Pseudo-Riemannian metrics on bicovariant bimodules

BLAISE PASCAL

Jyotishman Bhowmick & Sugato Mukhopadhyay Pseudo-Riemannian metrics on bicovariant bimodules Volume 27, n^o2 (2020), p. 159-180.

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Pseudo-Riemannian metrics on bicovariant bimodules

Jyotishman Bhowmick Sugato Mukhopadhyay

Abstract

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.

1. Introduction

The notion of metrics on covariant bimodules on Hopf algebras have been studied by a number of authors including Heckenberger and Schmüdgen ([6, 7, 8]) as well as Beggs, Majid and their collaborators ([11] and references therein). The goal of this article is to characterize bicovariant pseudo-Riemannian metrics on a cocycle-twisted bicovariant bimodule. As in [8], the symmetry of the metric comes from Woronowicz’s braiding map𝜎on the bicovariant bimodule. However, since our notion of non-degeneracy of the metric is slightly weaker than that in [8], we consider a slightly larger class of metrics than those in [8]. 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 [8]. 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.

These results will be used in the companion article [2].

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 𝑉₁ and 𝑉₂,

Keywords:bicovariant bimodules, pseudo-Riemannian metric, cocycle deformations.

𝜎^can :𝑉₁ ⊗_C𝑉₂ →𝑉₂⊗_C𝑉₁will denote the canonical flip map, i.e,𝜎^can(𝑣₁⊗_C𝑣₂) = 𝑣₂⊗_C𝑣₁.For the rest of the article,(A,Δ)will denote a Hopf algebra. We will use the Sweedler notation for the coproductΔ. Thus, we will write

Δ(𝑎)=𝑎₍₁₎⊗_C𝑎₍₂₎. (1.1) For a rightA-module𝑉 ,the notation𝑉^∗will stand for the set HomA(𝑉 ,A).

Following [16], the comodule coaction on a leftA-comodule𝑉will be denoted by the symbolΔ𝑉.Thus,Δ𝑉 is aC-linear mapΔ𝑉 :𝑉→ A ⊗_C𝑉such that

(Δ⊗_Cid)Δ𝑉 =(id⊗_CΔ𝑉)Δ𝑉, (𝜖⊗_Cid)Δ𝑉(𝑣)=𝑣 for all𝑣in𝑉(here𝜖is the counit ofA). We will use the notation

Δ𝑉(𝑣)=𝑣₍₋₁₎⊗_C𝑣₍₀₎. (1.2) Similarly, the comodule coaction on a rightA-comodule will be denoted by𝑉Δand we will write

𝑉Δ(𝑣)=𝑣₍₀₎⊗_C𝑣₍₁₎. (1.3) Finally, for a Hopf algebraA,^A_AMA,_AM_A^A,^A

AM_A^A will denote the categories of various types of mixed Hopf-bimodules as in [13, Subsection 1.9].

2. Covariant bimodules on quantum groups

In this section we recall and prove some basic facts on covariant bimodules. These objects were studied by many Hopf-algebraists (as Hopf-bimodules) including Abe ([1]) and Sweedler ([15]). During the 1980’s, they were re-introduced by Woronowicz ([16]) for studying differential calculi over Hopf algebras. Schauenburg ([14]) proved a categorical equivalence between bicovariant bimodules and Yetter–Drinfeld modules over a Hopf algebra, the latter being introduced by Yetter in [17].

We start by recalling the notions on covariant bimodules from Section 2 of [16].

Suppose 𝑀 is a bimodule over A such that (𝑀 ,Δ𝑀) is a left A-comodule. Then (𝑀 ,Δ𝑀)is called a left-covariant bimodule if this tuplet is an object of the category

A

AMA, i.e, for all𝑎inAand𝑚in𝑀, the following equation holds.

Δ𝑀(𝑎𝑚)= Δ(𝑎)Δ𝑀(𝑚), Δ𝑀(𝑚 𝑎)= Δ𝑀(𝑚)Δ(𝑎).

Similarly, if 𝑀Δis a right comodule coaction on𝑀 , then (𝑀 ,_𝑀Δ) is called a right covariant bimodule if it is an object of the categoryAM_A^A, i.e, for any𝑎inAand𝑚 in𝑀,

𝑀Δ(𝑎𝑚)= Δ(𝑎)𝑀Δ(𝑚), _𝑀Δ(𝑚 𝑎)=𝑀Δ(𝑚)Δ(𝑎).

Finally, let𝑀be a bimodule overAandΔ𝑀 :𝑀→ A ⊗_C𝑀 and𝑀Δ:𝑀 →𝑀 ⊗_CA beC-linear maps. Then we say that(𝑀 ,Δ𝑀,_𝑀Δ)is a bicovariant bimodule if this triplet is an object of^A_AM_A^A.Thus,

(i) (𝑀 ,Δ𝑀)is left-covariant bimodule, (ii) (𝑀 ,_𝑀Δ)is a right-covariant bimodule, (iii) (id⊗_C𝑀Δ)Δ𝑀 =(Δ𝑀 ⊗_Cid)𝑀Δ.

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.

Definition 2.1. Let (𝑀 ,Δ𝑀) be a left-covariant bimodule overA. The subspace of left-invariant elements of𝑀is defined to be the vector space

0𝑀 :={𝑚∈𝑀 :Δ𝑀(𝑚)=1⊗_C𝑚}.

Similarly, if(𝑀 ,_𝑀Δ)is a right-covariant bimodule overA, the subspace of right-invariant elements of𝑀 is the vector space

𝑀₀:={𝑚∈𝑀 :𝑀Δ(𝑚)=𝑚⊗_C1}.

Remark 2.2. We will say that a bicovariant bimodule(𝑀 ,Δ𝑀,_𝑀Δ)is finite if0𝑀 is a finite dimensional vector space. Throughout this article, we will only work with bicovariant bimodules which are finite.

Let us note the immediate consequences of the above definitions.

Lemma 2.3([16, Theorem 2.4]). Suppose𝑀is a bicovariantA-A-bimodule. Then

𝑀Δ(₀𝑀) ⊆₀𝑀⊗_CA, Δ𝑀(𝑀₀) ⊆ A ⊗_C𝑀₀.

More precisely, if{𝑚_𝑖}𝑖is a basis of₀𝑀 ,then there exist elements{𝑎_{𝑗 𝑖}}𝑖 , 𝑗inAsuch that

𝑀Δ(𝑚_𝑖)=∑︁

𝑗

𝑚_𝑗 ⊗_C𝑎_{𝑗 𝑖}. (2.1)

Proof. This is a simple consequence of the fact that^𝑀Δcommutes withΔ𝑀. The category^A_AMAhas a natural monoidal structure. Indeed, if(𝑀 ,Δ𝑀)and(𝑁 ,Δ𝑁) are left-covariant bimodules overA,then we have a left coactionΔ𝑀⊗A𝑁 ofAon𝑀⊗A𝑁 defined by the following formula:

Δ𝑀⊗_A𝑁(𝑚⊗A𝑛)=𝑚₍₋₁₎𝑛₍₋₁₎⊗_C𝑚₍₀₎⊗A𝑛₍₀₎.

