Fubini-Study metrics and Levi-Civita connections on quantum projective spaces

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 introduced by Heckenberger and Kolb. We define connections on these calculi and show that they are torsion free and cotorsion free, where the latter condition uses the quantum metric and is a weaker notion of metric compatibility. Finally we show that these connections are bimodule connections 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 differential geometry is built, hence it is desirable to extend these notions to the realm of quantum spaces.By quantum spaces, we mean a class of appropriately defined non-commutative algebras, 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 metrics and the corresponding Levi-Civita connections for the quantum projective spaces.This generalizes 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 metric is the theory of compact quantum metric spaces [Rie04], developed by Rieffel following the ideas of Connes.In this theory one introduces a metric on the state space of A in terms of an appropriately defined Dirac operator, which should satisfy some properties.Such Dirac operators 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 commutative situation the points can be identified with the pure states.Instead we are looking for a quantization of the metric tensor, since we want to have some notion of compatibility between 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 Riemannian geometry.Given an algebra A, we begin by introducing a differential calculus Ω • over A, with its degree-one part denoted by Ω 1 .Then a quantum metric can be defined as an element g ∈ Ω 1 ⊗ A Ω 1 satisfying an appropriate invertibility condition.Using the differential calculus, we can also define connections in the standard algebraic sense.In particular, given a connection ∇ on Ω 1 , there is a standard notion of torsion as well.To formulate an analogue of the compatibility of ∇ with the metric g there are two possibilities: 1) a weak version which uses the notion of cotorsion, due to Majid; 2) a strong version that requires ∇ to be a bimodule connection.In the classical case the second version coincides with the usual metric compatibility, 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 family within 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 and Kolb in [HeKo04,HeKo06].We refer to these calculi as canonical since, as soon as some natural conditions are imposed, they are uniquely defined.These quantum spaces and their differential calculi admit a uniform description, which we adopt in this paper, making simplifications relative to the quantum projective spaces only when needed.We expect that the results obtained 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 on them.We denote by B the algebra of a generic quantum projective space and write Ω = Ω 1 .Our first main result is the existence of quantum metrics in the sense of Definition 3.5, which also 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 existence 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 of cotorsion freeness should be seen as a weaker notion of compatibility with the metric (see Definition 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 to the 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-Civita connection 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 the metric.It is natural to ask whether ∇ is a bimodule connection and if the condition of metric compatibility 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 is compatible 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 with the classical description.Hence we find that, in the case of quantum projective spaces, the classical 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 a quantum projective space), with the notable difference that the conditions of being a bimodule connection and metric compatibility in the strong form were not investigated.
A quantum metric and a connection are the main ingredients needed to study further aspects of quantum Riemannian geometry, as discussed in [BeMa20].This program is carried out further in [Maj05], where it is shown for instance that the quantum two-sphere satisfies an analogue of the Einstein condition: this means that the quantum metric is proportional to an appropriately defined Ricci tensor, defined using the curvature of the connection.We conjecture that this will hold for all quantum projective spaces, and more generally for all quantum irreducible flag manifolds.We plan to tackle this problem in future research.
Let us also discuss how our results compare to the existing literature on connections for quantum 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.More relevant for us is the paper [ÓBu12], where the theory of quantum principal bundles is used to introduce a connection on the cotangent bundle, using a non-canonical calculus on the total space (the quantum special unitary group, in this case).We point out that no further properties of these connections are explored, and extensive use is made of the explicit algebraic relations, making it hard to generalize to arbitrary quantum flag manifolds.
We should mention that the connections introduced in [ÓBu12] turn out to coincide with those we describe here.This follows from the recent results of [DKÓ+20], where representationtheoretic methods are used to prove the following result: there exists a unique covariant connection on the Heckenberger-Kolb calculus Ω over a quantum irreducible flag manifold.Moreover they show that this connection is torsion free.It should be possible to extend these techniques 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 not give explicit formulae for the connections.On the other hand, in this paper we provide explicit formulae, which for instance allow us to straightforwardly check the classical limit.Another bonus is that our approach is essentially self-contained, since we only use the relations in the Heckenberger-Kolb calculus Ω, plus general identities of categorical nature.
Finally let us say something about uniqueness of the structures presented in this paper, since in the classical case the Levi-Civita connection is the unique connection which is torsion free and compatible with the metric.This is also the case here, as long as we insist that everything should be covariant, that is compatible with the quantum group (co)action.In this case uniqueness follows from representation theory, essentially as in the classical case: as already mentioned, the results from [DKÓ+20] show that there is a unique covariant connection on Ω; by similar arguments, one shows that coinvariant quantum metrics are unique up to a scalar.However, what is not clear from this point of view is why the connection and the quantum metric should be compatible, which is one of the goals we achieve in this paper.
Let us now discuss the organization of this paper.The first four sections contain various background material, presented in a form suitable for our needs.In Section 1 we recall some basic facts about compact quantum groups, while in Section 2 we recall various identities holding 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 metrics and 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 the description of the Heckenberger-Kolb calculi for quantum irreducible flag manifolds, but we also 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 introduce the quantum metrics, discuss some of their properties and finally prove the existence of appropriate inverse metrics.In Section 7 we introduce two connections on the holomorphic and antiholomorphic part of the calculi.Their direct sum gives a connection which we show to be torsion free and cotorsion free.In Section 8 we show that this is a bimodule connection and verify the property of metric compatibility in the strong form.
Many technical computations are relegated to the appendices, to make the main text more readable.In Appendix A we recall various results about projective spaces, to facilitate the comparison with the quantum case.In Appendix B we prove various properties satisfied by the maps S and S, which we use to rewrite some of the relations of the Heckenberger-Kolb calculus.In Appendix C we prove many of the technical identities that are used in the main text.Finally in Appendix D we introduce various bimodule maps, some used to define the inverse metrics and some to check the bimodule property of the connections.
Acknowledgements.I would like to thank Réamonn Ó Buachalla for various discussions and his comments on a preliminary version of this paper.

