Pseudo-Riemannian metrics on bicovariant bimodules

We study pseudo-Riemannian invariant metrics on bicovariant bimodules over Hopf algebras. We clarify some properties of such metrics and prove that pseudo-Riemannian invariant metrics on a bicovariant bimodule and its cocycle deformations are in one to one correspondence.


Introduction
The notion of metrics on covariant bimodules on Hopf algebras have been studied by a number of authors including Heckenberger and Schmüdgen ( [2], [3], [4] ) as well as Beggs, Majid and their collaborators ( [6] and references therein ). The goal of this article is to characterize bicovariant pseudo-Riemannian metrics on a cocycle-twisted bicovariant bimodule. As in [2], the symmetry of the metric comes from Woronowicz's braiding map σ on the bicovariant bimodule. However, since our notion of non-degeneracy of the metric is slightly weaker than that in [2], we consider a slightly larger class of metrics than those in [2]. The positive-definiteness of the metric does not play any role in what we do.
We refer to the later sections for the definitions of pseudo-Riemannian metrics and cocycle deformations. Our strategy is to exploit the covariance of the various maps between bicovariant bimodules to view them as maps between the finite-dimensional vector spaces of left-invariant elements of the respective bimodules. This was already observed and used crucially by Heckenberger and Schmüdgen in the paper [2]. We prove that bi-invariant pseudo-Riemannian metrics are automatically bicovariant maps and compare our definition of pseudo-Riemannian metric with some of the other definitions available in the literature. Finally, we prove that the pseudo-Riemannian bi-invariant metrics on a bicovariant bimodule and its cocycle deformation are in one to one correspondence.
In Section 2, we discuss some generalities on bicovariant bimodules. In Section 3, we define and study pseudo-Riemannian left metrics on a bicovariant differential calculus. Finally, in Section 4, we prove our main result on bi-invariant metrics on cocycle-deformations.
Let us set up some notations and conventions that we are going to follow. All vector spaces will be assumed to be over the complex field. For vector spaces V 1 and V 2 , σ can : V 1 ⊗ C V 2 → V 2 ⊗ C V 1 will denote the canonical flip map, i.e, σ can (v 1 ⊗ C v 2 ) = v 2 ⊗ C v 1 . For the rest of the article, (A, ∆) will denote a Hopf algebra. We will use the Sweedler notation for the coproduct ∆. Thus, we will write (1) ∆(a) = a (1) ⊗ C a (2) .
Following [7], the comodule coaction on a left A-comodule V will be denoted by the symbol ∆ V . Thus, ∆ V is a C-linear map ∆ V : V → A ⊗ C V such that for all v in V (here ǫ is the counit of A). We will use the notation Similarly, the comodule coaction on a right A-comodule will be denoted by V ∆ and we will write (1) .

Covariant bimodules on quantum groups
In this section we recall and prove some basic facts on covariant bimodules. We start by recalling the notions on covariant bimodules from Section 2 of [7]. Suppose M is a bimodule over A such that ( The vector space of left ( respectively, right ) invariant elements of a left ( respectively, right ) covariant bimodules will play a crucial role in the sequel and we introduce notations for them here.
Let us note the immediate consequences of the above definitions.
Proof. This is a simple consequence of the fact that M ∆ commutes with ∆ M .
If (M, ∆ M ) and (N, ∆ N ) are left-covariant bimodules over A, then we have a left coaction ∆ M⊗AN of A on M ⊗ A N defined by the following formula: Here Woronowicz ( [7] ) proved that if V is a left-covariant bimodule over A, then V is a free as a left ( as well as a right ) A-module. In fact, one has the following result:

Moreover, we have vector space isomorphisms
Proof. It is easy to see that For elements m in 0 M and n in 0 N , we get If ρ in 0 (M ⊗ A N ), Corollary 2.4 implies that ρ can be expressed as ρ = ij a ij m i ⊗ A n j , where the elements a ij of A are uniquely determined and {m i } i is a basis of 0 M and {n j } j is a basis of 0 N. Then by using (6), we have that By the uniqueness of the elements a ij , we get that a ij = ǫ(a ij ), i.e a ij are elements of C.1. Hence, we get 0 (M ⊗ A N ) ⊆ span C {m ⊗ A n : m ∈ 0 M, n ∈ 0 N }. This prove (5). For the isomorphism 0 (M ⊗ A N ) ∼ = 0 M ⊗ C 0 N , we define the following map: This map is well-defined, because each element of 0 (M ⊗ A N ) can be written uniquely as ij a ij m i ⊗ A n j where the elements a ij are in C.1, i.e {m i ⊗ A n j } ij forms a vector space basis of 0 (M ⊗ A N ). Clearly, the map is a vector space isomorphism.
The corresponding results for (M, M ∆) and (N, N ∆) are proved similarly and we omit the proofs.
Remark 2.6. In the light of Proposition 2.5, we are allowed to use the notations 0 M ⊗ C 0 N and We recall now the definition of covariant maps between bimodules. T is called right-covariant if for all m ∈ M, n ∈ N, a ∈ A,

