Documenta Math. 607 Moduli Spaces of Flat Connections and Morita Equivalence of Quantum Tori

We study moduli spaces of flat connections on surfaces with boundary, with boundary conditions given by Lagrangian Lie subalgebras. The resulting symplectic manifolds are closely related with Poisson-Lie groups and their algebraic structure (such as symplectic groupoid structure) gets a geometrical explanation via 3-dimensional cobordisms. We give a formula for the symplectic form in terms of holonomies, based on a central extension of the gauge group by closed 2-forms. This construction is finally used for a certain extension of the Morita equivalence of quantum tori to the world of Poisson-Lie groups.


Introduction
Let g be a Lie algebra with an invariant inner product •, • (of any signature).It gives rise to two interesting types of symplectic manifolds.The first type are moduli spaces of flat g-connections on oriented surfaces.The second type are symplectic manifolds connected with Poisson-Lie groups such as the Lu-Weinstein double symplectic groupoid [9] (the symplectic groupoid integrating a Poisson-Lie group) corresponding to a Manin triple We shall notice that these "Poisson-Lie type" symplectic manifolds are, in fact, themselves moduli spaces of flat connections, if we allow surfaces with boundary and impose boundary conditions on the flat connections.To get the Lu-Weinstein double groupoid, the surface is a square, with boundary conditions as on the picture: We shall provide formulas for symplectic forms using holonomies of the flat connections, in the spirit of Alekseev-Malkin-Meinrenken [1].The basic idea is that the symplectic form can be interpreted as the integral over the surface of the curvature of a certain connection.The integral is then readily computed in terms of parallel transport.Moreover we shall describe how 3dim bodies give rise to Lagrangian submanifolds; for example, this picture gives one of the two products in Lu-Weinstein double groupoid: This is a symplectic version of Chern-Simons TQFT in the sense of D. Freed [5], with appropriate boundary conditions.The motivation for this work was to give a symplectic description of Morita equivalence of quantum tori, and moreover, to extend this Morita equivalence from Abelian T-duality to Poisson-Lie T-duality [7] (though just on the symplectic level, without performing geometrical quantization).This is done in the final section.

Aknowledgements
I am grateful to Anton Alekseev, David Li-Bland, Štefan Sakáloš and András Szenes for useful discussion and suggestions.I am also grateful to Albert Schwarz and Alan Weinstein for asking me (independently) about extending the Morita equivalence of non-commutative tori to the case of Poisson-Lie Tduality.

Colored surfaces and moduli spaces
Let G be a connected Lie group and , an Ad-invariant inner product (of any signature) on its Lie algebra g.We shall consider compact oriented surfaces Σ with corners (i.e.locally looking like (R ≥0 ) 2 ).We shall assume that none of the components of Σ is closed and that on each component of ∂Σ there is at least one corner. 1The boundary of Σ is thus split by the corners into a finite number of arcs.
For each arc a we choose a Lie subalgebra h a ⊂ g which is Lagrangian w.r.t., (i.e.h ⊥ a = h a ).Let H a ⊂ G be the corresponding connected Lie subgroup.We demand for every corner x of Σ that if a and b are the arcs meeting at x then h a ∩ h b = 0. We shall call such a Σ (together with the choice of subalgebras) a colored surface.
For every colored surface Σ we define a symplectic manifold M Σ .Let us give three equivalent definitions of the manifold M Σ ; its symplectic form is defined in Section 3.

M Σ via cuts of Σ
Let us first describe M Σ in a way which depends on a choice of certain cuts of Σ.We keep cutting Σ along paths connecting corners until we get a polygon.For every side s of the polygon we choose an element g s ∈ G such that: 1. if s is an arc of the boundary of Σ then g s ∈ H s 2. if s and s ′ are the two sides which are the result of a cut then g s ′ = g −1 s 3. the product of all g s 's along the boundary of the polygon (in their natural cyclic order) is equal to 1.
An assignment s → g s satisfying these properties is, by definition, a point in

Pavol Ševera
M Σ can thus be described as the preimage of 1 under a map The map is a submersion and thus M Σ is a manifold.

M Σ as a space of groupoid morphisms
Let us now describe M Σ without using cuts.Let X ⊂ Σ be the set of corners of Σ and let Π 1 (Σ, X) be the fundamental groupoid of Σ (the set of objects of Π 1 (Σ, X) is X and morphisms are homotopy classes of paths between corners).Every arc of the boundary can be seen as a morphism in Π 1 (Σ, X).Then ∈ H a for every arc a}.

