≡ k and the other ¯ D generators,which is finally the Kazhdan–Lusztig-dual quantum group. This was how the Kazhdan–Lusztig-dual quantum groups, together with the crucial structures on them, were derivedin [9, 13].For U q sℓ (2) , for example, the M -matrix is explicitly expressed in terms of the PBWbasis as(4.5) M = p p − X m =0 p − X n =0 2 p − X i =0 2 p − X j =0 ( q − q − ) m + n [ m ]![ n ]! q m ( m − / n ( n − / × q − m − mj +2 nj − ni − ij + mi F m E n K j ⊗ E m F n K i . In a factorizable Hopf algebra,it follows that the Drinfeld-map images of the traces over irreducible representations forman algebra isomorphic to the Grothendieck ring. Thus, there are central elements χ ± r = χ (Tr X ± r ( g − ?)) , r p for U q sℓ (2) and χ ± r,r ′ = χ (Tr X ± r,r ′ ( g − ?)) , r p, r ′ p ′ for U p,p ′ , which satisfy the respective algebra (3.1) and (3.3). In U q sℓ (2) , for example, Eq. (4.5) allows us to calculate the χ αs explicitly,(4.6) χ αs = α p +1 ( − s +1 s − X n =0 n X m =0 ( q − q − ) m q − ( m +1)( m + s − − n ) ×× (cid:20) s − n + m − m (cid:21)(cid:20) nm (cid:21) E m F m K s − βp − n + m (where β = 0 if α = +1 and β = 1 if α = − ). In particular, χ +2 = − C , where C is theCasimir element, Eq. (3.4). The fact that the χ ± r given by (4.6) satisfy Grothendieck-ringrelations (3.1) implies a certain q -binomial identity, see [9]. The U q sℓ (2) Casimir element satisfies the minimal polynomial relation Ψ p ( C ) = 0 , with Ψ p in (3.5). This relation, with p − multiplicity- roots of Ψ p ,allows constructing a basis in the center Z of U q sℓ (2) consisting of primitive idempotents e r and elements w r in the radical of the associative commutative algebra Z [9] (see [23]and also [43, Ch. V.2]). For this, we define the polynomials ψ ( x ) = ( x − β p ) p − Y r =1 ( x − β r ) , ψ p ( x ) = ( x − β ) p − Y r =1 ( x − β r ) , UANTUM GROUPS IN LOGARITHMIC CFT 19 ψ s ( x ) = ( x − β ) ( x − β p ) p − Y r =1 r = s ( x − β r ) , s p − , where we recall that all β j are distinct. Then the canonical elements in the radical of Z are w ± s = π ± s w s , s p − , where w s = ψ s ( β s ) (cid:0) C − β s (cid:1) ψ s ( C ) and we introduce the projectors π + s = p s − X n =0 2 p − X j =0 q (2 n − s +1) j K j , π − s = p p − X n = s p − X j =0 q (2 n − s +1) j K j , and the canonical central idempotents are given by e s = ψ s ( β s ) (cid:0) ψ s ( C ) − ψ ′ s ( β s ) w s (cid:1) , s p, where we formally set w = w p = 0 . A similar construction exists for the center of U p,p ′ [11],where, in particular, there are not two but four types of projectors π ↑ r,r ′ , π ← r,r ′ , π → r,r ′ , and π ↓ r,r ′ ;for either algebra, these are projectors on the weights occurring in irreducible modules in the fullsubcategory labeled by the subscript . Applied to the basis γ A of traces and pseudotraces in Ch , theDrinfeld map gives a basis χ A = χ ( γ A ) in the center Z .This “Drinfeld” basis (which is not defined uniquely because pseudotraces are not de-fined uniquely) specifies an explicit splitting of the associative commutative algebra Z into the Grothendieck ring and its linear complement. The products of the Grothendieckring elements with elements from the complement may also be of significance in theKazhdan–Lusztig context. The full algebra of q -characters (traces and pseudotraces) for U q sℓ (2) , mapped into the center by the Drinfeld map, is evaluated in [44]; it can be un-derstood as a generalized fusion, to be compared with a recent calculation to this effect inthe logarithmic ( p, models in [34].Under S ∈ SL (2 , Z ) acting as in (1.7), clearly, the Drinfeld basis elements are mappedinto the Radford basis, S : χ A b φφφ A . Realizing T ∈ SL (2 , Z ) on the center requires yet another structure, the ribbon element. A ribbon element [45] is a v ∈ Z such that ∆( v ) = M − ( v ⊗ v ) , with ǫ ( v ) = 1 and S ( v ) = v (and v = u S ( u ) , see A.3 ). The procedure for finding theribbon element involves two steps: we first find the canonical element (A.7) (which in-volves the universal R -matrix for the larger, quasitriangular quantum group ¯ D mentionedin ) and then evaluate the