Here, we have made use of the Sweedler notation introduced in (1.2). This makes𝑀⊗A𝑁 into a left covariant A-A-bimodule. Similarly, there is a right coaction 𝑀⊗_A𝑁Δon 𝑀 ⊗A 𝑁if(𝑀 ,_𝑀Δ)and(𝑁 ,_𝑁Δ)are right covariant bimodules.

The fundamental theorem of Hopf modules (Theorem 1.9.4 of [13]) states that if𝑉 is a left-covariant bimodule overA,then𝑉is a free as a left (as well as a right)A-module.

This was reproved by Woronowicz in [16]. In fact, one has the following result:

Proposition 2.4([16, Theorem 2.1 and Theorem 2.3]). Let(𝑀 ,Δ𝑀)be a bicovariant bimodule over A. Then the multiplication maps ₀𝑀 ⊗_CA → 𝑀, A ⊗_C0𝑀 → 𝑀, 𝑀₀⊗_CA →𝑀andA ⊗_C𝑀₀→ 𝑀are isomorphisms.

Corollary 2.5. Let(𝑀 ,Δ𝑀)and(𝑁 ,Δ𝑁)be left-covariant bimodules overAand{𝑚_𝑖}𝑖

and{𝑛_𝑗}𝑗be vector space bases of₀𝑀and₀𝑁respectively. Then each element of𝑀⊗A𝑁 can be written asÍ

𝑖 𝑗𝑎_{𝑖 𝑗}𝑚_𝑖⊗A𝑛_𝑗andÍ

𝑖 𝑗𝑚_𝑖⊗A𝑛_𝑗𝑏_{𝑖 𝑗}, where𝑎_{𝑖 𝑗}and𝑏_{𝑖 𝑗}are uniquely determined.

A similar result holds for right-covariant bimodules(𝑀 ,_𝑀Δ)and(𝑁 ,_𝑁Δ)overA.

Finally, if(𝑀 ,Δ𝑀)is a left-covariant bimodule overAwith basis{𝑚_𝑖}𝑖 of0𝑀, and (𝑁 ,_𝑁Δ)is a right-covariant bimodule overAwith basis{𝑛_𝑗}𝑗 of𝑁₀, then any element of𝑀⊗A 𝑁can be written uniquely asÍ

𝑖 𝑗𝑎_{𝑖 𝑗}𝑚_𝑖⊗A𝑛_𝑗 as well asÍ

𝑖 𝑗𝑚_𝑖 ⊗A𝑛_𝑗𝑏_{𝑖 𝑗}. Proof. The proof of this result is an adaptation of [16, Lemma 3.2] and we omit it.

The next proposition will require the definition of right Yetter–Drinfeld modules for which we refer to [17] and [14, Definition 4.1].

Proposition 2.6 ([14, Theorem 5.7]). The functor 𝑀 ↦→ ₀𝑀 induces a monoidal equivalence of categories ^A_AM_A^A and the category of right Yetter–Drinfeld modules.

Therefore, if(𝑀 ,Δ𝑀)and(𝑁 ,Δ𝑀)be left-covariant bimodules overA,then

0(𝑀⊗A 𝑁)=span_C{𝑚⊗A𝑛:𝑚∈₀𝑀 , 𝑛∈₀𝑁}. (2.2) Similarly, if(𝑀 ,_𝑀Δ)and(𝑁 ,_𝑁Δ)are right-covariant bimodules overA, then we have that

(𝑀⊗A 𝑁)₀=span_C{𝑚⊗A𝑛:𝑚∈𝑀₀, 𝑛∈𝑁₀}. Thus,₀(𝑀 ⊗A 𝑁)=₀𝑀⊗_C0𝑁and(𝑀⊗A𝑁)₀=𝑀₀⊗_C𝑁₀.

Remark 2.7. In the light of Proposition 2.6, we are allowed to use the notations0𝑀⊗_C0𝑁 and0(𝑀⊗A𝑁)interchangeably.

We recall now the definition of covariant maps between bimodules.

Definition 2.8. Let(𝑀 ,Δ𝑀,_𝑀Δ)and(𝑁 ,Δ𝑁,_𝑁Δ)be bicovariantA-bimodules and𝑇 be aC-linear map from𝑀 to𝑁 .

𝑇is called left-covariant if𝑇 is a morphism in the category^AM, i.e, for all𝑚∈𝑀 , (id⊗_C𝑇) (Δ𝑀(𝑚))= Δ𝑁(𝑇(𝑚)).

𝑇is called right-covariant if𝑇 is a morphism in the categoryM^A.Thus, for all𝑚∈𝑀 , (𝑇 ⊗_Cid)𝑀Δ(𝑚)=𝑁Δ(𝑇(𝑚)).

Finally, a map which is both left and right covariant will be called a bicovariant map. In other words, a bicovariant map is a morphism in the category^AM^A.

We end this section by recalling the following fundamental result of Woronowicz.

Proposition 2.9([16, Proposition 3.1]). Given a bicovariant bimoduleEthere exists a unique bimodule homomorphism

𝜎:E ⊗A E → E ⊗A E such that𝜎(𝜔⊗A𝜂)=𝜂⊗A𝜔 (2.3) for any left-invariant element𝜔and right-invariant element𝜂inE.𝜎is invertible and is a bicovariantA-bimodule map fromE ⊗AEto itself. Moreover,𝜎satisfies the following braid equation onE ⊗_A E ⊗_A E:

(id⊗A𝜎) (𝜎⊗Aid) (id⊗A𝜎)=(𝜎⊗Aid) (id⊗A𝜎) (𝜎⊗Aid). 3. 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 [8]. 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 ofEare 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 ([8]) as well as by Beggs and Majid.

Definition 3.1([8]).SupposeEis a bicovariantA-bimoduleEand𝜎:E ⊗_AE → E ⊗_AE be the map as in Proposition 2.9. A pseudo-Riemannian metric for the pair(E, 𝜎)is a rightA-linear map𝑔:E ⊗A E → Asuch that the following conditions hold:

(i) 𝑔◦𝜎=𝑔.

(ii) If𝑔(𝜌⊗A𝜈)=0 for all𝜈inE,then𝜌=0.

For other notions of metrics on covariant differential calculus, we refer to [11] and references therein.

Definition 3.2([8]). A pseudo-Riemannian metric𝑔on a bicovariantA-bimoduleEis said to be left-invariant if for all𝜌, 𝜈inE,

(id⊗_C𝜖 𝑔) (Δ( E ⊗_AE)(𝜌⊗A 𝜈))=𝑔(𝜌⊗A𝜈).

Similarly, a pseudo-Riemannian metric𝑔on a bicovariantA-bimoduleEis said to be right-invariant if for all𝜌, 𝜈inE,

(𝜖 𝑔⊗_Cid) (( E ⊗_AE)Δ(𝜌⊗A 𝜈))=𝑔(𝜌⊗A 𝜈).