M Σ as a moduli space of flat connections
Finally, let us describe M Σ as a moduli space of flat connections.Let π : P → Σ be a principal G-bundle.For every arc a we choose a reduction of P | a to H a ⊂ G, i.e. a submanifold P a ⊂ π −1 (a) which is a principal H a -bundle over a.For every corner x ∈ Σ we choose a point p x ∈ P a ∩ P b where a and b are the arcs meeting at x. Let us call π : P → Σ with its additional structure a colored G-bundle over Σ.Let us consider connections on P which restrict to connections (i.e. to h a -valued 1-forms) on every P a ; we shall call such a connection a colored connection.M Σ can then be described the moduli space of colored flat connections on colored G-bundles over Σ.The groupoid morphism Π 1 (Σ, X) → G corresponding to a colored flat connection is given by parallel transport (the fiber of P over any corner x is trivialized by the choice of the point p x ).In the opposite direction, if F : Π 1 (Σ, X) → G satisfies F (a) ∈ H a for every arc a then the corresponding flat colored G-bundle is construced as follows.Let p : Σ → Σ be a universal cover of Σ with a chosen corner y 0 ∈ Σ and let x 0 = p(y 0 ).Let P = Σ × G → Σ be the trivial G-bundle, with the trivial flat connection.By restriction of F we have a homomorphism π 1 (Σ, x 0 ) → G and we define the flat G-bundle P → Σ as P = P /G.The reduction of P over a corner y ∈ Σ is (y, F ([p • γ y0y ])), where γ y0y is a path (unique up to homotopy) from y 0 to y.These reductions then extend uniquely to a coloring of P , and the coloring descends to a coloring of P .It is clear that every flat colored G-bundle P → Σ arises in this way, as the flat connection on p * P can be used to trivialize it.Notice that M Σ is the disjoint union over the isomorphism classes of colored G-bundles of the moduli spaces with fixed colored G-bundle class.
If P is the trivial G-bundle P = Σ × G and its coloring is also trivial (i.e.P a = a × H a , p x = (x, 1)) then a colored connection can be described as a 1form A ∈ Ω 1 (Σ) ⊗ g such that the restriction of A to any arc a is in Ω 1 (a) ⊗ h a .
The space of these flat connections modulo the gauge transformations (by maps g : Σ → G such that g(x) = 1 for every corner x and g(a) ⊂ H a for every arc a) is a connected component of M Σ .
3 Symplectic form in terms of holonomies

Symplectic form on moduli spaces of flat connections
Let P → Σ be a colored G-bundle.Colored connections on P form an affine space A col (P ) modeled on Ω 1 col (Σ, Ad P ), where Ω col (Σ, Ad P ) ⊂ Ω(Σ, Ad P ) is the space of forms that restrict to Ω(a, Ad Pa ) on every arc a ⊂ ∂Σ.If A is a flat colored connection on P then the covariant differential d A makes Ω col (Σ, Ad P ) to a complex and we have a natural isomorphism where [P, A] ∈ M Σ denotes the isomorphism class of (P, A).The antisymmetric pairing ) is non-degenerate by Poincaré-Verdier duality.The moduli space M Σ becomes in this way a symplectic manifold.To see that ω is smooth and closed (we already checked that it is non-degenerate), let us choose an open subset U ⊂ M Σ which admits a smooth family of colored flat connections φ : U → A col (P ), φ : where on the affine space A col (P ); ω is therefore closed.The symplectic form ω is a straightforward generalization of the symplectic form of Atiyah-Bott [2] and Goldman [6] who considered closed surfaces.
Remark 3.1.The symplectic manifold (M Σ , ω) is best described as the symplectic reduction of (A col (P ), ω A ) by the group of the automorphisms of P preserving the coloring.Making this statement precise is, however, rather technical.Here I present the formal part of the story, ignoring the problems with infinite-dimensional manifolds: To simplify notations, let us discuss the case when the colored G-bundle P → Σ is trivial.Let us recall how symplectic forms appear on moduli spaces of flat connections.
The group of smooth maps Σ → G acts affinely on Ω 1 (Σ) ⊗ g by gauge transformations (g : and this action preserves ω A .
The infinitesimal action of a t : Σ → g is generated by the Hamiltonian where F = dα + α 2 is the curvature of the g-connection α; the Poisson bracket of two such Hamiltonians is Notice that the cocycle (4) vanishes on the Lie algebra {t : Σ → g; t(a) ⊂ h a for every arc a} of infinitesimal gauge transformations preserving the coloring.The moment map ( 3) is 0 at α ∈ Ω 1 (Σ) ⊗ g iff α is flat and α| a ∈ Ω 1 (a) ⊗ h a for every arc a.
The symplectic reduction is thus the part of the moduli space M Σ coming from the trivial colored G-bundle.