balancing element g (see A.4 ) in accordance with Drinfeld’sLemma (A.11), from the comodulus obtained from the explicit expression for the integral(this is the job done by the comodulus). Then(4.7) v = ug − . It follows, again, that v is an element of a Hopf subalgebra in ¯ D , which is U q sℓ (2) or U p,p ′ . For U q sℓ (2) , where g = K p +1 , we have [9](4.8) v = p X s =0 ( − s +1 q − ( s − e s + p − X s =1 ( − p q − ( s − [ s ] q − q − √ p b ϕϕϕ s , where e s are the canonical idempotents in the center and(4.9) b ϕϕϕ s = p − sp b φφφ + s − sp b φφφ − p − s , s p − , are nilpotent central elements expressed through the Radford-map images b φφφ ± s of the(traces over) irreducible representations X ± s . The above form of v implies that [10] v = e iπL (where L is the zero-mode Virasoro generator in the ( p, logarithmic conformal model);in particular, the exponents involving s in (4.8) are simply related to conformal dimen-sions of primary fields. Rather interestingly, the nonsemisimple action of L on the latticevertex operator algebra underlying the construction of the logarithmic ( p, model is thuscorrelated with the decomposition of the ribbon element with respect to the central idem-potents and nilpotents. For U p,p ′ , the ribbon element is given by v = X ( r,r ′ ) ∈I e iπ ∆ r,r ′ e r,r ′ + nilpotent terms , where e r,r ′ are the ( p + 1)( p ′ + 1) primitive idempotents in the associative commutative algebra Z (the explicit form of the nilpotent terms being not very illuminating at this level of detail, see [11]for the full formula), and ∆ r,r ′ = ( pr ′ − p ′ r ) − ( p − p ′ ) pp ′ are conformal dimensions of primary fields borrowed from the logarithmic model [13]. UANTUM GROUPS IN LOGARITHMIC CFT 21 M ODULAR GROUP ACTION
In defining the modular group action on the center we fol-low [21, 22, 23] with an insignificant variation in the definition of T , introduced in [9, 13]in order to simplify comparison with the modular group representation generated fromcharacters of the chiral algebra in the corresponding logarithmic conformal model. Onthe quantum group center, the SL (2 , Z ) -action is defined by S : x b φφφ (cid:0) χ − ( x ) (cid:1) , T : x e − iπ c S ( v S − ( x )) , (5.1)where c is the central charge of the conformal model, e.g., c = 13 − pp ′ − p ′ p for the ( p, p ′ ) model . The result of evaluating (5.1) in each case gives the structure ofthe SL (2 , Z ) representation of the type that was first noted in [23] for the small quantum sℓ (2) . On the center of U q sℓ (2) , the SL (2 , Z ) representation is given by [9](5.2) Z U q sℓ (2) = R p +1 ⊕ C ⊗ R p − , where C is the defining two-dimensional representation, R p − is a ( p − -dimensional SL (2 , Z ) -representation (the “ sin πrsp ” representation, in fact, the one on the unitary b sℓ (2) k -characters at the level k = p − ), and R p +1 is a “ cos πrsp ” ( p + 1) -dimensional represen-tation. On the center of U p,p ′ , the SL (2 , Z ) -representation structure is given by [11](5.3) Z U p,p ′ = R min ⊕ R proj ⊕ C ⊗ ( R (cid:28) ⊕ R (cid:27) ) ⊕ C ⊗ R min , where C is the symmetrized square of C , R min is the ( p − p ′ − -dimensional SL (2 , Z ) -representation on the characters of the rational ( p, p ′ ) Virasoro model, and R proj , R (cid:28) , and R (cid:27) arecertain SL (2 , Z ) representations of the respective dimensions ( p + 1)( p ′ + 1) , ( p + 1)( p ′ − ,and ( p − p ′ + 1) . As noted above, (5.2) and (5.3) coincide with the respective SL (2 , Z ) -representationson generalized characters of ( p, and ( p, p ′ ) logarithmic conformal field models evalu-ated in [9, 13]. Reversing the argument, for a factorizable ribbon quantum group that can be expected to correspondto a conformal field model, the normalization of T (i.e., the factor accompanying the ribbon element) maythus indicate the central charge, and the decomposition of the ribbon element into the basis of primitiveidempotents and elements in the radical is suggestive about the conformal dimensions. The small quantum groups have been the subject of some constant interest, see, e.g., [46, 47, 48] andthe references therein.