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], since they are used in our main reference [HeKo06].Let g be a complex simple Lie algebra.Given a real number q such that 0 < q < 1, the quantized enveloping algebra U q (g) is a certain Hopf algebra deformation of the enveloping algebra U(g), defined as follows.It has generators {K i , E i , F i } r i=1 with r := rank(g) and relations as in [KlSc97, Section 6.1.2].In particular, the comultiplication, antipode and counit are given by α>0 α be the half-sum of the positive roots of g.Then we have S 2 (X) = K 2ρ XK −1 2ρ for any X ∈ U q (g).We also consider a * -structure on U q (g), which in the classical case corresponds to the compact real form u of g.We can take for instance The precise formulae are not very important here, as any equivalent * -structure works equally well 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 a subspace of the linear dual U q (g) * .We take the span of all the matrix coefficients of the finite-dimensional irreducible representations V (λ) (see below).It becomes a Hopf algebra by duality in the following manner: given X, Y ∈ U q (g) and a, b ∈ C q [G] we define Moreover it becomes a Hopf * -algebra by setting a * (X) := a(S(X) * ).
We have a left action ⊲ and a right action ⊳ of U q (g) on C q [G] given by 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 that of U(g), hence of g.In particular we have analogues of the highest weight modules V (λ) for any dominant weight λ, which we denote by the same symbol.Given a finite-dimensional representation V , we define its matrix coefficients by ).These elements span C q [G], according to the description given above.
We say that an inner product Here we use the * -structure of U q (u).It is well-known that an U q (u)-invariant inner product exists 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) * .