Central extension by closed 2-forms
Let M be a manifold and let Ω 2 cl (M ) denote the space of closed 2-forms on M .Let us recall that the Lie algebra g(M ) of smooth maps M → g has a central extension g(M ) by Ω 2 cl (M ): as a vector space, and the bracket is The corresponding group G(M ), a central extension of G(M ) (the group of smooth maps M → G) by Ω 2 cl (M ), can be described as follows: its elements are pairs Documenta Mathematica 17 (2012) 607-625 where the invariant 3-form η on G is given by The product in the group is and the inverse (g, ω Finally, let us also introduce an auxiliary group Gbig (M ) ⊃ G(M ): with the product and inverse given by the same formulas ( 7), (8).The map is a group morphism and G(M ) is its kernel.

Symplectic form in terms of holonomies
Let us cut Σ until we get a polygon (as in Section 2).For each side s of the polygon we have a map γ s : M Σ → G (the holonomy along the side).
Theorem 3.1.The symplectic form ω on M Σ is given by where the product is taken in the group Gbig (M Σ ) (see Equation (7)) and the sides of the polygon are taken in their natural (cyclic) order.
The idea of the proof is that ω is the integral of the curvature of a g(M Σ )-valued connection on Σ, and hence can be expressed in terms of the holonomies g s 's.
The proof is in Section 3.6.The formula for ω is a generalization of a similar formula of Alekseev-Malkin-Meinrenken [1] for the case of closed surfaces.

Integral of curvature
be a central extension of Lie groups.Let P → D be a principal K-bundle over a disk D and let P → D be the corresponding K-bundle, P = P /C.Suppose that A is a flat connection on P and Ã is a (non-flat) connection on P lifting A. The curvature F of Ã is a c-valued 2-form on D and its integral is The proof of this simple claim is obvious: trivialize P → D (and hence P → D) in such a way that the connection A on P = D × K becomes trivial.Such a trivialization can be achieved e.g. by by the parallel transport of Ã along straight lines starting at the center of the disc D. Formula (10) then becomes Stokes theorem.We shall use (10) for the central extension The disk will be the result of cutting Σ and M will run over certain open subsets of M Σ .

