Fubini-Study metrics and Levi-Civita connections on quantum projective spaces
aa r X i v : . [ m a t h . QA ] N ov FUBINI-STUDY METRICS AND LEVI-CIVITA CONNECTIONS ONQUANTUM PROJECTIVE SPACES
MARCO MATASSA
Abstract.
We introduce analogues of the Fubini-Study metrics and the corresponding Levi-Civita connections on quantum projective spaces. We define the quantum metrics as two-tensors, symmetric in the appropriate sense, in terms of the differential calculi introducedby Heckenberger and Kolb. We define connections on these calculi and show that they aretorsion free and cotorsion free, where the latter condition uses the quantum metric and is aweaker notion of metric compatibility. Finally we show that these connections are bimoduleconnections and that the metric compatibility also holds in a stronger sense.
Introduction
Metrics and connections are two of the cornerstones upon which our description of differ-ential geometry is built, hence it is desirable to extend these notions to the realm of quantumspaces. By quantum spaces, we mean a class of appropriately defined non-commutative al-gebras, which we interpret as quantizations of functions on the underlying classical spaces.There are various possible perspectives on this problem and we recall some of them below.The goal of this paper is to introduce certain appropriate analogues of the Fubini-Study met-rics and the corresponding Levi-Civita connections for the quantum projective spaces . Thisgeneralizes certain results of [Maj05] obtained in the case of the quantum two-sphere.Given a (unital) non-commutative algebra A , one possible approach to introduce a metricis the theory of compact quantum metric spaces [Rie04], developed by Rieffel following theideas of Connes. In this theory one introduces a metric on the state space of A in terms of anappropriately defined Dirac operator, which should satisfy some properties. Such Dirac oper-ators are readily available for quantum projective spaces, see [DąDA10]. Roughly speaking,what is being quantized in this approach is the distance between points, since in the com-mutative situation the points can be identified with the pure states. Instead we are lookingfor a quantization of the metric tensor, since we want to have some notion of compatibilitybetween a connection and a metric. For this reason we adopt a more algebraic approach,which is explained for instance in the recent book [BeMa20] by Beggs and Majid.Let us recall some of the ideas of this approach, which we refer to as quantum Riemanniangeometry . Given an algebra A , we begin by introducing a differential calculus Ω • over A , withits degree-one part denoted by Ω . Then a quantum metric can be defined as an element g ∈ Ω ⊗ A Ω satisfying an appropriate invertibility condition. Using the differential calculus, wecan also define connections in the standard algebraic sense. In particular, given a connection ∇ on Ω , there is a standard notion of torsion as well. To formulate an analogue of thecompatibility of ∇ with the metric g there are two possibilities: 1) a weak version whichuses the notion of cotorsion, due to Majid; 2) a strong version that requires ∇ to be abimodule connection. In the classical case the second version coincides with the usual metriccompatibility, while the first version is a weaker property (to be recalled later).This setup can be applied to the quantum projective spaces, which we regard as a familywithin the class of quantum irreducible flag manifolds . It turns out that all the quantum spaces in this class admit canonical differential calculi Ω • , introduced by Heckenberger andKolb in [HeKo04, HeKo06]. We refer to these calculi as canonical since, as soon as some naturalconditions are imposed, they are uniquely defined. These quantum spaces and their differentialcalculi admit a uniform description, which we adopt in this paper, making simplificationsrelative to the quantum projective spaces only when needed. We expect that the resultsobtained in this paper will hold more generally for all quantum irreducible flag manifolds,with those obtained here providing important steps in this direction.Having the calculi Ω • at our disposal, we can discuss quantum metrics and connections onthem. We denote by B the algebra of a generic quantum projective space and write Ω = Ω .Our first main result is the existence of quantum metrics in the sense of Definition 3.5, whichalso requires the existence of appropriate inverse metrics. Theorem (Theorem 6.11) . Any quantum projective space B admits a quantum metric g ∈ Ω ⊗ B Ω . Moreover, in the classical limit it reduces to the Fubini-Study metric. Next, we look at connections on the first-order differential calculi Ω . We show the exis-tence of some particular connections and investigate the properties of torsion and cotorsion.The latter involves the quantum metric g introduced above. In particular, the condition ofcotorsion freeness should be seen as a weaker notion of compatibility with the metric (seeDefinition 3.11 and the remarks after that). Our second main result is the following. Theorem (Theorem 7.7) . Any quantum projective space B admits a connection ∇ : Ω → Ω ⊗ B Ω which is torsion free and cotorsion free. Moreover, in the classical limit it reduces tothe Levi-Civita connection for the Fubini-Study metric on the cotangent bundle. A connection which is torsion and cotorsion free is called a weak quantum Levi-Civitaconnection in [BeMa20], since the ordinary Levi-Civita connection (on the tangent bundle)can be characterized as the unique connection which is torsion free and compatible with themetric. It is natural to ask whether ∇ is a bimodule connection and if the condition of metriccompatibility holds in the strong form. Indeed, this turns out to be the case. Theorem (Theorem 8.4) . The connection ∇ : Ω → Ω ⊗ B Ω is a bimodule connection and iscompatible with the quantum metric, in the sense that ∇ g = 0 . In this case we say that ∇ is a quantum Levi-Civita connection, in perfect agreement withthe classical description. Hence we find that, in the case of quantum projective spaces, theclassical theory can be lifted to the quantum realm in a fairly satisfactory way.Our results generalize those of [Maj05] for the quantum two-sphere (the simplest case of aquantum projective space), with the notable difference that the conditions of being a bimoduleconnection and metric compatibility in the strong form were not investigated.A quantum metric and a connection are the main ingredients needed to study furtheraspects of quantum Riemannian geometry, as discussed in [BeMa20]. This program is carriedout further in [Maj05], where it is shown for instance that the quantum two-sphere satisfiesan analogue of the Einstein condition: this means that the quantum metric is proportionalto an appropriately defined Ricci tensor, defined using the curvature of the connection. Weconjecture that this will hold for all quantum projective spaces, and more generally for allquantum irreducible flag manifolds. We plan to tackle this problem in future research.Finally, let us see how our results compare to the existing literature on connections forquantum projective spaces. A lot of attention has been reserved to the case of line bundles,for instance we mention [KLvS11, KhMo11] for their focus on complex geometry. Morerelevant for us is the paper [ÓBu12], where the theory of quantum principal bundles is used ETRICS AND CONNECTIONS ON QUANTUM PROJECTIVE SPACES 3 to introduce a connection on the cotangent bundle, using a non-canonical calculus on thetotal space (the quantum special unitary group, in this case). We point out that no furtherproperties of these connections are explored, and extensive use is made of the explicit algebraicrelations, making it hard to generalize to arbitrary quantum flag manifolds.We should mention that the connections introduced in [ÓBu12] turn out to coincide withthose we describe here. This follows from the recent results of [GKÓ+20], where representation-theoretic methods are used to prove the following result: there exists a unique covariant con-nection on the Heckenberger-Kolb calculus Ω over a generic quantum irreducible flag manifold.Moreover they show that this connection is torsion free. It should be possible to extend thesetechniques to study some further aspects, a plan which is currently under investigation.However, one notable drawback of the representation-theoretic approach is that it does notgive explicit formulae for the connections. On the other hand, in this paper we provide explicitformulae, which for instance allow us to straightforwardly check the classical limit. Anotherbonus is that our approach is essentially self-contained, since we only use the relations in theHeckenberger-Kolb calculus Ω , plus general identities of categorical nature.Let us now discuss the organization of this paper. The first four sections contain variousbackground material, presented in a form suitable for our needs. In Section 1 we recall somebasic facts about compact quantum groups, while in Section 2 we recall various identitiesholding in the setting of rigid braided monoidal categories, which we use throughout the text.In Section 3 we give the precise definitions involving differential calculi, quantum metricsand connections. In Section 4 we describe the quantum irreducible flag manifolds following[HeKo06], with some small changes. Section 5 is also largely explanatory, as we recall thedescription of the Heckenberger-Kolb calculi for quantum irreducible flag manifolds, but wealso prove various alternative expressions for some of the relations of the calculi.The next three sections contain the proofs of our main results. In Section 6 we introducethe quantum metrics, discuss some of their properties and finally prove the existence ofappropriate inverse metrics. In Section 7 we introduce two connections on the holomorphicand antiholomorphic part of the calculi. Their direct sum gives a connection which we showto be torsion free and cotorsion free. In Section 8 we show that this is a bimodule connectionand verify the property of metric compatibility in the strong form.Many technical computations are relegated to the appendices, to make the main text morereadable. In Appendix A we recall various results about projective spaces, to facilitate thecomparison with the quantum case. In Appendix B we prove various properties satisfied bythe maps S and ˜ S , which we use to rewrite some of the relations of the Heckenberger-Kolbcalculus. In Appendix C we prove many of the technical identities that are used in the maintext. Finally in Appendix D we introduce various bimodule maps, some used to define theinverse metrics and some to check the bimodule property of the connections. Acknowledgements.
I would like to thank Réamonn Ó Buachalla for various discussionsand his comments on a preliminary version of this paper.1.
Quantum groups
In this section we review some background material on compact quantum groups.1.1.
Quantized enveloping algebras.
We use the conventions of the book [KlSc97], sincethey are used in our main reference [HeKo06]. Let g be a complex simple Lie algebra. Givena real number q such that < q < , the quantized enveloping algebra U q ( g ) is a certain Hopfalgebra deformation of the enveloping algebra U ( g ) , defined as follows. It has generators MARCO MATASSA { K i , E i , F i } ri =1 with r := rank( g ) and relations as in [KlSc97, Section 6.1.2]. In particular,the comultiplication, antipode and counit are given 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 ( K i ) = K − i , S ( E i ) = − E i K − i , S ( F i ) = − K i F i ,ε ( K i ) = 1 , ε ( E i ) = 0 , ε ( F i ) = 0 . Given λ = P ri =1 n i α i we write K λ := K n · · · K n r r . Let ρ := P α> α be the half-sum of thepositive roots of g . Then we have S ( X ) = K ρ XK − ρ for any X ∈ U q ( g ) .We also consider a ∗ -structure on U q ( g ) , which in the classical case corresponds to thecompact real form u of g . We can take for instance K ∗ i = K i , E ∗ i = K i F i , F ∗ i = E i K − i . The precise formulae are not very important here, as any equivalent ∗ -structure works equallywell for our purposes. We write U q ( u ) := ( U q ( g ) , ∗ ) when we consider U q ( g ) endowed with the ∗ -structure corresponding to the compact real form.1.2. Quantized coordinate rings.
The quantized coordinate ring C q [ G ] is defined as asubspace of the linear dual U q ( g ) ∗ . We take the span of all the matrix coefficients of thefinite-dimensional irreducible representations V ( λ ) (see below). It becomes a Hopf algebra byduality in the following manner: given X, Y ∈ U q ( g ) and a, b ∈ C q [ G ] we define ( ab )( X ) := ( a ⊗ b )∆( X ) , ∆( a )( X ⊗ Y ) := a ( XY ) ,S ( a )( X ) := a ( S ( X )) , X ) := ε ( X ) , ε ( a ) := a (1) . Moreover it becomes a Hopf ∗ -algebra by setting a ∗ ( X ) := a ( S ( X ) ∗ ) . We write C q [ U ] := ( C q [ G ] , ∗ ) for C q [ G ] endowed with this ∗ -structure.We have a left action ⊲ and a right action ⊳ of U q ( g ) on C q [ G ] given by ( X ⊲ a )( Y ) := a ( Y X ) , ( a ⊳ X )( Y ) := a ( XY ) . Using the action of U q ( g ) on C q [ G ] we can define quantum homogeneous spaces.1.3. Matrix coefficients.
The representation theory of U q ( g ) is essentially the same as thatof U ( g ) , hence of g . In particular we have analogues of the highest weight modules V ( λ ) forany dominant weight λ , which we denote by the same symbol. Given a finite-dimensionalrepresentation V , we define its matrix coefficients by ( c Vf,v )( X ) := f ( Xv ) , f ∈ V ∗ , v ∈ V, X ∈ U q ( g ) . These elements span C q [ G ] , according to the description given above.We say that an inner product ( · , · ) on V is U q ( u ) -invariant if it satisfies ( Xv, w ) = ( v, X ∗ w ) , ∀ v, w ∈ V, ∀ X ∈ U q ( u ) . Here we use the ∗ -structure of U q ( u ) . It is well-known that an U q ( u ) -invariant inner productexists on every representation V ( λ ) , and it is unique up to a constant. We typically write { v i } i for an orthonormal weight basis of V ( λ ) with respect to ( · , · ) , and write λ i for the weight of v i . We also denote by { f i } i the corresponding dual basis of V ( λ ) ∗ . ETRICS AND CONNECTIONS ON QUANTUM PROJECTIVE SPACES 5 Categorical preliminaries
The category of finite-dimensional U q ( g ) -modules is braided monoidal, that is we have atensor product and an analogue of the flip map. We use some of the language of tensorcategories to make our computations more natural, with [EGNO16] as our main reference.2.1. The braiding. A braiding on a monoidal category is the choice of a natural isomorphism X ⊗ Y ∼ = Y ⊗ X for each pair of objects X and Y , satisfying the hexagon relations [EGNO16,Definition 8.1.1]. It is a generalization of the flip map in the category of vector spaces.For the category of finite-dimensional U q ( g ) -modules we write the braiding as ˆ R V,W : V ⊗ W → W ⊗ V. An important relation satisfied by the braiding is the braid equation , which is (ˆ R W,Z ⊗ id V )(id W ⊗ ˆ R V,Z )(ˆ R V,W ⊗ id Z ) = (id Z ⊗ ˆ R V,W )(ˆ R V,Z ⊗ id W )(id V ⊗ ˆ R W,Z ) , acting on V ⊗ W ⊗ Z for any modules V, W, Z . In the following we employ a leg-notation forthe action on tensor products, in terms of which the braid equation reads (ˆ R W,Z ) (ˆ R V,Z ) (ˆ R V,W ) = (ˆ R V,W ) (ˆ R V,Z ) (ˆ R W,Z ) . (2.1)A braiding on the category of finite-dimensional U q ( g ) -modules is not quite unique. Weadopt the same choice as [HeKo06], which is described as follows. Consider two simple modules V ( λ ) and V ( µ ) and choose a highest weight vector v λ for the first and a lowest weight vector v w µ for the second. Then the braiding is completely determined by ˆ R V ( λ ) ,V ( µ ) ( v λ ⊗ v w µ ) = q ( λ,w µ ) v w µ ⊗ v λ . Here ( · , · ) denotes the usual non-degenerate symmetric bilinear form on the dual of the Cartansubalgebra of g (rescaled so that ( α, α ) = 2 for short roots α , for definiteness). Indeed, v λ ⊗ v w µ is a cyclic vector for V ( λ ) ⊗ V ( µ ) , hence ˆ R V ( λ ) ,V ( µ ) is completely determined by the action onthis vector and the fact that it is a U q ( g ) -module map.2.2. Duality.
The notion of duality in a monoidal category is captured by the existence of evaluation and coevaluation morphisms. In our setting these are maps ev V : V ∗ ⊗ V → C , coev V : C → V ⊗ V ∗ , ev ′ V : V ⊗ V ∗ → C , coev ′ V : C → V ∗ ⊗ V, satisfying certain duality relations to be recalled below. Here V is a finite-dimensional U q ( g ) module and V ∗ its linear dual. The maps ev V and coev V are related to the existence of a leftdual , while ev ′ V and coev ′ V to the existence of a right dual . In the case of U q ( g ) , the property S ( X ) = K ρ XK − ρ guarantees that the two duals can be identified.Let us now discuss the explicit formulae for the category of finite-dimensional U q ( g ) -modules. Take a weight basis { v i } i of V , with λ i the weight of v i , and a dual basis { f i } i of V ∗ . Then the evaluation and coevaluation maps are given by ev V ( f i ⊗ v j ) = δ ij , coev V = X i v i ⊗ f i , ev ′ V ( v i ⊗ f j ) = q (2 ρ,λ i ) δ ji , coev ′ V = X i q − (2 ρ,λ i ) f i ⊗ v i . The factor q (2 ρ,λ i ) comes from the action of K ρ and is related to the square of the antipode. MARCO MATASSA
In the following we are going to fix a simple module V and write E := ev V , E ′ := ev ′ V , C := coev V , C ′ := coev ′ V . We use the leg-notation for the action of these morphisms on tensor products. We write E i,i +1 ( w ⊗ · · · ⊗ w i − ⊗ f ⊗ v ⊗ w i +2 ⊗ · · · ⊗ w n ):= E ( f ⊗ v ) w ⊗ · · · ⊗ w i − ⊗ w i +2 ⊗ · · · w n , with v ∈ V and f ∈ V ∗ , while for the coevaluation we write C i ( w ⊗ · · · ⊗ w n ) := w ⊗ · · · ⊗ w i − ⊗ X j v j ⊗ f j ⊗ w i ⊗ · · · ⊗ w n . Similarly in the case of E ′ and C ′ . Using the leg-notation, the duality relations of [EGNO16,Section 2.10] for the evaluation and coevaluation morphisms can be written as E C = id , E C = id , E ′ C ′ = id , E ′ C ′ = id . (2.2)We also have various compatibility relations with the braiding ˆ R V,W , since the latter is anatural isomorphism in both entries. For the evaluation morphisms we have E = E (ˆ R V ∗ ,W ) (ˆ R V,W ) , E = E (ˆ R W,V ) (ˆ R W,V ∗ ) , E ′ = E ′ (ˆ R V,W ) (ˆ R V ∗ ,W ) , E ′ = E ′ (ˆ R W,V ∗ ) (ˆ R W,V ) . (2.3)Similarly, for the coevaluations morphisms we have C = (ˆ R W,V ∗ ) (ˆ R W,V ) C , C = (ˆ R V,W ) (ˆ R V ∗ ,W ) C , C ′ = (ˆ R W,V ) (ˆ R W,V ∗ ) C ′ , C ′ = (ˆ R V ∗ ,W ) (ˆ R V,W ) C ′ . (2.4)Finally we need the following identity, valid for a simple module V . Lemma 2.1.
Let V = V ( λ ) be a simple module. Then we have E ◦ ˆ R V,V ∗ = q − ( λ,λ +2 ρ ) E ′ . (2.5) Proof.
