Holomorphic Relative Hopf Modules over the Irreducible Quantum Flag Manifolds
Fredy Díaz García, Andrey Krutov, Réamonn Ó Buachalla, Petr Somberg, Karen R. Strung
aa r X i v : . [ m a t h . QA ] J un HOLOMORPHIC RELATIVE HOPF MODULES OVER THEIRREDUCIBLE QUANTUM FLAG MANIFOLDS
FREDY D´IAZ GARC´IA, ANDREY KRUTOV, R´EAMONN ´O BUACHALLA,PETR SOMBERG, AND KAREN R. STRUNG
Abstract.
We construct covariant q -deformed holomorphic structures for allfinitely generated relative Hopf modules over the irreducible quantum flag man-ifolds endowed with their Heckenberger–Kolb calculi. In the classical limit thesereduce to modules of sections of holomorphic homogeneous vector bundles overirreducible flag manifolds. For the case of simple relative Hopf modules, we showthat this covariant holomorphic structure is unique. This generalises earlier workof Majid, Khalkhali, Landi, and van Suijlekom for line modules of the Podle´ssphere, and subsequent work of Khalkhali and Moatadelro for general quantumprojective space. Introduction
In this paper we consider noncommutative generalisations of homogeneous holo-morphic vector bundles for the irreducible quantum flag manifolds. Ideas fromclassical complex geometry have, to a greater or lesser extent, always played arole in noncommutative geometry. This is not surprising, given that no exam-ples can claim to be more influential than the noncommutative torus T θ and thePodle´s sphere O q ( S ). Both are noncommutative deformations of manifolds carry-ing a complex geometry in the classical limit. In fact, both T and S are K¨ahlermanifolds, the former being a Fano manifold and the latter a Calabi–Yau manifold.Much of the classical complex geometry of these examples survives deformationintact. Of particular relevance to this paper is the work of Polishchuk and Schwarzon θ -deformed holomorphic vector bundles over the noncommutative torus [34, 33], Mathematics Subject Classification.
Key words and phrases. quantum groups, noncommutative geometry, quantum principal bun-dles, quantum flag manifolds, complex geometry, holomorphic vector bundles.FDG is partially funded by Conacyt (Consejo Nacional de Ciencia y Tecnolog´ıa, M´exico). AKwas supported by the QuantiXLie Centre of Excellence, a project cofinanced by the Croatian Gov-ernment and European Union through the European Regional Development Fund — the Compet-itiveness and Cohesion Operational Programme (KK.01.1.1.01.0004). R ´OB acknowledges FNRSsupport through a postdoctoral fellowship within the framework of the MIS Grant “Antipode”grant number F.4502.18. R ´OB and PS are partially supported from the Eduard ˇCech Institutewithin the framework of the grant GAˇCR 19 − X , and by the grant GAˇCR 306 − / and Majid’s description of the noncommutative complex geometry of the Podle´ssphere in [26].The notion of a noncommutative complex structure was introduced in [20] and [4]as an abstract framework in which to study the noncommutative complex geom-etry of both the noncommutative torus and the Podle´s sphere. In particular,holomorphic modules, often called noncommutative holomorphic vector bundles,were introduced. The definition of a holomorphic module was motivated by theclassical Koszul–Malgrange equivalence between holomorphic vector bundles andsmooth vector bundles endowed with a flat (0 , q -deformation of theclassical Borel–Weil theorem [26, 20]. An extension of these results to the case ofgeneral projective space was established by Khalkhali and Moatadelro [21] usinga q -deformed Dolbeault anti-holomorphic complex originally introduced in [10] toconstruct spectral triples.In [2], Beggs and Majid later introduced the notion of an Hermitian holomor-phic module over an algebra (where they were called Hermitian holomorphic vectorbundles) and showed that any such module admits a Chern connection [2, Theorem8.53], see also [3, § O q ( G/L S ), the quantum counterpart of classical flag man-ifolds G/L S , where G is a compact Lie group and L S is a subgroup of G indexedby some subset S of the simple roots of G . The quantum flag manifolds forma far more general class of quantum homogeneous spaces whose theory is deeplyrooted in the representation theory of Drinfeld–Jimbo quantum groups. For the ir-reducible case, which is to say, those quantum flag manifolds which are irreducible(or equivalently Hermitian symmetric spaces) in the classical limit, Heckenbergerand Kolb established that each quantum space has an essentially unique covari-ant q -deformed de Rham complex Ω • q ( G/L S ) [15, 16]. These remarkable differential OLOMORPHIC RELATIVE HOPF MODULES 3 calculi are some of the most important objects in the study of the noncommutativegeometry of quantum groups. As shown in [31, 27] each Ω • q ( G/L S ) comes endowedwith a unique covariant noncommutative K¨ahler structure. For the special caseof quantum projective space, the q -deformed anti-holomorphic Dolbeault complexconstructed in [10] can be realised as a subcomplex of the Heckenberger–Kolbcalculus.This raises the question of whether we can construct holomorphic structures forevery finitely generated relative Hopf module over each irreducible quantum flagmanifold. Our main result, Theorem 4.5, shows that this is indeed possible. To es-tablish the result, we need to establish the existence of covariant (0 , , O q ( CP n ) were explicitly constructed and flatness was verified by directcalculation. Extending this approach to all irreducible quantum flag manifoldswould be prohibitively lengthy and tedious. Instead, we show existence and flat-ness by turning to the general theory of principal comodule algebras, following anapproach closer to the original constructions of Brzezinski and Majid [7, 26].While the existence of holomorphic structures is highly interesting in its ownright, we are especially interested in the implications of our main result. In partic-ular, the existence of holomorphic structures for line modules is an essential ingre-dient in a number of associated works. In [13] the holomorphic line modules overthe irreducible quantum flag manifolds are shown to satisfy a direct q -deformationof the classical Borel–Weil theorem. This extends the case of quantum projec-tive space discussed above, as well as the more general quantum Grassmannianpicture established in [28]. Building on the q -defomed Borel–Weil theorem, allnon-trivial line modules over the irreducible quantum flag manifolds were identi-fied in [12] as either positive or negative. This provides valuable information aboutthe behaviour of their q -deformed Chern curvatures. These results in turn allowedDas and the third and fourth authors to establish the Fredholm property for anyDolbeault–Dirac operator twisted by a negative line module [11]. These operatorsthen become natural candidates for spectral triples in the sense of Connes andMoscovici [9]. The existence of holomorphic structures also suggests a number ofimportant new directions of research, such as the extension of Kostant’s theoremto the quantum setting as conjectured in § § § B = A co( H ) , cosemisimplicity of H impliesthe existence of a left A -covariant strong principal connection. Using this result, OLOMORPHIC RELATIVE HOPF MODULES 4 we are able to produce covariant connections for finitely generated relative Hopfmodules with respect to any covariant calculus.In § § O q ( G/L S ), and using a repre-sentation theoretic argument, prove flatness of the (0 , Preliminaries
In this section we recall the necessary preliminaries on differential calculi, com-plex structures, connections, holomorphic structures, and strong principal connec-tions. Note that all algebras are defined over C and assumed to be unital, and allalgebra maps are assumed to be unital.2.1. Calculi, Connections, and Holomorphic Modules.