The role of the subrepresentations identified in (5.2) and (5.3) is yet to be under-stood from the quantum-group standpoint, but it is truly remarkable in the context of theKazhdan–Lusztig correspondence. The occurrence of the C n tensor factors is rigorouslycorrelated with the fact that the ψ b ′ ( τ ) functions in (1.1)–(1.2) are given by (certain linearcombinations of) characters times polynomials in τ of degree n − . In the quantum group center, the subrepresentations in (5.2) and (5.3) are describedas the span of certain combinations of the elements of “Radford” and “Drinfeld” bases b φφφ A and χ A . For U q sℓ (2) , in particular, the central elements (4.9), together with their S -images p − sp χ + s − sp χ − p − s , s p − , span the C ⊗ R p − representation; in thelogarithmic ( p, model, the same representation is realized on the p − functions τ (cid:0) p − sp χ + s ( τ ) − sp χ − p − s ( τ ) (cid:1) , p − sp χ + s ( τ ) − sp χ − p − s ( τ ) , where χ ± r ( τ ) are the triplet algebra characters [9]. On the other hand, the ( p + 1) -dimen-sional representation R p +1 in the center is linearly spanned by χ ± p and χ + s + χ − p − s , s p − (the ideal already mentioned after (3.2)); in the ( p, model, the same rep-resentation is realized on the linear combinations of characters χ ± p ( τ ) , χ + s ( τ ) + χ − p − s ( τ ) . The U p,p ′ setting in [11, Sec. ] gives rather an abundant picture of how the various traces andpseudotraces , mapped into the center, combine to produce the subrepresentations and how pre-cisely these linear combinations correspond to the characters and extended characters in the loga-rithmic ( p, p ′ ) model. Here, we only note the R proj representation, of dimension ( p + 1)( p ′ + 1) ,linearly spanned by χ + r,r ′ + χ − p − r,r ′ + χ − r,p ′ − r ′ + χ + p − r,p ′ − r ′ (with ( r, r ′ ) ∈ I , where |I | = ( p − p ′ − ), χ + r,p ′ + χ − p − r,p ′ (with r p − ), χ + p,r ′ + χ − p,p ′ − r ′ ( r ′ p ′ − ), and χ ± p,p ′ . In the logarithmic ( p, p ′ ) model, the same SL (2 , Z ) -representation is realized on the linearcombinations of W -algebra characters χ r,r ′ ( τ ) + 2 χ + r,r ′ ( τ ) + 2 χ − r,p ′ − r ′ ( τ ) + 2 χ − p − r,r ′ ( τ ) + 2 χ + p − r,p ′ − r ′ ( τ ) , ( r, r ′ ) ∈ I , χ + p,p ′ − r ′ ( τ ) + 2 χ − p,r ′ ( τ ) , r ′ p ′ − , χ + p − r,p ′ ( τ ) + 2 χ − r,p ′ ( τ ) , r p − , χ ± p,p ′ ( τ ) (with the same size | I | = ( p − p ′ − of the index set), where χ r,r ′ ( τ ) are the characters ofthe Virasoro rational model and χ ± r,r ′ ( τ ) are the other pp ′ characters of the W -algebra [13]. Theabove combinations do not involve generalized characters (which occur where the C n factors areinvolved in the SL (2 , Z ) -representation isomorphic to the one in (5.3) and which are in fact theorigin of these C n factors from the conformal field theory standpoint). Once again: C -linear combinations of the (3 p − p ′ − traces and pseudotraces (mapped tothe center) carry the same SL (2 , Z ) -representations as certain C [ τ ] -linear combinations of the pp ′ + ( p − p ′ − characters of the W -algebra; the total dimension is (3 p − p ′ − in either case. UANTUM GROUPS IN LOGARITHMIC CFT 23