Finally, a pseudo-Riemannian metric𝑔on a bicovariantA-AbimoduleEis 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. Let𝑔be a pseudo-Riemannian metric on the bicovariant bimoduleE.

Then𝑔is left-invariant if and only if𝑔:E ⊗AE → Ais a left-covariant map. Similarly, 𝑔is right-invariant if and only if𝑔:E ⊗AE → Ais a right-covariant map.

Proof. Let𝑔be a left-invariant metric onE, and𝜌,𝜈be elements ofE. Then the following computation shows that𝑔is a left-covariant map.

Δ𝑔(𝜌⊗A 𝜈)= Δ( (id⊗_C𝜖 𝑔) (Δ( E ⊗_AE)(𝜌⊗A𝜈)))

= Δ(id⊗_C𝜖 𝑔) (𝜌₍₋₁₎𝜈₍₋₁₎⊗_C𝜌₍₀₎⊗A𝜈₍₀₎)

= Δ(𝜌₍₋₁₎𝜈₍₋₁₎)𝜖 𝑔(𝜌₍₀₎⊗_A 𝜈₍₀₎)

=(𝜌₍₋₁₎)₍₁₎(𝜈₍₋₁₎)₍₁₎⊗_C (𝜌₍₋₁₎)₍₂₎(𝜈₍₋₁₎)₍₂₎𝜖 𝑔(𝜌₍₀₎⊗A 𝜈₍₀₎)

=(𝜌₍₋₁₎)(1)(𝜈₍₋₁₎)(1)⊗_C( (id⊗_C𝜖 𝑔) ( (𝜌₍₋₁₎)(2)(𝜈₍₋₁₎)(2)⊗_C𝜌₍₀₎⊗A𝜈₍₀₎))

=𝜌₍₋₁₎𝜈₍₋₁₎⊗_C( (id⊗_C𝜖 𝑔) (Δ_{( E ⊗AE)}(𝜌₍₀₎⊗A𝜈₍₀₎)))

(where we have used co associativity of comodule coactions)

=𝜌₍₋₁₎𝜈₍₋₁₎⊗_C𝑔(𝜌₍₀₎⊗A𝜈₍₀₎)

=(id⊗_C𝑔) (Δ( E ⊗AE)(𝜌⊗A𝜈)).

On the other hand, suppose𝑔 :E ⊗A E → Ais a left-covariant map. Then we have (id⊗_C𝜖 𝑔)Δ( E ⊗AE)(𝜌⊗A𝜈)=(id⊗_C𝜖) (id⊗_C𝑔)Δ( E ⊗AE)(𝜌⊗A 𝜈)

=(id⊗_C𝜖)Δ𝑔(𝜌⊗A𝜈)=𝑔(𝜌⊗A𝜈).

The proof of the right-covariant case is similar.

The following key result will be used throughout the article.

Lemma 3.4 ([8]). If 𝑔 is a pseudo-Riemannian metric which is left-invariant on a left-covariantA-bimoduleE, then𝑔(𝜔₁⊗A𝜔₂) ∈C.1for all𝜔₁, 𝜔₂in0E.Similarly, if 𝑔is a right-invariant pseudo-Riemannian metric on a right-covariantA-bimodule, then 𝑔(𝜂₁⊗A𝜂₂) ∈C.1for all𝜂₁, 𝜂₂inE₀.

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 which makes sense as we always work with finite bicovariant bimodules (see Remark 2.2). The notations used in the next definition will be used throughout the article.

Definition 3.5. LetEand𝑔be as above. For a fixed basis{𝜔₁, . . . , 𝜔_𝑛}of0E,we define 𝑔_{𝑖 𝑗} =𝑔(𝜔_𝑖 ⊗_A𝜔_𝑗).Moreover, we define𝑉_𝑔 :E → E^∗ =HomA(E,A)to be the map defined by the formula

𝑉_𝑔(𝑒) (𝑓)=𝑔(𝑒⊗_A 𝑓).

Proposition 3.6. Let 𝑔 be a left-invariant pseudo-Riemannian metric for E as in Definition 3.1. Then the following statements hold:

(i) The map𝑉_𝑔is a one-one rightA-linear map fromEtoE^∗.

(ii) If𝑒∈ Eis such that𝑔(𝑒⊗A 𝑓)=0for all 𝑓 ∈₀E,then𝑒=0.In particular, the map𝑉_𝑔|₀Eis one-one and hence an isomorphism from₀Eto(₀E)^∗.

(iii) The matrix( (𝑔_{𝑖 𝑗}))𝑖 𝑗is invertible.

(iv) Let𝑔^{𝑖 𝑗}denote the(𝑖, 𝑗)-th entry of the inverse of the matrix( (𝑔_{𝑖 𝑗}))𝑖 𝑗. Then𝑔^{𝑖 𝑗} is an element ofC.1for all𝑖, 𝑗.

(v) If𝑔(𝑒⊗A 𝑓)=0for all𝑒in₀E, then 𝑓 =0.

Proof. The rightA-linearity of𝑉_𝑔 follows from the fact that𝑔is a well-defined map fromE ⊗A EtoA.The condition (ii) of Definition 3.1 forces𝑉_𝑔 to be one-one. This proves (i).

For proving (ii), note that𝑉_𝑔|₀E is the restriction of a one-one map to a subspace.

Hence, it is a one-oneC-linear map. Since𝑔is left-invariant, by Lemma 3.4, for any𝑒in

0E,𝑉_𝑔(𝑒) (₀E)is contained inC. Therefore,𝑉_𝑔maps0Einto(₀E)^∗. Since,0Eand(₀E)^∗ have the same finite dimension as vector spaces,𝑉_𝑔|₀E :0E → (₀E)^∗is an isomorphism.

This proves (ii).

Now we prove (iii). Let {𝜔_𝑖}𝑖 be a basis of 0E and {𝜔^∗

𝑖}𝑖 be a dual basis, i.e, 𝜔^∗

𝑖(𝜔_𝑗)=𝛿_{𝑖 𝑗}.Since𝑉_𝑔|₀Eis a vector space isomorphism from0Eto(₀E)^∗by part (ii),

there exist complex numbers𝑎_{𝑖 𝑗}such that (𝑉_𝑔)⁻¹(𝜔^∗

𝑖)=∑︁

𝑗

𝑎_{𝑖 𝑗}𝜔_𝑗.

This yields

𝛿_{𝑖 𝑘}=𝜔^∗

𝑖(𝜔_𝑘)=𝑔

∑︁

𝑗

𝑎_{𝑖 𝑗}𝜔_𝑗⊗A 𝜔_𝑘

!

=∑︁

𝑗

𝑎_{𝑖 𝑗}𝑔_{𝑗 𝑘}.

Therefore,( (𝑎_{𝑖 𝑗}))𝑖 𝑗is the left-inverse and hence the inverse of the matrix( (𝑔_{𝑖 𝑗}))𝑖 𝑗. This proves (iii).