Observe that both E ◦ ˆ R V,V ∗ and E ′ are morphisms from V ⊗ V ∗ to C . Since V is asimple module, we must have E ◦ ˆ R V,V ∗ = c E ′ for some c ∈ C . To find the constant we evaluateboth sides at v ⊗ f , where v is a highest weight vector of V = V ( λ ) and f is its dual. Inour conventions for the braiding we have ˆ R V,V ∗ ( v ⊗ f ) = q − ( λ,λ ) f ⊗ v . Then E ◦ ˆ R V,V ∗ ( v ⊗ f ) = q − ( λ,λ ) E ( f ⊗ v ) = q − ( λ,λ ) . On the other hand we have E ′ ( v ⊗ f ) = q (2 ρ,λ ) = q (2 ρ,λ ) . Hence c = q − ( λ,λ +2 ρ ) . (cid:3) Differential calculi, metric and connections
In this section we collect various definitions about differential calculi and connections.
ETRICS AND CONNECTIONS ON QUANTUM PROJECTIVE SPACES 7
Differential calculi.
In this section A denotes an arbitrary algebra. The definitionsrecalled here are fairly standard and one possible reference is [KlSc97]. Definition 3.1. A differential calculus over A is a differential graded algebra (Ω • , d) suchthat Ω = A and which is generated by the elements a, d b with a, b ∈ A .If A is a ∗ -algebra, we say that (Ω • , d) is a ∗ -differential calculus if in addition the ∗ -structure of A extends to an involutive conjugate-linear map on Ω • , such that d a ∗ = (d a ) ∗ and ( ω ∧ χ ) ∗ = ( − pq χ ∗ ∧ ω ∗ for all ω ∈ Ω p and χ ∈ Ω q .The concrete definition of a differential calculus usually begins with the description of itsdegree-one part. This leads to the following definition. Definition 3.2. A first order differential calculus (FODC) over A is an A -bimodule Ω witha linear map d : A → Ω which obeys the Lebnitz rule d( ab ) = d ab + a d b, a, b ∈ A, and such that Ω is generated as a left A -module by the elements d a with a ∈ A .Given any FODC (Ω , d) , there exists a universal differential calculus such that its degree-onepart is Ω . The universal property in this case is the following. Definition 3.3.
The universal differential calculus associated to a FODC (Ω , d) over A isthe unique differential calculus (Ω • u , d u ) over A with Ω u = Ω , d u | A = d and such that thefollowing property is satisfied: for any differential calculus (Γ • , d ′ ) with Γ = Ω and d ′ | A = d ,there exists a map of differential graded algebras φ : Ω • u → Γ • such that φ | A ⊕ Ω = id .The universal differential calculus can be constructed as a quotient of the tensor algebraof the A -bimodule Ω , with differential d u ( a d a ∧ · · · ∧ d a n ) = d a d a ∧ · · · ∧ d a n . Anydifferential calculus can be obtained as a quotient of the universal differential calculus.Finally we recall the notion of induced calculus over a subalgebra. Definition 3.4.
Let B ⊂ A be a subalgebra and (Ω , d) a FODC over A . Then the induced FODC over B is defined by Ω | B = span { b d b : b , b ∈ B } and with differential d | B .3.2. Metrics.
We now recall the notion of quantum metric as stated in [BeMa20, Definition1.15]. Notice that invertibility is part of the definition.
Definition 3.5. A (generalized) quantum metric is an element g ∈ Ω ⊗ A Ω which is invert-ible, in the sense that there exists a bimodule map ( · , · ) : Ω ⊗ A Ω → A such that ( ω, g (1) ) g (2) = ω = g (1) ( g (2) , ω ) for all ω ∈ Ω , where we write g = g (1) ⊗ g (2) . Remark . From the categorical point of view, a quantum metric makes Ω into a self-dualobject in the monoidal category of A -bimodules.Notice that the definition of a quantum metric only uses Ω , the degree-one part of Ω • . Toimpose an analogue of the symmetry condition we also use Ω . Definition 3.7.
A quantum metric g ∈ Ω ⊗ A Ω is symmetric if we have ∧ ( g ) = 0 , where ∧ : Ω ⊗ Ω → Ω denotes the wedge product of one-forms.Finally, in the case when A is a ∗ -algebra and Ω • is a ∗ -differential calculus, we can requirethe metric to be real in the following sense. Definition 3.8.
A quantum metric g ∈ Ω ⊗ A Ω is real if we have g † = g , where † :=flip ◦ ( ∗ ⊗ ∗ ) is given by the ∗ -structure composed with the flip map. MARCO MATASSA
Connections.
The notion of connection on a module is quite standard. We are onlygoing to consider left connections, so we omit "left" after the definition.
Definition 3.9.
A (left) connection on a (left) A -module E is a linear map ∇ E : E → Ω ⊗ A E which obeys the (left) Leibnitz rule, that is ∇ E ( ae ) = d a ⊗ e + a ∇ E ( e ) , a ∈ A, e ∈ E. For the left A -module E = Ω we can define additional properties. Definition 3.10.
The torsion of a connection ∇ : Ω → Ω ⊗ A Ω is the left A -module map T ∇ : Ω → Ω defined by T ∇ := ∧ ◦ ∇ − d . A connection is called torsion free if T ∇ = 0 .Now suppose that Ω admits a quantum metric g ∈ Ω ⊗ A Ω as in Definition 3.5. Thenwe can consider the cotorsion of the connection ∇ with respect to g , a notion introduced byMajid in [Maj99] as a weaker version of metric compatibility. Definition 3.11.
The cotorsion of a connection ∇ : Ω → Ω ⊗ A Ω with quantum metric g ∈ Ω ⊗ A Ω is the element co T ∇ ∈ Ω ⊗ A Ω defined by co T ∇ := (d ⊗ id − ( ∧ ⊗ id) ◦ (id ⊗ ∇ )) g. A connection is called cotorsion free if co T ∇ = 0 . Remark . Let M be a smooth manifold with metric g . Consider a connection ∇ on M ,defined in the usual sense as acting on vector fields. Denote by ∇ ∗ its dual connection actingon one-forms, defined in terms of the metric g . As discussed in [BeMa20, Corollary 5.70], thecotorsion of the connection ∇ can be identified with the torsion of the dual connection ∇ ∗ .In particular, if ∇ is torsion free, then the cotorsion free condition gives ( ∇ X g )( Y, Z ) = ( ∇ Y g )( X, Z ) for all vector fields X, Y, Z . This is weaker condition than ( ∇ X g )( Y, Z ) = 0 , which is thestandard metric compatibility condition with respect to the metric g .In the classical case, the Levi-Civita connection is the unique connection on the tangentbundle of a smooth manifold which is torsion free and compatible with the metric. Thismotivates the following definition, see [BeMa20, Definition 8.2]. Definition 3.13. A weak quantum Levi-Civita connection is a connection ∇ : Ω → Ω ⊗ A Ω which is torsion free and cotorsion free.To introduce a strong version we need bimodule connections, which we now recall.3.4. Bimodule connections.
Classically a connection on Ω naturally extends to a connec-tion on the tensor product Ω ⊗ A Ω . In the quantum case this lifting requires the connectionto be a bimodule connection, defined as in [BeMa20, Definition 3.66]. Definition 3.14.
A (left) bimodule connection on an A -bimodule E is a (left) connection ∇ E : E → Ω ⊗ A E together with an A -bimodule map σ E : E ⊗ A Ω → Ω ⊗ A E such that ∇ E ( ea ) = σ E ( e ⊗ d a ) + ∇ E ( e ) a, e ∈ E, a ∈ A. ETRICS AND CONNECTIONS ON QUANTUM PROJECTIVE SPACES 9
The bimodule map σ E , called the generalized braiding , is not additional data for the con-nection. Indeed, if it exists it is uniquely determined by the condition above.A bimodule connection ∇ : Ω → Ω ⊗ A Ω can be extended to a connection on Ω ⊗ A Ω by the Leibnitz rule and the generalized braiding, see [BeMa20, Theorem 3.78]. In particular,given a quantum metric g ∈ Ω ⊗ A Ω we can consider ∇ g = ( ∇ ⊗ id) g + ( σ ⊗ id)(id ⊗ ∇ ) g. This naturally leads to the following definition.
Definition 3.15.
Let ∇ be a bimodule connection on Ω . We say that it is quantum metriccompatible with a quantum metric g ∈ Ω ⊗ A Ω if ∇ g = 0 . Remark . As discussed before, in the classical case cotorsion freeness is a weaker propertythan metric compatibility. In the quantum case we need an extra assumption to compare thetwo notions, namely the condition ∧ ◦ ( σ + id) = 0 for the generalized braiding. Under thiscondition and torsion freeness of ∇ we obtain co T ∇ = ( ∧ ⊗ id) ∇ g , see [BeMa20, Section 8.1].This shows that cotorsion freeness is a weaker property, in this case.Finally we can formulate the notion of Levi-Civita connection, as in the classical case. Definition 3.17.
Let ∇ : Ω → Ω ⊗ A Ω be a bimodule connection and let g ∈ Ω ⊗ A Ω bea quantum metric. Then we say that ∇ is a quantum Levi-Civita connection if it is torsionfree and quantum metric compatible with g .4. Quantum flag manifolds
The quantum projective spaces can be regarded as the easiest family to describe withinthe class of quantum (irreducible) flag manifolds. All these quantum spaces admit a uniformdescription, which we recall here. Even though the focus of this paper is on the projectivespaces, many of our computations also work for general irreducible flag manifolds. We takesome care in explaining the index-free notation we are going to employ in the following, as itsimplifies the computations tremendously (once one gets the hang of it).4.1.
Geometrical description.
We start by quickly recalling the definition of a flag manifoldin the classical case, for precise definitions see for instance [ČaSl09]. Let G be a complex simpleLie group, with compact real form U . Corresponding to any subset of simple roots, denoted by S , we can define a parabolic subgroup P S ⊂ G and a Levi subgroup L S ⊂ P S . A (generalized)flag manifold is a homogeneous space of the form G/P S . In terms of the compact real form,we have the subgroup K S := P S ∩ U = L S ∩ U and the isomorphism G/P S ∼ = U/K S .In the quantum case, we begin by introducing an analogue of the Levi factor l S (the Liealgebra of L S ), following [StDi99]. The quantized Levi factor U q ( l S ) is defined by U q ( l S ) := h K i , E j , F j : i ∈ I, j ∈ S i ⊂ U q ( g ) . Here h·i denotes the subalgebra generated by the given elements in U q ( g ) . It is easily verifiedthat U q ( l S ) is a Hopf subalgebra. Moreover it is a Hopf ∗ -subalgebra with ∗ correspondingto the compact real form. Taking the ∗ -structure into account we write U q ( k S ) := ( U q ( l S ) , ∗ ) .The quantum flag manifold C q [ U/K S ] is then defined as C q [ U/K S ] := C q [ U ] U q ( k S ) = { a ∈ C q [ U ] : X ⊲ a = ε ( X ) a, ∀ X ∈ U q ( k S ) } . In the following we restrict to the case of irreducible flag manifolds. At the Lie algebralevel these can be characterized as follows: the set S consists of all the simple roots exceptfor α s , where α s is a simple root appearing with multiplicity one in the highest root of g . Generators and relations.
The quantum flag manifolds C q [ U/K S ] admit a uniformdescription in terms of generators and relations. We follow the presentation in [HeKo06].Consider the simple U q ( g ) -module V := V ( ω s ) , where ω s is the fundamental weight corre-sponding to the simple root α s described above, and write N := dim V .We define the algebra A with generators { v i , f i } Ni =1 and relations f i f j = q − ( ω s ,ω s ) X k,l (ˆ R V,V ) ijkl f k f l , v i v j = q − ( ω s ,ω s ) X k,l (ˆ R V ∗ ,V ∗ ) ijkl v k v l ,v i f j = q ( ω s ,ω s ) X k,l (ˆ R V,V ∗ ) ijkl f k v l , X i v i f i = 1 . (4.1)These generators should be interpreted as follows: after fixing a weight basis { v i } Ni =1 of V = V ( ω s ) , we have the dual basis { f i } Ni =1 of V ∗ ∼ = V ( − w ω s ) and the double dual basis { v i } Ni =1 of V ∗∗ ∼ = V ( ω s ) (defined by v i ( f j ) = δ ij ). One can check that this identification makes A into a U q ( g ) -module algebra. Then we have the following result. Lemma 4.1.
The U q ( g ) -module algebra A is isomorphic to the U q ( g ) -module subalgebra of C q [ G ] generated by the matrix coefficients c ω s f i ,v and c − w ω s v i ,f , where v is a fixed highest weightvector of V ( ω s ) . The isomorphism is given by f i c ω s f i ,v , v i c − w ω s v i ,f . The algebra A is Z -graded by deg f i := 1 and deg v i := − . We write B := A for itsdegree-zero subalgebra, which is generated by the elements p ij := f i v j . Proposition 4.2.
The algebra B is isomorphic to the quantum irreducible flag manifold C q [ U/K S ] as a U q ( g ) -module under the isomorphism above. The relations for the generators p ij of B are given in [HeKo06, Section 3.1.3]. Write P V,V := ˆ R V,V − q ( ω s ,ω s ) , P V ∗ ,V ∗ := ˆ R V ∗ ,V ∗ − q ( ω s ,ω s ) . Then the relations can be written as X a,b,c,d ( P V,V ) ijab (ˆ R − V,V ∗ ) nkcd p ac p dl = 0 , X a,b,c,d ( P V ∗ ,V ∗ ) klab (ˆ R − V,V ∗ ) jacd p ic p db = 0 , X i q − (2 ρ,λ i ) p ii = q ( ω s , ρ ) . (4.2) Remark . The last relation appears as q ( ω s ,ω s ) P i,j,k (ˆ R V,V ∗ ) kkij p ij = 1 in [HeKo06]. Werewrite it using the identity E ◦ ˆ R V,V ∗ = q − ( ω s ,ω s +2 ρ ) E ′ from Lemma 2.1, which leads to P k (ˆ R V,V ∗ ) kkij = q − ( ω s ,ω s +2 ρ ) q − (2 ρ,λ i ) δ ij . Using this we obtain P i q − (2 ρ,λ i ) p ii = q ( ω s , ρ ) .As shown in [Mat19, Proposition 3.3], the algebras A and B can be made into ∗ -algebrasas follows. Choosing an orthonormal basis for V = V ( ω s ) with respect to a U q ( u ) -invariantinner product, the ∗ -structure is given by ( f i ) ∗ = v i . In this case, the isomorphism fromLemma 4.1 becomes a ∗ -isomorphism. For the generators p ij of B we have ( p ij ) ∗ = p ji .4.3. Index-free notation.
In the following we adopt an index-free notation, as done in[HeKo06], since it makes computations significantly clearer. The basic idea is very simple: forinstance, with { v i } i the basis of V (note the lower index) we write X a,b (ˆ R V,V ) abjk v i v a v b v l ←→ (ˆ R V,V ) vvvv. ETRICS AND CONNECTIONS ON QUANTUM PROJECTIVE SPACES 11
We want to use a similar notation for the generators { f i , v i } Ni =1 of A . What complicatesmatters here is that we want to consider f i as a linear functional on V and v i as a linearfunctional on V ∗ , since this is how we have defined the U q ( g ) -module structure on A (andthe reason why we use upper indices for the generators). The bottom line is that we need toconsider the action of U q ( g ) -module maps on these elements via the transpose.To give a concrete example, in this notation the first relation of (4.1) becomes f i f j = q − ( ω s ,ω s ) X k,l (ˆ R V,V ) ijkl f k f l ←→ f f = q − ( ω s ,ω s ) (ˆ R V,V ) f f. To give a different example, the last relation P i v i f i = 1 of (4.1) becomes E vf = 1 , sincethe LHS is P i,j E ij v i f j and we have E ij = δ ij . As a final example, the expression C f carriesthree indices and corresponds to ( C f ) ijk = δ ij f k .With this notation, the relations of A can be rewritten in the condensed form f f = q − ( ω s ,ω s ) (ˆ R V,V ) f f, vv = q − ( ω s ,ω s ) (ˆ R V ∗ ,V ∗ ) vv,vf = q ( ω s ,ω s ) (ˆ R V,V ∗ ) f v, E vf = 1 . (4.3)The situation is similar for the flag manifold B ⊂ A . In this case we have the generators p ij = f i v j , which carry two indices, and the relations can be written as ( P V,V ) (ˆ R − V,V ∗ ) pp = 0 , ( P V ∗ ,V ∗ ) (ˆ R − V,V ∗ ) pp = 0 , E ′ p = q ( ω s , ρ ) . (4.4)5. Heckenberger-Kolb calculus
In this section we describe the Heckenberger-Kolb calculus associated to an irreduciblequantum flag manifold, as introduced in [HeKo06]. We also give a slightly different presenta-tion of some of the relations, which turns out to be more convenient for our purposes. Finallywe focus on a particular situation, which we refer to as the quadratic case, which geometricallycorresponds to the quantum projective spaces.5.1.
Definitions.
We start by describing the FODC (Ω , d) associated to the Heckenberger-Kolb calculus. We have Ω := Ω + ⊕ Ω − and d := ∂ + ¯ ∂ , where the two FODCs Ω + and Ω − are generated as left B -modules by ∂p and ¯ ∂p respectively (we use the index-free notationfrom now on). To describe the relations we need some additional notation. Recall that P V,V = ˆ R V,V − q ( ω s ,ω s ) and P V ∗ ,V ∗ = ˆ R V ∗ ,V ∗ − q ( ω s ,ω s ) . We also write Q V,V := ˆ R V,V + q ( ω s ,ω s ) − ( α s ,α s ) , Q V ∗ ,V ∗ := ˆ R V ∗ ,V ∗ + q ( ω s ,ω s ) − ( α s ,α s ) . Then Ω + is generated by ∂p , as a left B -module, with relations ( P V,V ) ( Q V,V ) (ˆ R − V,V ∗ ) p∂p = 0 , ( P V ∗ ,V ∗ ) (ˆ R − V,V ∗ ) p∂p = 0 , E ′ ∂p = 0 . (5.1)Similarly, Ω − is generated by ¯ ∂p with relations ( P V ∗ ,V ∗ ) ( Q V ∗ ,V ∗ ) (ˆ R − V,V ∗ ) p ¯ ∂p = 0 , ( P V,V ) (ˆ R − V,V ∗ ) p ¯ ∂p = 0 , E ′ ¯ ∂p = 0 . (5.2)To define the right B -module structure, let us introduce the notation T := (ˆ R V,V ∗ ) (ˆ R V,V ) (ˆ R − V ∗ ,V ∗ ) (ˆ R − V,V ∗ ) . (5.3)Then the right B -module structure of Ω + and Ω − is defined by ∂pp = q ( α s ,α s ) T p∂p, ¯ ∂pp = q − ( α s ,α s ) T p ¯ ∂p. (5.4)Finally, the Heckenberger-Kolb calculus (Ω • , d) is the universal differential calculus associ-ated to (Ω , d) . It turns out that we have the decomposition d = ∂ + ¯ ∂ also in higher degrees. In particular, this implies that ∂ = ¯ ∂ = 0 and ∂ ¯ ∂ = − ¯ ∂∂ . Moreover, as shown in [Mat19,Theorem 4.2], the calculus (Ω • , d) becomes a ∗ -calculus upon setting ( ∂p ij ) ∗ = ¯ ∂p ji .5.2. Induced calculi.