Finally, a map which is both left and right covariant will be called a bicovariant map.
We end this section by recalling the following fundamental result of Woronowicz.

Pseudo-Riemannian metrics on bicovariant bimodules
In this section, we study pseudo-Riemannian metrics on bicovariant differential calculus on Hopf algebras. After defining pseudo-Riemannian metrics, we recall the definitions of left and right invariance of a pseudo-Riemannian metrics from [2]. We prove that a pseudo-Riemannian metric is left ( respectively, right ) invariant if and only if it is left ( respectively, right ) covariant. The coefficients of a left-invariant pseudo-Riemannian metric with respect to a left-invariant basis of E are scalars. We use this fact to clarify some properties of pseudo-Riemannian invariant metrics. We end the section by comparing our definition with those by Heckenberger and Schmüdgen ( [2] ) as well as by Beggs and Majid.
For other notions of metrics on covariant differential calculus, we refer to [6] and references therein.

Finally, a pseudo-Riemannian metric g on a bicovariant A-A bimodule E is said to be bi-invariant if it is both left-invariant as well as right-invariant.
We observe that a pseudo-Riemannian metric is invariant if and only if it is covariant.

Proposition 3.3. Let g be a pseudo-Riemannian metric on the bicovariant bimodule E. Then g is left-invariant if and only if
Proof. Let g be a left-invariant metric on E, and ρ, ν be elements of E. Then the following computation shows that g is a left-covariant map.
(where we have used coassociativity of comodule coactions) On the other hand, suppose g : The proof of the right-covariant case is similar.
The following key result will be used throughout the article.
Let us clarify some of the properties of a left-invariant and right-invariant pseudo-Riemannian metrics. To that end, we make the next definition.The notations used in the next definition will be used throughout the article.
Definition 3.5. Let E and g be as above. For a fixed basis Proposition 3.6. Let g be a left-invariant pseudo-Riemannian metric for a pair (E, d) as in Definition 3.1. Then the following statements hold: Proof. The right A-linearity of V g follows from the right A-linearity of g. The condition (2) of Definition 3.1 forces V g to be one-one. This proves (i).
For proving (ii), note that V g | 0 E is the restriction of a one-one map to a subspace. Hence, it is a one-one C-linear map. Since g is left-invariant, by Lemma 3.4, for any e in 0 E, . This yields Therefore, ((a ij )) ij is the left-inverse and hence the inverse of the matrix ((g ij )) ij . This proves (iii). For proving (iv), we use the fact that g ij is an element of C.1 for all i, j. Since So, the matrix ((ǫ(g ij ))) ij is also an inverse to the matrix ((g(ω i ⊗ A ω j ))) ij and hence g ij = ǫ(g ij ) and g ij is in C.1.
Finally, we prove (v) using (iv). Suppose f be an element in E such that g(e ⊗ A f ) = 0 for all e in 0 E. Let f = k ω k a k for some elements a k in A. Then for any fixed index i 0 , we obtain Hence, we have that f = 0. This finishes the proof.
We apply the results in Proposition 3.6 to exhibit a basis of the free right A-module V g (E). This will be used in making Definition 4.9 which is needed to prove our main Theorem 4.13.
Lemma 3.7. Suppose {ω i } i is a basis of 0 E and {ω * i } i be the dual basis as in the proof of Proposition 3.6. If g is a pseudo-Riemannian left-invariant metric on E, then V g (E) is a free right A-module with basis {ω * i } i . Proof. We will use the notations (g ij ) ij and g ij from of Proposition 3.6. Since V g is a right A-linear map, V g (E) is a right A-module. Since j and the inverse matrix (g ij ) ij has scalar entries ( Proposition 3.6 ), we get and so ω * k belongs to V g (E) for all k. By the right A-linearity of the map V g , we conclude that the set As V g is one-one and {ω i } i is a basis of the free module E, we get k g ki a k = 0 ∀ i.
Multiplying by g ij and summing over i yields a j = 0. This proves that {ω * i } i is a basis of V g (E) and finishes the proof. Now we state a result on bi-invariant (i.e both left and right-invariant) pseudo-Riemannian metric.
Proposition 3.8. Let g be a pseudo-Riemannian metric on E and the symbols {ω i } i , {g ij } ij be as above. If (4)), then g is bi-invariant if and only if the elements g ij are scalar and Proof. Since g is left-invariant, the elements g ij are in C.1. Moreover, the right-invariance of g implies that g is right-covariant (Proposition 3.3), i.e.
Conversely, if g ij = g(ω i ⊗ A ω j ) are scalars and (10) is satisfied, then g is left-invariant and right-covariant. By Proposition 3.3, g is right-covariant.
We end this section by comparing our definition of pseudo-Riemannian metrics with some of the other definitions available in the literature. Proposition 3.6 shows that our notion of pseudo-Riemannian metric coincides with the right Alinear version of a "symmetric metric" introduced in Definition 2.1 of [2] if we impose the condition of left-invariance.
Next, we compare our definition with the one used by Beggs and Majid in Proposition 4.2 of [5] ( also see [6] and references therein ). To that end, we need to recall the construction of the two forms by Woronowicz ( [7] ).
If E is a bicovariant A-bimodule and σ be the map as in Proposition 2.8, Woronowicz defined the space of two forms as: The symbol ∧ will denote the quotient map Thus, However, by equation ( 3.15 ) of [7], we know that for some scalars σ kl ij . Therefore, we have We claim that the element h = i,j g(ω i ⊗ A ω j )ω i ⊗ A ω j satisfies ∧(h) = 0. Indeed, by virtue of (12), it is enough to prove that (σ − 1)(h) = 0. But this directly follows from (13) using the left A-linearity of σ. This argument is reversible and hence starting from h ∈ E ⊗ A E satisfying ∧(h) = 0, we get an element g ∈ Hom A (E ⊗ A E, A) such that for all i, j, Since {ω i ⊗ A ω j : i, j} is right A-total in E ⊗ A E ( Corollary 2.4 ) and the maps g, σ are right A-linear, we get that g • σ = g. This proves our claim. Let us note that since we did not assume g to be left invariant, the quantities g(ω i ⊗ A ω j ) need not be scalars. However, the proof goes through since the elements σ ij kl are scalars.