For proving (iv), we use the fact that𝑔_{𝑖 𝑗}is an element ofC.1 for all𝑖, 𝑗. Since

∑︁

𝑘

𝑔(𝜔_𝑖⊗A𝜔_𝑘)𝑔^{𝑘 𝑗} =𝛿_{𝑖 𝑗}.1=∑︁

𝑘

𝑔^{𝑖 𝑘}𝑔(𝜔_𝑘⊗A𝜔_𝑗)=𝛿_{𝑖 𝑗},

we have

∑︁

𝑘

𝑔(𝜔_𝑖⊗A 𝜔_𝑘)𝜖(𝑔^{𝑘 𝑗})=𝛿_{𝑖 𝑗} =∑︁

𝑘

𝜖(𝑔^{𝑖 𝑘})𝑔(𝜔_𝑘 ⊗A𝜔_𝑗).

So, the matrix( (𝜖(𝑔^{𝑖 𝑗})))𝑖 𝑗is also an inverse to the matrix( (𝑔(𝜔_𝑖 ⊗A𝜔_𝑗)))𝑖 𝑗and hence 𝑔^{𝑖 𝑗}=𝜖(𝑔^{𝑖 𝑗})and𝑔^{𝑖 𝑗}is inC.1.

Finally, we prove (v) using (iv). Suppose 𝑓 be an element inEsuch that𝑔(𝑒⊗_A 𝑓)=0 for all𝑒in0E. Let 𝑓 =Í

𝑘𝜔_𝑘𝑎_𝑘for some elements𝑎_𝑘inA. Then for any fixed index𝑖₀, we obtain

0=𝑔

∑︁

𝑗

𝑔^𝑖0𝑗𝜔_𝑗⊗A

∑︁

𝑘

𝜔_𝑘𝑎_𝑘

!

=∑︁

𝑘

∑︁

𝑗

𝑔^𝑖0𝑗𝑔_{𝑗 𝑘}𝑎_𝑘 =∑︁

𝑘

𝛿_𝑖

0𝑘𝑎_𝑘 =𝑎_𝑖

0.

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 rightA-module 𝑉_𝑔(E).This will be used in making Definition 4.11 which is needed to prove our main Theorem 4.15.

Lemma 3.7. Suppose{𝜔_𝑖}𝑖 is a basis of₀Eand{𝜔^∗

𝑖}𝑖be the dual basis as in the proof of Proposition 3.6. If𝑔is a pseudo-Riemannian left-invariant metric onE,then𝑉_𝑔(E)is a free rightA-module with basis{𝜔^∗

𝑖}𝑖.

Proof. We will use the notations(𝑔_{𝑖 𝑗})𝑖 𝑗and𝑔^{𝑖 𝑗}from of Proposition 3.6. Since𝑉_𝑔is a rightA-linear map,𝑉_𝑔(E)is a rightA-module. Since

𝑉_𝑔(𝜔_𝑖)=∑︁

𝑗

𝑔_{𝑖 𝑗}𝜔^∗_𝑗 (3.1)

and the inverse matrix(𝑔^{𝑖 𝑗})𝑖 𝑗has scalar entries (Proposition 3.6), we get 𝜔^∗

𝑘 =∑︁

𝑖

𝑔^{𝑘 𝑖}𝑉_𝑔(𝜔_𝑖)

and so𝜔^∗

𝑘belongs to𝑉_𝑔(E)for all𝑘 .By the rightA-linearity of the map𝑉_𝑔,we conclude that the set{𝜔^∗

𝑖}𝑖 is rightA-total in𝑉_𝑔(E). Finally, if𝑎_𝑖are elements inAsuch thatÍ

𝑘𝜔^∗

𝑘𝑎_𝑘 =0,then by (3.1), we have 0=∑︁

𝑖 , 𝑘

𝑔^{𝑘 𝑖}𝑉_𝑔(𝜔_𝑖)𝑎_𝑘 =𝑉_𝑔

∑︁

𝑖

𝜔_𝑖

∑︁

𝑘

𝑔^{𝑘 𝑖}𝑎_𝑘

! ! .

As𝑉_𝑔is one-one and{𝜔_𝑖}𝑖 is a basis of the free moduleE,we get

∑︁

𝑘

𝑔^{𝑘 𝑖}𝑎_𝑘 =0 ∀ 𝑖 .

Multiplying by𝑔_{𝑖 𝑗}and summing over𝑖yields𝑎_𝑗 =0.This proves that{𝜔^∗

𝑖}𝑖 is a basis of

𝑉_𝑔(E)and finishes the proof.

Remark 3.8. Let us note that for all𝑒∈ E,the following equation holds:

𝑒=∑︁

𝑖

𝜔_𝑖𝜔^∗_𝑖(𝑒). (3.2)

The following proposition was kindly pointed out to us by the referee for which we will need the notion of a left dual of an object in a monoidal category. We refer to Definition 2.10.1 of [5] or Definition XIV.2.1 of [9] for the definition.

Proposition 3.9. Suppose𝑔is a pseudo-RiemannianA-bilinear pseudo-Riemannian metric on a finite bicovariantA-bimodule. LeteEdenote the left dual of the objectEin the category^A_AM_A^A.ThenEeis isomorphic toEas objects in the category^A_AM^A_A via the morphism𝑉_𝑔.

Proof. It is well-known thatEeandE^∗are isomorphic objects in the category^A_AM^A_A. This follows by using the bicovariantA-bilinear maps

ev :E ⊗e AE −→ A; coev :A −→ E ⊗A E;e 𝜙⊗A𝑒↦−→𝜙(𝑒), 1↦−→∑︁

𝑖

𝜔_𝑖⊗A 𝜔^∗

𝑖

We define ev𝑔 :E ⊗AE → Aand coev𝑔:A → E ⊗AEby the following formulas:

ev𝑔(𝑒⊗A 𝑓)=𝑔(𝑒⊗A 𝑓), coev𝑔(1)=∑︁

𝑖

𝜔_𝑖⊗A𝑉_𝑔⁻¹(𝜔^∗

𝑖).

Then since𝑔is both left and rightA-linear, ev𝑔and coev𝑔areA-A-bilinear maps. The bicovariance of𝑔 implies the bicovariance of ev𝑔 while the bicovariance of coev𝑔 = (id⊗A𝑉_𝑔⁻¹) ◦coev follows from the bicovariance of𝑉_𝑔and coev.

Since the left dual of an object is unique upto isomorphism, we need to check the following identities for all𝑒inE:

(ev𝑔⊗Aid) (id⊗Acoev𝑔) (𝑒)=𝑒, (id⊗Aev𝑔) (coev𝑔⊗Aid) (𝑒)=𝑒 . But these follow by a simple computation using the fact that0Eis rightA-total inEand the identity (3.2).