In [HeKo06] the FODCs Ω + and Ω − over B are constructed as inducedcalculi from some auxiliary FODCs Γ + and Γ − over the larger algebra A . This description isalso useful for us, so we recall the details below.Consider the algebra A introduced before, generated by f and v . We define the left A -modules Γ + and Γ − , generated respectively by ∂f and ¯ ∂v , with relations ( P V,V ) ( Q V,V ) f ∂f = 0 , ( P V ∗ ,V ∗ ) ( Q V ∗ ,V ∗ ) v ¯ ∂v = 0 , E v∂f = 0 , E ′ f ¯ ∂v = 0 . (5.5)The last two relations come from E vf = 1 and E ′ f v = q ( ω s , ρ ) .The right A -module relations are as follows. For Γ + we have ∂f f = q ( α s ,α s ) − ( ω s ,ω s ) (ˆ R V,V ) f ∂f, ∂f v = q − ( ω s ,ω s ) (ˆ R − V,V ∗ ) v∂f. (5.6)The relations for Γ − are ¯ ∂vf = q ( ω s ,ω s ) (ˆ R V,V ∗ ) f ¯ ∂v, ¯ ∂vv = q ( ω s ,ω s ) − ( α s ,α s ) (ˆ R − V ∗ ,V ∗ ) v ¯ ∂v. (5.7)The differentials of the two FODCs are specified by requiring that ∂v = 0 and ¯ ∂f = 0 .The FODCs Ω + and Ω − are the induced calculi over B obtained from Γ + and Γ − . Thisdescription, together with the relation E vf = 1 and p = f v , leads to the relations E p∂p = 0 , E ∂pp = ∂p, E p ¯ ∂p = ¯ ∂p, E ¯ ∂pp = 0 . (5.8)For more details see for instance [Mat19, Lemma 5.2] (with different notation).5.3. Different presentation.
It is convenient to work with the relations of the FODC Ω ina slightly different form, which we now derive. We begin by defining the maps S := (ˆ R V,V ∗ ) (ˆ R V,V ) (ˆ R − V,V ∗ ) , ˜ S := (ˆ R V,V ∗ ) (ˆ R − V ∗ ,V ∗ ) (ˆ R − V,V ∗ ) . (5.9)Observe that T = S ˜ S , where T is the map defined in (5.3) and related to the right B -module structure. The proof of the following result is given in Proposition B.1. Proposition 5.1.
The maps S and ˜ S satisfy the following properties. (1) We have the commutation relations S ˜ S = ˜ S S , ˜ S S = S ˜ S . (5.10)(2) We have the "braid equations" S S S = S S S , ˜ S ˜ S ˜ S = ˜ S ˜ S ˜ S . (5.11)In particular, observe that we can also write T = ˜ S S .We now use the maps S and ˜ S to rewrite some of the relations of the FODC Ω . We beginwith the relations that involve P V,V and P V ∗ ,V ∗ . ETRICS AND CONNECTIONS ON QUANTUM PROJECTIVE SPACES 13
Lemma 5.2.
The relations ( P V ∗ ,V ∗ ) (ˆ R − V,V ∗ ) p∂p = 0 , ( P V,V ) (ˆ R − V,V ∗ ) p ¯ ∂p = 0 , of the FODC Ω are equivalent to (˜ S − q − ( ω s ,ω s ) ) p∂p = 0 , ( S − q ( ω s ,ω s ) ) p ¯ ∂p = 0 . (5.12) Using the right B -module structure, they are also equivalent to (˜ S − q − ( ω s ,ω s ) ) ∂pp = 0 , ( S − q ( ω s ,ω s ) ) ¯ ∂pp = 0 . (5.13) Proof.
We consider the first relation ( P V,V ) (ˆ R − V,V ∗ ) p ¯ ∂p = 0 , the other one is treated in asimilar way. Applying (ˆ R V,V ∗ ) and using P V,V = ˆ R V,V − q ( ω s ,ω s ) we get (ˆ R V,V ∗ ) (ˆ R V,V ) (ˆ R − V,V ∗ ) p ¯ ∂p − q ( ω s ,ω s ) p ¯ ∂p = 0 . Since S = (ˆ R V,V ∗ ) (ˆ R V,V ) (ˆ R − V,V ∗ ) we obtain ( S − q ( ω s ,ω s ) ) p ¯ ∂p = 0 .Next, using the right B -module structure from (5.4), we have p ¯ ∂p = q ( α s ,α s ) T − ¯ ∂pp . Thenthe identity ( S − q ( ω s ,ω s ) ) p ¯ ∂p = 0 can be rewritten in the form ( S − q ( ω s ,ω s ) ) T − ¯ ∂pp = 0 . As S commutes with T − , the identity ( S − q ( ω s ,ω s ) ) ¯ ∂pp = 0 follows by applying T . (cid:3) Note that the previous identities also hold with pp instead of p∂p or p ¯ ∂p , since we have therelations (4.4). Next, we rewrite the right B -module relations. Lemma 5.3.
The right B -module relations (5.4) of Ω are equivalent to ∂pp = q ( α s ,α s ) − ( ω s ,ω s ) S p∂p, ¯ ∂pp = q ( ω s ,ω s ) − ( α s ,α s ) ˜ S p ¯ ∂p. (5.14) Proof.
By Lemma 5.2, the relation ( P V ∗ ,V ∗ ) (ˆ R − V,V ∗ ) p∂p = 0 is equivalent to ˜ S p∂p = q − ( ω s ,ω s ) p∂p . Using this and T = S ˜ S we compute ∂pp = q ( α s ,α s ) T p∂p = q ( α s ,α s ) S ˜ S p∂p = q ( α s ,α s ) − ( ω s ,ω s ) S p∂p. Similarly, using S p ¯ ∂p = q ( ω s ,ω s ) p ¯ ∂p we obtain ¯ ∂pp = q − ( α s ,α s ) T p ¯ ∂p = q − ( α s ,α s ) ˜ S S p ¯ ∂p = q ( ω s ,ω s ) − ( α s ,α s ) ˜ S p ¯ ∂p. (cid:3) Finally, we rewrite the relations involving P V,V Q V,V and P V ∗ ,V ∗ Q V ∗ ,V ∗ in terms the maps S and ˜ S , as well as using the bimodule structure of Ω . Lemma 5.4.
The relations ( P V,V ) ( Q V,V ) (ˆ R − V,V ∗ ) p∂p = 0 , ( P V ∗ ,V ∗ ) ( Q V ∗ ,V ∗ ) (ˆ R − V,V ∗ ) p ¯ ∂p = 0 , of the FODC Ω are equivalent to ( S − q ( ω s ,ω s ) )( p∂p + ∂pp ) = 0 , (˜ S − q − ( ω s ,ω s ) )( p ¯ ∂p + ¯ ∂pp ) = 0 . (5.15) Proof.
We only prove the first identity, the second one is treated similarly. Consider ( P V,V ) (ˆ R − V,V ∗ ) (ˆ R V,V ∗ ) ( Q V,V ) (ˆ R − V,V ∗ ) p∂p = 0 . Using Lemma 5.2 for the element (ˆ R V,V ∗ ) ( Q V,V ) (ˆ R − V,V ∗ ) p∂p , this is equivalent to ( S − q ( ω s ,ω s ) )(ˆ R V,V ∗ ) ( Q V,V ) (ˆ R − V,V ∗ ) p∂p = 0 . Next, using Q V,V = ˆ R V,V + q ( ω s ,ω s ) − ( α s ,α s ) we rewrite (ˆ R V,V ∗ ) ( Q V,V ) (ˆ R − V,V ∗ ) = (ˆ R V,V ∗ ) (ˆ R V,V ) (ˆ R − V,V ∗ ) + q ( ω s ,ω s ) − ( α s ,α s ) = q ( ω s ,ω s ) − ( α s ,α s ) ( q ( α s ,α s ) − ( ω s ,ω s ) S + 1) . Using ∂pp = q ( α s ,α s ) − ( ω s ,ω s ) S p∂p from (5.13) we obtain (ˆ R V,V ∗ ) ( Q V,V ) (ˆ R − V,V ∗ ) p∂p = q ( ω s ,ω s ) − ( α s ,α s ) ( ∂pp + p∂p ) . Hence we conclude that ( S − q ( ω s ,ω s ) )( p∂p + ∂pp ) = 0 . (cid:3) The quadratic case.
In this paper we consider the situation when ˆ R V,V satisfies aquadratic relation, and refer to this as the quadratic case . When V = V ( ω s ) , this correspondsto a tensor product decomposition with only two simple factors, that is V ( ω s ) ⊗ V ( ω s ) ∼ = V (2 ω s ) ⊕ V (2 ω s − α s ) . The eigenvalues of the braiding in this case are q ( ω s ,ω s ) and − q ( ω s ,ω s ) − ( α s ,α s ) , corresponding to V (2 ω s ) and V (2 ω s − α s ) respectively. The quadratic relation satisfied by the braiding ˆ R V,V ,also known as the Hecke relation in this context, is given by P V,V Q V,V = (ˆ R V,V − q ( ω s ,ω s ) )(ˆ R V,V + q ( ω s ,ω s ) − ( α s ,α s ) ) = 0 . The situation is completely analogous for ˆ R V ∗ ,V ∗ . Geometrically, the quadratic case of theHeckenberger-Kolb calculus corresponds to the quantum projective spaces. Indeed this holdsfor U q ( g ) = U q ( sl r +1 ) and the choice ω s = ω or ω s = ω r , corresponding to the fundamentalrepresentation or its dual, which satisfies the quadratic decomposition above.In the quadratic case the relations for Ω + and Ω − can be simplified. Indeed, the firstrelation of (5.1) is automatically satisfied, due to the quadratic relation for ˆ R V,V . Similarlyfor the first relation of (5.2), due to the quadratic relation for ˆ R V ∗ ,V ∗ . Taking into accountthe presentation in terms of S and ˜ S discussed above, we obtain the following description.In the quadratic case, Ω + is generated as a left B -module by ∂p with relations (˜ S − q − ( ω s ,ω s ) ) p∂p = 0 , E ′ ∂p = 0 . (5.16)Similarly, Ω − is generated by ¯ ∂p with relations ( S − q ( ω s ,ω s ) ) p ¯ ∂p = 0 , E ′ ¯ ∂p = 0 . (5.17)Let us also consider the FODCs Γ + and Γ − over the larger algebra A in the quadratic case.The first two relations in (5.5) are identically satisfied and we are left with E v∂f = 0 , E ′ f ¯ ∂v = 0 . (5.18)Finally, as a consequence of the quadratic relation for the braiding, we have the followingrelations for the maps S and ˜ S , as shown in Lemma B.2. Lemma 5.5.
In the quadratic case we have the relations S = q ω s ,ω s ) − ( α s ,α s ) S − + q ( ω s ,ω s ) (1 − q − ( α s ,α s ) ) , (5.19) ˜ S = q ( α s ,α s ) − ω s ,ω s ) ˜ S − + q − ( ω s ,ω s ) (1 − q ( α s ,α s ) ) . (5.20) ETRICS AND CONNECTIONS ON QUANTUM PROJECTIVE SPACES 15 Quantum metrics
In this section we define quantum metrics for the quantum projective spaces, reducing tothe Fubini-Study metrics in the classical case. Our main result here is Theorem 6.11, whichshows that these are quantum metrics according to Definition 3.5, that is they are invertiblein a suitable sense. We also discuss various properties they satisfy.6.1.
Definition and properties.
For this first part there is no particular need to restrictto the case of quantum projective spaces, hence Ω denotes the Heckenberger-Kolb FODCcorresponding to a generic quantum irreducible flag manifold.We define g := g + − + g − + where we write g + − := E ′ E ∂p ⊗ ¯ ∂p ∈ Ω + ⊗ B Ω − ,g − + := E ′ E ¯ ∂p ⊗ ∂p ∈ Ω − ⊗ B Ω + . (6.1) Remark . For the quantum projective spaces, g reduces to the Fubini-Study metric in theclassical limit. This can be seen from the formula (A.1) in terms of the projection p .Before tackling the issue of invertibility, we show some properties satisfied by g . We beginby showing that g is symmetric (Definition 3.7) and real (Definition 3.8). Proposition 6.2.
We have that g is symmetric and real.Proof. To show that g is symmetric consider the identity (C.8), that is ∂ ¯ ∂p = E ∂p ∧ ¯ ∂p + E ¯ ∂p ∧ ∂p. Applying E ′ and using E ′ ¯ ∂p = 0 from (5.2) we obtain E ′ E ∂p ∧ ¯ ∂p + E ′ E ¯ ∂p ∧ ∂p. Now observe that the right-hand side is equal to ∧ ( g ) = ∧ ( g + − ) + ∧ ( g − + ) , where here weconsider the wedge product as a map ∧ : Ω ⊗ B Ω → Ω .To show that g is real it is convenient to employ the usual index notation. We have g + − = P i,j q (2 ρ,λ i ) ∂p ij ⊗ ¯ ∂p ji and g − + = P i,j q (2 ρ,λ i ) ¯ ∂p ij ⊗ ∂p ji . Using ( ∂p ij ) ∗ = ¯ ∂p ji and ( ¯ ∂p ij ) ∗ = ∂p ji we easily check that g † + − = g − + and hence g † = g . (cid:3) Remark . It is also possible to show that g is left C q [ G ] -coinvariant, which amounts to acomputation similar to that of [Mat19, Lemma 5.4].The next property we want to discuss is related to Kähler metrics. First, we need thefollowing technical result on the vanishing of certain terms of degree . This is essentialyproven in [Mat19, Lemma 5.3], but we revisit it here using the index-free notation.From now on all tensor products are over B (omitted), except where specified. Lemma 6.4.
We have E ′ E E ∂p ⊗ ¯ ∂p ⊗ ∂p = 0 , E ′ E E ¯ ∂p ⊗ ∂p ⊗ ¯ ∂p = 0 . Proof.
Write A = E ′ E E ∂p ⊗ ¯ ∂p ⊗ ∂p for the first term. Using the identity ∂p = E ∂pp from (5.8) and the right B -module relations (5.4) we compute A = E ′ E E E ∂p ⊗ ¯ ∂p ⊗ ∂pp = q ( α s ,α s ) E ′ E E E T T T p∂p ⊗ ¯ ∂p ⊗ ∂p = q ( α s ,α s ) E ′ E E T E T T p∂p ⊗ ¯ ∂p ⊗ ∂p. In the last step we have used E E T = E T E , as E is an evaluation. Now considerthe identity E T T = T E from (C.1). Using it twice we get A = q ( α s ,α s ) E ′ E E T T E p∂p ⊗ ¯ ∂p ⊗ ∂p = q ( α s ,α s ) E ′ E E T T E p∂p ⊗ ¯ ∂p ⊗ ∂p = q ( α s ,α s ) E ′ E T E E p∂p ⊗ ¯ ∂p ⊗ ∂p. Next, using E ′ E T = E ′ E from (C.3) we have A = q ( α s ,α s ) E ′ E E E p∂p ⊗ ¯ ∂p ⊗ ∂p. Finally using E E E = E E E and E p∂p = 0 we obtain A = q ( α s ,α s ) E ′ E E E p∂p ⊗ ¯ ∂p ⊗ ∂p = 0 . The second identity is similar. Write B = E ′ E E ¯ ∂p ⊗ ∂p ⊗ ¯ ∂p . Using the identity ¯ ∂p = E p ¯ ∂p from (5.8) and the right B -module relations (5.4) we compute B = E ′ E E E p ¯ ∂p ⊗ ∂p ⊗ ¯ ∂p = q ( α s ,α s ) E ′ E E E T − T − T − ¯ ∂p ⊗ ∂p ⊗ ¯ ∂pp = q ( α s ,α s ) E ′ E E E T − T − T − ¯ ∂p ⊗ ∂p ⊗ ¯ ∂pp. We have E T − T − = T − E , again from (C.1). Then B = q ( α s ,α s ) E ′ E E T − E T − ¯ ∂p ⊗ ∂p ⊗ ¯ ∂pp = q ( α s ,α s ) E ′ E E T − T − E ¯ ∂p ⊗ ∂p ⊗ ¯ ∂pp = q ( α s ,α s ) E ′ E E T − T − E ¯ ∂p ⊗ ∂p ⊗ ¯ ∂pp = q ( α s ,α s ) E ′ E T − E E ¯ ∂p ⊗ ∂p ⊗ ¯ ∂pp. Using E ′ E T − = E ′ E from (C.3) we rewrite B = q ( α s ,α s ) E ′ E E E ¯ ∂p ⊗ ∂p ⊗ ¯ ∂pp. Finally using E E E = E E E and E ¯ ∂pp = 0 we obtain B = q ( α s ,α s ) E ′ E E E ¯ ∂p ⊗ ∂p ⊗ ¯ ∂pp = 0 . (cid:3) We are now ready to prove the Kähler property of the metric.
Proposition 6.5.
The metric g satisfies (d ⊗ id) g = (id ⊗ d) g = 0 . Proof.
The metric can be written in the form g = g + − + g − + = E ′ E ( ∂p ⊗ ¯ ∂p + ¯ ∂p ⊗ ∂p ) . We apply d to the first leg. Using d = ∂ + ¯ ∂ and ∂ = ¯ ∂ = 0 we obtain (d ⊗ id) g = E ′ E ( ¯ ∂∂p ⊗ ¯ ∂p + ∂ ¯ ∂p ⊗ ∂p ) . We have ∂ ¯ ∂p = E ( ∂p ∧ ¯ ∂p + ¯ ∂p ∧ ∂p ) by (C.8). Using this and ∂ ¯ ∂ = − ¯ ∂∂ we get (d ⊗ id) g = − E ′ E E ( ∂p ∧ ¯ ∂p ⊗ ¯ ∂p + ¯ ∂p ∧ ∂p ⊗ ¯ ∂p )+ E ′ E E ( ∂p ∧ ¯ ∂p ⊗ ∂p + ¯ ∂p ∧ ∂p ⊗ ∂p ) . ETRICS AND CONNECTIONS ON QUANTUM PROJECTIVE SPACES 17
The second and third term of the previous expression vanish by Lemma 6.4. The other twoterms also vanish, since we have the relations E ∂p ⊗ ∂p = E ¯ ∂p ⊗ ¯ ∂p = 0 , which can beeasily proven using (5.8) and keeping in mind that the tensor product is over B .Similarly, applying d to the second leg we get (id ⊗ d) g = E ′ E ( ∂p ⊗ ∂ ¯ ∂p + ¯ ∂p ⊗ ¯ ∂∂p ) . Using again the expression for ∂ ¯ ∂p we obtain (id ⊗ d) g = E ′ E E ( ∂p ⊗ ∂p ∧ ¯ ∂p + ∂p ⊗ ¯ ∂p ∧ ∂p ) − E ′ E E ( ¯ ∂p ⊗ ∂p ∧ ¯ ∂p + ¯ ∂p ⊗ ¯ ∂p ∧ ∂p ) . This is easily seen to vanish as in the previous case. (cid:3)
Remark . The definition of the metric g and its properties discussed above are valid forany quantum irreducible flag manifold, not just in the quadratic case (we never used this inthe proofs). This is completely analogous to the definition of the Kähler forms discussed in[Mat19]. See also [Mat20] for a discussion of general quantum flag manifolds, not necessarilyirreducibles, from the point of view of (twisted) Hochschild homology.6.2. Inverse metric.