Pseudo-Riemannian metrics for cocycle deformations
This section concerns the braiding map and pseudo-Riemannian metrics of bicovariant bimodules on cocycle deformations of Hopf algebras. This section contains three main results. Firstly, we prove that a bicovariant bimodule E over a Hopf algebra A can be twisted in the presence of a normalized dual 2-cocycle Ω on A to a bicovariant A Ω -bimodule E Ω . Secondly, the canonical braiding map of the bicovariant bimodule E Ω ( Proposition 2.8 ) is a cocycle deformation of the canonical braiding map of E. Finally, we prove that pseudo-Riemannian bi-invariant metrics on E and E Ω are in one to one correspondence.
Throughout this section, we will make heavy use of the Sweedler notations as spelled out in (1), (2) and (3). The coassociativity of ∆ will be expressed by the following equation: Also, when m is an element of a bicovariant bimodule, we will use the notation For a Hopf algebra (A, ∆), we will denote its restricted dual by the symbol ( A, ∆).

Definition 4.1. A normalized dual 2-cocycle on a Hopf algebra A is an invertible element Ω ∈
A ⊗ C A such that the following equations hold: Here, ǫ denotes the counit of the Hopf algebra A.

Associated to a normalized dual (left) 2-cocycle Ω
Then γ is convolution invertible, γ is unital, i.e, and for all a, b, c in A, Given a Hopf algebra (A, ∆) and such a cocycle Ω as above, we have a new Hopf algebra (A Ω , ∆ Ω ) which is equal to A as a vector space, the coproduct ∆ Ω is equal to ∆ while the algebra structure * Ω on A Ω is defined by the following equation: Here, γ is the convolution inverse to γ which is unital and satisfies the following equation: We refer to [1] for more details. Suppose M is a bicovariant A-A-bimodule. Then M can also be deformed in the presence of a cocycle. This is the content of the next proposition whose proof is probably known to experts but since we could not find an explicit reference, we sketch the proof.  Let us prove the first equation of (21). By an abuse of notation, we will denote the A Ω ⊗ C A Ωbimodule structures on M Ω ⊗ C A Ω , A Ω ⊗ C M Ω and A Ω ⊗ C A Ω by the same symbol * Ω . (2) .m (−2) ǫ(a (3) )ǫ(m (−1) ) ⊗ C a (4) .m (0) γ(a (5) ⊗ C m (1) ) (where we have used the fact that γ is the convolution inverse of γ) =γ(a (1) ⊗ C m (−2) )a (2) .m (−1) ⊗ C a (3) .m (0) γ(a (4) using (18)).
The proofs of the second equation of (21) and the equations of (22) are similar and we omit them.
We end this subsection by recalling the following result on the deformation of bicovariant maps. implies that we have deformed maps σ Ω and g Ω . In this subsection, we study the map σ Ω . The map g Ω will be discussed in the next subsection. We will need the following result: As an illustration, we make the following computation which will be needed later in this subsection: Then the following equation holds: Proof. Let us first clarify that we view γ(η (−1) ⊗ C 1)η (0) ⊗ A ω (0) γ(1 ⊗ C ω (1) ) as an element in (E ⊗ A E) Ω . Then the equation holds because of the following computation: =ǫ(η (−2) )ǫ(η (−1) )η (0) ⊗ AΩ ω (0) ǫ(ω (1) )ǫ(ω (2) ) (since γ and γ are normalised) =η ⊗ AΩ ω. Now, we are in a position to study the map σ Ω . By Proposition 4.2, E Ω is a bicovariant A Ωbimodule. Then Proposition 2.8 guarantees the existence of a canonical braiding from E Ω ⊗ AΩ E Ω to itself. We show that this map is nothing but the deformation σ Ω of the map σ associated with the bicovariant A-bimodule E. By the definition of σ Ω , it is a map from (E ⊗ A E) Ω to (E ⊗ A E) Ω . However, by virtue of Proposition 4.4, the map ξ defines an isomorphism from E Ω ⊗ AΩ E Ω to (E ⊗ A E) Ω . By an abuse of notation, we will denote the map Theorem 4.6. Let E be a bicovariant A-bimodule and Ω be a cocycle as above. Then the deformation σ Ω of σ is the unique bicovariant A Ω -bimodule braiding map on E Ω given by Proposition 2.8.