Categorical preliminaries
The category of finite-dimensional U q (g)-modules is braided monoidal, that is we have a tensor product and an analogue of the flip map.We use some of the language of tensor categories to make our computations more natural, with [EGNO16] as our main reference.
2.1.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 An important relation satisfied by the braiding is the braid equation, which is acting on V ⊗ W ⊗ Z for any modules V, W, Z.In the following we employ a leg-notation for the action on tensor products, in terms of which the braid equation reads (2.1) A braiding on the category of finite-dimensional U q (g)-modules is not quite unique.We adopt 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 0 µ for the second.Then the braiding is completely determined by Here (•, •) denotes the usual non-degenerate symmetric bilinear form on the dual of the Cartan subalgebra of g (rescaled so that (α, α) = 2 for short roots α, for definiteness).Indeed, v λ ⊗v w 0 µ is a cyclic vector for V (λ) ⊗ V (µ), hence RV (λ),V (µ) is completely determined by the action on this 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 , satisfying certain duality relations to be recalled below.Here V is a finite-dimensional U q (g)module and V * is its linear dual.The maps ev V and coev V are related to the existence of a left dual, while ev ′ V and coev ′ V to the existence of a right dual.In the case of U q (g), the property S 2 (X) = K 2ρ XK −1 2ρ 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 The factor q (2ρ,λ i ) comes from the action of K 2ρ and is related to the square of the antipode.
In the following we are going to fix a simple module V and write We use the leg-notation for the action of these morphisms on tensor products.We write while for the coevaluation we write 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 (2.2) We also have various compatibility relations with the braiding RV,W , since the latter is a natural isomorphism in both entries.For the evaluation morphisms we have (2.3) Similarly, for the coevaluations morphisms we have (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 To find the constant we evaluate both sides at v 1 ⊗ f 1 , where v 1 is a highest weight vector of V = V (λ) and f 1 is its dual.In our conventions for the braiding we have RV, λ) .On the other hand we have E ′ (v 1 ⊗ f 1 ) = q (2ρ,λ 1 ) = q (2ρ,λ) .Hence c = q −(λ,λ+2ρ) .

Differential calculi, metric and connections
In this section we collect various definitions about differential calculi and connections.
3.1.Differential calculi.In this section A denotes an arbitrary algebra.The definitions recalled here are fairly standard and one possible reference is [KlSc97].
and which is generated by the elements a, db with a, b ∈ A.
If A is a * -algebra, we say that The concrete definition of a differential calculus usually begins with the description of its degree-one part.This leads to the following definition.Definition 3.2.A first order differential calculus (FODC) over A is an A-bimodule Ω with a linear map d : A → Ω which obeys the Leibniz rule and such that Ω is generated as a left A-module by the elements da with a ∈ A.
Given any FODC (Ω, d), there exists a universal differential calculus such that its degree-one part is Ω.The universal property in this case is the following.
and such that the following property is satisfied: for any differential calculus