Differential Calculi. A differential calculus (cid:0) Ω • ≃ L k ∈ N Ω k , d (cid:1) is a differ-ential graded algebra (dg-algebra) which is generated in degree 0 as a dg-algebra,that is to say, it is generated as an algebra by the elements a, d b , for a, b ∈ Ω .A differential ∗ -calculus is a differential calculus equipped with a conjugate linearinvolutive map ∗ : Ω • → Ω • satisfying d( ω ∗ ) = (d ω ) ∗ , and (cid:0) ω ∧ ν (cid:1) ∗ = ( − kl ν ∗ ∧ ω ∗ , for all ω ∈ Ω k , ν ∈ Ω l . For a given algebra B , a differential calculus over B is a differential calculus (Ω • , d)such that Ω = B . Note that if (Ω • , d) is a differential ∗ -calculus over B , then B is a ∗ -algebra. We say that ω ∈ Ω • is closed if d ω = 0. See [3, §
1] for a moredetailed discussion of differential calculi.2.1.2.
First-Order Differential Calculi. A first-order differential calculus over analgebra B is a pair (Ω , d), where Ω is a B -bimodule and d : B → Ω is a linearmap for which the Leibniz rule holdsd( ab ) = a (d b ) + (d a ) b, a, b ∈ B, and for which Ω is generated as a left B -module by those elements of the form d b ,for b ∈ B . The universal first-order differential calculus over B is the pair(Ω u ( B ) , d u ), where Ω u ( B ) is the kernel of the multiplication map m B : B ⊗ B → B endowed with the obvious bimodule structure, and d u is the map defined byd u : B → Ω u ( B ) , b ⊗ b − b ⊗ . OLOMORPHIC RELATIVE HOPF MODULES 5
Every first-order differential calculus over B is of the form (Ω u ( B ) /N, proj ◦ d u ),where N is a B -sub-bimodule of Ω u ( B ) andproj : Ω u ( B ) → Ω u ( B ) /N is the canonical quotient map. This gives a bijective correspondence between first-order differential calculi and sub-bimodules of Ω u ( B ).We say that a differential calculus (Γ • , d Γ ) extends a first-order differential cal-culus (Ω , d Ω ) if there exists a bimodule isomorphism φ : Ω → Γ such thatd Γ = φ ◦ d Ω . It can be shown that any first-order differential calculus admits anextension Ω • which is maximal in the sense that there exists a unique dg-algebramorphism from Ω • onto any other extension of Ω , see [3, § maximal prolongation of the first-order differential calculus.2.1.3. Connections.
Motivated by the Serre–Swan theorem, we think of a finitelygenerated projective left B -module F as a noncommutative generalisation of a vec-tor bundle. For Ω • a differential calculus over an algebra B and F a finitelygenerated projective left B -module, a connection on F is a C -linear map ∇ : F → Ω ⊗ B F satisfying ∇ ( bf ) = d b ⊗ f + b ∇ f, for all b ∈ B, f ∈ F . (1)An immediate but important consequence of the definition is that the differenceof two connections ∇ − ∇ ′ is a left B -module map.Any connection can be extended to a map ∇ : Ω • ⊗ B F → Ω • ⊗ B F uniquelydefined by ∇ ( ω ⊗ f ) = d ω ⊗ f + ( − | ω | ω ∧ ∇ f, where f ∈ F , and ω is a homogeneous element of Ω • of degree | ω | . The curvature of a connection is the left B -module map ∇ : F → Ω ⊗ B F . A connection is saidto be flat if ∇ = 0. Since ∇ ( ω ⊗ f ) = ω ∧ ∇ ( f ), a connection is flat if and onlyif the pair (Ω • ⊗ B F , ∇ ) is a cochain complex.2.1.4. Complex Structures.
In this subsection we recall the definition of a complexstructure for a differential calculus, as introduced in [20, 4], see also [3]. This givesan abstract characterisation of the properties of the de Rham complex of a classicalcomplex manifold [18].
Definition 2.1. A complex structure Ω ( • , • ) , for a differential ∗ -calculus (Ω • , d), isan N -algebra grading L ( a,b ) ∈ N Ω ( a,b ) for Ω • such that, for all ( a, b ) ∈ N :1. Ω k = L a + b = k Ω ( a,b ) ,2. (cid:0) Ω ( a,b ) (cid:1) ∗ = Ω ( b,a ) ,3. dΩ ( a,b ) ⊆ Ω ( a +1 ,b ) ⊕ Ω ( a,b +1) . OLOMORPHIC RELATIVE HOPF MODULES 6
An element of Ω ( a,b ) is called an ( a, b ) -form . For proj Ω ( a +1 ,b ) , and proj Ω ( a,b +1) , theprojections from Ω a + b +1 to Ω ( a +1 ,b ) , and Ω ( a,b +1) respectively, we write ∂ | Ω ( a,b ) := proj Ω ( a +1 ,b ) ◦ d , ∂ | Ω ( a,b ) := proj Ω ( a,b +1) ◦ d . It follows from Definition 2.1.3 that for any complex structure,d = ∂ + ∂, ∂ ◦ ∂ = − ∂ ◦ ∂, ∂ = ∂ = 0 . Thus (cid:0) L ( a,b ) ∈ N Ω ( a,b ) , ∂, ∂ (cid:1) is a double complex. Both ∂ and ∂ satisfy the gradedLeibniz rule. Moreover, ∂ ( ω ∗ ) = (cid:0) ∂ω (cid:1) ∗ , ∂ ( ω ∗ ) = (cid:0) ∂ω (cid:1) ∗ , for all ω ∈ Ω • . (2)Associated to any complex structure Ω ( • , • ) we have a second complex structure,called its opposite complex structure , defined asΩ ( • , • ) := M ( a,b ) ∈ N Ω ( a,b ) , where Ω ( a,b ) := Ω ( b,a ) . See [3, §
1] or [30] for a more detailed discussion of complex structures.2.1.5.