From the above discussion, we have thatE andE^∗ are two left duals of the object E in the category ^A_AM_A^A.Then by the proof of Proposition 2.10.5 of [5], we know that(ev^𝑔⊗AidE^∗) (idE⊗Acoev)is an isomorphism fromEtoE^∗.But it can be easily checked that(ev𝑔⊗AidE^∗) (idE⊗Acoev)=𝑉_𝑔.This completes the proof.

Now we state a result on bi-invariant (i.e both left and right-invariant) pseudo- Riemannian metric.

Proposition 3.10. Let𝑔be a pseudo-Riemannian metric onEand the symbols{𝜔_𝑖}𝑖, {𝑔_{𝑖 𝑗}}𝑖 𝑗be as above. If

EΔ(𝜔_𝑖)=∑︁

𝑗

𝜔_𝑗 ⊗_C𝑅_{𝑗 𝑖} (3.3)

(see(2.1)), then𝑔is bi-invariant if and only if the elements𝑔_{𝑖 𝑗}are scalar and 𝑔_{𝑖 𝑗}=∑︁

𝑘 𝑙

𝑔_{𝑘 𝑙}𝑅_{𝑘 𝑖}𝑅_{𝑙 𝑗}. (3.4)

Proof. Since𝑔is left-invariant, the elements𝑔_{𝑖 𝑗}are inC.1. Moreover, the right-invariance of𝑔implies that𝑔is right-covariant (Proposition 3.3), i.e.

1⊗_C𝑔_{𝑖 𝑗}= Δ(𝑔_{𝑖 𝑗})=(𝑔⊗Aid)E ⊗_AEΔ(𝜔_𝑖 ⊗_C𝜔_𝑗)

=(𝑔⊗Aid) ∑︁

𝑘 𝑙

𝜔_𝑘 ⊗A𝜔_𝑙⊗_C𝑅_{𝑘 𝑖}𝑅_{𝑙 𝑗}

!

=1⊗_C∑︁

𝑘 𝑙

𝑔_{𝑘 𝑙}𝑅_{𝑘 𝑖}𝑅_{𝑙 𝑗},

so that

𝑔_{𝑖 𝑗}=∑︁

𝑘 𝑙

𝑔_{𝑘 𝑙}𝑅_{𝑘 𝑖}𝑅_{𝑙 𝑗}. (3.5)

Conversely, if𝑔_{𝑖 𝑗} =𝑔(𝜔_𝑖⊗A𝜔_𝑗)are scalars and (3.4) is satisfied, then𝑔is left-invariant and right-covariant. By Proposition 3.3,𝑔is right-invariant.

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 rightA-linear version of a “symmetric metric” introduced in Definition 2.1 of [8] 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 [10] (also see [11] and references therein). To that end, we need to recall the construction of the two forms by Woronowicz ([16]).

IfEis a bicovariantA-bimodule and𝜎be the map as in Proposition 2.9, Woronowicz defined the space of two forms as:

Ω²(A):=(E ⊗_A E)

Ker(𝜎−1). The symbol∧will denote the quotient map

∧:E ⊗A E →Ω²(A). Thus,

Ker(∧)=Ker(𝜎−1). (3.6)

In Proposition 4.2 of [10], the authors define a metric on a bimoduleEover a (possibly) noncommutative algebraAas an elementℎofE ⊗AEsuch that∧(ℎ)=0.We claim that metrics in the sense of Beggs and Majid are in one to one correspondence with elements 𝑔∈HomA(E ⊗AE,A)(not necessarily left-invariant) such that𝑔◦𝜎=𝑔 .Thus, modulo the nondegeneracy condition (ii) of Definition 3.1, our notion of pseudo-Riemannian metric matches with the definition of metric by Beggs and Majid.

Indeed, if𝑔 ∈ HomA(E ⊗A E,A) as above and{𝜔_𝑖}𝑖,is a basis of0E,then the equation𝑔◦𝜎=𝑔implies that

𝑔◦𝜎(𝜔_𝑖⊗A𝜔_𝑗)=𝑔(𝜔_𝑖⊗A𝜔_𝑗). However, by equation (3.15) of [16], we know that

𝜎(𝜔_𝑖 ⊗A𝜔_𝑗)=∑︁

𝑘 ,𝑙

𝜎^{𝑘 𝑙}

𝑖 𝑗𝜔_𝑘 ⊗A𝜔_𝑙

for some scalars𝜎^{𝑘 𝑙}

𝑖 𝑗.Therefore, we have

∑︁

𝑘 ,𝑙

𝜎^{𝑘 𝑙}

𝑖 𝑗𝑔(𝜔_𝑘 ⊗A𝜔_𝑙)=𝑔(𝜔_𝑖 ⊗A𝜔_𝑗). (3.7) We claim that the elementℎ=Í

𝑖 , 𝑗𝑔(𝜔_𝑖 ⊗A𝜔_𝑗)𝜔_𝑖⊗A𝜔_𝑗 satisfies∧(ℎ)=0.Indeed, by virtue of (3.6), it is enough to prove that (𝜎−1) (ℎ) =0.But this directly follows from (3.7) using the leftA-linearity of𝜎 .

This argument is reversible and hence starting fromℎ∈ E ⊗A Esatisfying∧(ℎ)=0, we get an element𝑔 ∈HomA(E ⊗AE,A)such that for all𝑖, 𝑗 ,

𝑔◦𝜎(𝜔_𝑖⊗A𝜔_𝑗)=𝑔(𝜔_𝑖⊗A𝜔_𝑗).

Since{𝜔_𝑖 ⊗A𝜔_𝑗 :𝑖, 𝑗}is rightA-total inE ⊗A E(Corollary 2.5) and the maps𝑔, 𝜎 are rightA-linear, we get that𝑔◦𝜎=𝑔 .This proves our claim. Let us note that since we did not assume𝑔to be left invariant, the quantities𝑔(𝜔_𝑖 ⊗A𝜔_𝑗)need not be scalars.

However, the proof goes through since the elements𝜎

𝑖 𝑗

𝑘 𝑙are scalars.

4. Pseudo-Riemannian metrics for cocycle deformations

This section concerns the braiding map and pseudo-Riemannian metrics of bicovariant bimodules under cocycle deformations of Hopf algebras. This section contains two main results. We start by recalling that a bicovariant bimoduleEover a Hopf algebraAcan be deformed in the presence of a 2-cocycle𝛾onAto a bicovariantA𝛾-bimoduleE𝛾.We prove that the canonical braiding map of the bicovariant bimoduleE𝛾(Proposition 2.9) is a cocycle deformation of the canonical braiding map of E.Finally, we prove that pseudo-Riemannian bi-invariant metrics onEandE𝛾are in one to one correspondence.