The universal differential calculus can be constructed as a quotient of the tensor algebra of the
Any differential calculus can be obtained as a quotient of the universal differential calculus.
Finally we recall the notion of induced calculus over a subalgebra.Definition 3.5.A (generalized) quantum metric is an element g ∈ Ω 1 ⊗ A Ω 1 which is invertible, in the sense that there exists a bimodule map (•, , ω) for all ω ∈ Ω 1 , where we write g = g (1) ⊗ g (2) .Remark 3.6.From the categorical point of view, a quantum metric makes Ω 1 into a self-dual object in the monoidal category of A-bimodules.
Notice that the definition of a quantum metric only uses Ω 1 , the degree-one part of Ω • .To impose an analogue of the symmetry condition we also use Ω 2 .Definition 3.7.A quantum metric g ∈ Ω 1 ⊗ A Ω 1 is symmetric if we have ∧(g) = 0, where ∧ : Ω 1 ⊗ Ω 1 → Ω 2 denotes the wedge product of one-forms.
Finally, in the case when A is a * -algebra and Ω • is a * -differential calculus, we can require the metric to be real in the following sense.
3.3.Connections.The notion of connection on a module is quite standard.We are only going 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 For the left A-module E = Ω 1 we can define additional properties.
Definition 3.10.The torsion of a connection ∇ : Now suppose that Ω 1 admits a quantum metric g ∈ Ω 1 ⊗ A Ω 1 as in Definition 3.5.Then we can consider the cotorsion of the connection ∇ with respect to g, a notion introduced by Majid in [Maj99] as a weaker version of metric compatibility.
Definition 3.11.The cotorsion of a connection ∇ : Remark 3.12.Let M be a smooth manifold with metric g.Consider a connection ∇ on M, defined in the usual sense as acting on vector fields.Define a connection ∇ * by Here X, Y, Z are vector fields on M. As discussed in [BeMa20, Corollary 5.70], the cotorsion of the connection ∇ can be identified with the torsion of the connection ∇ * defined above.In particular, if ∇ is torsion free then the cotorsion free condition gives for all vector fields X, Y, Z.This is a weaker condition than (∇ X g)(Y, Z) = 0, which is the standard metric compatibility condition with respect to the metric g.
In the classical case, the Levi-Civita connection is the unique connection on the tangent bundle of a smooth manifold which is torsion free and compatible with the metric.This motivates the following definition, see [BeMa20, Definition 8.2].Definition 3.13.A weak quantum Levi-Civita connection is a connection ∇ : Ω 1 → Ω 1 ⊗ A Ω 1 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 1 naturally extends to a connection on the tensor product Ω 1 ⊗ A Ω 1 .In the quantum case this lifting requires the connection to 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 The bimodule map σ E , called the generalized braiding, is not additional data for the connection.Indeed, if it exists it is uniquely determined by the condition above.
A bimodule connection ∇ : Ω 1 → Ω 1 ⊗ A Ω 1 can be extended to a connection on Ω 1 ⊗ A Ω 1 by the Leibniz rule and the generalized braiding, see [BeMa20, Theorem 3.78].In particular, given a quantum metric g ∈ Ω 1 ⊗ A Ω 1 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 1 .We say that it is quantum metric compatible with a quantum metric g ∈ Ω Remark 3.16.As discussed before, in the classical case cotorsion freeness is a weaker property than metric compatibility.In the quantum case we need an extra assumption to compare the two notions, namely the condition ∧ • (σ + id) = 0 for the generalized braiding.Under this condition and torsion freeness of ∇ we obtain coT ∇ = (∧ ⊗ 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 1 → Ω 1 ⊗ A Ω 1 be a bimodule connection and let g ∈ Ω 1 ⊗ A Ω 1 be a quantum metric.Then we say that ∇ is a quantum Levi-Civita connection if it is torsion free and quantum metric compatible with g.