Holomorphic Modules.
In this subsection we present the notion a holomor-phic left B -module for an algebra B . Such a module should be thought of as anoncommutative holomorphic vector bundle, as has been considered in a number ofprevious papers, see for example [4], [34], and [20]. Indeed, the definition for holo-morphic modules is motivated by the classical Koszul–Malgrange characterisationof holomorphic bundles [24]. See [32] for a more detailed discussion.With respect to a choice Ω ( • , • ) of complex structure on Ω • , a (0 , -connectionon F is a connection with respect to the differential calculus (Ω (0 , • ) , ∂ ). Definition 2.2.
Let (Ω • , d) be a differential ∗ -calculus over a ∗ -algebra B , equippedwith a complex structure Ω ( • , • ) . A holomorphic left B -module is a pair ( F , ∂ F ),where F is a finitely generated projective left B -module, and ∂ F : F → Ω (0 , ⊗ B F is a flat (0 , ∂ F the holomorphic structure of the holomorphicleft B -module.In the classical setting the kernel of the holomorphic structure map coincideswith the space of holomorphic sections of a holomorphic vector bundle. Thismotivates us to call H ∂ ( F ) = ker (cid:0) ∂ F : F → Ω (0 , ⊗ B F (cid:1) , the space of holomorphic sections of ( F , ∂ F ). OLOMORPHIC RELATIVE HOPF MODULES 7
Quantum Homogeneous Spaces and Holomorphic Relative HopfModules.
From this point in the paper A and H will always denote Hopf alge-bras defined over C , with coproduct, counit, and antipode denoted by ∆ , ǫ , and S respectively, without explicit reference to the Hopf algebra in question. Moreover,all antipodes are assumed to be invertible. Hence we always have an equivalencebetween the categories of right and left comodules of any Hopf algebra.2.2.1. Comodule Algebras and Quantum Homogeneous Spaces.
For H a Hopf alge-bra, and V a right H -comodule with coaction ∆ R , we say that an element v ∈ V is coinvariant if ∆ R ( v ) = v ⊗
1. We denote the subspace of all coinvariant elementsby V co( H ) and call it the coinvariant subspace of the coaction ∆ R .A right H -comodule algebra P is a right H -comodule which is also an algebra,such that the comodule structure map ∆ R : P → P ⊗ H is an algebra map.Equivalently, it is a monoid object in Mod H , the category of right H -comodules.Note that for a right H -comodule algebra P , its coinvariant subspace B := P co( H ) is a subalgebra of P . In what follows we will always use B in this sense.If the functor P ⊗ B − : B Mod → C Mod, from the category of left B -modulesto the category of complex vector spaces, preserves and reflects exact sequences,then we say that P is faithfully flat as a right module over B . Faithful flatnessfor P as a left B -module is defined analogously.In this paper we are interested in a particular type of comodule algebra. Let π : A → H be a surjective Hopf algebra map between Hopf algebras A and H .Then a homogeneous right H -coaction is given by the map∆ R := (id ⊗ π ) ◦ ∆ : A → A ⊗ H. Note that ∆ R gives A the structure of a right H -comodule algebra. The as-sociated quantum homogeneous space is defined to be the space of coinvariantelements A co( H ) .2.2.2. Takeuchi’s Equivalence.
Let B := A co( H ) be a quantum homogeneous space.We denote by AB mod the category of finitely generated relative Hopf modules , thatis, the category whose objects are left A -comodules ∆ L : F → A ⊗ F , endowedwith a finitely generated left B -module structure, such that∆ L ( bf ) = ∆ L ( b )∆ L ( f ) , for all f ∈ F , b ∈ B, (3)and whose morphism sets AB Hom( − , − ) consist of left A -comodule, left B -module,maps. It is important to note that B is naturally an object in AB mod.We denote by H mod the category whose objects are finite-dimensional left H -comodules, and whose morphisms are left H -comodule maps. For a quantumhomogeneous space B := A co( H ) , we denote B + := B ∩ ker( ǫ ). Consider the functorΦ : AB mod → H mod , F 7→ F /B + F , OLOMORPHIC RELATIVE HOPF MODULES 8 where the left H -comodule structure of Φ( F ) is given by ∆ L [ f ] := π ( f ( − ) ⊗ [ f (0) ](with square brackets denoting the coset of an element in Φ( F )). If V ∈ H modwith coaction ∆ L : V → H ⊗ V , then the cotensor product of A and V is given by A (cid:3) H V := ker(∆ R ⊗ id − id ⊗ ∆ L : A ⊗ V → A ⊗ H ⊗ V ) . Using the cotensor product we can define the functorΨ : H mod → AB mod , V A (cid:3) H V, where the left B -module and left A -comodule structures of Ψ( V ) are defined onthe first tensor factor, and if γ is a morphism in H mod, then Ψ( γ ) := id ⊗ γ . Thefollowing equivalence was established in [37, Theorem 1]. Theorem 2.3 (Takeuchi’s Equivalence) . Let B = A co( H ) be a quantum homoge-neous space such that A is faithfully flat as a right B -module. An adjoint equiva-lence of categories between AB mod and H mod is given by the functors Φ and Ψ andunit, and counit, natural isomorphisms U :
F → Ψ ◦ Φ( F ) , f f ( − ⊗ [ f (0) ] , C : Φ ◦ Ψ( V ) → V, h X i a i ⊗ v i i X i ε ( a i ) v i . The usual tensor product of comodules gives H mod the structure of a monoidalcategory. Every object F ∈ AB mod admits a right B -module structure uniquelydefined by F × B → F , ( f, b ) f ( − bS ( f ( − ) f (0) , giving F the structure of a bimodule. The usual tensor product of bimodules thenendows AB mod with the structure of a monoidal category. It forms a monoidalsubcategory of the category of B -bimodules, which for sake of clarity we denoteby AB mod . Takeuchi’s equivalence can now be given the structure of a monodialequivalence in the obvious way. In particular, this means that for any monoidobject M ∈ AB mod the corresponding Φ( M ) ∈ H mod also has the structure of amonoid object. We will use this fact tacitly throughout the paper.2.2.3. Relative Hopf Modules and Covariant Connections.