Corollary 4.7. If the unique bicovariant A-bimodule braiding map σ for a bicovariant A-bimodule E satisfies the equation σ 2 = 1, then the bicovariant A Ω -bimodule braiding map σ Ω for the bicovariant A Ω -bimodule E Ω also satisfies σ 2 Ω = 1. In particular, if A is the commutative Hopf algebra of regular functions on a compact semisimple Lie group G and E is its canonical space of one-forms, then the braiding map σ Ω for E Ω satisfies σ 2 Ω = 1. Proof. By Theorem 4.6, σ Ω is the unique braiding map for the bicovariant A Ω -bimodule E Ω . Since, by our hypothesis, σ 2 = 1, the deformed map σ Ω also satisfies σ 2 Ω = 1 by part (iii) of Proposition 4.3. Next, if A is a commutative Hopf algebra as in the statement of the corollary and E is its canonical space of one-forms, then we know that the braiding map σ is just the flip map, i.e. for all e 1 , e 2 in E, σ(e 1 ⊗ A e 2 ) = e 2 ⊗ A e 1 , and hence it satisfies σ 2 = 1. Therefore, for every cocycle deformation E Ω of E, the corresponding braiding map satisfies σ 2 Ω = 1. 4.2. Pseudo-Riemannian bi-invariant metrics on E Ω . Suppose E is a bicovariant A-bimodule and E Ω be its cocycle deformation as above. The goal of this subsection is to prove that a pseudo-Riemannian bi-invariant metric on E naturally deforms to a pseudo-Riemannian bi-invariant metric on E Ω . Since g is a bicovariant ( i.e, both left and right covariant ) map from the bicovariant bimodule E ⊗ A E to itself, then by Proposition 4.3, we have a right A Ω -linear bicovariant map g Ω from E Ω ⊗ AΩ E Ω to itself. We need to check the conditions (i) and (ii) of Definition 3.1 for the map g Ω .
The proof of the equality g Ω = g Ω σ Ω is straightforward. However, checking condition (ii), i.e, verifying that the map V gΩ is an isomorphism onto its image needs some work. The root of the problem is that we do not yet know whether E * = V g (E Ω ). Our strategy to verify condition (ii) is the following: we show that the right A-module V g (E Ω ) is a bicovariant right A-module ( see Definition 4.8 ) in a natural way. Let us remark that since the map g ( hence V g ) is not left A-linear, V g (E Ω ) need not be a left A-module. Since bicovariant right A-modules and bicovariant maps can be deformed ( Proposition 4.11 ), the map V g deforms to a right A Ω -linear isomorphism from E Ω to (V g (E)) Ω . Then in Theorem 4.13, we show that (V g ) Ω coincides with the map V gΩ and the latter is an isomorphism onto its image. This is the only subsection where we use the theory of bicovariant right modules (as opposed to bicovariant bimodules). For the rest of the subsection, E will denote a bicovariant A-bimodule. Moreover, {ω i } i will denote a basis of 0 E and {ω * i } i the dual basis, i.e, ω * i (ω j ) = δ ij . Let us recall that (4) implies the existence of elements R ij in A such that We want to show that V g (E) is a bicovariant right A-module. To this end, we recall that ( Lemma 3.7 ) V g (E) is a free right A-module with basis {ω * i } i . This allows us to make the following definition. Definition 4.9. Let {ω i } i and {ω * i } i be as above and g a bi-invariant pseudo-Riemannian metric on E. Then we can endow where the elements R ij are as in (23).
Then we have the following result.
Proof. The fact that (V g (E), ∆ Vg (E) , Vg(E) ∆) is a bicovariant right A-module follows immediately from the definition of the maps ∆ Vg (E) and Vg (E) ∆. So we are left with proving (25). Let e ∈ E. Then there exist elements a i in A such that e = i ω i a i . Hence, by (8), we obtain This proves the first equation of (25). For the second equation, we begin by making an observation.
Therefore, multiplying (11) by S(R jm ) and summing over j, we obtain