Quantum flag manifolds
The quantum projective spaces can be regarded as the easiest family to describe within the class of quantum (irreducible) flag manifolds.All these quantum spaces admit a uniform description, which we recall here.Even though the focus of this paper is on the projective spaces, many of our computations also work for general irreducible flag manifolds.We take some care in explaining the index-free notation we are going to employ in the following, as it simplifies the computations tremendously (once one gets the hang of it).4.1.Geometrical description.We start by quickly recalling the definition of a flag manifold in the classical case, for precise definitions see for instance [ČaSl09].Let G be a complex simple Lie 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 Lie algebra of L S ), following [StDi99].The quantized Levi factor U q (l S ) is defined by U q (l S ) := K i , E j , F j : i ∈ I, j ∈ S ⊂ U q (g).Here • denotes the subalgebra generated by the given elements in U q (g).It is easily verified that U q (l S ) is a Hopf subalgebra.Moreover it is a Hopf * -subalgebra with * corresponding to 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 In the following we restrict to the case of irreducible flag manifolds.At the Lie algebra level these can be characterized as follows: the set S consists of all the simple roots except for α s , where α s is a simple root appearing with multiplicity one in the highest root of g. 4.2.Generators and relations.The quantum flag manifolds C q [U/K S ] admit a uniform description 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 corresponding to the simple root α s described above, and write N := dim V .
We define the algebra A with generators {v i , f i } N i=1 and relations These generators should be interpreted as follows: after fixing a weight basis {v i } N i=1 of V = V (ω s ), we have the dual basis {f i } N i=1 of V * ∼ = V (−w 0 ω s ) and the double dual basis {v i } N i=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 ωs f i ,v 1 and c −w 0 ωs v i ,f 1 , where v 1 is a fixed highest weight vector of V (ω s ).The isomorphism is given by The algebra A is Z-graded by deg f i := 1 and deg v i := −1.We write B := A 0 for its degree-zero subalgebra, which is generated by the elements 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 := RV,V − q (ωs,ωs) , P V * ,V * := RV * ,V * − q (ωs,ωs) .Then the relations can be written as a,b,c,d (4.2) Remark 4.3.The last relation appears as q (ωs,ωs) i,j,k ( RV,V * ) kk ij p ij = 1 in [HeKo06].We rewrite it using the identity E • RV,V * = q −(ωs,ωs+2ρ) E ′ from Lemma 2.1, which leads to k ( RV,V * ) kk ij = q −(ωs,ωs+2ρ) q (2ρ,λ i ) δ ij .Using this we obtain i q (2ρ,λ i ) p ii = q (ωs,2ρ) .As shown in [Mat19, Proposition 3.3], the algebras A and B can be made into * -algebras as follows.Choosing an orthonormal basis for V = V (ω s ) with respect to a U q (u)-invariant inner product, the * -structure is given by (f i ) * = v i .In this case, the isomorphism from Lemma 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: for instance, with {v i } i the basis of V (note the lower index) we write We want to use a similar notation for the generators {f i , v i } N i=1 of A. What complicates matters here is that we want to consider f i as a linear functional on V and v i as a linear functional on V * , since this is how we have defined the U q (g)-module structure on A (and the reason why we use upper indices for the generators).The bottom line is that we need to consider 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 To give a different example, the last relation i v i f i = 1 of (4.1) becomes E 12 vf = 1, since the LHS is i,j E ij v i f j and we have E ij = δ ij .As a final example, the expression C 1 f carries three indices and corresponds to With this notation, the relations of A can be rewritten in the condensed form 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 2ρ) .(4.4)

Heckenberger-Kolb calculus
In this section we describe the Heckenberger-Kolb calculus associated to an irreducible quantum flag manifold, as introduced in [HeKo06].We also give a slightly different presentation of some of the relations, which turns out to be more convenient for our purposes.Finally we focus on a particular situation, which we refer to as the quadratic case, which geometrically corresponds 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 notation from now on).To describe the relations we need some additional notation.Recall that P V,V = RV,V − q (ωs,ωs) and P V * ,V * = RV * ,V * − q (ωs,ωs) .We also write Q V,V := RV,V + q (ωs,ωs)−(αs,αs) , Q V * ,V * := RV * ,V * + q (ωs,ωs)−(αs,αs) .
Then Ω + is generated by ∂p, as a left B-module, with relations Similarly, Ω − is generated by ∂p with relations To define the right B-module structure, let us introduce the notation (5.3) Then the right B-module structure of Ω + and Ω − is defined by ∂pp = q (αs,αs) T 1234 p∂p, ∂pp = q −(αs,αs) T 1234 p ∂p. (5.4) Finally, the Heckenberger-Kolb calculus (Ω • , d) is the universal differential calculus associated to (Ω, d).It turns out that we have the decomposition d = ∂ + ∂ also in higher degrees.In particular, this implies that ∂ 2 = ∂2 = 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 induced calculi from some auxiliary FODCs Γ + and Γ − over the larger algebra A. This description is also useful for us, so we recall the details below.