Throughout this section, we will make heavy use of the Sweedler notations as spelled out in (1.1), (1.2) and (1.3). The coassociativity ofΔwill be expressed by the following equation:

(Δ⊗_Cid)Δ(𝑎)=(id⊗_CΔ)Δ(𝑎)=𝑎₍₁₎⊗_C𝑎₍₂₎⊗_C𝑎₍₃₎. Also, when𝑚is an element of a bicovariant bimodule, we will use the notation

(id⊗_C𝑀Δ)Δ𝑀(𝑚)=(Δ𝑀 ⊗_Cid)𝑀Δ(𝑚)=𝑚₍₋₁₎⊗_C𝑚₍₀₎⊗_C𝑚₍₁₎. (4.1) Definition 4.1. A cocycle𝛾on a Hopf algebra(A,Δ)is aC-linear map𝛾:A ⊗_CA →C such that it is convolution invertible, unital, i.e,

𝛾(𝑎⊗_C1)=𝜖(𝑎)=𝛾(1⊗_C𝑎) and for all𝑎, 𝑏, 𝑐inA,

𝛾(𝑎₍₁₎⊗_C𝑏₍₁₎)𝛾(𝑎₍₂₎𝑏₍₂₎⊗_C𝑐)=𝛾(𝑏₍₁₎⊗_C𝑐₍₁₎)𝛾(𝑎⊗_C𝑏₍₂₎𝑐₍₂₎). (4.2) Given a Hopf algebra (A,Δ)and such a cocycle𝛾as above, we have a new Hopf algebra(A𝛾,Δ𝛾)which is equal toAas a vector space, the coproductΔ𝛾is equal toΔ while the algebra structure∗𝛾onA𝛾is defined by the following equation:

𝑎∗𝛾𝑏=𝛾(𝑎₍₁₎⊗_C𝑏₍₁₎)𝑎₍₂₎𝑏₍₂₎𝛾(𝑎₍₃₎⊗_C𝑏₍₃₎). (4.3) Here,𝛾is the convolution inverse to𝛾which is unital and satisfies the following equation:

𝛾(𝑎₍₁₎𝑏₍₁₎⊗_C𝑐)𝛾(𝑎₍₂₎⊗_C𝑏₍₂₎)=𝛾(𝑎⊗_C𝑏₍₁₎𝑐₍₁₎)𝛾(𝑏₍₂₎⊗_C𝑐₍₂₎). (4.4) We refer to [4, Theorem 1.6] for more details.

Suppose𝑀 is a bicovariantA-A-bimodule. Then𝑀 can also be deformed in the presence of a cocycle. This is the content of the next proposition.

Proposition 4.2 ([12, Theorem 2.5]). Suppose 𝑀 is a bicovariant A-bimodule and 𝛾is a2-cocycle onA.Then we have a bicovariantA𝛾-bimodule𝑀_𝛾 which is equal to𝑀 as a vector space but the left and rightA𝛾-module structures are defined by the following formulas:

𝑎∗𝛾𝑚=𝛾(𝑎₍₁₎⊗_C𝑚₍₋₁₎)𝑎₍₂₎.𝑚₍₀₎𝛾(𝑎₍₃₎⊗_C𝑚₍₁₎) (4.5) 𝑚∗𝛾𝑎 =𝛾(𝑚₍₋₁₎⊗_C𝑎₍₁₎)𝑚₍₀₎.𝑎₍₂₎𝛾(𝑚₍₁₎⊗_C𝑎₍₃₎), (4.6) for all elements𝑚of𝑀and for all elements𝑎ofA. Here,∗𝛾 denotes the right and left A𝛾-module actions, and.denotes the right and leftA-module actions.

TheA𝛾-bicovariant structures are given by

Δ𝑀_𝛾 := Δ𝑀 :𝑀_𝛾→ A𝛾⊗_C𝑀_𝛾and𝑀_𝛾Δ:=𝑀Δ:𝑀_𝛾→ 𝑀_𝛾⊗_CA𝛾. (4.7) Remark 4.3. From Proposition 4.2, it is clear that if𝑀is a finite bicovariant bimodule (see Remark 2.2), then𝑀_𝛾is also a finite bicovariant bimodule.

We end this subsection by recalling the following result on the deformation of bicovariant maps.

Proposition 4.4([12, Theorem 2.5]). Let(𝑀 ,Δ𝑀,_𝑀Δ)and(𝑁 ,Δ𝑁,_𝑁Δ)be bicovariant A-bimodules,𝑇 :𝑀 →𝑁be aC-linear bicovariant map and𝛾be a cocycle as above.

Then there exists a map𝑇_𝛾 :𝑀_𝛾→𝑁_𝛾defined by𝑇_𝛾(𝑚)=𝑇(𝑚)for all𝑚in𝑀. Thus, 𝑇_𝛾 =𝑇 asC-linear maps. Moreover, we have the following:

(i) the deformed map𝑇_𝛾:𝑀_𝛾→ 𝑁_𝛾is anA𝛾bicovariant map,

(ii) if𝑇is a bicovariant right (respectively left)A-linear map, then𝑇_𝛾is a bicovariant right (respectively left)A𝛾-linear map,

(iii) if(𝑃,Δ𝑃,_𝑃Δ)is another bicovariantA-bimodule, and𝑆:𝑁→𝑃is a bicovari- ant map, then(𝑆◦𝑇)𝛾:𝑀_𝛾 →𝑃_𝛾is a bicovariant map and𝑆_𝛾◦𝑇_𝛾=(𝑆◦𝑇)𝛾. 4.1. Deformation of the braiding map

SupposeEis a bicovariantA-bimodule,𝜎be the bicovariant braiding map of Propo- sition 2.9 and 𝑔 be a bi-invariant metric. Then Proposition 4.4 implies that we have deformed maps𝜎_𝛾and𝑔_𝛾.In this subsection, we study the map𝜎_𝛾.The map𝑔_𝛾will be discussed in the next subsection. We will need the following result:

Proposition 4.5([12, Theorem 2.5]). Let(𝑀 ,Δ𝑀,_𝑀Δ)and(𝑁 ,Δ𝑁,_𝑁Δ)be bicovariant bimodules over a Hopf algebraA and 𝛾be a cocycle as above. Then there exists a bicovariantA𝛾- bimodule isomorphism

𝜉:𝑀_𝛾⊗A𝛾 𝑁_𝛾→ (𝑀⊗A𝑁)𝛾. The isomorphism𝜉and its inverse are respectively given by

𝜉(𝑚⊗_A𝛾𝑛)=𝛾(𝑚₍₋₁₎⊗_C𝑛₍₋₁₎)𝑚₍₀₎⊗_A𝑛₍₀₎𝛾(𝑚₍₁₎⊗_C𝑛₍₁₎) 𝜉⁻¹(𝑚⊗A𝑛)=𝛾(𝑚₍₋₁₎⊗_C𝑛₍₋₁₎)𝑚₍₀₎⊗A𝛾𝑛₍₀₎𝛾(𝑚₍₁₎⊗_C𝑛₍₁₎)

As an illustration, we make the following computation which will be needed later in this subsection:

Lemma 4.6. Suppose𝜔∈₀E, 𝜂∈ E₀.Then the following equation holds:

𝜉⁻¹(𝛾(𝜂₍₋₁₎⊗_C1)𝜂₍₀₎⊗A𝜔₍₀₎𝛾(1⊗_C𝜔₍₁₎))=𝜂⊗A𝛾𝜔.