From now on we focus on the case of quantum projective spaces. Thismeans that we take U q ( g ) = U q ( sl r +1 ) and choose either ω s = ω or ω s = ω r , in such a waythat the quadratic condition discussed in Section 5.4 is satisfied.Our goal is to prove that g is a quantum metric according to Definition 3.5. To show this,we need an "inverse metric", which is an appropriate B -bimodule map ( · , · ) : Ω ⊗ B Ω → B .We begin by defining a certain B -bimodule map on Ω − ⊗ B Ω + . Lemma 6.7.
We have a B -bimodule map ( · , · ) − + : Ω − ⊗ B Ω + → B defined by ( ¯ ∂p, ∂p ) − + = C ′ p − q − ( ω s , ρ ) pp. Proof.
We use the FODCs Γ + and Γ − over A introduced in Section 5.2 and some results fromAppendix D. According to Proposition D.3 we have an A -bimodule map Φ − + − q − ( ω s , ρ ) Ψ − + : Γ − ⊗ B Γ + → A , where Φ − + and Ψ − + are the A -bimodule maps given by Φ − + ( ¯ ∂v ⊗ ∂f ) = C ′ , Ψ − + ( ¯ ∂v ⊗ ∂f ) = vf. Recall that Ω + and Ω − are induced from Γ + and Γ − over A . We now show that Φ − + − q − ( ω s , ρ ) Ψ − + restricts to a map Ω − ⊗ B Ω + → B , which coincides with ( · , · ) − + .Starting with Φ − + we compute Φ − + ( ¯ ∂p ⊗ ∂p ) = Φ − + ( f ¯ ∂v ⊗ ∂f v ) = f C ′ v = C ′ f v = C ′ p. Similarly for Ψ − + we compute Ψ − + ( ¯ ∂p ⊗ ∂p ) = Ψ − + ( f ¯ ∂v ⊗ ∂f v ) = f vf v = pp. Therefore ( ¯ ∂p, ∂p ) − + = (Φ − + − q − ( ω s , ρ ) Ψ − + )( ¯ ∂p ⊗ ∂p ) , which gives the result. (cid:3) Similarly we define a B -bimodule map on Ω + ⊗ B Ω − , which is more involved. Lemma 6.8.
We have a B -bimodule map ( · , · ) + − : Ω + ⊗ B Ω − → B defined by ( ∂p, ¯ ∂p ) + − = q ( α s ,α s ) q − ( ω s ,ω s +2 ρ ) S C p − q ( α s ,α s ) q − ( ω s , ρ ) pp. Proof.
According to Proposition D.3 we have an A -bimodule map Φ + − − Ψ + − : Γ + ⊗ B Γ − → A , where Φ + − and Ψ + − are the A -bimodule maps given by Φ + − ( ∂f ⊗ ¯ ∂v ) = C , Ψ + − ( ∂f ⊗ ¯ ∂v ) = f v. We now show that Φ + − − Ψ + − restricts to a map Ω + ⊗ B Ω − → B .Consider first Φ + − . Using the right A -module relations (5.6) and (5.7) we compute Φ + − ( ∂p ⊗ ¯ ∂p ) = q − ω s ,ω s ) (ˆ R − V,V ∗ ) (ˆ R − V,V ∗ ) Φ + − ( v∂f ⊗ ¯ ∂vf )= q − ω s ,ω s ) (ˆ R − V,V ∗ ) (ˆ R − V,V ∗ ) C ( vf )= q − ( ω s ,ω s ) (ˆ R − V,V ∗ ) (ˆ R − V,V ∗ ) C (ˆ R V,V ∗ ) p. In the last step we have used vf = q ( ω s ,ω s ) (ˆ R V,V ∗ ) f v from (4.3). Next, we use the identity (ˆ R − V,V ∗ ) C = (ˆ R V,V ) C from (2.4). Then Φ + − ( ∂p ⊗ ¯ ∂p ) = q − ( ω s ,ω s ) (ˆ R − V,V ∗ ) (ˆ R V,V ) C (ˆ R V,V ∗ ) p = q − ( ω s ,ω s ) (ˆ R − V,V ∗ ) (ˆ R V,V ) (ˆ R V,V ∗ ) C p = q − ( ω s ,ω s ) (ˆ R V,V ∗ ) (ˆ R V,V ) (ˆ R − V,V ∗ ) C p. Finally using the braid equation (2.1) we obtain Φ + − ( ∂p ⊗ ¯ ∂p ) = q − ( ω s ,ω s ) (ˆ R V,V ∗ ) (ˆ R V,V ) (ˆ R − V,V ∗ ) C p = q − ( ω s ,ω s ) S C p. Similarly, for Ψ + − we compute Ψ + − ( ∂p ⊗ ¯ ∂p ) = q − ω s ,ω s ) (ˆ R − V,V ∗ ) (ˆ R − V,V ∗ ) Ψ + − ( v∂f ⊗ ¯ ∂vf )= q − ω s ,ω s ) (ˆ R − V,V ∗ ) (ˆ R − V,V ∗ ) ( vf vf )= f vf v = pp. Using the identities above we find that q − ( α s ,α s ) q ( ω s , ρ ) ( ∂p, ¯ ∂p ) + − = (Φ + − − Ψ + − )( ∂p ⊗ ¯ ∂p ) . This proves the claim about ( · , · ) + − . (cid:3) Remark . The normalization factor in this map is chosen for later convenience.Hence we can define a B -bimodule map ( · , · ) : Ω ⊗ B Ω → B by ( · , · ) := ( · , · ) + − on Ω + ⊗ Ω − ( · , · ) − + on Ω − ⊗ Ω + otherwise . Remark . In the classical limit the map ( · , · ) : Ω ⊗ B Ω → B reduces to the inverse of theFubini-Study metric, see the explicit formulae in (A.2).We are now ready to show that g is a quantum metric according to Definition 3.5. Theorem 6.11.
Write g = g (1) ⊗ g (2) . Then for any ω ∈ Ω we have g (1) ( g (2) , ω ) = ω = ( ω, g (1) ) g (2) . Hence g ∈ Ω ⊗ B Ω is a quantum metric. ETRICS AND CONNECTIONS ON QUANTUM PROJECTIVE SPACES 19
Proof.
Since ( · , · ) is a B -bimodule map, it suffices to prove the claim for the generators ∂p and ¯ ∂p of Ω . We have to consider four different cases. • The case g (1) ( g (2) , ∂p ) . Since ( ∂p, ∂p ) = 0 we have g (1) ( g (2) , ∂p ) = E ′ E ∂p ( ¯ ∂p, ∂p ) = E ′ E ∂p (cid:0) C ′ p − q − ( ω s , ρ ) pp (cid:1) . We have E ′ E ∂ppp = E ′ ∂pp = 0 . Hence g (1) ( g (2) , ∂p ) = E ′ E ∂p C ′ p = E ′ E C ′ ∂pp. Using the duality relation E ′ C ′ = id from (2.2) we obtain g (1) ( g (2) , ∂p ) = E ′ C ′ E ∂pp = E ∂pp = ∂p. • The case ( ¯ ∂p, g (1) ) g (2) . Since ( ¯ ∂p, ¯ ∂p ) = 0 we have ( ¯ ∂p, g (1) ) g (2) = E ′ E ( ¯ ∂p, ∂p ) ¯ ∂p = E ′ E (cid:0) C ′ p − q − ( ω s , ρ ) pp (cid:1) ¯ ∂p. We have E ′ E pp ¯ ∂p = E ′ p ¯ ∂p = 0 . Hence ( ¯ ∂p, g (1) ) g (2) = E ′ E C ′ p ¯ ∂p. Using the duality relation E ′ C ′ = id from (2.2) we obtain ( ¯ ∂p, g (1) ) g (2) = E ′ C ′ E p ¯ ∂p = ¯ ∂p. • The case ( ∂p, g (1) ) g (2) . Since ( ∂p, ∂p ) = 0 we have ( ∂p, g (1) ) g (2) = E ′ E ( ∂p, ¯ ∂p ) ∂p = E ′ E (cid:0) q ( α s ,α s ) q − ( ω s ,ω s +2 ρ ) S C p − q ( α s ,α s ) q − ( ω s , ρ ) pp (cid:1) ∂p = q ( α s ,α s ) q − ( ω s ,ω s +2 ρ ) E ′ E S C p∂p, where in the last step we have used E ′ E pp∂p = 0 . Since E commutes with S , using theduality relation E C = id from (2.2) we get ( ∂p, g (1) ) g (2) = q ( α s ,α s ) q − ( ω s ,ω s +2 ρ ) E ′ S p∂p. Then using p∂p = q ( ω s ,ω s ) − ( α s ,α s ) S − ∂pp from (5.14) we conclude that ( ∂p, g (1) ) g (2) = q − ( ω s , ρ ) E ′ ∂pp = ∂p. • The case g (1) ( g (2) , ¯ ∂p ) . Since ( ¯ ∂p, ¯ ∂p ) = 0 we have g (1) ( g (2) , ¯ ∂p ) = E ′ E ¯ ∂p ( ∂p, ¯ ∂p )= E ′ E ¯ ∂p (cid:0) q ( α s ,α s ) q − ( ω s ,ω s +2 ρ ) S C p − q ( α s ,α s ) q − ( ω s , ρ ) pp (cid:1) . Using again E ′ E ∂ppp = 0 we rewrite g (1) ( g (2) , ¯ ∂p ) = q ( α s ,α s ) q − ( ω s ,ω s +2 ρ ) E ′ E S C ¯ ∂pp. Next, consider the expression E S C = E (ˆ R V,V ∗ ) (ˆ R V,V ) (ˆ R − V,V ∗ ) C = (ˆ R V,V ∗ ) E (ˆ R V,V ) (ˆ R − V,V ∗ ) C . By naturality of the braiding, equations (2.3) and (2.4) give E (ˆ R V,V ) = E (ˆ R − V,V ∗ ) , (ˆ R − V,V ∗ ) C = (ˆ R V ∗ ,V ∗ ) C . Using these and the duality relation E C = id from (2.2) we get E S C = (ˆ R V,V ∗ ) E (ˆ R − V,V ∗ ) (ˆ R V ∗ ,V ∗ ) C = (ˆ R V,V ∗ ) (ˆ R V ∗ ,V ∗ ) E C (ˆ R − V,V ∗ ) = (ˆ R V,V ∗ ) (ˆ R V ∗ ,V ∗ ) (ˆ R − V,V ∗ ) = ˜ S − . Therefore we have g (1) ( g (2) , ¯ ∂p ) = q ( α s ,α s ) q − ( ω s ,ω s +2 ρ ) E ′ ˜ S − ¯ ∂pp. Finally using ¯ ∂pp = q ( ω s ,ω s ) − ( α s ,α s ) ˜ S p ¯ ∂p from (5.14) we conclude that g (1) ( g (2) , ¯ ∂p ) = q − ( ω s , ρ ) E ′ p ¯ ∂p = ¯ ∂p. (cid:3) Corollary 6.12.
We have that g is central, in the sense that bg = gb for all b ∈ B .Proof. This is a general property of quantum metrics, see [BeMa20, Lemma 1.16]. Of course,this can also be proven directly using the relations of the FODC Ω . (cid:3) Since g = g + − + g − + this also implies that g + − and g − + are central.7. Connections and properties
In this section we introduce two connections ∇ + and ∇ − on the FODCs Ω + and Ω − , forthe case of quantum projective spaces. Their direct sum ∇ = ∇ + + ∇ − is a connectionon the FODC Ω , which in the classical limit reduces to the Levi-Civita connection on thecotangent bundle. As for the quantum case, we show that this connection is torsion free andcotorsion free. In other words, ∇ is a weak quantum Levi-Civita connection in the sense ofDefinition 3.13, which is our main result from Theorem 7.7.7.1. Connections.
We begin with the connection on Ω − , which is easier. Proposition 7.1.
We have a connection ∇ − : Ω − → Ω ⊗ B Ω − defined by ∇ − ( ¯ ∂p ) = E ∂p ⊗ ¯ ∂p − q − ( ω s , ρ ) pg + − . Proof.
Recall that, in the quadratic case, Ω − is generated as a left B -module by ¯ ∂p with therelations ( S − q ( ω s ,ω s ) ) p ¯ ∂p = 0 and E ′ ¯ ∂p = 0 , as described in (5.16). Hence to show that ∇ − is well-defined we need to check the relations ( S − q ( ω s ,ω s ) ) ∇ − ( p ¯ ∂p ) = 0 , E ′ ∇ − ( ¯ ∂p ) = 0 . Let us begin with the second relation. Recall that g + − = E ′ E ∂p ⊗ ¯ ∂p and we have theidentity E ′ p = q ( ω s , ρ ) . Using these we compute E ′ ∇ − ( ¯ ∂p ) = g + − − g + − = 0 . Next, we want to show that ( S − q ( ω s ,ω s ) ) ∇ − ( p ¯ ∂p ) = 0 . Using the Leibnitz rule we have ∇ − ( p ¯ ∂p ) = d p ⊗ ¯ ∂p + p ∇ − ( ¯ ∂p )= ∂p ⊗ ¯ ∂p + ¯ ∂p ⊗ ¯ ∂p + E ( p∂p ⊗ ¯ ∂p ) − q − ( ω s , ρ ) ppg + − . First we claim that ( S − q ( ω s ,ω s ) ) ¯ ∂p ⊗ ¯ ∂p = 0 . To see this, we use (5.8) to write ¯ ∂p ⊗ ¯ ∂p = E ¯ ∂p ⊗ p ¯ ∂p = E ¯ ∂pp ⊗ ¯ ∂p. Then using ( S − q ( ω s ,ω s ) ) ¯ ∂pp = 0 from (5.13) we obtain the claim. Similarly one shows thatthe term ppg + − vanishes under S − q ( ω s ,ω s ) . Hence let us consider A = ∂p ⊗ ¯ ∂p + E p∂p ⊗ ¯ ∂p, ETRICS AND CONNECTIONS ON QUANTUM PROJECTIVE SPACES 21 which we want to vanish under S − q ( ω s ,ω s ) . Using ¯ ∂p = E p ¯ ∂p we rewrite it as A = E ∂p ⊗ p ¯ ∂p + E p∂p ⊗ ¯ ∂p = E ( ∂pp + p∂p ) ⊗ ¯ ∂p. Clearly S − q ( ω s ,ω s ) commutes with E . Finally, using ( S − q ( ω s ,ω s ) )( ∂pp + p∂p ) = 0 from(5.15), we conclude that ( S − q ( ω s ,ω s ) ) A = 0 . (cid:3) Next we consider the case of Ω + , which is more complicated. Proposition 7.2.
We have a connection ∇ + : Ω + → Ω ⊗ B Ω + defined by ∇ + ( ∂p ) = q ( α s ,α s ) E T ¯ ∂p ⊗ ∂p − q ( α s ,α s ) q − ( ω s , ρ ) pg − + . Proof.
Recall that, in the quadratic case, Ω + is generated as a left B -module by ∂p with therelations (˜ S − q − ( ω s ,ω s ) ) p∂p = 0 and E ′ ∂p = 0 , as described in (5.17). Hence to show that ∇ + is well-defined we need to check the relations (˜ S − q − ( ω s ,ω s ) ) ∇ + ( p∂p ) = 0 , E ′ ∇ + ( ∂p ) = 0 . Let us begin with the second relation. We can use the identity E ′ E T = E ′ E from(C.4). Then we obtain the expression E ′ E T ¯ ∂p ⊗ ∂p = E ′ E ¯ ∂p ⊗ ∂p = g − + . Using this and E ′ p = q ( ω s , ρ ) we conclude that E ′ ∇ + ( ∂p ) = q ( α s ,α s ) g − + − q ( α s ,α s ) g − + = 0 . Next we want to show that (˜ S − q − ( ω s ,ω s ) ) ∇ + ( p∂p ) = 0 . We have ∇ + ( p∂p ) = d p ⊗ ∂p + p ∇ + ( ∂p )= ∂p ⊗ ∂p + ¯ ∂p ⊗ ∂p + q ( α s ,α s ) p E T ¯ ∂p ⊗ ∂p − q ( α s ,α s ) q − ( ω s , ρ ) ppg − + . The terms ∂p ⊗ ∂p and pp can be shown to vanish under ˜ S − q − ( ω s ,ω s ) , exactly as in theproof of Proposition 7.1. Hence let us consider the term A = ¯ ∂p ⊗ ∂p + q ( α s ,α s ) E T p ¯ ∂p ⊗ ∂p. We want to show that it vanishes under ˜ S − q − ( ω s ,ω s ) . We can rewrite it using ¯ ∂p ⊗ ∂p = E ¯ ∂p ⊗ ∂pp = q ( α s ,α s ) E T ¯ ∂pp ⊗ ∂p. Hence we obtain the expression A = q ( α s ,α s ) E T ( ¯ ∂pp + p ¯ ∂p ) ⊗ ∂p. Now consider the identity ˜ S E ˜ S = E ˜ S ˜ S from (C.5). Multiplying by S on theright and using the fact that S and ˜ S commute we get ˜ S E T = E T ˜ S . Then we can commute ˜ S with E T to obtain (˜ S − q − ( ω s ,ω s ) ) A = q ( α s ,α s ) E T (˜ S − q − ( ω s ,ω s ) )( ¯ ∂pp + p ¯ ∂p ) ⊗ ∂p. Finally, we can use the identity (˜ S − q − ( ω s ,ω s ) )( ¯ ∂pp + p ¯ ∂p ) = 0 from (5.15) to conclude that (˜ S − q − ( ω s ,ω s ) ) A = 0 , which finishes the proof. (cid:3) Remark . The algebra B is easily seen to be a left C q [ G ] -comodule, while the FODCs Ω + and Ω − are shown to be left covariant in [HeKo06]. It is possible to show that theconnections ∇ + and ∇ − introduced here are left C q [ G ] -covariant. This means that the map ∇ + : Ω + → Ω ⊗ B Ω + (and similarly ∇ − ) is a map of left C q [ G ] -comodules, where Ω ⊗ B Ω + isgiven the usual structure of tensor product of left comodules. We are not going to give thedetails here, as this is not one of the main goals of this paper.In the following we write ∇ := ∇ + + ∇ − : Ω → Ω ⊗ B Ω for the direct sum of the two connections on Ω = Ω + ⊕ Ω − . Remark . In the classical limit, the connection ∇ reduces to the Levi-Civita connection onthe cotangent bundle, as can be seen from the formulae in (A.3).7.2. Torsion.