Let π : A → H be asurjective Hopf map and B = A co( H ) a quantum homogeneous space. A differentialcalculus (Ω • , d) over B is said to be covariant if the coaction ∆ L : B → A ⊗ B extends to a (necessarily unique) map ∆ L : Ω • → A ⊗ Ω • giving Ω • the structureof a monoid object in AB mod, and such that d is a left A -comodule map. For any F ∈ AB mod, a connection ∇ : F → Ω ⊗ B F is said to be covariant if it is a left A -comodule map.We say that a first-order differential calculus Ω ( B ) over B is left covariant ifthere exists a (necessarily unique) left A -coaction ∆ L : Ω ( B ) → A ⊗ Ω ( B ) givingΩ ( B ) the structure of an object in AB mod and such that d is a left A -comodule map. OLOMORPHIC RELATIVE HOPF MODULES 9
Note the universal calculus over B is left A -covariant. Moreover, any other first-order differential calculus over B , with corresponding B -sub-bimodule N ⊆ Ω u ( B ),is covariant if and only if N is a left A -sub-comodule of Ω u ( B ). In particular, wenote that the maximal prolongation of a covariant first-order differential calculusis covariant.A complex structure Ω ( • , • ) for Ω • is said to be covariant if the N -decompositionof Ω • is a decomposition in the category AB mod, or explicitly if the homogeneoussubspace Ω ( a,b ) is a left A -sub-comodule of Ω • , for each ( a, b ) ∈ N . (Note thatthe grading implies Ω ( a,b ) is automatically a B -sub-bimodule.) For any covariantcomplex structure the differentials ∂ and ∂ are left A -comodule maps. Definition 2.4. A holomorphic relative Hopf module is a pair ( F , ∂ F ) where F ∈ AB mod, ∂ F : F → Ω (0 , ⊗ B F is a covariant (0 , F , ∂ F ) isa holomorphic left B -module.2.3. Principal Comodule Algebras and Strong Principal Connections.
Inthis subsection we recall the basic theory of principal comodule algebras, structuresof central importance in the paper.2.3.1.
General Case.
We say that a right H -comodule algebra ( P, ∆ R ) is a H -Hopf–Galois extension of B := P co( H ) if for m P : P ⊗ B P → P the multiplicationof P , the map can := ( m P ⊗ id) ◦ (id ⊗ ∆ R ) : P ⊗ B P → P ⊗ H, is a bijection. Definition 2.5. A principal right H -comodule algebra is a right H -comodule al-gebra ( P, ∆ R ) such that P is a Hopf–Galois extension of B := P co( H ) and P isfaithfully flat as a right and left B -module.We next recall the notion of a strong principal connection for a right H -comodulealgebra, and its relationship with the definition of principal comodule algebras. Definition 2.6.
Let H be a Hopf algebra, P a right H -comodule algebra, and B := P co( H ) . A principal connection for P is a left P -module right H -comoduleprojection Π : Ω u ( P ) → Ω u ( P ) satisfyingker(Π) = P Ω u ( B ) P. A principal connection is said to be strong if(id − Π)d P ⊆ Ω ( B ) P. As we now recall, the existence of a strong principal connection for a comodulealgebra is equivalent to the comodule algebra being principal, see [6, § Theorem 2.7.
A comodule algebra is principal if and only if it admits a strongprincipal connection.
OLOMORPHIC RELATIVE HOPF MODULES 10
The Case of Quantum Homogeneous Spaces.
In this subsection we restrictto the special case of a quantum homogeneous space B = A co( H ) associated to aHopf algebra surjection π : A → H . First we present a natural construction forstrong principal connections. Consider H as a H -bicomodule in the obvious way,and consider A as a H -bicomodule with respect to the left and right H -coactions∆ L = ( π ⊗ id) ◦ ∆ and ∆ R = (id ⊗ π ) ◦ ∆. Suppose that there exists a H -bicomodulemap i : H → A splitting π , and such that i (1 H ) = 1 A . Then a left A -covariantstrong principal connection is given byΠ := m ◦ (id ⊗ ω ) ◦ can : Ω u ( A ) → Ω u ( A ) , (4)where can is the restriction of can to Ω u ( A ), and ω : H → Ω u ( A ) , h S ( i ( h ) (1) )d u ( i ( h ) (2) ) . (See [25, §
24] for further details.) Moreover, as shown in [7, Proposition 4.4], thisgives an equivalence between left A -covariant strong principal connections and H -bicomodule splittings of π which send the unit of H to the unit of A .We are interested in strong principal connections because they allow us to con-struct left A -covariant connections for any F ∈ AB mod. Consider first the isomor-phism j : Ω u ( B ) ⊗ B F ≃ Ω u ( B ) A (cid:3) H Φ( F ) , ω ⊗ f ωf ( − ⊗ [ f (0) ] . We claim that a strong principal connection Π defines a connection ∇ on F by ∇ : F → Ω u ( B ) ⊗ B F , f j − (cid:0)(cid:0) (id − Π)d u f ( − (cid:1) ⊗ [ f (0) ] (cid:1) . Indeed, since d u and Π are both right H -comodule maps, a right H -comodule mapis also given by the composition (id − Π) ◦ d u . Hence (cid:0) (id − Π)d u f ( − (cid:1) ⊗ [ f (0) ] ∈ j (cid:0) Ω u ( B ) ⊗ B F (cid:1) , for all f ∈ F , meaning that ∇ defines a left A -covariant connection. We call ∇ the connectionfor F associated to Π.3.
Covariant Connections and Holomorphic Structures
In this section we use Takeuchi’s equivalence to convert questions about exis-tence and uniqueness of connections into representation-theoretic statements. Wealso discuss principal comodule algebras and show how cosemisimplicity of a Hopfalgebra H can be used to construct left A -covariant strong principal connections.This sets up a general framework in terms of which we prove the main results ofthe paper in Section 4. Recall that A and H denote Hopf algebras and B = A co( H ) a quantum homogeneous space. OLOMORPHIC RELATIVE HOPF MODULES 11
Quotients of Connections.
In this subsection we present some elementarytechnical results about producing connections for non-universal calculi from con-nections for universal calculi.
Proposition 3.1.