A remarkable feature of the SL (2 , Z ) representation on the U p,p ′ center is the occurrence of R min , the SL (2 , Z ) -representation on the characters of the rational Virasoro model, even thoughthe U p,p ′ -representations X ± r,r ′ are in a correspondence not with all the primary fields of the W -algebra in the logarithmic model but just with those except the rational-model ones. Two algebraic structures on the quantum group cen-ter are most important from the standpoint of the Kazhdan–Lusztig correspondence: themodular group action and the Grothendieck ring (the latter is a subring in the centerspanned by Drinfeld-map images of the irreducible representations). The resulting Groth-endieck rings, or Verlinde algebras are nonsemisimple.A classification of Verlinde algebras has been proposed in a totally different approach,that of double affine Hecke algebras (Cherednik algebras) [49], where Verlinde algebrasoccur as certain representations of Cherednik algebras; an important point is that a mod-ular group action is built into the structure of Cherednik algebras. It can thus be expectedthat the ( p, -model fusion (the U q sℓ (2) Grothendieck ring) (3.1), of dimension p , ad-mits a realization associated with a Cherednik algebra representation. But because anisomorphic image of the Grothendieck ring is contained in the center, a natural furtherquestion is whether the entire U q sℓ (2) center, of dimension p − , endowed with the SL (2 , Z ) action, is also related to Cherednik algebras.It was shown in [50] that the center Z of U q sℓ (2) , as an associative commutative algebraand as an SL (2 , Z ) representation, is indeed extracted from a representation space of thesimplest Cherednik algebra H , defined by the relations T XT = X − , T Y − T = Y,XY = q Y XT , ( T − q )( T + q − ) = 0 on the generators T , X , Y , and their inverse. In these terms, the P SL (2 , Z ) action isdefined by the elements τ + = (cid:0) (cid:1) and τ − = (cid:0) (cid:1) being realized as the H automor-phisms [49] τ + : X X, Y q − / XY, T T,τ − : X q / Y X, Y Y, T T. For each p > , the authors of [50] construct a (6 p − -dimensional (reducible butindecomposable) representation of H in which the eigensubspace of T with eigenvalue q (as before, q = e iπ/p ) is (3 p − -dimensional. The associative commutative algebrastructure induced on this eigensubspace in accordance with Cherednik’s theory then co-incides with the associative commutative algebra structure on the center Z of U q sℓ (2) .Furthermore, the SL (2 , Z ) representations constructed on this space `a la Cherednik and`a la Lyubashenko coincide. Also, the Radford- and Drinfeld-map images of irreducible representations in the center can be “lifted” to the level of H (as eigenvectors of X + X − and Y + Y − respectively) [50]. C ONCLUSIONS
Without a doubt, it would be extremely useful to rederive the results such as the equiv-alence of modular group representations in a more “categorical” approach; this wouldimmediately suggest generalizations. But the quantum group “next in the queue” after U q sℓ (2) and U p,p ′ is a quantum sℓ (2 | (cf. the remarks in [51]), which already requiresextending many basic facts (e.g., those in [21]) to the case of quantum super groups.The center of the Kazhdan–Lusztig-dual quantum group is to be regarded as the centerof the corresponding logarithmic conformal field model; this calls for applications toboundary states in logarithmic models. Acknowledgments.
I am grateful to M. Finkelberg and J. Fuchs for useful comments andto V. Schomerus for discussions. Special thanks go to A. Gainutdinov for his criticism.This paper was supported in part by the RFBR grant 07-01-00523.A
PPENDIX
A.A.1. The center and q -characters. The center Z of a Hopf algebra U can be character-ized as the set(A.1) Z = (cid:8) y ∈ U (cid:12)(cid:12) Ad x ( y ) = ǫ ( x ) y ∀ x ∈ U (cid:9) . The space of q -characters Ch = Ch ( U ) ⊂ U ∗ is defined as(A.2) Ch = (cid:8) β ∈ U ∗ (cid:12)(cid:12) Ad ∗ x ( β ) = ǫ ( x ) β ∀ x ∈ U (cid:9) = (cid:8) β ∈ U ∗ (cid:12)(cid:12) β ( xy ) = β (cid:0) S ( y ) x (cid:1) ∀ x, y ∈ U (cid:9) , where the coadjoint action Ad ∗ a : U ∗ → U ∗ is Ad ∗ a ( β ) = β (cid:0) S ( a ′ )? a ′′ (cid:1) , a ∈ U , β ∈ U ∗ . A.2. Quasitriangular Hopf algebras.