The first property of the connection ∇ we want to explore is its torsion, whichaccording to Definition 3.10 is the left B -module map T ∇ = ∧ ◦ ∇ − d : Ω → Ω . Proposition 7.5.
We have T ∇ = 0 , that is ∇ is torsion free.Proof. It suffices to show that T ∇ vanishes on the generators ∂p and ¯ ∂p , since T ∇ is a B -module map. For the rest of this proof we write κ + − = ∧ ( g + − ) and κ − + = ∧ ( g − + ) . Moreoverobserve that κ + − = − κ − + , since g is symmetric by Proposition 6.2.Consider first T ∇ ( ¯ ∂p ) . We have d ¯ ∂p = ∂ ¯ ∂p , since d = ∂ + ¯ ∂ . Moreover we can write ∂ ¯ ∂p = E ( ∂p ∧ ¯ ∂p + ¯ ∂p ∧ ∂p ) , as in (C.8). Therefore T ∇ ( ¯ ∂p ) = E ∂p ∧ ¯ ∂p − q − ( ω s , ρ ) pκ + − − E ( ∂p ∧ ¯ ∂p + ¯ ∂p ∧ ∂p )= q − ( ω s , ρ ) pκ − + − E ¯ ∂p ∧ ∂p. Next, consider the identity E ¯ ∂p ⊗ ∂p = q − ( ω s , ρ ) pg − + from (C.7). Applying ∧ to it gives E ¯ ∂p ∧ ∂p = q − ( ω s , ρ ) pκ − + . Hence we conclude that T ∇ ( ¯ ∂p ) = q − ( ω s , ρ ) pκ − + − q − ( ω s , ρ ) pκ − + = 0 . Now consider T ∇ ( ∂p ) . Using d ∂p = ¯ ∂∂p = − E ( ∂p ∧ ¯ ∂p + ¯ ∂p ∧ ∂p ) we have T ∇ ( ∂p ) = q ( α s ,α s ) E T ¯ ∂p ∧ ∂p − q ( α s ,α s ) q − ( ω s , ρ ) pκ − + + E ( ∂p ∧ ¯ ∂p + ¯ ∂p ∧ ∂p ) . In Lemma C.8 we show that we have the identity q ( α s ,α s ) E T ¯ ∂p ∧ ∂p = − E ∂p ∧ ¯ ∂p + ( q ( α s ,α s ) − E ¯ ∂p ∧ ∂p. Plugging this in and simplifying we obtain T ∇ ( ∂p ) = q ( α s ,α s ) E ¯ ∂p ∧ ∂p − q ( α s ,α s ) q − ( ω s , ρ ) pκ − + . Using again E ¯ ∂p ∧ ∂p = q − ( ω s , ρ ) pκ − + , we conclude that T ∇ ( ∂p ) = q ( α s ,α s ) q − ( ω s , ρ ) pκ − + − q ( α s ,α s ) q − ( ω s , ρ ) pκ − + = 0 . (cid:3) ETRICS AND CONNECTIONS ON QUANTUM PROJECTIVE SPACES 23
Cotorsion.
From Definition 3.11, the cotorsion corresponding to a connection ∇ : Ω → Ω ⊗ A Ω and a quantum metric g is the element coT ∇ ∈ Ω ⊗ A Ω given by coT ∇ = (d ⊗ id − ( ∧ ⊗ id) ◦ (id ⊗ ∇ )) g. Proposition 7.6.
We have coT ∇ = 0 , that is ∇ is cotorsion free.Proof. Using (d ⊗ id) g = 0 from Proposition 6.5, we can write the cotorsion as coT ∇ = − ( ∧ ⊗ id) ◦ (id ⊗ ∇ ) g. First we compute (id ⊗ ∇ ) g . We have (id ⊗ ∇ ) g = E ′ E ( ∂p ⊗ ∇ ( ¯ ∂p ) + ¯ ∂p ⊗ ∇ ( ∂p ))= E ′ E ( ∂p ⊗ E ( ∂p ⊗ ¯ ∂p ) − q − ( ω s , ρ ) ∂p ⊗ pg − + )+ q ( α s ,α s ) E ′ E ( ¯ ∂p ⊗ E T ( ¯ ∂p ⊗ ∂p ) − q − ( ω s , ρ ) ¯ ∂p ⊗ pg + − ) . It is easy to show that E ′ E ∂p ⊗ pg − + = 0 and E ′ E ¯ ∂p ⊗ pg + − = 0 by using the relationsin (5.8) (recall that the tensor product is over B ). Hence we have (id ⊗ ∇ ) g = E ′ E E ∂p ⊗ ∂p ⊗ ¯ ∂p + q ( α s ,α s ) E ′ E E T ¯ ∂p ⊗ ¯ ∂p ⊗ ∂p. The first term vanishes using E ∂p ⊗ ∂p = 0 , since it can be rewritten as E ′ E E ∂p ⊗ ∂p ⊗ ¯ ∂p = 0 . Then applying − ( ∧ ⊗ id) to (id ⊗ ∇ ) g we are left with coT ∇ = − q ( α s ,α s ) E ′ E E T ¯ ∂p ∧ ¯ ∂p ⊗ ∂p. We now show that this term vanishes. First, apply ¯ ∂ to ¯ ∂pp = q − ( α s ,α s ) T p ¯ ∂p to get theidentity − ¯ ∂p ∧ ¯ ∂p = q − ( α s ,α s ) T ¯ ∂p ∧ ¯ ∂p . Using this we rewrite coT ∇ = E ′ E E T T ¯ ∂p ∧ ¯ ∂p ⊗ ∂p. Noting that E E = E E allows us to use the identity E T T = T E from (C.1).Then the cotorsion takes the form coT ∇ = E ′ E T E ¯ ∂p ∧ ¯ ∂p ⊗ ∂p. Next, using E ′ E T = E ′ E from (C.4) we have coT ∇ = E ′ E E ¯ ∂p ∧ ¯ ∂p ⊗ ∂p = E ′ E E ¯ ∂p ∧ ¯ ∂p ⊗ ∂p. Finally this expression vanishes, since E ¯ ∂p ∧ ¯ ∂p = 0 . (cid:3) Levi-Civita connection.
Summarizing the results obtained so far, we obtain the fol-lowing theorem, which is the main result of this section.
Theorem 7.7.
The connection ∇ : Ω → Ω ⊗ B Ω is a weak quantum Levi-Civita connectionwith respect to the quantum metric g .Proof. According to Definition 3.13, this means that ∇ is torsion free and cotorsion free (thelatter involves g ). This is what we have proven in Proposition 7.5 and Proposition 7.6. (cid:3) In view of these properties, and the fact that it reduces to the Levi-Civita connection onthe cotangent bundle in the classical limit, it seems appropriate to consider ∇ as a quantumanalogue of the Levi-Civita connection for the quantum projective spaces.Having the quantum metric g and the weak Levi-Civita connection ∇ , one can investigatefurther aspects of quantum Riemannian geometry in the sense of [BeMa20]. These include the computation of the Riemann tensor and an appropriately defined Ricci tensor, for instance.For the case of the quantum 2-sphere, which corresponds to the easiest case of a quantumprojective space, such computations have been performed in [Maj05]. An important resultis that an analogue of the Einstein condition holds, that is the Ricci tensor is proportionalto the quantum metric. We conjecture that this will also be the case for general quantumprojective spaces, and we plan to investigate this aspect in future research.We can also ask for the stronger version of the property of compatibility with the metric,as opposed to the cotorsion free condition. We investigate this in the next section.8. Bimodule connections and metric compatibility
In this section we show that the connections ∇ + and ∇ − are bimodule connections. Thenwe use this fact to show that the connection ∇ satisfies the quantum metric compatibility ∇ g = 0 , which means that ∇ is a quantum Levi-Civita connection.8.1. Bimodule connections.
We investigate whether ∇ : Ω → Ω ⊗ B Ω is a bimodule con-nection by studying its components ∇ + and ∇ − . Recall from Definition 3.14 that this requiresthe existence of a B -bimodule map σ : Ω ⊗ B Ω → Ω ⊗ B Ω such that ∇ ( ωb ) = σ ( ω ⊗ d b ) + ∇ ( ω ) b, ω ∈ Ω , b ∈ B . We begin by obtaining a useful expression for ∇ − ( ¯ ∂pp ) . Lemma 8.1.
We have ∇ − ( ¯ ∂pp ) = σ −− + σ − + + ∇ − ( ¯ ∂p ) p , where σ −− = q − ( α s ,α s ) T ¯ ∂p ⊗ ¯ ∂pσ − + = q ω s ,ω s ) − α s ,α s ) S − ˜ S ∂p ⊗ ¯ ∂p − ( q − ( α s ,α s ) − q − ( ω s , ρ ) pg + − p. Proof.
We have ∇ − ( ¯ ∂pp ) = q − ( α s ,α s ) T ∇ − ( p ¯ ∂p ) by (5.4). Then we compute ∇ − ( ¯ ∂pp ) = q − ( α s ,α s ) T d p ⊗ ¯ ∂p + q − ( α s ,α s ) T p ∇ − ( ¯ ∂p )= q − ( α s ,α s ) T ¯ ∂p ⊗ ¯ ∂p + q − ( α s ,α s ) T ∂p ⊗ ¯ ∂p + q − ( α s ,α s ) T E p∂p ⊗ ¯ ∂p − q − ( α s ,α s ) q − ( ω s , ρ ) T ppg + − . We have T E = E T T by (C.1). Then T E p∂p ⊗ ¯ ∂p = E T T p∂p ⊗ ¯ ∂p = E ∂p ⊗ ¯ ∂pp. Also using the analogue of Lemma 5.2 for pp we have T pp = S ˜ S pp = q − ( ω s ,ω s ) S pp = pp. Using these relations and pg + − = g + − p we obtain ∇ − ( ¯ ∂pp ) = q − ( α s ,α s ) T ¯ ∂p ⊗ ¯ ∂p + q − ( α s ,α s ) T ∂p ⊗ ¯ ∂p + q − ( α s ,α s ) E ∂p ⊗ ¯ ∂pp − q − ( α s ,α s ) q − ( ω s , ρ ) pg + − p. The second line coincides with q − ( α s ,α s ) ∇ − ( ∂p ) p . Then we can write ∇ − ( ¯ ∂pp ) = σ −− + σ − + + ∇ − ( ¯ ∂p ) p, where we define σ −− = q − ( α s ,α s ) T ¯ ∂p ⊗ ¯ ∂p and σ − + = q − ( α s ,α s ) T ∂p ⊗ ¯ ∂p + ( q − ( α s ,α s ) − E ∂p ⊗ ¯ ∂pp − ( q − ( α s ,α s ) − q − ( ω s , ρ ) pg + − p. ETRICS AND CONNECTIONS ON QUANTUM PROJECTIVE SPACES 25
Now we rewrite σ − + in a more convenient form. In the quadratic case we have S = q ω s ,ω s ) − ( α s ,α s ) S − + q ( ω s ,ω s ) (1 − q − ( α s ,α s ) ) from (5.19). Then we get T ∂p ⊗ ¯ ∂p = q ω s ,ω s ) − ( α s ,α s ) S − ˜ S ∂p ⊗ ¯ ∂p + q ( ω s ,ω s ) (1 − q − ( α s ,α s ) )˜ S ∂p ⊗ ¯ ∂p. Using (5.8) and (5.14) we can rewrite ˜ S ∂p ⊗ ¯ ∂p = ˜ S E ∂pp ⊗ ¯ ∂p = E ˜ S ∂p ⊗ p ¯ ∂p = q ( α s ,α s ) − ( ω s ,ω s ) E ∂p ⊗ ¯ ∂pp. Hence we get T ∂p ⊗ ¯ ∂p = q ω s ,ω s ) − ( α s ,α s ) S − ˜ S ∂p ⊗ ¯ ∂p + (1 − q − ( α s ,α s ) ) q ( α s ,α s ) E ∂p ⊗ ¯ ∂pp. Finally plugging back into σ − + gives the expression σ − + = q ω s ,ω s ) − α s ,α s ) S − ˜ S ∂p ⊗ ¯ ∂p − ( q − ( α s ,α s ) − q − ( ω s , ρ ) ppg + − . This gives the result as claimed. (cid:3)
We proceed in the same way for ∇ + , which is a bit more involved. Lemma 8.2.
We have ∇ + ( ∂pp ) = σ ++ + σ + − + ∇ + ( ∂p ) p , where σ ++ = q ( α s ,α s ) T ∂p ⊗ ∂p,σ + − = q α s ,α s ) − ω s ,ω s ) S ˜ S − ¯ ∂p ⊗ ∂p − ( q ( α s ,α s ) − q ( α s ,α s ) q − ( ω s , ρ ) pg − + p. Proof.
We have ∇ + ( ∂pp ) = q ( α s ,α s ) T ∇ + ( p∂p ) by (5.4). Then we compute ∇ + ( ∂pp ) = q ( α s ,α s ) T d p ⊗ ∂p + q ( α s ,α s ) T p ∇ + ( ∂p )= q ( α s ,α s ) T ∂p ⊗ ∂p + q ( α s ,α s ) T ¯ ∂p ⊗ ∂p + q α s ,α s ) T E T p ¯ ∂p ⊗ ∂p − q α s ,α s ) q − ( ω s , ρ ) T ppg − + . We have T E = E T T from (C.1). Then consider T E T p ¯ ∂p ⊗ ∂p = E T T T p ¯ ∂p ⊗ ∂p. Using (5.11) we can derive an analogue of the braid equation for T , that is T T T = T T T . Using this identity we obtain T E T p ¯ ∂p ⊗ ∂p = E T T T p ¯ ∂p ⊗ ∂p = E T ¯ ∂p ⊗ ∂pp. Since we have T ppg − + = ppg − + = pg − + p , as in the proof of Lemma 8.1, we get ∇ + ( ∂pp ) = q ( α s ,α s ) T ∂p ⊗ ∂p + q ( α s ,α s ) T ¯ ∂p ⊗ ∂p + q α s ,α s ) E T ¯ ∂p ⊗ ∂pp − q α s ,α s ) q − ( ω s , ρ ) pg − + p. The second line coincides with q ( α s ,α s ) ∇ + ( ∂p ) p . Then we can write ∇ + ( ∂pp ) = σ ++ + σ + − + ∇ + ( ∂p ) p, where σ ++ = q ( α s ,α s ) T ∂p ⊗ ∂p and σ + − = q ( α s ,α s ) T ¯ ∂p ⊗ ∂p + ( q ( α s ,α s ) − q ( α s ,α s ) E T ¯ ∂p ⊗ ∂pp − ( q ( α s ,α s ) − q ( α s ,α s ) q − ( ω s , ρ ) pg − + p. We now rewrite σ + − in a more convenient form. In the quadratic case we have ˜ S = q ( α s ,α s ) − ω s ,ω s ) ˜ S − + q − ( ω s ,ω s ) (1 − q ( α s ,α s ) ) from (5.20). Then T ¯ ∂p ⊗ ∂p = q ( α s ,α s ) − ω s ,ω s ) S ˜ S − ¯ ∂p ⊗ ∂p + q − ( ω s ,ω s ) (1 − q ( α s ,α s ) ) S ¯ ∂p ⊗ ∂p. Consider the term S ¯ ∂p ⊗ ∂p . Using (5.8) and (5.14) we get S ¯ ∂p ⊗ ∂p = E S ¯ ∂p ⊗ ∂pp = q ( α s ,α s ) − ( ω s ,ω s ) E S S ¯ ∂pp ⊗ ∂p. Next, we use E = E ˜ S S from (C.3) and the "braid equation" for S (5.11). Then S ¯ ∂p ⊗ ∂p = q ( α s ,α s ) − ( ω s ,ω s ) E ˜ S S S S ¯ ∂pp ⊗ ∂p = q ( α s ,α s ) − ( ω s ,ω s ) E ˜ S S S S ¯ ∂pp ⊗ ∂p. Now we can use (5.13) and (5.14) to get S ¯ ∂p ⊗ ∂p = q ( α s ,α s ) E ˜ S S S ¯ ∂pp ⊗ ∂p = q ( ω s ,ω s ) E ˜ S S ¯ ∂p ⊗ ∂pp = q ( ω s ,ω s ) E T ¯ ∂p ⊗ ∂pp. Therefore we obtain T ¯ ∂p ⊗ ∂p = q ( α s ,α s ) − ω s ,ω s ) S ˜ S − ¯ ∂p ⊗ ∂p + (1 − q ( α s ,α s ) ) E T ¯ ∂p ⊗ ∂pp. Finally plugging this into σ + − gives the expression σ + − = q α s ,α s ) − ω s ,ω s ) S ˜ S − ¯ ∂p ⊗ ∂p − ( q ( α s ,α s ) − q ( α s ,α s ) q − ( ω s , ρ ) pg − + p. This gives the result as claimed. (cid:3)
The computations above suggest that the terms σ ab with a, b ∈ { + , −} might correspondto B -bimodule maps Ω a ⊗ B Ω b → Ω b ⊗ B Ω a . This is indeed the case, as we verify by lengthycomputations in Appendix D. Then we obtain the following result. Proposition 8.3.
We have that ∇ + and ∇ − are bimodule connections.Proof. From Proposition D.4, Proposition D.5, Proposition D.6 and Proposition D.7 we havefour B -bimodule maps Ω a ⊗ B Ω b → Ω b ⊗ B Ω a with appropriate a, b ∈ { + , −} given by σ ++ ( ∂p ⊗ ∂p ) = q ( α s ,α s ) T ∂p ⊗ ∂p,σ −− ( ¯ ∂p ⊗ ¯ ∂p ) = q − ( α s ,α s ) T ¯ ∂p ⊗ ¯ ∂p,σ + − ( ∂p ⊗ ¯ ∂p ) = q α s ,α s ) − ω s ,ω s ) S ˜ S − ¯ ∂p ⊗ ∂p − ( q ( α s ,α s ) − q ( α s ,α s ) q − ( ω s , ρ ) pg − + p,σ − + ( ¯ ∂p ⊗ ∂p ) = q ω s ,ω s ) − α s ,α s ) S − ˜ S ∂p ⊗ ¯ ∂p − ( q − ( α s ,α s ) − q − ( ω s , ρ ) pg + − p. They can be assembled into a B -bimodule map σ : Ω ⊗ B Ω → Ω ⊗ B Ω . Then using theexpressions from Lemma 8.1 and Lemma 8.2 we observe that ∇ − ( ¯ ∂pp ) = σ ( ¯ ∂p ⊗ d p ) + ∇ − ( ¯ ∂p ) p, ∇ + ( ∂pp ) = σ ( ∂p ⊗ d p ) + ∇ + ( ∂p ) p, which shows that ∇ − and ∇ + are bimodule connection with generalized braiding σ . (cid:3) ETRICS AND CONNECTIONS ON QUANTUM PROJECTIVE SPACES 27
Metric compatibility.