For an algebra B , let F be a finitely generated projective left B -module and let Ω • ( B ) be a differential calculus over B . Then the zero map F → Ω ( B ) ⊗ B F is a connection if and only if Ω • is the zero calculus.Proof. If the zero map were a connection, then we would necessarily haved b ⊗ B f = 0 , for all b ∈ B, f ∈ F . Since F is by assumption projective as a left B -module, this would imply thatd b = 0, for all b ∈ Ω ( B ), and hence that the calculus was trivial. The converse isclear, giving us the claimed equivalence. (cid:3) Corollary 3.2.
For any proper B -sub-bimodule N ⊆ Ω u ( B ) , let us denote proj N : Ω u ( B ) → Ω u ( B ) /N =: Ω , ω [ ω ] , where [ ω ] denotes the coset of ω in Ω u ( B ) /N . If ∇ : F → Ω u ( B ) ⊗ B F is aconnection with respect to the universal calculus, then a non-zero connection withrespect to Ω is given by ∇ ′ : F → Ω ⊗ B F , f (proj N ⊗ id) ◦ ∇ ( f ) . Proof.
It is clear from the definition of ∇ ′ that it is a linear map satisfying theLeibniz rule (1), which is to say, it is clear that ∇ ′ is a connection. The fact thatit is non-zero follows from Proposition 3.1 and the assumption that N is a proper B -sub-bimodule. (cid:3) Covariant Connections and Takeuchi’s Equivalence.
In this subsectionwe make some novel observations about the flatness and uniqueness for covariantconnections on a relative Hopf module
F ∈ AB mod. The idea is to produce sufficientcriteria in terms of the morphism sets of the category AB mod. In practical cases, thisallows these questions to be transferred to representation-theoretic form, allowingfor a solution by direct calculation. We first give a criteria for flatness. Proposition 3.3. If AB Hom( F , Ω ⊗ B F ) = 0 , then any left A -covariant connection ∇ : F → Ω ⊗ B F is necessarily flat.Proof. Since the curvature of any connection is a module map, the curvature of acovariant connection is a morphism. Thus if AB Hom( F , Ω ⊗ B F ) is trivial, ∇ mustbe flat. (cid:3) The second proposition gives an analogous criteria for uniqueness of a covariantconnection on a finitely generated relative Hopf module.
Proposition 3.4.
For
F ∈ AB mod such that AB Hom( F , Ω ⊗ B F ) = 0 , there existsat most one covariant connection for F . OLOMORPHIC RELATIVE HOPF MODULES 12
Proof.
Since the difference of any two connections is a module map and the differ-ence of two comodule maps is again a comodule map, the difference of two covariantconnections is a morphism in AB mod. Thus if AB Hom( F , Ω ⊗ B F ) is trivial, thenthere exists at most one covariant connection F → Ω ⊗ B F . (cid:3) We direct the interested reader to [12, § § Principal Comodule Algebras and Cosemisimple Hopf Algebras.
In this subsection we discuss comodule algebras B = A co( H ) for which H is acosemisimple Hopf algebra. We begin by recalling the definition of cosemisimplic-ity. Definition 3.5.
A Hopf algebra A is cosemisimple if it satisfies the following threeequivalent conditions:1. A is the direct sum of its cosimple subcoalgebras, that is subcoalgebras whichhave no proper subcoalgebras,2. in the abelian category A Mod of left A -comodules all short exact sequencessplit,3. there exists a unique linear map h : A → C , which we call the Haar func-tional , satisfying h (1) = 1, and(id ⊗ h ) ◦ ∆( a ) = h ( a )1 , ( h ⊗ id) ◦ ∆( a ) = h ( a )1 . For details about the equivalence of these three properties see [22, § A into its simple subcoalgebras, the associated Haar functional is givenby projection onto the trivial sub-coalgebra C A .Consider H Mod H the category whose objects are finite-dimensional H -bicomodulesand whose morphisms are H -bicomodule maps. In this paper, all Hopf algebras areassumed to have invertible antipodes, so we have an equivalence between H Modthe category of finite-dimensional right H -comodules, and Mod H the category offinite-dimensional left A -comodules. Hence we have an equivalence of categories H Mod H ≃ H ⊗ H Mod , where H ⊗ H is the usual tensor product of Hopf algebras. Denoting the Haar of H by h , the linear map defined on simple tensors by h H ⊗ H : H ⊗ H → C , g ⊗ g ′ h ( g ) h ( g ′ ) , is readily seen to be a Haar functional for H ⊗ H in the sense of Definition 3.5. Itfollows that H ⊗ H is a cosemisimple Hopf algebra. Hence H ⊗ H Mod is a semisimpleabelian category, meaning that H Mod H is a semisimple abelian category. OLOMORPHIC RELATIVE HOPF MODULES 13
Let π : A → H be a Hopf algebra surjection and B = A co( H ) the associatedquantum homogeneous space. It is well known that cosemisimplicity of H impliesthat ( A, ∆ R ) is a principal H -comodule algebra. For example, it was shown in [29,Corollary 1.5] that cosemisimplicity of H implies that A is faithfully flat as aleft and right B -module, and it follows from [35, Corollary 2.6] that A is a H -Hopf–Galois extension of B . In fact cosemisimplicity implies a stronger result,namely the existence of a left A -covariant strong principal connection. This easyobservation is undoubtedly well known to the experts, but we include a proof forsake of completeness. Lemma 3.6.
Let π : A → H be a Hopf algebra surjection and let ∆ R denotethe associated homogeneous right H -coaction on A . If H is a cosemisimple Hopfalgebra, then Ω u ( A ) admits a left A -covariant strong principal connection. Inparticular, ( A, ∆ R ) is a principal comodule algebra.Proof. Since π : A → H is a Hopf algebra map, it is necessarily a H -bicomodulemap. Since H is cosemisimple, H ⊗ H Mod is semisimple. Hence we can choosea H -bicomodule map i : H → A splitting π and satisfying i (1 H ) = 1 A . It nowfollows from the discussions of § A, ∆ R ) is a principal comodule algebraadmitting a left A -covariant strong principal connection. (cid:3) Proposition 3.7.