Symplectic form as integral of curvature
Let Σ be a colored surface.Let U ⊂ M Σ be an open subset, P → Σ a colored G-bundle and A x a smooth family of colored flat connections on P parametrized by x ∈ U , such that the class of (P, A x ) is x.M Σ can be covered by such open subsets U .
Using the inclusion G → G(U ), g → (g, 0), we lift P to a principal G(U )-bundle PU → Σ.Similarly, the inclusion G → G(U ) lifts P to a principal G(U )-bundle P U → Σ and P U = PU /Ω 2 closed (U ).The family A x can be seen as a flat connection A on P U , and Ã = (A, 0) as a (non-flat) connection on PU .The curvature F of Ã is a Ω 2 cl (U )-valued 2form on Σ, and the integral of F is (using (2) and ( 5)) the symplectic form on Remark 3.2.To speak properly about principal G(U )-and G(U ) bundles, we should understand in what sense G(U ) and G(U ) are Lie groups.However, we don't need to do it.A principal G(U )-bundle ober Σ is given by an open cover {V α } of Σ and by a cocycle In our case the maps are constant on U .A connection on such a bundle is given by 1-forms ) is by definition a family of 1-forms in Ω 1 (V α , g), smoothly parametrized by U .That is how the family A x is seen as a flat connection A on P U .Notice also that since P can be trivialized when we cut Σ to a dics, P (U ) and P (U ) are also trivialized.
Let us also remark that we are not constructing G(M Σ )-and G(M Σ )-bundles over Σ. Formula (11) gives the symplectic form only on our open subsets U ⊂ M Σ , but these open subsets cover M Σ .
Documenta Mathematica 17 (2012) 607-625 3.6 Symplectic form in terms of holonomies (proof) Proof of Theorem 3.1.It follows immediately from (11) and (10).We cut Σ to a polygon.If s ⊂ ∂Σ then (γ s , 0) is the holonomy of Ã along s (since h s is isotropic and thus the cocycle in (5) vanishes).If s comes from a cut then the holonomy of Ã along s is (γ s , β) for some 2-form β.However, the holonomy along the other side coming from the same cut is its inverse; we can thus replace β with 0 and the product of holonomies will not change.This proves Theorem Let us denote the holonomies as on the picture, i.e.
This symplectic manifold (M Σ , ω) is the Lu-Weinstein double symplectic groupoid [9] corresponding to the triple R, B ⊂ G.This fact was already noticed by the author in [12].A similar interpretation of the Lu-Weinstein double groupoid was found by P. Boalch in [3] using irregular connections.
In one of the groupoid structures, the space of objects is B and a point in M Σ is an arrow from b 2 to b 1 ; in the other groupoid structure, the roles of R and B are exchanged.The groupoid products are given by concatenation of squares (either horizontal or vertical); they will be explained more properly in the following section.
Example 3.2.Let now Σ be a square colored as follows: The symplectic form is The symplectic manifold (M Σ , ω) is again a well-known object: it is the symplectic groupoid integrating the homogeneous Poisson space given by R, B, V ⊂ G via Drinfeld's classification [4].This symplectic groupoid was discovered by Jiang-Hua Lu [8].
Example 3.3.Now let Σ be a triangle.In this case The symplectic form is This symplectic manifold is, up to covering, the big symplectic leaf in the homogeneous Poisson space given by R, B, V ⊂ G.It will play a role when we discuss Morita equivalence.
Example 3.4.Finally, let us discuss the simplest Σ that requires a cut.
The symplectic manifold (M Σ , ω) is the double symplectic groupoid integrating the Drinfeld double given by the triple R, B ⊂ G, i.e. the Lu-Weinstein groupoid of the triple R × B, G diag ⊂ G × G.

Painted bodies and Lagrangian submanifolds
In this section we shall discuss how cobordisms of painted surfaces give rise to Lagrangian submanifolds in the moduli spaces.These Lagrangian submanifolds will turn the moduli spaces into interesting algebraic objects, such as (double) groupoids, modules, etc.The Lagrangian submanifold will consist of those flat connection on the surface that can be extended to flat connections on the 3dim manifold (cobordism).This construction is a straightforward generalization of the symplectic Chern-Simons theory of D. Freed [5], who considered closed surfaces.

Painted bodies
Let us consider a compact oriented 3dim manifold with corners (i.e.locally looking as (R ≥0 ) 3 ).Its boundary is divided to vertices (corners), edges and faces.For some of the faces we choose a Lagrangian Lie subalgebra of g (we shall call such a face painted).We shall require the following.Whenever two faces meet along an edge then at least one of them is painted, and if both are painted, then the two subalgebras are transverse.At each vertex should meet two painted and one unpainted face.Finally, the unpainted part of the boundary should be a colored surface, i.e. each of its components should have boundary and on each of the boundary circles there should be a corner.We shall call such a manifold a painted body.The unpainted part of the boundary of a body X will be denoted Σ X .

Flat connections on a painted body
Let X be a painted body.We shall consider principal G-bundles P → X with a reduction to H a over every painted face a and with a section over every edge between painted faces.We then consider flat connections compatible with the reductions.Let L X ⊂ M ΣX denote the set of equivalence classes of flat colored connections on Σ X that are extensible to X.We shall call L X smooth if it is a submanifold and moreover it can be locally lifted to a smooth family of flat connections on X.
Proof.We shall prove that the formal tangent spaces to L X are Lagrangian in the tangent spaces of M ΣX .If L X is smooth then these formal tangent spaces are the actual tangent spaces.Let P be a painted G-bundle over X, Ad P → X the vector bundle associated to the adjoint representation of G on g, and let Ω col (X, Ad P ) ⊂ Ω(X, Ad P ) be the space of Ad P -valued differential forms that take values in the corresponding subalgebra of g when restricted to a painted face of X.Let A be a Pavol Ševera flat colored connection; then d A makes Ω col (X, Ad P ) into a complex.Let us denote this complex Ω A (X) and its cohomology H A (X).Let P ′ = P | ΣX and A ′ = A| ΣX .Let us consider the short exact sequence (where Ω A,0 (X) are the forms vanishing at Σ X ) and the following piece of the resulting long exact sequence: We have M ΣX , and the image of the first arrow is the formal By Poincaré duality the dual of ( 12) is obtained just by reversing the arrows: ) with its dual is via the symplectic form).As a consequence, the image of the first arrow in ( 12) is a Lagrangian subspace.
In all the examples that we consider below, L X is easily seen to be smooth.