We now investigate whether the connection ∇ is quantum met-ric compatible, in the sense that ∇ g = 0 . Here the action on ∇ on g is given by ∇ g = ( ∇ ⊗ id) g + ( σ ⊗ id)(id ⊗ ∇ ) g. Theorem 8.4.
We have ∇ g = 0 . Hence ∇ is a quantum Levi-Civita connection.Proof. We compute separately the action of ∇ on the two legs of g + − and g − + . • The term ( ∇ ⊗ id) g − + . We have ( ∇ ⊗ id) g − + = E ′ E ∇ ( ¯ ∂p ) ⊗ ∂p = E ′ E E ∂p ⊗ ¯ ∂p ⊗ ∂p − q − ( ω s , ρ ) E ′ E pg + − ⊗ ∂p. The second term vanishes, since pg + − = g + − p and E p∂p = 0 . The first term vanishes dueto Lemma 6.4, which is related to the Kähler property of the metric. • The term ( ∇ ⊗ id) g + − . We have ( ∇ ⊗ id) g + − = E ′ E ∇ ( ∂p ) ⊗ ¯ ∂p = q ( α s ,α s ) E ′ E E T ¯ ∂p ⊗ ∂p ⊗ ¯ ∂p − q ( α s ,α s ) q − ( ω s , ρ ) E ′ E pg − + ⊗ ¯ ∂p. The second term vanishes due to E ′ E p ¯ ∂p = E ′ ¯ ∂p = 0 .Now write A = E E T ¯ ∂p ⊗ ∂p ⊗ ¯ ∂p . Then A = E E T E ¯ ∂p ⊗ ∂p ⊗ p ¯ ∂p = E E E ˜ S S ¯ ∂p ⊗ ∂p ⊗ p ¯ ∂p = E E E ˜ S S ¯ ∂p ⊗ ∂pp ⊗ ¯ ∂p. We proceed by using (5.13) and (5.14). We obtain A = q ( α s ,α s ) − ( ω s ,ω s ) E E E ˜ S S S ¯ ∂pp ⊗ ∂p ⊗ ¯ ∂p = q ( α s ,α s ) − ω s ,ω s ) E E E ˜ S S S S ¯ ∂pp ⊗ ∂p ⊗ ¯ ∂p. Now we use the "braid equation" S S S = S S S from (5.11) and the identity E ˜ S S = E from (C.3). We get A = q ( α s ,α s ) − ω s ,ω s ) E E E ˜ S S S S ¯ ∂pp ⊗ ∂p ⊗ ¯ ∂p = q ( α s ,α s ) − ω s ,ω s ) E E E S S ¯ ∂pp ⊗ ∂p ⊗ ¯ ∂p = q ( α s ,α s ) − ω s ,ω s ) E E S E S ¯ ∂pp ⊗ ∂p ⊗ ¯ ∂p. Next consider the identity S = q ω s ,ω s ) − ( α s ,α s ) S − + q ( ω s ,ω s ) (1 − q − ( α s ,α s ) ) from (5.19),valid in the quadratic case. Using E S − = q − ( ω s ,ω s +2 ρ ) E ′ from (C.2) we get E S = q − ( ω s , ρ ) q ( ω s ,ω s ) − ( α s ,α s ) E ′ + q ( ω s ,ω s ) (1 − q − ( α s ,α s ) ) E . (8.1)Plugging this into the previous identity for A we get A = q − ( ω s , ρ ) q − ( ω s ,ω s ) E E S E ′ ¯ ∂pp ⊗ ∂p ⊗ ¯ ∂p + (1 − q − ( α s ,α s ) ) q ( α s ,α s ) − ( ω s ,ω s ) E E S E ¯ ∂p ⊗ p∂p ⊗ ¯ ∂p = q − ( ω s ,ω s ) E E S ¯ ∂p ⊗ ∂p ⊗ ¯ ∂p. Above we used E ′ p = q ( ω s , ρ ) and E p∂p = 0 . Using (8.1) once more we get A = q − ( ω s , ρ ) q − ( α s ,α s ) E E ′ ¯ ∂p ⊗ ∂p ⊗ ¯ ∂p + (1 − q − ( α s ,α s ) ) E E ¯ ∂p ⊗ ∂p ⊗ ¯ ∂p = (1 − q − ( α s ,α s ) ) E E ¯ ∂p ⊗ ∂p ⊗ ¯ ∂p. Plugging this expression into ( ∇ ⊗ id) g + − we obtain ( ∇ ⊗ id) g + − = (1 − q − ( α s ,α s ) ) q ( α s ,α s ) E ′ E E ¯ ∂p ⊗ ∂p ⊗ ¯ ∂p. But then this term vanishes due to Lemma 6.4. • The term (id ⊗ ∇ ) g + − . We have (id ⊗ ∇ ) g + − = E ′ E ∂p ⊗ ∇ ( ¯ ∂p )= E ′ E E ∂p ⊗ ∂p ⊗ ¯ ∂p − q − ( ω s , ρ ) E ′ E ∂p ⊗ pg + − . The first term vanishes by E ∂p ⊗ ∂p = 0 and the second term by E ′ E ∂pp = 0 . • The term (id ⊗ ∇ ) g − + . We have (id ⊗ ∇ ) g − + = E ′ E ¯ ∂p ⊗ ∇ ( ∂p )= q ( α s ,α s ) E ′ E E T ¯ ∂p ⊗ ¯ ∂p ⊗ ∂p − q ( α s ,α s ) q − ( ω s , ρ ) E ′ E ¯ ∂p ⊗ pg − + . The second term vanishes due to E ¯ ∂pp = 0 . Now we apply σ ⊗ id to this expression. Since σ ( ¯ ∂p ⊗ ¯ ∂p ) = q − ( α s ,α s ) T ¯ ∂p ⊗ ¯ ∂p we get ( σ ⊗ id)(id ⊗ ∇ ) g − + = q ( α s ,α s ) E ′ E E T σ ( ¯ ∂p ⊗ ¯ ∂p ) ⊗ ∂p = E ′ E E T T ¯ ∂p ⊗ ¯ ∂p ⊗ ∂p. Next we use E T T = T E from (C.1). We obtain ( σ ⊗ id)(id ⊗ ∇ ) g − + = E ′ E E T T ¯ ∂p ⊗ ¯ ∂p ⊗ ∂p = E ′ E T E ¯ ∂p ⊗ ¯ ∂p ⊗ ∂p. Using the identity E ′ E T = E ′ E from (C.3) we get ( σ ⊗ id)(id ⊗ ∇ ) g − + = E ′ E E ¯ ∂p ⊗ ¯ ∂p ⊗ ∂p = E ′ E E ¯ ∂p ⊗ ¯ ∂p ⊗ ∂p = 0 . In the last step we have used E ¯ ∂p ⊗ ¯ ∂p = 0 . (cid:3) The quantum metric compatibility of ∇ shows that this connection has all the propertiesone would expect from the Levi-Civita connection on quantum projective spaces. Appendix A. Results on projective spaces
In this appendix we recall some results on (classical) projective spaces, to facilitate thecomparison between the classical and the quantum descriptions.From the point of view of differential geometry, the complex projective space C P N can beidentified with C N +1 \{ } modulo the relation ( Z , · · · , Z N ) ∼ λ ( Z , · · · , Z N ) with λ = 0 .Here { Z , · · · , Z N } are the global coordinates of C N +1 . Define the functions p ij = Z i ¯ Z j k Z k , i, j = 1 , · · · , N + 1 . These descend to C P N , as they are invariant under the equivalence relation. ETRICS AND CONNECTIONS ON QUANTUM PROJECTIVE SPACES 29
Consider the coordinate patch with Z N +1 = 0 and denote by z i = Z i /Z N +1 the corre-sponding homogeneous coordinates (the discussion is similar for the other patches). Thenwith respect to these local coordinates the Fubini-Study metric takes the form g = N X i,j =1 g i ¯ j d z i ⊙ d¯ z j = N X i,j =1 (1 + k z k ) δ ij − ¯ z i z j (1 + k z k ) d z i ⊙ d¯ z j , where we write d z i ⊙ d¯ z j = d z i ⊗ d¯ z j + d¯ z j ⊗ d z i for the symmetric product. The inversemetric can be seen to have components g ¯ ij = ( δ ij + ¯ z i z j )(1 + k z k ) .The metric can be rewritten in terms of the functions p ij , which can be seen as the entriesof a projection of rank one. An explicit computation shows that g = N +1 X i,j =1 ( ∂p ij ⊗ ¯ ∂p ji + ¯ ∂p ij ⊗ ∂p ji ) . (A.1)Similarly, the inverse metric can be seen as a map ( · , · ) on the cotangent bundle satisfying ( ∂p ij , ¯ ∂p kl ) = δ il p kj − p ij p kl , ( ¯ ∂p ij , ∂p kl ) = δ kj p il − p ij p kl , ( ∂p ij , ∂p kl ) = 0 , ( ¯ ∂p ij , ¯ ∂p kl ) = 0 . (A.2)Next, we describe the Levi-Civita connection on the cotangent bundle, defined with respectto the Fubini-Study metric. We have the formulae ∇ ( ∂p ij ) = N +1 X k =1 ¯ ∂p kj ⊗ ∂p ik − p ij g − + , ∇ ( ¯ ∂p ij ) = N +1 X k =1 ∂p ik ⊗ ∂p kj − p ij g + − . (A.3)Here g + − = P i,j ∂p ij ⊗ ¯ ∂p ji and g − + = P i,j ¯ ∂p ij ⊗ ∂p ji .The classical analogue of the relations (5.16) and (5.17) are p ij ∂p kl = p il ∂p kj , p ij ¯ ∂p kl = p kj ¯ ∂p il , N +1 X i =1 ∂p ii = 0 , N +1 X i =1 ¯ ∂p ii = 0 . Using these relations, one can prove directly that the Levi-Civita connection is given by (A.3).Indeed, one checks that ∇ is torsion free and metric compatible, that is ∇ g = 0 . Appendix B. The maps S and ˜ S In this appendix we prove various properties satisfied by the maps S = (ˆ R V,V ∗ ) (ˆ R V,V ) (ˆ R − V,V ∗ ) , ˜ S = (ˆ R V,V ∗ ) (ˆ R − V ∗ ,V ∗ ) (ˆ R − V,V ∗ ) . These were introduced in (5.9) to rewrite some of the relations of the differential calculus.The most important properties are the commutation relations among them.
Proposition B.1.
The maps S and ˜ S satisfy the following properties. (1) We have the commutation relations S ˜ S = ˜ S S , ˜ S S = S ˜ S . (2) We have the "braid equations" S S S = S S S , ˜ S ˜ S ˜ S = ˜ S ˜ S ˜ S . Proof.
These identities are valid more generally within the braid group with generators { σ i } i .Under the obvious identifications, we can consider the elements S = σ σ σ − , ˜ S = σ σ − σ − . (1) The first identity is straightforward. For the second we compute ˜ S S = ( σ σ − σ − )( σ σ σ − ) = ( σ − σ − σ )( σ − σ σ )= σ − σ − σ σ = ( σ − σ σ )( σ − σ − σ )= ( σ σ σ − )( σ σ − σ − ) = S ˜ S . (2) We consider the first identity. We compute S S S = ( σ σ σ − )( σ σ σ − )( σ σ σ − ) = σ ( σ σ σ − ) σ ( σ σ σ − ) σ − = σ ( σ − σ σ ) σ ( σ − σ σ ) σ − = σ σ − ( σ σ σ ) σ σ − = σ σ − ( σ σ σ ) σ σ − = σ σ ( σ − σ σ ) σ σ − = σ σ ( σ σ σ − ) σ σ − = ( σ σ σ − )( σ σ σ − )( σ σ σ − )= S S S . The second identity can be proven in a similar way. (cid:3)
Now we consider the case when ˆ R V,V satisfies the quadratic relation P V,V Q V,V = (ˆ R V,V − q ( ω s ,ω s ) )(ˆ R V,V + q ( ω s ,ω s ) − ( α s ,α s ) ) = 0 . Then an analogous relation also holds for the braiding ˆ R V ∗ ,V ∗ . Lemma B.2.
In the quadratic case we have the relations S = q ω s ,ω s ) − ( α s ,α s ) S − + q ( ω s ,ω s ) (1 − q − ( α s ,α s ) ) , ˜ S = q ( α s ,α s ) − ω s ,ω s ) ˜ S − + q − ( ω s ,ω s ) (1 − q ( α s ,α s ) ) . Proof.
The quadratic relation P V,V Q V,V = 0 can be rewritten as ˆ R V,V = q ω s ,ω s ) − ( α s ,α s ) ˆ R − V,V + q ( ω s ,ω s ) (1 − q − ( α s ,α s ) ) . Using this identity we can rewrite S in the following way S = (ˆ R V,V ∗ ) (ˆ R V,V ) (ˆ R − V,V ∗ ) = q ω s ,ω s ) − ( α s ,α s ) (ˆ R V,V ∗ ) (ˆ R − V,V ) (ˆ R − V,V ∗ ) + q ( ω s ,ω s ) (1 − q − ( α s ,α s ) )(ˆ R V,V ∗ ) (ˆ R − V,V ∗ ) = q ω s ,ω s ) − ( α s ,α s ) S − + q ( ω s ,ω s ) (1 − q − ( α s ,α s ) ) . The other identity is proven similarly. (cid:3)
ETRICS AND CONNECTIONS ON QUANTUM PROJECTIVE SPACES 31
Appendix C. Various identities
In this appendix we provide the proofs of various identities that have been used in themain text. We divide them into two groups, those that only depend on the (rigid) braidedmonoidal structure, and those that also involve the differential calculus Ω .C.1. Categorical identities.
The first identity we consider appears in the proof of [HeKo06,Proposition 3.11] in slightly different terms. We reprove it here for convenience.
Lemma C.1.
We have the identity E T T = T E . (C.1) Proof.
Using the definition of T we can write T T = (ˆ R V,V ∗ ) (ˆ R V,V ) (ˆ R V,V ∗ ) (ˆ R V,V ) (ˆ R − V ∗ ,V ∗ ) (ˆ R − V,V ∗ ) (ˆ R − V ∗ ,V ∗ ) (ˆ R − V,V ∗ ) . We have E (ˆ R V,V ∗ ) = (ˆ R V,V ∗ ) E . Moreover E (ˆ R V,V ) (ˆ R V,V ∗ ) = E by naturality ofthe braiding, as in (2.3). Then we obtain E (ˆ R V,V ∗ ) (ˆ R V,V ) (ˆ R V,V ∗ ) (ˆ R V,V ) = (ˆ R V,V ∗ ) (ˆ R V,V ) E . Hence we can write E T T = (ˆ R V,V ∗ ) (ˆ R V,V ) E (ˆ R − V ∗ ,V ∗ ) (ˆ R − V,V ∗ ) (ˆ R − V ∗ ,V ∗ ) (ˆ R − V,V ∗ ) . As above, using E (ˆ R − V,V ∗ ) (ˆ R − V ∗ ,V ∗ ) = E due to (2.3) we obtain E (ˆ R − V ∗ ,V ∗ ) (ˆ R − V,V ∗ ) (ˆ R − V ∗ ,V ∗ ) (ˆ R − V,V ∗ ) = (ˆ R − V ∗ ,V ∗ ) (ˆ R − V,V ∗ ) E . We conclude that E T T = (ˆ R V,V ∗ ) (ˆ R V,V ) (ˆ R − V ∗ ,V ∗ ) (ˆ R − V,V ∗ ) E = T E . (cid:3) In the following we suppose that V is a simple module. First we obtain some relations forthe evaluations E and E ′ applied to the maps S and ˜ S . Lemma C.2.
Let V = V ( λ ) be a simple module. Then we have E ′ S = q ( λ,λ +2 ρ ) E , E ′ ˜ S − = q ( λ,λ +2 ρ ) E . (C.2) From these identities we also obtain E ′ S ˜ S = E ′ , E ˜ S S = E . (C.3) Proof.
Consider the first identity. Using (2.3) we compute E ′ S = E ′ (ˆ R V,V ∗ ) (ˆ R V,V ) (ˆ R − V,V ∗ ) = E ′ (ˆ R − V,V ) (ˆ R V,V ) (ˆ R − V,V ∗ ) = E ′ (ˆ R − V,V ∗ ) = q ( λ,λ +2 ρ ) E . In the last step we have used (2.5). Similarly, for the second identity we have E ′ ˜ S − = E ′ (ˆ R V,V ∗ ) (ˆ R V ∗ ,V ∗ ) (ˆ R − V,V ∗ ) = E ′ (ˆ R − V ∗ ,V ∗ ) (ˆ R V ∗ ,V ∗ ) (ˆ R − V,V ∗ ) = E ′ (ˆ R − V,V ∗ ) = q ( λ,λ +2 ρ ) E . The other identities easily follow from these. (cid:3)
The next result, again in the case of a simple module V , shows that we can get rid of theterm T when performing the evaluations E ′ E . Lemma C.3.
Let V = V ( λ ) be a simple module. Then we have E ′ E T = E ′ E . (C.4) Proof.
We have E (ˆ R V,V ∗ ) = q − ( λ,λ +2 ρ ) E ′ from (2.5). Then E ′ E T = q − ( λ,λ +2 ρ ) E ′ E ′ (ˆ R − V ∗ ,V ∗ ) (ˆ R V,V ) (ˆ R − V,V ∗ ) . By (2.3) we have E ′ (ˆ R − V ∗ ,V ∗ ) = E ′ (ˆ R V,V ∗ ) . We obtain E ′ E T = q − ( λ,λ +2 ρ ) E ′ E ′ (ˆ R V,V ∗ ) (ˆ R V,V ) (ˆ R − V,V ∗ ) = q − ( λ,λ +2 ρ ) E ′ E ′ (ˆ R V,V ∗ ) (ˆ R V,V ) (ˆ R − V,V ∗ ) . Again by (2.3) we have E ′ (ˆ R V,V ∗ ) = E ′ (ˆ R − V,V ) . Then E ′ E T = q − ( λ,λ +2 ρ ) E ′ E ′ (ˆ R − V,V ) (ˆ R V,V ) (ˆ R − V,V ∗ ) = q − ( λ,λ +2 ρ ) E ′ E ′ (ˆ R − V,V ∗ ) = E ′ E . In the last step we have used again (2.5). (cid:3)
Finally we need the following identity in the proof of Proposition 7.2.