Assume that H is cosemisimple. Let F ∈ AB mod and let Ω • bea left A -covariant differential calculus over B = A co( H ) . There exists an associated left A -covariant connection ∇ : F → Ω ⊗ B F . If we additionally assume that Ω • is a differential ∗ -calculus endowed witha covariant complex structure Ω ( • , • ) , then there will exist a left A -covariant (0 , -connection ∂ F : F → Ω (0 , ⊗ B F .Proof. Since A is a principal comodule algebra, we know from the discussionsin § F admits an associated universal connection ∇ : F → Ω u ( B ) ⊗ B F .As discussed in § ∇ with the quotient map Ω u ( B ) ⊗ B F → Ω ⊗ B F will give a covariant connection ∇ : F → Ω • ⊗ B F for the non-universal calculus Ω • .Finally we note that if Ω • is endowed with a covariant complex structure Ω ( • , • ) ,then ∂ F := (proj Ω (0 , ⊗ id) ◦ ∇ : F → Ω (0 , ⊗ B F is a left A -covariant (0 , (cid:3) Holomorphic Relative Hopf Modules over Quantum FlagManifolds
In this section we present the primary results of the paper, namely the exis-tence of covariant holomorphic structures for any relative Hopf module over the
OLOMORPHIC RELATIVE HOPF MODULES 14 irreducible quantum flag manifolds, and uniqueness of such structures in the irre-ducible case. We first recall the necessary definitions and results about Drinfeld–Jimbo quantum groups, quantum flag manifolds, and the Heckenberger–Kolb dif-ferential calculi over the irreducible quantum flag manifolds, and then follow withthe existence and uniqueness results for holomorphic relative Hopf modules.4.1.
Drinfeld–Jimbo Quantum Groups.
Let g be a finite-dimensional complexsemisimple Lie algebra of rank r . We fix a Cartan subalgebra h with correspondingroot system ∆ ⊆ h ∗ , where h ∗ denotes the linear dual of h . With respect toa choice of simple roots Π = { α , . . . , α r } , denote by ( · , · ) the symmetric bilinearform induced on h ∗ by the Killing form of g , normalised so that any shortest simpleroot α i satisfies ( α i , α i ) = 2. The coroot α ∨ i of a simple root α i is defined by α ∨ i := α i d i = 2 α i ( α i , α i ) , where d i := ( α i , α i )2 . The Cartan matrix A = ( a ij ) ij of g is the ( r × r )-matrix defined by a ij := (cid:0) α ∨ i , α j (cid:1) . Let { ̟ , . . . , ̟ r } denote the corresponding set of fundamental weights of g , whichis to say, the dual basis of the coroots.Let q ∈ R such that q / ∈ {− , , } , and denote q i := q d i . The quantised universalenveloping algebra U q ( g ) is the noncommutative associative algebra generated bythe elements E i , F i , K i , and K − i , for i = 1 , . . . , r , subject to the relations K i E j = q a ij i E j K i , K i F j = q − a ij i F j K i , K i K j = K j K i , K i K − i = K − i K i = 1 ,E i F j − F j E i = δ ij K i − K − i q i − q − i , along with the quantum Serre relations X − a ij s =0 ( − s (cid:20) − a ij s (cid:21) q i E − a ij − si E j E si = 0 , for i = j, X − a ij s =0 ( − s (cid:20) − a ij s (cid:21) q i F − a ij − si F j F si = 0 , for i = j ;where we have used the q -binomial coefficients defined according to[ n ] q ! = [ n ] q [ n − q · · · [2] q [1] q , where [ m ] q := q m − q − m q − q − . A Hopf algebra structure is defined on U q ( g ) by∆( K i ) = K i ⊗ K i , ∆( E i ) = E i ⊗ K i + 1 ⊗ E i , ∆( F i ) = F i ⊗ K − i ⊗ F i ,S ( E i ) = − E i K − i , S ( F i ) = − K i F i , S ( K i ) = K − i ,ε ( E i ) = ε ( F i ) = 0 , ε ( K i ) = 1 . OLOMORPHIC RELATIVE HOPF MODULES 15
A Hopf ∗ -algebra structure, called the compact real form of U q ( g ), is defined by K ∗ i := K i , E ∗ i := K i F i , F ∗ i := E i K − i . Let P be the weight lattice of g , and P + its set of dominant integral weights.For each µ ∈ P + there exists an irreducible finite-dimensional U q ( g )-module V µ ,uniquely defined by the existence of a vector v µ ∈ V µ , which we call a highestweight vector , satisfying E i ⊲ v µ = 0 , K i ⊲ v µ = q ( α i ,µ ) v µ , for all i = 1 , . . . , r .Moreover, v µ is the unique such element up to scalar multiple. We call any finitedirect sum of such U q ( g )-representations a type- representation . In general, anon-zero vector v ∈ V µ is called a weight vector of weight wt( v ) ∈ P if K i ⊲ v = q (wt( v ) ,α i ) v, for all i = 1 , . . . , r. (5)Finally, we note that since U q ( g ) has an invertible antipode, we have an equivalencebetween U q ( g ) Mod, the category of left U q ( g )-modules, and Mod U q ( g ) , the categoryof right U q ( g )-modules, as induced by the antipode.For further details on Drinfeld–Jimbo quantised enveloping algebras, we referthe reader to the standard texts [8, 22], or to the seminal papers [14, 19].4.2. Quantum Coordinate Algebras.
In this subsection we recall some neces-sary material about quantised coordinate algebras. Let V be a finite-dimensionalleft U q ( g )-module, v ∈ V , and f ∈ V ∗ , the C -linear dual of V , endowed with itsright U q ( g )-module structure. Let us note that, with respect to the equivalencebetween type-1 U q ( g )-modules and finite-dimensional representations of g , the leftmodule corresponding to V ∗ µ is isomorphic to V − w ( µ ) , where w denotes the longestelement in the Weyl group of g .Consider the function c Vf,v : U q ( g ) → C defined by c Vf,v ( X ) := f (cid:0) X ⊲ v (cid:1) . The coordinate ring of V is the subspace C ( V ) := span C (cid:8) c Vf,v | v ∈ V, f ∈ V ∗ (cid:9) ⊆ U q ( g ) ∗ . A U q ( g )-bimodule structure on C ( V ) is given by(6) ( Y ⊲ c
Vf,v ⊳ Z )( X ) := f (( ZXY ) ⊲ v ) = c Vf⊳Z,Y ⊲v ( X ) . Let U q ( g ) ◦ denote the Hopf dual of U q ( g ). It is easily checked that a Hopf subal-gebra of U q ( g ) ◦ is given by(7) O q ( G ) := M µ ∈P + C ( V µ ) . We call O q ( G ) the quantum coordinate algebra of G , where G is the compact,connected, simply-connected, simple Lie group having g as its complexified Liealgebra. Note that O q ( G ) is a cosemisimple Hopf algebra by construction. OLOMORPHIC RELATIVE HOPF MODULES 16
Quantum Flag Manifolds.