Quasitriangular (or braided) Hopf algebras wereintroduced in [42] (also see [52]). A quasitriangular Hopf algebra U has an invertibleelement R ∈ U ⊗ U satisfying ∆ op ( x ) = R ∆( x ) R − , (A.3) (∆ ⊗ id)( R ) = R R , (A.4) (id ⊗ ∆)( R ) = R R , (A.5)the Yang–Baxter equation R R R = R R R , (A.6)and the relations ( ǫ ⊗ id)( R ) = 1 = (id ⊗ ǫ )( R ) , ( S ⊗ S )( R ) = R . UANTUM GROUPS IN LOGARITHMIC CFT 25
A.3. Square of the antipode [40] . In any quasitriangular Hopf algebra, the square ofthe antipode is represented by a similarity transformation S ( x ) = u x u − , where the canonical element u is given by u = · (cid:0) ( S ⊗ id) R (cid:1) , u − = · (cid:0) ( S − ⊗ S ) R (cid:1) (A.7)(where · ( a ⊗ b ) = ab ) and satisfies the property ∆( u ) = M − ( u ⊗ u ) = ( u ⊗ u ) M − (A.8)(where we recall that M = R R ). A.4. Balancing element.
We also need the so-called balancing element g ∈ U thatsatisfies [40](A.9) S ( x ) = g x g − ∀ x ∈ U, ∆( g ) = g ⊗ g , The balancing element g allows constructing the “canonical” q -character associatedwith any (irreducible, because traces are insensitive to indecomposability) representa-tion X as the (“quantum”) trace(A.10) Tr X ( g − ?) ∈ Ch ( U ) . For a Hopf algebra U with a right integral λ , we recall the definition of a comodulusin . Whenever a square root of the comodulus a can be calculated in U , a lemma ofDrinfeld [40] states that(A.11) g = a . R EFERENCES [1] L. Alvarez-Gaum´e, C. Gomez, and G. Sierra,
Quantum group interpretation of some conformal fieldtheories , Phys. Lett. B 220 (1989) 142–152;
Hidden quantum symmetry in rational conformal fieldtheories , Nucl. Phys. B310 (1989) 155.[2] G. Moore and N. Reshetikhin,
A comment on quantum symmetry in conformal field theory , Nucl.Phys. B328 (1989) 557–574.[3] V. Pasquier and H. Saleur,
Common structures between finite systems and conformal field theoriesthrough quantum groups , Nucl. Phys. B330 (1990) 523.[4] G. Mack and V. Schomerus,
Conformal field algebras with quantum symmetry from the theory ofsuperselection sectors , Commun. Math. Phys. 134 (1990) 139.[5] 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.[6] M. Finkelberg,
An equivalence of fusion categories , Geometric and Functional Analysis (GAFA) 6(1996) 249–267. [7] V. Gurarie,
Logarithmic operators in conformal field theory,
Nucl. Phys. B410 (1993) 535 [hep-th / A rational logarithmic conformal field theory,
Phys. Lett. B386(1996) 131–137 [hep-th / Modular group representationsand fusion in logarithmic conformal field theories and in the quantum group center , Commun. Math.Phys. 265 (2006) 47–93 [hep-th / Kazhdan–Lusztig correspon-dence for the representation category of the triplet W -algebra in logarithmic CFT , Theor. Math.Phys. 148 (2006) 1210–1235 [math.QA / Kazhdan–Lusztig-dual quan-tum group for logarithmic extensions of Virasoro minimal models , J. Math. Phys. 48 (2007) 032303[math.QA / Logarithmic extensions of min-imal models: characters and modular transformations , Nucl. Phys. B757 (2006) 303–343 [hep-th / b sℓ (2) ⊕ b sℓ (2) / b sℓ (2) Coset theory as a Hamiltonian reductionof b D (2 | α ) , Nucl. Phys. B610 (2001) 489–530 [hep-th / W (2) n Algebras , Nucl. Phys. B698 (2004) 409–449 [math.QA / BRST approach to minimal models , Nucl. Phys. B317 (1989) 215–236.[17] P. Bouwknegt, J. McCarthy, and K. Pilch,
Fock space resolutions of the Virasoro highest weight mod-ules with c , Lett. Math. Phys. 23 (1991) 193–204 [hep-th / On modular invariant partition functions of conformal field theories with logarithmicoperators , Int. J. Mod. Phys. A11 (1996) 4147 [hep-th/9509166].[19] J. Fuchs, S. Hwang, A.M. Semikhatov, and I.Yu. Tipunin,