Lemma C.4.
Let V = V ( λ ) be a simple module. Then we have ˜ S E ˜ S = E ˜ S ˜ S . (C.5) Proof.
Taking into account that T = ˜ S S we write ˜ S E ˜ S = ˜ S S S − E ˜ S = T E S − ˜ S . Using T E = E T T from (C.1) this becomes ˜ S E ˜ S = E T T S − ˜ S = E S ˜ S ˜ S ˜ S . Now we use the "braid equation" for ˜ S from (5.11). We get ˜ S E ˜ S = E S ˜ S ˜ S ˜ S . Finally we have E S ˜ S = E from (C.3), which gives the result. (cid:3) C.2.
Differential calculus identities.
We now derive various identities involving some el-ements of the differential calculus Ω . In the following V always denotes the simple module V ( ω s ) . We begin with some identities involving the metric. Lemma C.5.
We have the following identities for the metric g . (1) For g + − we have g + − p = q ( ω s ,ω s ) − ( α s ,α s ) E ′ ˜ S ∂p ⊗ ¯ ∂p = q ( ω s ,ω s ) − ( α s ,α s ) E ′ S − ∂p ⊗ ¯ ∂p. (C.6)(2) For g − + we have pg − + = q ( ω s , ρ ) E ¯ ∂p ⊗ ∂p. (C.7) ETRICS AND CONNECTIONS ON QUANTUM PROJECTIVE SPACES 33
Proof. (1) Using the right B -module relations (5.14) and (5.8) we compute E ∂p ⊗ ¯ ∂pp = q ( ω s ,ω s ) − ( α s ,α s ) E ˜ S ∂pp ⊗ ¯ ∂p = q ( ω s ,ω s ) − ( α s ,α s ) ˜ S E ∂pp ⊗ ¯ ∂p = q ( ω s ,ω s ) − ( α s ,α s ) ˜ S ∂p ⊗ ¯ ∂p. Applying E ′ we get g + − p = q ( ω s ,ω s ) − ( α s ,α s ) E ′ ˜ S ∂p ⊗ ¯ ∂p . The second expression can beobtained from the first one, since using (C.2) we can rewrite E ′ ˜ S = q ( ω s ,ω s +2 ρ ) E S − ˜ S = q ( ω s ,ω s +2 ρ ) E ˜ S S − = E ′ S − . (2) Using E ′ = q ( ω s ,ω s +2 ρ ) E ˜ S from (C.2) we write pg − + = E ′ E p ¯ ∂p ⊗ ∂p = q ( ω s ,ω s +2 ρ ) E ˜ S E p ¯ ∂p ⊗ ∂p. Since T = ˜ S S and S − p ¯ ∂p = q − ( ω s ,ω s ) p ¯ ∂p from (5.12), we get pg − + = q ( ω s ,ω s +2 ρ ) E T S − E p ¯ ∂p ⊗ ∂p = q ( ω s ,ω s +2 ρ ) E T E S − p ¯ ∂p ⊗ ∂p = q ( ω s , ρ ) E T E p ¯ ∂p ⊗ ∂p. Next, we can use the identity T E = E T T from (C.1). Finally, taking into accountthe right B -module relations (5.4) and E ∂pp = ∂p , we get pg − + = q ( ω s , ρ ) E E T T p ¯ ∂p ⊗ ∂p = q ( ω s , ρ ) E E ¯ ∂p ⊗ ∂pp = q ( ω s , ρ ) E E ¯ ∂p ⊗ ∂pp = q ( ω s , ρ ) E ¯ ∂p ⊗ ∂p. (cid:3) The next two identities we discuss appear in the proof of [HeKo06, Proposition 3.11]. Thefirst one lets us rewrite ∂ ¯ ∂p as a product of one-forms. Lemma C.6.
We have ∂ ¯ ∂p = E ( ∂p ∧ ¯ ∂p + ¯ ∂p ∧ ∂p ) . (C.8) Moreover we have E p∂ ¯ ∂p = E ∂ ¯ ∂pp = E ¯ ∂p ∧ ∂p. Proof.
Using the identity ¯ ∂p = E p ¯ ∂p from (5.8) and the Leibnitz rule we have ∂ ¯ ∂p = E ∂ ( p ¯ ∂p ) = E ( ∂p ∧ ¯ ∂p + p∂ ¯ ∂p ) . Now we apply ¯ ∂ to E p∂p = 0 . Using ∂ ¯ ∂ = − ¯ ∂∂ we get E p∂ ¯ ∂p = E ¯ ∂p ∧ ∂p . Plugging thisinto the expression above we get the result for ∂ ¯ ∂p .For the other identities, using the formula above we compute E p∂ ¯ ∂p = E ( E V ) ( p∂p ∧ ¯ ∂p + p ¯ ∂p ∧ ∂p )= E E ( p∂p ∧ ¯ ∂p + p ¯ ∂p ∧ ∂p )= E ¯ ∂p ∧ ∂p. In the last step we have used (5.8). The identity E ∂ ¯ ∂pp = E ¯ ∂p ∧ ∂p follows similarly. (cid:3) The second identity from [HeKo06, Proposition 3.11] is a right B -module relation for ∂ ¯ ∂p . Lemma C.7.
We have ∂ ¯ ∂pp = T p∂ ¯ ∂p . Proof.
Using the identity for ∂ ¯ ∂p from (C.8) and the bimodule relations (5.4) we get ∂ ¯ ∂pp = E T T p ( ∂p ∧ ¯ ∂p + ¯ ∂p ∧ ∂p ) . Then, using E T T = T E from (C.1), we obtain ∂ ¯ ∂pp = T E p ( ∂p ∧ ¯ ∂p + ¯ ∂p ∧ ∂p ) = T p∂ ¯ ∂p. (cid:3) Finally, the next identity lets us rewrite ∂p ∧ ¯ ∂p in terms of ¯ ∂p ∧ ∂p under the evaluation E . We have used it in the computation of the torsion in Proposition 7.5. Lemma C.8.
We have q ( α s ,α s ) E T ¯ ∂p ∧ ∂p = − E ∂p ∧ ¯ ∂p + ( q ( α s ,α s ) − E ¯ ∂p ∧ ∂p. Proof.
Applying ¯ ∂ to ∂pp = q ( α s ,α s ) T p∂p gives ¯ ∂∂pp − ∂p ∧ ¯ ∂p = q ( α s ,α s ) T ¯ ∂p ∧ ∂p + q ( α s ,α s ) T p ¯ ∂∂p. Using Lemma C.7 we rewrite this as q ( α s ,α s ) T ¯ ∂p ∧ ∂p = − ∂p ∧ ¯ ∂p + (1 − q ( α s ,α s ) ) ¯ ∂∂pp. Now we use the identity E ∂ ¯ ∂pp = E ¯ ∂p ∧ ∂p from Lemma C.6, taking into account that ∂ ¯ ∂ = − ¯ ∂∂ . This gives the result. (cid:3) Appendix D. Bimodule maps
In this appendix we introduce various bimodule maps, which in the main text were used todefine the inverse metric and check the bimodule property of the connections.D.1.
Inverse metric.
First we introduce certain A -bimodule maps involving the FODCs Γ + and Γ − over A . We assume that we are in the quadratic case , which means that the onlyrelations are as in (5.18). For this part the tensor products are taken over C , unless specified.We denote by ˜Γ + and ˜Γ − the free left A -modules generated by ∂f and ¯ ∂v respectively, with A -bimodule structures given by (5.6) and (5.7). Lemma D.1.
Define the A -bimodule maps Φ + − : ˜Γ + ⊗ ˜Γ − → A , Φ − + : ˜Γ − ⊗ ˜Γ + → A , by the formulae Φ + − ( ∂f ⊗ ¯ ∂v ) = C , Φ − + ( ¯ ∂v ⊗ ∂f ) = C ′ . Then they descend to maps on the tensor product over B .Proof. To prove that Φ + − descends to a map ˜Γ + ⊗ B ˜Γ − → A we need to show the equality Φ + − ( ∂f p ⊗ ¯ ∂v ) = Φ + − ( ∂f ⊗ p ¯ ∂v ) , where p = f v ∈ B are the generators of B .Using the relations from (5.6) and (5.7) we compute Φ + − ( ∂f f ⊗ ¯ ∂v ) = q ( α s ,α s ) − ( ω s ,ω s ) (ˆ R V,V ) Φ + − ( f ∂f ⊗ ¯ ∂v ) = q ( α s ,α s ) − ( ω s ,ω s ) (ˆ R V,V ) C f. We have (ˆ R V,V ) C = (ˆ R − V,V ∗ ) C from (2.4). Then Φ + − ( ∂f f ⊗ ¯ ∂v ) = q ( α s ,α s ) − ( ω s ,ω s ) (ˆ R − V,V ∗ ) C f = q ( α s ,α s ) − ( ω s ,ω s ) (ˆ R − V,V ∗ ) Φ + − ( ∂f ⊗ ¯ ∂vf )= q ( α s ,α s ) Φ + − ( ∂f ⊗ f ¯ ∂v ) . Similarly we compute Φ + − ( ∂f v ⊗ ¯ ∂v ) = q − ( ω s ,ω s ) (ˆ R − V,V ∗ ) Φ + − ( v∂f ⊗ ¯ ∂v ) = q − ( ω s ,ω s ) (ˆ R − V,V ∗ ) C v. ETRICS AND CONNECTIONS ON QUANTUM PROJECTIVE SPACES 35
We have (ˆ R − V,V ∗ ) C = (ˆ R V ∗ ,V ∗ ) C from (2.4). Then Φ + − ( ∂f v ⊗ ¯ ∂v ) = q − ( ω s ,ω s ) (ˆ R V ∗ ,V ∗ ) C v = q − ( ω s ,ω s ) (ˆ R V ∗ ,V ∗ ) Φ + − ( ∂f ⊗ ¯ ∂vv )= q − ( α s ,α s ) Φ + − ( ∂f ⊗ v ¯ ∂v ) . Using these identities and p = f v , we obtain that Φ + − ( ∂f p ⊗ ¯ ∂v ) = Φ + − ( ∂f ⊗ p ¯ ∂v ) . Thecomputations for the map Φ − + are very similar and we omit the details. (cid:3) We proceed similarly for the following "multiplication" maps.
Lemma D.2.
Define the A -bimodule maps Ψ + − : ˜Γ + ⊗ ˜Γ − → A , Ψ − + : ˜Γ − ⊗ ˜Γ + → A , by the formulae Ψ + − ( ∂f ⊗ ¯ ∂v ) = f v, Ψ − + ( ¯ ∂v ⊗ ∂f ) = vf. Then they descend to maps on the tensor product over B .Proof. Consider the map Ψ + − . Taking into account the relations (4.3) we compute Ψ + − ( ∂f f ⊗ ¯ ∂v ) = q ( α s ,α s ) − ( ω s ,ω s ) (ˆ R V,V ) Ψ − + ( f ∂f ⊗ ¯ ∂v ) = q ( α s ,α s ) − ( ω s ,ω s ) (ˆ R V,V ) f f v = q ( α s ,α s ) f f v = q ( α s ,α s ) − ( ω s ,ω s ) (ˆ R − V,V ∗ ) f vf = q ( α s ,α s ) − ( ω s ,ω s ) (ˆ R − V,V ∗ ) Ψ + − ( ∂f ⊗ ¯ ∂vf )= q ( α s ,α s ) Ψ + − ( ∂f ⊗ f ¯ ∂v ) . Similarly we compute Ψ + − ( ∂f v ⊗ ¯ ∂v ) = q − ( ω s ,ω s ) (ˆ R − V,V ∗ ) Ψ − + ( v∂f ⊗ ¯ ∂v ) = q − ( ω s ,ω s ) (ˆ R − V,V ∗ ) vf v = f vv = q − ( ω s ,ω s ) (ˆ R V ∗ ,V ∗ ) f vv = q − ( ω s ,ω s ) (ˆ R V ∗ ,V ∗ ) Ψ + − ( ∂f ⊗ ¯ ∂vv )= q − ( α s ,α s ) Ψ + − ( ∂f ⊗ v ¯ ∂v ) . Since p = f v , these identities show that Ψ + − ( ∂f p ⊗ ¯ ∂v ) = Ψ + − ( ∂f ⊗ p ¯ ∂v ) . The computationsfor the map Ψ − + are completely analogous and we omit them. (cid:3) Finally, we show that certain linear combinations of the maps Φ and Ψ defined abovedescend to the FODCs Γ + and Γ − . Proposition D.3.
Let Φ and Ψ be the maps defined above. (1) The map Φ + − − Ψ + − descends to a map Γ + ⊗ B Γ − → A . (2) The map Φ − + − q − ( ω s , ρ ) Ψ + − descends to a map Γ − ⊗ B Γ + → A .Proof. (1) We need to check that the relations of Γ + and Γ − are preserved under these maps.According to (5.18) we need to consider E v∂f = 0 and E ′ f ¯ ∂v = 0 .Using the duality relations (2.2) and the relations of A from (4.3) we compute E (Φ + − − Ψ + − )( v∂f ⊗ ¯ ∂v ) = E C v − E vf v = v − v = 0 . Similarly, using the same relations together with (5.7) and (2.5), we compute E ′ (Φ + − − Ψ + − )( ∂f ⊗ f ¯ ∂v ) = q − ( ω s ,ω s ) E ′ (ˆ R − V,V ∗ ) (Φ + − − Ψ + − )( ∂f ⊗ ¯ ∂vf )= q ( ω s , ρ ) E (Φ + − − Ψ + − )( ∂f ⊗ ¯ ∂vf )= q ( ω s , ρ ) E C f − q ( ω s , ρ ) E f vf = q ( ω s , ρ ) f − q ( ω s , ρ ) f = 0 . This shows that Φ + − − Ψ + − descends to a map Γ + ⊗ B Γ − → A .(2) The computations for Φ − + − q − ( ω s , ρ ) Ψ + − are very similar and we omit them. (cid:3) D.2.
Bimodule connections.
Consider the terms σ ab with a, b ∈ { + , −} appearing inLemma 8.1 and Lemma 8.2. Our goal is to show that they correspond to B -bimodule maps Ω a ⊗ B Ω b → Ω b ⊗ B Ω a defined by the same expressions. Proposition D.4.
We have a B -bimodule map σ ++ : Ω + ⊗ B Ω + → Ω + ⊗ B Ω + given by σ ++ ( ∂p ⊗ ∂p ) = q ( α s ,α s ) T ∂p ⊗ ∂p. Proof.
First we check that σ ++ is well-defined as a map Ω + ⊗ Ω + → Ω + ⊗ Ω + . We treat thefirst factor as a left B -module using (5.12) and the second factor as a right B -module using(5.13). Using this description we need to check the relations E ′ σ ++ ( ∂p ⊗ ∂p ) = 0 , E ′ σ ++ ( ∂p ⊗ ∂p ) = 0 , (˜ S − q − ( ω s ,ω s ) ) σ ++ ( p∂p ⊗ ∂p ) = 0 , (˜ S − q − ( ω s ,ω s ) ) σ ++ ( ∂p ⊗ ∂pp ) = 0 . Once this is done, we check that σ ++ descends to a map Ω + ⊗ B Ω + → Ω + ⊗ B Ω + .It is convenient to rewrite σ ++ in a slightly different form. Using ˜ S p∂p = q − ( ω s ,ω s ) p∂p from (5.12) allows us to obtain the identity ∂p ⊗ ∂p = E ∂pp ⊗ ∂p = E ∂p ⊗ p∂p = q ( ω s ,ω s ) E ˜ S ∂p ⊗ p∂p = q ( ω s ,ω s ) ˜ S E ∂pp ⊗ ∂p = q ( ω s ,ω s ) ˜ S ∂p ⊗ ∂p. Using this identity we can rewrite σ ++ ( ∂p ⊗ ∂p ) = q ( α s ,α s ) S ˜ S ∂p ⊗ ∂p = q ( α s ,α s ) − ( ω s ,ω s ) S ∂p ⊗ ∂p. Now we proceed with the verifications. • Action of E ′ . Consider the expression σ ++ ( ∂p ⊗ ∂p ) = q ( α s ,α s ) T ∂p ⊗ ∂p . We havethe identity E ′ S ˜ S = E ′ from (C.3). Then E ′ σ ++ ( ∂p ⊗ ∂p ) = q ( α s ,α s ) E ′ ∂p ⊗ ∂p = 0 . • Action of E ′ . Consider the expression σ ++ ( ∂p ⊗ ∂p ) = q ( α s ,α s ) − ( ω s ,ω s ) S ∂p ⊗ ∂p . Using ∂p ⊗ ∂p = q ( ω s ,ω s ) ˜ S ∂p ⊗ ∂p we rewrite this as σ ++ ( ∂p ⊗ ∂p ) = q ( α s ,α s ) − ω s ,ω s ) ˜ S − S ∂p ⊗ ∂p. Next, using E ′ ˜ S − = q ( ω s ,ω s +2 ρ ) E from (C.2) we get E ′ σ ++ ( ∂p ⊗ ∂p ) = q ( ω s , ρ ) q ( α s ,α s ) − ( ω s ,ω s ) E S ∂p ⊗ ∂p. Now we use the quadratic condition S = q ω s ,ω s ) − ( α s ,α s ) S − + q ( ω s ,ω s ) (1 − q − ( α s ,α s ) ) from(5.19). Then we obtain E ′ σ ++ ( ∂p ⊗ ∂p ) = q ( ω s , ρ ) q ( ω s ,ω s ) E S − ∂p ⊗ ∂p + q ( ω s , ρ ) q ( α s ,α s ) (1 − q − ( α s ,α s ) ) E ∂p ⊗ ∂p. The second term vanishes due to E ∂p ⊗ ∂p = 0 . Finally using E S − = q − ( ω s ,ω s +2 ρ ) E ′ from(C.2) we conclude that E ′ σ ++ ( ∂p ⊗ ∂p ) = E ′ ∂p ⊗ ∂p = 0 . • Action of ˜ S . Using the expression σ ++ ( ∂p ⊗ ∂p ) = q ( α s ,α s ) − ( ω s ,ω s ) S ∂p ⊗ ∂p and theidentity ˜ S ∂p ⊗ ∂p = q − ( ω s ,ω s ) ∂p ⊗ ∂p we easily get ˜ S σ ++ ( p∂p ⊗ ∂p ) = q − ( ω s ,ω s ) σ ++ ( p∂p ⊗ ∂p ) . ETRICS AND CONNECTIONS ON QUANTUM PROJECTIVE SPACES 37 • Action of ˜ S . As for the case of ˜ S we immediately get ˜ S σ ++ ( ∂p ⊗ ∂pp ) = q − ( ω s ,ω s ) σ ++ ( ∂p ⊗ ∂pp ) . • Tensor product . Consider the expression σ ++ ( ∂p ⊗ ∂p ) = q ( α s ,α s ) − ( ω s ,ω s ) S ∂p ⊗ ∂p .Then using ∂pp = q ( α s ,α s ) − ( ω s ,ω s ) S p∂p from (5.14) we compute σ ++ ( ∂pp ⊗ ∂p ) = q ( α s ,α s ) − ( ω s ,ω s ) S σ ++ ( p∂p ⊗ ∂p ) = q α s ,α s ) − ω s ,ω s ) S S p∂p ⊗ ∂p = S S S − S − ∂p ⊗ ∂pp. Using the "braid equation" for S from (5.11) we obtain σ ++ ( ∂pp ⊗ ∂p ) = S − S S S − ∂p ⊗ ∂pp = q ( ω s ,ω s ) − ( α s ,α s ) S − σ ++ ( ∂p ⊗ ∂pp )= σ ++ ( ∂p ⊗ p∂p ) . This shows that σ ++ descends to a map Ω + ⊗ B Ω + → Ω + ⊗ B Ω + . (cid:3) The case of σ −− is fairly similar. Proposition D.5.