Now by using (8), we compute
This finishes the proof.
Now we recall that bicovariant right A-modules can be deformed too. Proof. Parts (i) and (ii) follow from Proposition 2.27 of [1]. Part (iii) follows by noting that since the map T is a bicovariant right A-linear map, its inverse T −1 is also a bicovariant right A-linear map. Thus, the deformation (T −1 ) Ω of T −1 exists and is the inverse of the map T Ω .
As an immediate corollary, we make the following observation. (V g ) Ω : E Ω → (V g (E)) Ω = (V g ) Ω (E Ω ) Proof. Since both E and V g (E) are bicovariant right A-modules, and V g is a right A-linear bicovariant map (Proposition 4.10), Proposition 4.11 guarantees the existence of (V g ) Ω . Since g is a pseudo-Riemannian metric, by (ii) of Definition 3.1, V g : E → V g (E) is an isomorphism. Then, by (iii) of Proposition 4.11, (V g ) Ω is also an isomorphism. Now we are in a position to state and prove the main result of this section which shows that there is an abundant supply of bi-invariant pseudo-Riemannian metrics on E Ω . Since g is a map from E ⊗ A E to A, g Ω is a map from (E ⊗ A E) Ω to A Ω . But we have the isomorphism ξ from E Ω ⊗ AΩ E Ω to (E ⊗ A E) Ω ( Proposition 4.4 ). As in Subsection 4.1, we will make an abuse of notation to denote the map g Ω ξ −1 by the symbol g Ω . Theorem 4.13. If g is a bi-invariant pseudo-Riemannian metric on a bicovariant A-bimodule E and Ω is a normalised dual 2-cocycle on A, then g deforms to a right A Ω -linear map g Ω from E Ω ⊗ AΩ E Ω to itself. Moreover, g Ω is a bi-invariant pseudo-Riemannian metric on E Ω . Finally, any bi-invariant pseudo-Riemannian metric on E Ω is a deformation (in the above sense) of some bi-invariant pseudo-Riemannian metric on E.
Proof. Sicne g is a right A-linear bicovariant map ( Proposition 3.3), g indeed deforms to a right A Ω -linear map g Ω from (E ⊗ A E) Ω ∼ = E Ω ⊗ AΩ E Ω (see Proposition 4.4) to A Ω . The second assertion of Proposition 4.3 implies that g Ω is bicovariant. Then Proposition 3.3 implies that g Ω is bi-invariant. Since gσ = g, part (iii) of Proposition 4.3 implies that