For { α i } i ∈ S ⊆ Π a subset of simple roots, con-sider the Hopf ∗ -subalgebra U q ( l S ) := (cid:10) K i , E j , F j | i = 1 , . . . , r ; j ∈ S (cid:11) . Just as for U q ( g ), see for example [22, § U q ( l S )-modules issemisimple. The Hopf ∗ -algebra embedding ι S : U q ( l S ) ֒ → U q ( g ) induces the dualHopf ∗ -algebra map ι ◦ S : U q ( g ) ◦ → U q ( l S ) ◦ . By construction O q ( G ) ⊆ U q ( g ) ◦ , sowe can consider the restriction map π S := ι ◦ S | O q ( G ) : O q ( G ) → U q ( l S ) ◦ , and the Hopf ∗ -subalgebra O q ( L S ) := π S (cid:0) O q ( G ) (cid:1) ⊆ U q ( l S ) ◦ . The quantum flagmanifold associated to S is the quantum homogeneous space associated to thesurjective Hopf ∗ -algebra map π S : O q ( G ) → O q ( L S ). We denote it by O q (cid:0) G/L S (cid:1) := O q (cid:0) G (cid:1) co( O q ( L S )) . Since the category of U q ( l S )-modules is semisimple, O q ( L S ) must be a cosemisim-ple Hopf algebra. Thus by Proposition 3.6, the pair ( O q ( G ) , ∆ R ) is a principalcomodule algebra.Denoting µ S := P s / ∈ S ̟ s , choose for V µ S a weight basis { v i } i , with correspondingdual basis { f i } i . As shown in [15, Proposition 3.2], writing N := dim( V µ S ), a setof generators for O q ( G/L S ) is given by z ij := c V µS f i ,v N c V − w µS ) v j ,f N for i, j = 1 , . . . , N, where v N , and f N , are the highest weight basis elements of V µ S , and V − w ( µ S ) ,respectively.4.4. The Heckenberger–Kolb Calculi and their Complex Structures.
Theconstruction and classification of covariant differential calculi over the quantum flagmanifolds poses itself as a very important and challenging question. At presentthis question has only been addressed for the irreducible quantum flag manifolds,a distinguished sub-family whose definition we now recall.
Definition 4.1.
A quantum flag manifold is irreducible if the defining subset ofsimple roots is of the form S = { , . . . , r } \ { s } where α s has coefficient 1 in the expansion of the highest root of g .In the classical limit of q = 1, these homogeneous spaces reduce to the compactHermitian symmetric spaces, see for example [1, Table 10.1] or [17, § X.3]. For aconvenient diagrammatic presentation of the explicit simple roots identified by thiscondition, as well as the dimensions of the classical differential manifolds, see [12,Appendix B].
OLOMORPHIC RELATIVE HOPF MODULES 17
The irreducible quantum flag manifolds are distinguished by the existence ofan essentially unique q -deformation of their classical de Rham complexes. Theexistence of such a canonical deformation is one of the most important results inthe noncommutative geometry of quantum groups, establishing it as a solid basefrom which to investigate more general classes of quantum spaces. The followingtheorem is a direct consequence of results established in [15], [16], and [27]. See [11, §
10] for a more detailed presentation.
Theorem 4.2.
Over any irreducible quantum flag manifold O q ( G/L S ) , there existsa unique finite-dimensional left O q ( G ) -covariant differential ∗ -calculus Ω • q ( G/L S ) ∈ O q ( G ) O q ( G/L S ) mod , of classical dimension, that is to say, satisfying dim Φ (cid:0) Ω kq ( G/L S ) (cid:1) = (cid:18) Mk (cid:19) , for all k = 0 , . . . , M, where M is the complex dimension of the corresponding classical manifold. The calculus Ω • q ( G/L S ), which we refer to as the Heckenberger–Kolb calculus of O q ( G/L S ), has many remarkable properties. We recall here only the existenceof a unique covariant complex structure, following from the results of [15], [16],and [27]. Proposition 4.3.
Let O q ( G/L S ) be an irreducible quantum flag manifold, and Ω • q ( G/L S ) its Heckenberger–Kolb differential ∗ -calculus. Then the following hold:
1. Ω • q ( G/L S ) admits precisely two left O q ( G ) -covariant complex structures, eachof which is opposite to the other, for each complex structure Ω (1 , and Ω (0 , are simple objects in O q ( G ) O q ( G/L S ) mod . Complementing this abstract characterisation of the calculus is the original pre-sentation of Heckenberger and Kolb in terms of the generators z ij ∈ O q ( G/L S )given in [16]. Here we need only recall the following: Consider the subset of theindex set J := { , . . . , dim( V ̟ s ) } given by J (1) := { i ∈ J | ( ̟ s , ̟ s − α s − wt( v i )) = 0 } , where { v i } i ∈ J is a weight basis of V ̟ s . It follows from [16, Proposition 3.6] thatwe can make an explicit choice of left A -covariant complex structureΩ • q ( G/L S ) ≃ M ( a,b ) ∈ N Ω ( a,b ) =: Ω ( • , • ) uniquely defined by the fact that a basis of Φ(Ω (0 , ) is given by (cid:8) [ ∂z Ni ] | for i ∈ J (1) (cid:9) . (8) OLOMORPHIC RELATIVE HOPF MODULES 18
Holomorphic Modules.
Here we establish existence and uniqueness resultsof holomorphic structures for relative Hopf modules over the irreducible quantumflag manifolds. We begin by observing that the general theory of principal comod-ule algebras, together with cosemisimplicity of O q ( L S ), implies the existence ofcovariant connections. Lemma 4.4.