Nonsemisimple fusion algebras and theVerlinde formula , Commun. Math. Phys. 247 (2004) 713–742 [hep-th / Logarithmic torus amplitudes , J. Phys. A39 (2006) 1955–1968 [hep-th / Invariants of -manifolds and projective representations of mapping class groupsvia quantum groups at roots of unity , Commun. Math. Phys. 172 (1995) 467–516 [hep-th / Modular properties of ribbon abelian categories , Symposia Gaussiana, Proc. of the 2nd Gauss Sym-posium, Munich, 1993, Conf. A (Berlin, New York), Walter de Gruyter, (1995) 529–579 [hep-th / Modular Transformations for Tensor Categories , J. Pure Applied Algebra 98 (1995) 279–327.[22] V. Lyubashenko and S. Majid,
Braided groups and quantum Fourier transform , J. Algebra 166 (1994)506–528.[23] T. Kerler,
Mapping class group action on quantum doubles , Commun. Math. Phys. 168 (1995) 353–388 [hep-th / Indecomposable fusion products , Nucl. Phys. B477 (1996) 293[hep-th / An algebraic approach to logarithmic conformal field theory [hep-th / Virasoro representations and fusion for general augmented minimal models ,J. Phys. A39 (2006) 15245–15286 [hep-th / UANTUM GROUPS IN LOGARITHMIC CFT 27 [27] M.R. Gaberdiel and H.G. Kausch,
A local logarithmic conformal field theory,
Nucl. Phys. B538 (1999)631–658 [hep-th / Logarithmic conformal fieldtheories via logarithmic deformations , Nucl. Phys. B633 (2002) 379–413 [hep-th / Quantum Groups , Springer-Verlag, New York, 1995.[31] D.E. Radford,
The trace function and Hopf algebras , J. Alg. 163 (1994) 583–622.[32] D.E. Radford,
Minimal quasitriangular Hopf algebras , J. of Algebra 157 (1993) 285–315.[33] J. Fuchs,
On non-semisimple fusion rules and tensor categories , Contemp. Math., to appear [hep-th / On Verlinde-like formulas in c p, logarithmic conformal field theories ,arXiv:0705.0545.[35] K. Erdmann, E.L. Green, N. Snashall, and R. Taillefer, Representation theory of the Drinfeld doublesof a family of Hopf algebras,
J. Pure and Applied Algebra 204 (2006) 413–454 [math.RT / Representation theory of sl (2 | , J. of Algebra 312 (2007)829–848 [hep-th / Free fermion resolution of supergroup WZNW models , arXiv:0706.0744.[38] H.G. Kausch,
Extended conformal algebras generated by a multiplet of primary fields,
Phys. Lett.B 259 (1991) 448.[39] N.Yu. Reshetikhin and M.A. Semenov-Tian-Shansky,
Quantum R -matrices and factorization prob-lems , J. Geom. Phys. 5 (1988) 533–550.[40] V.G. Drinfeld, On almost cocommutative Hopf algebras , Leningrad Math. J. 1 (1990) No. 2, 321–342.[41] H.-J. Schneider,
Some properties of factorizable Hopf algebras .[42] V.G. Drinfeld,
Quantum groups , in: Proc. Int. Cong. Math. (Berkeley, 1986) 798–820. Amer. Math.Soc., Providence, RI (1987).[43] F.R. Gantmakher,
Teoriya Matrits [in Russian], Nauka, Moscow (1988).[44] A.M. Gainutdinov and I.Yu. Tipunin,
Logarithmic models: boundary states and fusion algebra .[45] N.Yu. Reshetikhin and V.G. Turaev,
Ribbon graphs and their invariants derived from quantum groups ,Commun. Math. Phys. 127 (1990) 1–26;
Invariants of 3-manifolds via link polynomials and quantumgroups , Invent. Math. 103 (1991) 547–598.[46] A. Lachowska,
On the center of the small quantum group , math.QA / The small quantum group and the Springer resolution , math / Decomposition of the adjoint representation of the small quantum sl Commun. Math. Phys.186 (1997) 253–264 [q-alg/9512026].[49] I. Cherednik, Double Affine Hecke Algebras (London Mathematical Society Lecture Note SeriesNo. 319), Cambridge University Press (2005);
Introduction to double Hecke algebras , math.QA / Double affine Hecke algebra in logarithmic conformal field theory .[51] A.M. Semikhatov,
Toward logarithmic extensions of b sℓ (2) k conformal field models , hep-th / On the antipode of a quasitriangular Hopf algebra , J. of Algebra 151 (1992) 1–11.L
EBEDEV P HYSICS I NSTITUTE
A M S @ S C I . L E B E D E V ..