Proof. Let us first clarify that we view𝛾(𝜂₍₋₁₎ ⊗_C1)𝜂₍₀₎ ⊗A 𝜔₍₀₎𝛾(1⊗_C𝜔₍₁₎)as an element in(E ⊗AE)𝛾.Then the equation holds because of the following computation:

𝜉⁻¹(𝛾(𝜂₍₋₁₎⊗_C1)𝜂₍₀₎ ⊗A𝜔₍₀₎𝛾(1⊗_C𝜔₍₁₎))

=𝛾(𝜂₍₋₁₎⊗_C1)𝜉⁻¹(𝜂₍₀₎⊗A𝜔₍₀₎)𝛾(1⊗_C𝜔₍₁₎)

=𝛾(𝜂₍₋₂₎⊗_C1)𝛾(𝜂₍₋₁₎⊗_C1)𝜂₍₀₎⊗A𝛾𝜔₍₀₎𝛾(1⊗_C𝜔₍₁₎)𝛾(1⊗_C𝜔₍₂₎) (since𝜔∈₀E, 𝜂∈ E₀)

=𝜖(𝜂₍₋₂₎)𝜖(𝜂₍₋₁₎)𝜂₍₀₎⊗A𝛾𝜔₍₀₎𝜖(𝜔₍₁₎)𝜖(𝜔₍₂₎) (since𝛾and𝛾are normalised)

=𝜂⊗_A𝛾𝜔.

Now, we are in a position to study the map𝜎_𝛾.By Proposition 4.2,E𝛾is a bicovariant A𝛾-bimodule. Then Proposition 2.9 guarantees the existence of a canonical braiding fromE𝛾⊗A𝛾E𝛾to itself. We show that this map is nothing but the deformation𝜎_𝛾of the map𝜎associated with the bicovariantA-bimoduleE. By the definition of𝜎_𝛾,it is a map from(E ⊗_AE)𝛾to(E ⊗_A E)𝛾.However, by virtue of Proposition 4.5, the map𝜉 defines an isomorphism fromE𝛾⊗A𝛾 E𝛾 to(E ⊗AE)𝛾.By an abuse of notation, we will denote the map

𝜉⁻¹𝜎_𝛾𝜉:E𝛾⊗A𝛾 E𝛾→ E𝛾⊗A𝛾E𝛾

by the symbol𝜎_𝛾again.

Theorem 4.7([12, Theorem 2.5]). LetEbe a bicovariantA-bimodule and𝛾be a cocycle as above. Then the deformation𝜎_𝛾of𝜎is the unique bicovariantA𝛾-bimodule braiding map onE𝛾given by Proposition 2.9.

Proof. Since𝜎 is a bicovariantA-bimodule map fromE ⊗A E to itself, part (ii) of Proposition 4.4 implies that𝜎_𝛾is a bicovariantA𝛾-bimodule map from(E ⊗A E)𝛾 E𝛾⊗A𝛾E𝛾to itself. By Proposition 2.9, there exists a uniqueA𝛾-bimodule map𝜎⁰from E𝛾⊗A𝛾 E𝛾to itself such that𝜎⁰(𝜔⊗A𝛾 𝜂)=𝜂⊗A𝛾𝜔for all𝜔in0(E𝛾),𝜂in(E𝛾)₀.

Since0(E𝛾)=₀Eand(E𝛾)₀ =E₀, it is enough to prove that𝜎_𝛾(𝜔⊗_A𝛾𝜂)=𝜂⊗_A𝛾𝜔 for all𝜔in0E,𝜂inE₀.

We will need the concrete isomorphism betweenE𝛾⊗A𝛾E𝛾 and(E ⊗AE)𝛾defined in Proposition 4.5. Since𝜔is in0Eand𝜂is inE₀, this isomorphism maps the element 𝜔⊗A𝛾 𝜂to𝛾(1⊗_C𝜂₍₋₁₎)𝜔₍₀₎⊗A𝜂₍₀₎𝛾(𝜔₍₁₎⊗_C1). Then, by the definition of𝜎_𝛾, we compute the following:

𝜎_𝛾(𝜔⊗A𝛾𝜂)=𝜎(𝛾(1⊗_C𝜂₍₋₁₎)𝜔₍₀₎⊗A𝜂₍₀₎𝛾(𝜔₍₁₎⊗_C1))

=𝜎(𝜖(𝜂₍₋₁₎)𝜔₍₀₎⊗A 𝜂₍₀₎𝜖(𝜔₍₁₎))=𝜖(𝜂₍₋₁₎)𝜂₍₀₎⊗A𝜔₍₀₎𝜖(𝜔₍₁₎)

=𝛾(𝜂₍₋₁₎⊗_C1)𝜂₍₀₎⊗A𝜔₍₀₎𝛾(1⊗_C𝜔₍₁₎)=𝜂⊗A𝛾𝜔,

where, in the last step we have used Lemma 4.6.

Remark 4.8. Proposition 4.2, Proposition 4.4, Proposition 4.5 and Theorem 4.7 together imply that the categories ^A_AM_A^A and ^A_A^𝛾

𝛾

M^A_A^𝛾

𝛾 are isomorphic as braided monoidal categories. This was the content of Theorem 2.5 of [12]. The referee has pointed out that this is a special case of a much more generalized result of Bichon ([3, Theorem 6.1]) which says that if two Hopf algebras are monoidally equivalent, then the corresponding categories of right-right Yetter Drinfeld modules are also monoidally equivalent.

However, in Theorem 4.7, we have proved in addition that the braiding on^A_A^𝛾

𝛾

M^A_A^𝛾

𝛾 is precisely the Woronowicz braiding of Proposition 2.9.

Corollary 4.9. If the unique bicovariantA-bimodule braiding map𝜎for a bicovariant A-bimoduleEsatisfies the equation𝜎²=1, then the bicovariantA𝛾-bimodule braiding map𝜎_𝛾for the bicovariantA𝛾-bimoduleE𝛾 also satisfies𝜎²

𝛾 =1.

In particular, ifAis the commutative Hopf algebra of regular functions on a compact semisimple Lie group𝐺andEis its canonical space of one-forms, then the braiding map 𝜎_𝛾forE𝛾 satisfies𝜎_𝛾²=1.

Proof. By Theorem 4.7,𝜎_𝛾is the unique braiding map for the bicovariantA𝛾-bimodule E𝛾. Since, by our hypothesis,𝜎² =1, the deformed map𝜎_𝛾 also satisfies𝜎²

𝛾 =1 by part (ii) of Proposition 4.4.

Next, ifAis a commutative Hopf algebra as in the statement of the corollary andEis its canonical space of one-forms, then we know that the braiding map𝜎is just the flip map, i.e. for all𝑒₁, 𝑒₂inE,

𝜎(𝑒₁⊗A 𝑒₂)=𝑒₂⊗A𝑒₁,

and hence it satisfies𝜎² =1. Therefore, for every cocycle deformationE𝛾 ofE, the corresponding braiding map satisfies𝜎²

𝛾 =1.