Let O q ( G/L S ) be an irreducible quantum flag manifold endowed withits Heckenberger–Kolb calculus Ω • q ( G/L S ) . Every F ∈ O q ( G ) O q ( G/L S ) mod admits a left O q ( G ) -covariant connection ∇ : F → Ω q ( G/L S ) ⊗ O q ( G/L S ) F .Proof. As observed in § § F ∈ O q ( G ) O q ( G/L S ) mod . Corollary 3.2 nowimplies that we can quotient this connection to produce a covariant connectionwith respect to the Heckenberger–Kolb calculus Ω • q ( G/L S ). (cid:3) We now use Proposition 3.4 to show uniqueness for covariant connections when-ever F is simple. This is most easily done by considering Φ( F ) as a module overthe centre of U q ( l S ). Recalling that the transpose of the Cartan matrix A is thechange of basis matrix taking fundamental weights to simple roots, we see thatdet( A ) ̟ s is contained in the root lattice of g . Denotingdet( A ) ̟ s =: a α + · · · + a r α r , it follows directly from the commutation relations of U q ( g ) that Z := K a · · · K a r r is a central element of U q ( l S ). Recall that the elements of the centre z ( U q ( l S )) of U q ( l S ) act on any irreducible U q ( l S )-module V by a corresponding central charac-ter χ V , which is to say, an element of Hom( z ( U q ( l S )) , C ) the set of algebra mapsfrom z ( U q ( l S )) to C . For the explicit case of Φ(Ω (0 , ), it follows from the proofof [12, Theorem 4.11] that(9) χ Φ(Ω (0 , ) ( Z ) = 1 , − . Note also that for V and W two irreducible U q ( l S )-modules, since elements of thecentre z ( U q ( l S )) are grouplike, z ( U q ( l S )) will act on V ⊗ W by a central character χ V ⊗ W according to χ V ⊗ W ( x ) = χ V ( x ) χ W ( x ) , for any x ∈ z ( U q ( l S )) . Theorem 4.5.
Let O q ( G/L S ) be an irreducible quantum flag manifold endowedwith its Heckenberger–Kolb calculus, and F ∈ O q ( G ) O q ( G/L S ) mod . It holds that F admits a left O q ( G ) -covariant connection ∇ : F → Ω q ( G/L S ) ⊗ O q ( G/L S ) F ,and this is the unique such connection if F is simple, ∂ F := proj (0 , ◦ ∇ is a left O q ( G ) -covariant holomorphic structure for F ,and this is the unique such holomorphic structure if F is simple. OLOMORPHIC RELATIVE HOPF MODULES 19
Proof.
1. By Lemma 4.4 a covariant connection exists. Assuming that F is simple,it follows from (9) that χ Φ(Ω (0 , ) ⊗ Φ( F ) ( Z ) = χ Φ(Ω (0 , ) ( Z ) χ Φ( F ) ( Z ) = χ Φ( F ) ( Z ) . By Schur’s lemma we conclude that there are no non-zero U q ( l S )-module mapsfrom Φ( F ) to Φ(Ω (0 , ) ⊗ Φ( F ). Moreover, since we have a non-degenerate dualpairing between U q ( l S ) and O q ( L S ), there are no non-zero O q ( L S )-comodule mapsfrom Φ( F ) to Φ(Ω (0 , ) ⊗ Φ( F ). This in turn implies that there can exist nonon-zero morphisms from F to Ω (0 , ⊗ O q ( G/L S ) F . Proposition 3.4 now impliesthat there exists at most one left O q ( G )-covariant connection on F for the cal-culus Ω (0 , • ) . An analogous argument shows that there exists at most one left O q ( G )-covariant connection on F for the calculus Ω ( • , . Hence if F is simple,then ∇ : F → Ω q ( G/L S ) ⊗ O q ( G/L S ) F is the unique such covariant connection.2. Since Ω • is a monoid object in O q ( G ) O q ( G/L S ) mod , we see that Φ(Ω • ) has thestructure of a monoid object in O q ( L S ) mod, or in other words, it has the structureof a left O q ( L S )-comodule algebra. In particular, for any two forms ω, ν ∈ Ω • , itholds that ([ ω ] ∧ [ ν ]) ⊳ Z = ([ ω ] ⊳ Z ) ∧ ([ ν ] ⊳ Z ) . Thus we see that χ Φ(Ω (0 , ) ( Z ) = (cid:0) χ Φ(Ω (0 , ) ( Z ) (cid:1) . From this we see that, for any irreducible F , χ Φ(Ω (0 , ) ⊗ Φ( F ) ( Z ) = (cid:0) χ Φ(Ω (0 , ) ( Z ) (cid:1) χ Φ( F ) ( Z ) = χ Φ( F ) ( Z ) , where we have used (9). Following the same argument as for (0 , O q ( G )-comodule maps from F to Ω (0 , ⊗ O q ( G/L S ) F . Flatness of the (0 , ∂ F now follows from Propo-sition 3.3. Uniqueness was already established in 1.For the case of a non-simple F , cosemisimplicity of O q ( L S ) implies that F isa direct sum of a finite number of simple objects F ≃ L i F i . The direct sumof the covariant holomorphic structures of the summands F i gives a covariantholomorphic structure for F . (cid:3) Remark 4.6.
The proof that ∂ F is a flat (0 , ∂ F := proj (0 , ◦ ∇ satisfies ∂ F = 0 . (10)This can be described more formally in terms of opposite complex structures.Indeed, (10) says that ∂ is a holomorphic structure with respect to the oppositecomplex structure of Ω q ( G/L S ). OLOMORPHIC RELATIVE HOPF MODULES 20
A Quantum Kostant Conjecture.
The existence of a holomorphic struc-ture ∂ F , for each F ∈ O q ( G ) O q ( G/L S ) mod , gives a complex ∂ F : Ω k ⊗ O q ( G/L S ) F → Ω k +1 ⊗ O q ( G/L S ) F , with associated cohomology groups H k∂ ( F ) ≃ M ( a,b ) ∈ N H ( a,b ) ∂ ( F ) . In the classical setting, efforts to calculate these groups—a major undertaking—resulted in the celebrated Borel–Weil theorem for zero-th cohomology for line mod-ules [36], the Bott–Borel–Weil theorem for anti-holomorphic cohomologies for linemodules [5], and finally Kostant’s beautiful description of the general case [23]. Inthe quantum setting, for the irreducible quantum flag manifolds, the Borel–Weiltheorem will be shown in [13] to hold for all line modules, that is, those relativeHopf modules E satisfying dim(Φ( E )) = 1. The Bott–Borel–Weil theorem [12,Theorem 4.18] holds for all positive line modules over the irreducible quantum flagmanifolds (see [32, 12] for the definition of a positive line module). This motivatesus to make the following general conjecture. Conjecture 4.7.
For every
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