We have a B -bimodule map σ −− : Ω − ⊗ B Ω − → Ω − ⊗ B Ω − given by σ −− ( ¯ ∂p ⊗ ¯ ∂p ) = q − ( α s ,α s ) T ¯ ∂p ⊗ ¯ ∂p. Proof.
We skip the computations, as they are quite similar to the case of σ ++ . To verify that σ −− is well-defined as a map Ω − ⊗ Ω − → Ω − ⊗ Ω − we have to check the relations E ′ σ −− ( ¯ ∂p ⊗ ¯ ∂p ) = 0 , E ′ σ −− ( ¯ ∂p ⊗ ¯ ∂p ) = 0 , ( S − q ( ω s ,ω s ) ) σ −− ( p ¯ ∂p ⊗ ¯ ∂p ) = 0 , ( S − q ( ω s ,ω s ) ) σ −− ( ¯ ∂p ⊗ ¯ ∂pp ) = 0 . One can show the identity S ¯ ∂p ⊗ ¯ ∂p = q ( ω s ,ω s ) ¯ ∂p ⊗ ¯ ∂p . This allows us to rewrite σ −− ( ¯ ∂p ⊗ ¯ ∂p ) = q ( ω s ,ω s ) − ( α s ,α s ) ˜ S ¯ ∂p ⊗ ¯ ∂p. The various verifications are then performed as in Proposition D.4. (cid:3)
Next we consider the term σ + − , which is more involved. Proposition D.6.
We have a B -bimodule map σ + − : Ω + ⊗ B Ω − → Ω − ⊗ B Ω + given by σ + − ( ∂p ⊗ ¯ ∂p ) = q α s ,α s ) − ω s ,ω s ) S ˜ S − ¯ ∂p ⊗ ∂p − ( q ( α s ,α s ) − q ( α s ,α s ) q − ( ω s , ρ ) pg − + p. Proof.
First we check that σ + − is well-defined as a B -bimodule map Ω + ⊗ Ω − → Ω − ⊗ Ω + .In other words, we need to check that the following identities hold E ′ σ + − ( ∂p ⊗ ¯ ∂p ) = 0 , E ′ σ + − ( ∂p ⊗ ¯ ∂p ) = 0 , (˜ S − q − ( ω s ,ω s ) ) σ + − ( p∂p ⊗ ¯ ∂p ) = 0 , ( S − q ( ω s ,ω s ) ) σ + − ( ∂p ⊗ ¯ ∂pp ) = 0 . Then we check that σ + − descends to a map Ω + ⊗ B Ω − → Ω − ⊗ B Ω + . • Action of E ′ . We use E ′ S = q ( ω s ,ω s +2 ρ ) E from (C.2) and the identity ˜ S − = q ω s ,ω s ) − ( α s ,α s ) ˜ S + (1 − q − ( α s ,α s ) ) q ( ω s ,ω s ) from (5.20). We get E ′ S ˜ S − ¯ ∂p ⊗ ∂p = q ( ω s ,ω s +2 ρ ) E ˜ S − ¯ ∂p ⊗ ∂p = q ω s ,ω s ) − ( α s ,α s ) q ( ω s ,ω s +2 ρ ) E ˜ S ¯ ∂p ⊗ ∂p + (1 − q − ( α s ,α s ) ) q ( ω s ,ω s ) q ( ω s ,ω s +2 ρ ) E ¯ ∂p ⊗ ∂p. The first term vanishes since, E ˜ S = q − ( ω s ,ω s +2 ρ ) E ′ by (C.2) and E ′ ∂p = 0 . For thesecond term we use E ¯ ∂p ⊗ ∂p = q − ( ω s , ρ ) pg − + from (C.7). Then E ′ S ˜ S − ¯ ∂p ⊗ ∂p = (1 − q − ( α s ,α s ) ) q ω s ,ω s ) pg − + . Finally using E ′ p = q ( ω s , ρ ) we obtain E ′ σ + − ( ∂p ⊗ ¯ ∂p ) = q α s ,α s ) (1 − q − ( α s ,α s ) ) pg − + − ( q ( α s ,α s ) − q ( α s ,α s ) g − + p = 0 . • Action of E ′ . We use E ′ ˜ S − = q ( ω s ,ω s +2 ρ ) E from (C.2) and the identity S = q ω s ,ω s ) − ( α s ,α s ) S − + (1 − q − ( α s ,α s ) ) q ( ω s ,ω s ) from (5.19). We get E ′ ˜ S − S ¯ ∂p ⊗ ∂p = q ( ω s ,ω s +2 ρ ) E S ¯ ∂p ⊗ ∂p = q ω s ,ω s ) − ( α s ,α s ) q ( ω s ,ω s +2 ρ ) E S − ¯ ∂p ⊗ ∂p + (1 − q − ( α s ,α s ) ) q ( ω s ,ω s ) q ( ω s ,ω s +2 ρ ) E ¯ ∂p ⊗ ∂p. The first term vanishes, since E S − = q − ( ω s ,ω s +2 ρ ) E ′ and E ′ ¯ ∂p = 0 . Then E ′ ˜ S − S ¯ ∂p ⊗ ∂p = (1 − q − ( α s ,α s ) ) q ω s ,ω s ) pg − + . Therefore we obtain E ′ σ + − ( ∂p ⊗ ¯ ∂p ) = q α s ,α s ) (1 − q − ( α s ,α s ) ) pg − + − ( q ( α s ,α s ) − q ( α s ,α s ) pg − + = 0 . • Action of ˜ S . Let us write M L − + = ˜ S − p ¯ ∂p ⊗ ∂p . Using the relations (5.12), (5.14) andthe "braid equation" for ˜ S (5.11) we compute ˜ S − M L − + = ˜ S − ˜ S − p ¯ ∂p ⊗ ∂p = q ( α s ,α s ) − ( ω s ,ω s ) ˜ S − ˜ S − ˜ S − ¯ ∂p ⊗ p∂p = q ( α s ,α s ) − ( ω s ,ω s ) ˜ S − ˜ S − ˜ S − ¯ ∂p ⊗ p∂p = q ( α s ,α s ) ˜ S − ˜ S − ¯ ∂pp ⊗ ∂p = q ( ω s ,ω s ) ˜ S − p ¯ ∂p ⊗ ∂p = q ( ω s ,ω s ) M L − + . Now we rewrite σ + − ( p∂p ⊗ ¯ ∂p ) in terms of M L − + as σ + − ( p∂p ⊗ ¯ ∂p ) = q α s ,α s ) − ω s ,ω s ) S M L − + − ( q ( α s ,α s ) − q ( α s ,α s ) q − ( ω s , ρ ) ppg − + p. Then using ˜ S M L − + = q − ( ω s ,ω s ) M L − + and ˜ S pp = q − ( ω s ,ω s ) pp we obtain (˜ S − q − ( ω s ,ω s ) ) σ + − ( p∂p ⊗ ¯ ∂p ) = 0 . • Action of S . Let us write M R − + = S ¯ ∂p ⊗ ∂pp . As above we compute S M R − + = S S ¯ ∂p ⊗ ∂pp = q ( α s ,α s ) − ( ω s ,ω s ) S S S ¯ ∂pp ⊗ ∂p = q ( α s ,α s ) − ( ω s ,ω s ) S S S ¯ ∂pp ⊗ ∂p = q ( α s ,α s ) S S ¯ ∂p ⊗ p∂p = q ( ω s ,ω s ) S ¯ ∂p ⊗ ∂pp = q ( ω s ,ω s ) M R − + . Now we rewrite σ + − ( ∂p ⊗ ¯ ∂pp ) in terms of M R − + as σ + − ( ∂p ⊗ ¯ ∂pp ) = q α s ,α s ) − ω s ,ω s ) ˜ S − M R − + − ( q ( α s ,α s ) − q ( α s ,α s ) q − ( ω s , ρ ) pg − + pp. Then using S M R − + = q ( ω s ,ω s ) M R − + and S pp = q ( ω s ,ω s ) pp we obtain ( S − q ( ω s ,ω s ) ) σ + − ( ∂p ⊗ ¯ ∂pp ) = 0 . ETRICS AND CONNECTIONS ON QUANTUM PROJECTIVE SPACES 39 • Tensor product . We want to show that σ + − ( ∂pp ⊗ ¯ ∂p ) = σ + − ( ∂p ⊗ p ¯ ∂p ) . Using theright B -module relations (5.14) we compute σ + − ( ∂pp ⊗ ¯ ∂p ) = q ( α s ,α s ) − ( ω s ,ω s ) S σ + − ( p∂p ⊗ ¯ ∂p )= q α s ,α s ) − ω s ,ω s ) S S ˜ S − p ¯ ∂p ⊗ ∂p − q ( α s ,α s ) − ( ω s ,ω s ) ( q ( α s ,α s ) − q ( α s ,α s ) q − ( ω s , ρ ) S ppg − + p. We focus on the first term. Using again (5.14) we have S S ˜ S − p ¯ ∂p ⊗ ∂p = S S ˜ S − ˜ S − S − ¯ ∂p ⊗ ∂pp = ˜ S − S ˜ S − ¯ ∂p ⊗ ∂pp. We also used that the terms S and ˜ S commute. For the second term we have S ppg − + p = q ( ω s ,ω s ) ppg − + p = q ( ω s ,ω s ) pg − + pp = ˜ S − pg − + pp. Using these identities we obtain σ + − ( ∂pp ⊗ ¯ ∂p ) = q ( α s ,α s ) − ( ω s ,ω s ) ˜ S − σ + − ( ∂p ⊗ ¯ ∂pp ) = σ + − ( ∂p ⊗ p ¯ ∂p ) . (cid:3) Finally we consider σ − + , which is similar to σ + − . However, since the details are fairlyinvolved, we include the necessary steps also in this case. Proposition D.7.
We have a B -bimodule map σ − + : Ω − ⊗ B Ω + → Ω + ⊗ B Ω − given by σ − + ( ¯ ∂p ⊗ ∂p ) = q ω s ,ω s ) − α s ,α s ) S − ˜ S ∂p ⊗ ¯ ∂p − ( q − ( α s ,α s ) − q − ( ω s , ρ ) pg + − p. Proof.
We need to verify the following relations E ′ σ − + ( ¯ ∂p ⊗ ∂p ) = 0 , E ′ σ − + ( ¯ ∂p ⊗ ∂p ) = 0 , ( S − q ( ω s ,ω s ) ) σ − + ( p ¯ ∂p ⊗ ∂p ) = 0 , (˜ S − q − ( ω s ,ω s ) ) σ − + ( ¯ ∂p ⊗ ∂pp ) = 0 . Then we check that σ − + descends to a map Ω − ⊗ B Ω + → Ω + ⊗ B Ω − . • Action of E ′ . We have the identity S − = q ( α s ,α s ) − ω s ,ω s ) S + (1 − q ( α s ,α s ) ) q − ( ω s ,ω s ) from (5.19). Then we can rewrite E ′ S − ˜ S ∂p ⊗ ¯ ∂p = q ( α s ,α s ) − ω s ,ω s ) E ′ S ˜ S ∂p ⊗ ¯ ∂p + (1 − q ( α s ,α s ) ) q − ( ω s ,ω s ) E ′ ˜ S ∂p ⊗ ¯ ∂p. The first term vanishes, since E ′ S ˜ S = E ′ from (C.3) and E ′ ¯ ∂p = 0 . For the secondterm we use g + − p = q ( ω s ,ω s ) − ( α s ,α s ) E ′ ˜ S ∂p ⊗ ¯ ∂p from (C.7). Then E ′ S − ˜ S ∂p ⊗ ¯ ∂p = ( q − ( α s ,α s ) − q α s ,α s ) − ω s ,ω s ) g + − p. Using this identity we conclude that E ′ σ − + ( ¯ ∂p ⊗ ∂p ) = ( q − ( α s ,α s ) − g + − p − ( q − ( α s ,α s ) − g + − p = 0 . • Action of E ′ . We have the identity ˜ S = q ( α s ,α s ) − ω s ,ω s ) ˜ S − + (1 − q ( α s ,α s ) ) q − ( ω s ,ω s ) from (5.20). Then we can rewrite E ′ ˜ S S − ∂p ⊗ ¯ ∂p = q ( α s ,α s ) − ω s ,ω s ) E ′ ˜ S − S − ∂p ⊗ ¯ ∂p + (1 − q ( α s ,α s ) ) q − ( ω s ,ω s ) E ′ S − ∂p ⊗ ¯ ∂p. The first term vanishes, since E ′ ˜ S − S − = E ′ from (C.3) and E ′ ∂p = 0 . For the secondterm we use pg + − = q ( ω s ,ω s ) − ( α s ,α s ) E ′ S − ∂p ⊗ ¯ ∂p from (C.6). Then E ′ ˜ S S − ∂p ⊗ ¯ ∂p = ( q − ( α s ,α s ) − q α s ,α s ) − ω s ,ω s ) pg + − . Using this identity we conclude that E ′ σ − + ( ¯ ∂p ⊗ ∂p ) = ( q − ( α s ,α s ) − pg + − − ( q − ( α s ,α s ) − pg + − = 0 . • Action of S . Let us write M L + − = S − p∂p ⊗ ¯ ∂p . Then using the relations (5.12),(5.14) and the "braid equation" for S (5.11) we compute S − M L + − = S − S − p∂p ⊗ ¯ ∂p = q ( ω s ,ω s ) − ( α s ,α s ) S − S − S − ∂p ⊗ p ¯ ∂p = q ( ω s ,ω s ) − ( α s ,α s ) S − S − S − ∂p ⊗ p ¯ ∂p = q − ( α s ,α s ) S − S − ∂p ⊗ p ¯ ∂p = q − ( ω s ,ω s ) S − ∂p ⊗ p ¯ ∂p = q − ( ω s ,ω s ) M L + − . Now we rewrite σ − + ( p ¯ ∂p ⊗ ∂p ) in terms of M L + − as σ − + ( p ¯ ∂p ⊗ ∂p ) = q ω s ,ω s ) − α s ,α s ) ˜ S M L + − − ( q − ( α s ,α s ) − q − ( ω s , ρ ) ppg + − p. Then using S M L + − = q ( ω s ,ω s ) M L + − and S pp = q ( ω s ,ω s ) pp we obtain ( S − q ( ω s ,ω s ) ) σ − + ( p ¯ ∂p ⊗ ∂p ) = 0 . • Action of ˜ S . Let us write M R + − = ˜ S ∂p ⊗ ¯ ∂pp . As above we compute ˜ S M R + − = ˜ S ˜ S ∂p ⊗ ¯ ∂pp = q ( ω s ,ω s ) − ( α s ,α s ) ˜ S ˜ S ˜ S ∂pp ⊗ ¯ ∂p = q ( ω s ,ω s ) − ( α s ,α s ) ˜ S ˜ S ˜ S ∂pp ⊗ ¯ ∂p = q − ( α s ,α s ) ˜ S ˜ S ∂p ⊗ p ¯ ∂p = q − ( ω s ,ω s ) ˜ S ∂p ⊗ ¯ ∂pp = q − ( ω s ,ω s ) M R + − . Now we rewrite σ − + ( ¯ ∂p ⊗ ∂pp ) in terms of M R + − as σ − + ( ¯ ∂p ⊗ ∂pp ) = q ω s ,ω s ) − α s ,α s ) S − M R + − − ( q − ( α s ,α s ) − q − ( ω s , ρ ) pg + − pp. Then using ˜ S M R + − = q − ( ω s ,ω s ) M R + − and ˜ S pp = q − ( ω s ,ω s ) pp we obtain (˜ S − q − ( ω s ,ω s ) ) σ − + ( ¯ ∂p ⊗ ∂pp ) = 0 . • Tensor product . We want to show that σ − + ( ¯ ∂pp ⊗ ∂p ) = σ − + ( ¯ ∂p ⊗ p∂p ) . Using theright B -module relations (5.14) we compute σ − + ( ¯ ∂pp ⊗ ∂p ) = q ( ω s ,ω s ) − ( α s ,α s ) ˜ S σ − + ( p ¯ ∂p ⊗ ∂p )= q ω s ,ω s ) − α s ,α s ) ˜ S S − ˜ S p∂p ⊗ ¯ ∂p − q ( ω s ,ω s ) − ( α s ,α s ) ( q − ( α s ,α s ) − q − ( ω s , ρ ) ˜ S ppg + − p. We focus on the first term. Using again (5.14) we have ˜ S S − ˜ S p∂p ⊗ ¯ ∂p = ˜ S S − ˜ S S − ˜ S − ∂p ⊗ ¯ ∂pp = S − S − ˜ S ∂p ⊗ ¯ ∂pp. For the second term we have ˜ S ppg − + p = q − ( ω s ,ω s ) ppg − + p = q − ( ω s ,ω s ) pg − + pp = S − pg − + pp. Using these identities we obtain σ − + ( ¯ ∂pp ⊗ ∂p ) = q ( ω s ,ω s ) − ( α s ,α s ) S − σ − + ( ¯ ∂p ⊗ ∂pp ) = σ − + ( ¯ ∂p ⊗ p∂p ) . (cid:3) ETRICS AND CONNECTIONS ON QUANTUM PROJECTIVE SPACES 41
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