Examples
As we noticed above, Lu-Weinstein's double symplectic groupoid corresponding to a Manin triple B, R ⊂ G is the moduli space for the surface The graph of one of the products in this double groupoid is L X where X is In other words, the product is given by gluing squares along the adjacent sides on the picture Similar pictures can be drawn for the double symplectic groupoid integrating the Drinfeld double and also for its R-matrix in the sense of Weinstein and Xu; see [12] for details.

Morita equivalence of quantum tori and beyond
This last section is a bit speculative.On the other hand, it describes the motivation for the constructions described above, so it is included anyway.

Morita equivalence of quantum tori
Recall that two algebras A and B are said to be Morita equivalent if their categories of modules are linearly equivalent.Equivalently, there exist a A ⊗ Let θ ij = −θ ji , 1 ≤ i, j ≤ n, be a skew-symmetric matrix with real elements.We suppose that the graph of the corresponding linear map R n → R n intersects Z 2n ⊂ R 2n only in 0 ∈ Z 2n .To the matrix θ we associate the algebra T n θ (a quantum torus) generated by elements u i (1 ≤ i ≤ n) and their inverses, modulo relations u i u j = exp(2π √ −1θ ij )u j u i .A famous result of Rieffel and Schwarz [10] says that the algebra T n θ is Morita equivalent to T n θ −1 .2The quantum torus T n θ can be seen as a quantization of the n-dimensional torus T n with the constant Poisson structure given by θ.The following natural questions are due to A. Schwarz and A. Weinstein (motivated by an extension of T -duality [11] to Poisson-Lie T -duality [7]).We shall give an answer to Question 2. It will provide a conjectural answer to Question 1.

H-Morita equivalence
Let H be a Hopf algebra.Let A be an associative algebra in the (monoidal) category H-mod of left H-modules.In other words, A is an H-module and the product A ⊗ A → A is a morphism of H-modules.
Let A-mod H be the category of A-modules in H-mod, i.e. the category of vector spaces V which are modules of both A and H, such that A ⊗ V → V is a morphism of H-modules.Let A-mod be the category of A-modules in the category of vector spaces.We have the forgetful functor res : A-mod  Let us notice that we prove the symplectic version of the conjecture only under the additional assumption that v is transverse to both r and b.The general case would require colored surfaces where we allow adjacent subalgebras to have non-trivial intersection.The corresponding colored G-bundles would have a reduction over the corresponding corner to the group exponentiating the intersection.We shall treat this generalization elsewhere.

Symplectic H-Morita equivalence
In this final section we provide a symplectic analogue of our conjectural answer to Question 1. Vector spaces are replaced by symplectic manifolds of the form M Σ as follows: Let us recall the definition of Morita equivalence of symplectic groupoids [15].Let I denote the groupoid with two objects 0 and 1 and with a unique morphism 0 → 1.Let Γ be a Lie groupoid with a groupoid morphism Γ → I. Γ splits naturally to 4 components: Γ ij (i, j ∈ {0, 1}) is the space of arrows lying over the unique morphism i → j.Let X i denote the space of objects of Γ lying over i ∈ {0, 1}.