4.2. Pseudo-Riemannian bi-invariant metrics onE𝛾

SupposeE is a bicovariantA-bimodule and E𝛾 be its cocycle deformation as above.

The goal of this subsection is to prove that a pseudo-Riemannian bi-invariant metric onEnaturally deforms to a pseudo-Riemannian bi-invariant metric onE𝛾.Since𝑔is a bicovariant (i.e, both left and right covariant) map from the bicovariant bimoduleE ⊗_AE to itself, then by Proposition 4.4, we have a rightA𝛾-linear bicovariant map𝑔_𝛾 from E𝛾⊗A𝛾 E𝛾to itself. We need to check the conditions (i) and (ii) of Definition 3.1 for the map𝑔_𝛾.

The proof of the equality 𝑔_𝛾 = 𝑔_𝛾 ◦𝜎_𝛾 is straightforward. However, checking condition (ii), i.e, verifying that the map𝑉_𝑔

𝛾 is an isomorphism onto its image needs some work. The root of the problem is that we do not yet know whetherE^∗=𝑉_𝑔(E).Our strategy to verify condition (ii) is the following: we show that the rightA-module𝑉_𝑔(E) is a bicovariant rightA-module (see Definition 4.10) in a natural way. Let us remark that since the map𝑔(hence𝑉_𝑔) is not leftA-linear,𝑉_𝑔(E)need not be a leftA-module.

Since bicovariant rightA-modules and bicovariant maps between them can be deformed (Proposition 4.13), the map𝑉_𝑔deforms to a rightA𝛾-linear isomorphism fromE𝛾 to (𝑉_𝑔(E))𝛾.Then in Theorem 4.15, we show that(𝑉_𝑔)𝛾 coincides with the map𝑉_𝑔

𝛾and 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).

Definition 4.10. Let𝑀 be a rightA-module, andΔ𝑀 :𝑀 → A ⊗_C𝑀 and𝑀Δ:𝑀 → 𝑀⊗_CAbeC-linear maps. We say that(𝑀 ,Δ𝑀,_𝑀Δ)is a bicovariant rightA-module if the triplet is an object of the category ^AM^A_A,i.e,

(i) (𝑀 ,Δ𝑀)is a leftA-comodule, (ii) (𝑀 ,_𝑀Δ)is a rightA-comodule, (iii) (id⊗_C𝑀Δ)Δ𝑀 =(Δ𝑀 ⊗_Cid)𝑀Δ, (iv) For any𝑎inAand𝑚in𝑀,

Δ𝑀(𝑚 𝑎)= Δ𝑀(𝑚)Δ(𝑎), _𝑀Δ(𝑚 𝑎)=𝑀Δ(𝑚)Δ(𝑎).

For the rest of the subsection,E will denote a bicovariantA-bimodule. Moreover, {𝜔_𝑖}𝑖will denote a basis of0Eand{𝜔^∗

𝑖}𝑖the dual basis, i.e,𝜔^∗

𝑖(𝜔_𝑗)=𝛿_{𝑖 𝑗}.

Let us recall that (2.1) implies the existence of elements𝑅_{𝑖 𝑗}inAsuch that

EΔ(𝜔_𝑖)=∑︁

𝑖 𝑗

𝜔_𝑗⊗_C𝑅_{𝑗 𝑖}. (4.8)

We want to show that𝑉_𝑔(E)is a bicovariant rightA-module. To this end, we recall that (Lemma 3.7)𝑉_𝑔(E)is a free rightA-module with basis{𝜔^∗

𝑖}𝑖.This allows us to make the following definition.

Definition 4.11. Let{𝜔_𝑖}𝑖and{𝜔^∗

𝑖}𝑖be as above and𝑔a bi-invariant pseudo-Riemannian metric on E. Then we can endow𝑉_𝑔(E) with a left-coaction Δ𝑉_𝑔( E) : 𝑉_𝑔(E) → A ⊗_C𝑉_𝑔(E) and a right-coaction 𝑉_𝑔( E)Δ : 𝑉_𝑔(E) → 𝑉_𝑔(E) ⊗_C A, defined by the formulas

Δ𝑉_𝑔( E)

∑︁

𝑖

𝜔^∗_𝑖𝑎_𝑖

!

=∑︁

𝑖

(1⊗_C𝜔^∗_𝑖)Δ(𝑎_𝑖), _𝑉

𝑔( E)Δ ∑︁

𝑖

𝜔^∗_𝑖𝑎_𝑖

!

=∑︁

𝑖 𝑗

(𝜔^∗

𝑗 ⊗_C𝑆(𝑅_{𝑖 𝑗}))Δ(𝑎_𝑖),

(4.9)

where the elements𝑅_{𝑖 𝑗}are as in (4.8).

Then we have the following result.

Proposition 4.12. The triplet(𝑉_𝑔(E),Δ𝑉_𝑔( E),_𝑉

𝑔( E)Δ)is a bicovariant rightA-module.

Moreover, the map𝑉_𝑔:E →𝑉_𝑔(E)is bicovariant, i.e, we have Δ𝑉_𝑔( E)(𝑉_𝑔(𝑒))=(id⊗_C𝑉_𝑔)ΔE(𝑒), _𝑉

𝑔( E)Δ(𝑉_𝑔(𝑒))=(𝑉_𝑔⊗_Cid)EΔ(𝑒). (4.10) Proof. The fact that(𝑉_𝑔(E),Δ𝑉_𝑔( E),_𝑉

𝑔( E)Δ)is a bicovariant rightA-module follows immediately from the definition of the mapsΔ𝑉_𝑔( E) and𝑉_𝑔( E)Δ.So we are left with proving (4.10). Let𝑒 ∈ E.Then there exist elements 𝑎_𝑖 inAsuch that 𝑒= Í

𝑖𝜔_𝑖𝑎_𝑖. Hence, by (3.1), we obtain

Δ𝑉_𝑔( E)(𝑉_𝑔(𝑒))= Δ𝑉_𝑔( E) 𝑉_𝑔

∑︁

𝑖

𝜔_𝑖𝑎_𝑖

! !

= Δ𝑉_𝑔( E)

∑︁

𝑖 𝑗

𝑔_{𝑖 𝑗}𝜔^∗

𝑗𝑎_𝑖

!

=∑︁

𝑖 𝑗

(1⊗_C𝑔_{𝑖 𝑗}𝜔^∗

𝑗)Δ(𝑎_𝑖)=∑︁

𝑖

( (id⊗_C𝑉_𝑔) (1⊗_C𝜔_𝑖))Δ(𝑎_𝑖)

=∑︁

𝑖

(id⊗_C𝑉_𝑔) (ΔE(𝜔_𝑖))Δ(𝑎_𝑖)

=∑︁

𝑖

(id⊗_C𝑉_𝑔)ΔE(𝜔_𝑖𝑎_𝑖)=(id⊗_C𝑉_𝑔)ΔE(𝑒).

This proves the first equation of (4.10).