Different presentation.
It is convenient to work with the relations of the FODC Ω in a slightly different form, which we now derive.We begin by defining the maps (5.9) Observe that T 1234 = S 123 S234 , 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 123 S234 = S234 S 123 , S234 S 345 = S 345 S234 . (5.10) (2) We have the "braid equations" S 123 S 345 S 123 = S 345 S 123 S 345 , S234 S456 S234 = S456 S234 S456 . (5.11) In particular, observe that we can also write T 1234 = S234 S 123 .We now use the maps S and S to rewrite some of the relations of the FODC Ω.We begin with the relations that involve P V,V and P V * ,V * .
Finally, we rewrite the relations involving P V,V Q V,V and P V * ,V * Q V * ,V * in terms of S and S.
For the second identity, using S−1 V,V * ) 23 = S−1 234 + q (ωs,ωs)−(αs,αs) .The result then easily follows.5.4.The quadratic case.In this paper we consider the situation when RV,V satisfies a quadratic relation, and refer to this as the quadratic case.When V = V (ω s ), this corresponds to a tensor product decomposition with only two simple factors, that is The eigenvalues of the braiding in this case are q (ωs,ωs) and −q (ωs,ωs)−(αs,αs) , corresponding to V (2ω s ) and V (2ω s − α s ) respectively.The quadratic relation satisfied by the braiding RV,V , also known as the Hecke relation in this context, is given by P V,V Q V,V = ( RV,V − q (ωs,ωs) )( RV,V + q (ωs,ωs)−(αs,αs) ) = 0.
The situation is completely analogous for RV * ,V * .Geometrically, the quadratic case of the Heckenberger-Kolb calculus corresponds to the quantum projective spaces.Indeed this holds for U q (g) = U q (sl r+1 ) and the choice ω s = ω 1 or ω s = ω r , corresponding to the fundamental representation or its dual, which satisfies the quadratic decomposition above.
In the quadratic case the relations for Ω + and Ω − can be simplified.Indeed, the first relation of (5.1) is automatically satisfied, due to the quadratic relation for RV,V .Similarly for the first relation of (5.2), due to the quadratic relation for RV * ,V * .Taking into account the presentation in terms of S and S discussed above, we obtain the following description.

Quantum metrics
In this section we define quantum metrics for the quantum projective spaces, reducing to the Fubini-Study metrics in the classical case.Our main result here is Theorem 6.11, which shows that these are quantum metrics according to Definition 3.5, that is they are invertible in 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 restrict to the case of quantum projective spaces, hence Ω denotes the Heckenberger-Kolb FODC corresponding to a generic quantum irreducible flag manifold.
We define g := g +− + g −+ where we write (6.1) Remark 6.1.For the quantum projective spaces, g reduces to the Fubini-Study metric in the classical 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 begin by showing that g is symmetric (Definition 3.7) and real (Definition 3.8).Proposition 6.2.We have that g is symmetric and real.