3 . 1
for open U ⊂ M Σ satisfying the condition of Section 3.5, and since they cover M Σ , it proves it for entire M Σ 3.7 Examples Example 3.1.Let Σ be a square colored by a Manin triple r, b ⊂ g: Documenta Mathematica 17 (2012) 607-625The other product is obtained when we exchange the colors.The moduli space of It is a symplectic groupoid integrating the Poisson B-homogeneous space corresponding to the quadruple R, B, V ⊂ G.It is also a module of In other words, A is replaced by the symplectic groupoid integrating the Poisson homogeneous space R v , B by the symplectic groupoid integrating B v , H by the double symplectic groupoid integrating both R and B, and M by the moduli space of the displayed triangle.Documenta Mathematica 17 (2012) 607-625

Definition 5 . 2 . 11 ∼
If the maps Γ 01 → X 1 and Γ 10 → X 0 are surjective then the groupoids Γ 00 ⇒ X 0 and Γ 11 ⇒ X 1 are said to be Morita equivalent via the bimodules Γ 01 and Γ 10 If the groupoid Γ is symplectic then we have a Morita equivalence of symplectic groupoids.Let r, b, v ⊂ g be as above and let R, B, V ⊂ G be the corresponding groups.Let X 0 = R × B and let the arrows (r1 , b 1 ) → (r 2 , b 2 ) in Γ 00 be (v, r) ∈ V × R such that r 1 b 1 v = b r 2 b 2 ; composition of arrows is by (v, r)(v ′ , r ′ ) = (vv ′ , rr ′ ).The groupoid Γ 00 can be seen as M Σ for the surface which makes it to a symplectic groupoid.The groupoid composition is by (horizontal) gluing of rectangles.The symplectic groupoid Γ 00 integrates the following Poisson structure on R × B. We have the Poisson action B × B v → B v .The forgetful functor Poisson manifolds with a moment map to R → Poisson b-manifolds has a right adjoint F (see [13]), and F (B v ) = R×B is our Poisson manifold.It is a semi-classical analog of the crossed product A ⋊ H, where A is a quantization of B v (an associative algebra) and H a quantization of the Lie bialgebra b (a Hopf algebra).Notice that A ⋊ H-mod is equivalent to A-mod H . Let us suppose for simplicity that the map R × B → G, (r, b) → rb, is a diffeomorphism.The symplectic groupoid Γ 00 is Morita equivalent to the symplectic groupoid Γ 11 integrating the Poisson manifold R v , i.e.M Σ for the surface Documenta Mathematica 17 (2012) 607-625 1. Is there a generalization of Morita equivalence when T n is replaced by a quantum group H and T n θ by a torsor of H? 2. Is there a symplectic/Poisson version of Morita equivalence for tori with constant Poisson structure?Can it be extended to Poisson-Lie groups, giving a symplectic/Poisson analog of Question 1?
H → A-mod and its left adjoint ind : A-mod → A-mod H . Let now B be an algebra in the (monoidal) category H-comod of right Hcomodules.We have the category B-mod H of B-modules in H-comod and the category B-mod.Now we have the forgetful functor cores : B-mod H → B-mod and its right adjoint coind : B-mod → B-mod H . Definition 5.1.We shall say that A and B are H-Morita equivalent if there are equivalences of linear categories A-mod → B-mod H and A-mod H → B-modThe simplest example is when A = k is trivial (k is the base field) and B = H.The category H-mod H is called the category of Hopf modules of H.A linear equivalence F : k-mod → H-mod H making the diagram Proposition 5.1.An H-Morita equivalence is equivalent to a vector space M which is a right A-module and left B-module, satisfying the compatibility relation [4]where F is the functor A-mod → B-mod H . Suppose again that G is a connected Lie group and its Lie algebra g has an invariant inner product , .Let r, b ⊂ g be a Manin triple and let R and B be the corresponding Poisson-Lie groups.Suppose also that v is another Lagrangian subalgebra with the property that v ∩ r is the Lie algebra of a closed connected subgroup R v ⊂ R and similarly, v ∩ b is the Lie algebra of a closed connected subgroup B v ⊂ B. By Drinfeld's classification of Poisson homogeneous spaces[4]the homogeneous space R/R v has a Poisson structure such that the map R ×(R/R v ) → R/R v isPoisson, and similarly for B/B v .Below we shall prove a symplectic version of the following loosely stated conjecture: If Hopf algebra H is a (suitable) quantization of the Lie bialgebra corresponding to the Manin triple r, b ⊂ g and algebras A and B are (suitable) quantizations of the Poisson manifolds R v and B v respectively, then A and B are H-Morita equivalent.In particular, if v = r we get A = k and B = H, i.e.Sweedler's example of H-Morita equivalence.