Inverse metric.
From now on we focus on the case of quantum projective spaces.This means that we take U q (g) = U q (sl r+1 ) and choose either ω s = ω 1 or ω s = ω r , in such a way that the quadratic condition discussed in Section 5.4 is satisfied.
Proof.According to Proposition D.3 we have an A-bimodule map where Φ +− and Ψ +− are the A-bimodule maps given by Consider first Φ +− .Using the right A-module relations (5.6) and (5.7) we compute In the last step we have used vf = q (ωs,ωs) ( RV,V * ) 12 f v from (4.3).Next, we use the identity Finally using the braid equation (2.1) we obtain ωs,ωs) S 123 C 3 p.Similarly, for Ψ +− we compute Using the identities above we find that q −(αs,αs) q (ωs,2ρ) (∂p, ∂p) This proves the claim about (•, •) +− .Remark 6.9.The normalization factor in this map is chosen for later convenience.
Hence we can define a B-bimodule map (•, Remark 6.10.In the classical limit the map (•, •) : Ω ⊗ B Ω → B reduces to the inverse of the Fubini-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.
Hence g ∈ Ω ⊗ B Ω is a quantum metric.
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.
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.
Since g = g +− + g −+ this also implies that g +− and g −+ are central.7.2.Torsion.The first property of the connection ∇ we want to explore is its torsion, which according to Definition 3.10 is the left B-module map Proposition 7.5.We have T ∇ = 0, that is ∇ is torsion free.
Proof.Using (d ⊗ id)g = 0 from Proposition 6.5, we can write the cotorsion as First we compute (id ⊗ ∇)g.We have It is easy to show that E ′ 12 E 23 ∂p ⊗ pg −+ = 0 and E ′ 12 E 23 ∂p ⊗ pg +− = 0 by using the relations in (5.8) (recall that the tensor product is over B).Hence we have 8.1.Bimodule connections.We investigate whether ∇ : Ω → Ω ⊗ B Ω is a bimodule connection by studying its components ∇ + and ∇ − .Recall from Definition 3.14 that this requires the existence of a B-bimodule map σ : We begin by obtaining a useful expression for ∇ − ( ∂pp).
Lemma C.4.Let V = V (λ) be a simple module.C.2. Differential calculus identities.We now derive various identities involving some elements 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.
The next two identities we discuss appear in the proof of [ For the other identities, using the formula above we compute In the last step we have used (5.Finally, the next identity lets us rewrite ∂p ∧ ∂p in terms of ∂p ∧ ∂p under the evaluation E 23 .We have used it in the computation of the torsion in Proposition 7.5. Lemma C.8.We have q (αs,αs) E 23 T 1234 ∂p ∧ ∂p = −E 23 ∂p ∧ ∂p + (q (αs,αs) − 1)E 23 ∂p ∧ ∂p.

Appendix D. Bimodule maps
In this appendix we introduce various bimodule maps, which in the main text were used to define 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 only relations are as in (5.17).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).Using the relations from (5.6) and (5.7) we compute Φ +− (∂f f ⊗ ∂v) = q (αs,αs)−(ωs,ωs) ( RV,V ) 12 Φ +− (f ∂f ⊗ ∂v) = q (αs,αs)−(ωs,ωs) ( RV,V ) 12 C 2 f.
Using these identities and p = f v, we obtain that Φ +− (∂f p ⊗ ∂v) = Φ +− (∂f ⊗ p ∂v).The computations for the map Φ −+ are very similar and we omit the details.
Finally, we show that certain linear combinations of the maps Φ and Ψ defined above descend to the FODCs Γ + and Γ − .Proposition D.3.Let Φ and Ψ be the maps defined above.
(1) We need to check that the relations of Γ + and Γ − are preserved under these maps.

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 1 db 2 : b 1 , b 2 ∈ B} and with differential d| B .3.2.Metrics.We now recall the notion of quantum metric as stated in [BeMa20, Definition 1.15].Notice that invertibility is part of the definition.
Now we use the identity E 23 ∂ ∂pp = E 23 ∂p ∧ ∂p from Lemma C.6, taking into account that ∂ ∂ = − ∂∂.This gives the result.

D. 2 .
Bimodule connections.Consider the terms σ ab with a, b ∈ {+, −} appearing in Lemma 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.
have T 1234 E 45 = E 23 T 3456 T 1234 from (C.1).Then consider T 1234 E 45 T 3456 p ∂p ⊗ ∂p = E 23 T 3456 T 1234 T 3456 p ∂p ⊗ ∂p.Using (5.11) we can derive an analogue of the braid equation for T, that is T 1234 T 3456 T 1234 = T 3456 T 1234 T 3456 .Using this identity we obtain T 1234 E 45 T 3456 p ∂p ⊗ ∂p = E 23 T 1234 T 3456 T 1234 p ∂p ⊗ ∂p = E 23 T 1234 ∂p ⊗ ∂pp.