First-order diﬀerential calculi over multi-braided quantum groups

A diﬀerential calculus of the ﬁrst order over multi-braided quantum groups is developed. In analogy with the standard theory, left/right-covariant and bicovariant diﬀerential structures are introduced and investigated. Furthermore, antipodally covariant calculi are studied. The concept of the *-structure on a multi-braided quantum group is formulated, and in particular the structure of left-covariant *-covariant calculi is analyzed. These structures naturally incorporate the idea of the quantum Lie algebra associated to a given multibraded quantum group, the space of left-invariant forms corresponding to the dual of the Lie algebra itself. A special attention is given to diﬀerential calculi covariant with respect to the action of the associated braid system. In particular it is shown that the left/right braided-covariance appears as a consequence of the left/right-covariance relative to the group action. Braided counterparts of all basic results of the standard theory are found.


Introduction
The basic theme of this study is the analysis of the first-order differential structures over multibraided quantum groups. Standard braided quantum groups are included as a special case into the theory of multi-braided quantum groups [3]. The difference between two types of braided quantum groups is in the behavior of the coproduct map. In the standard theory, the coproduct φ : A → A ⊗ A is interpretable as a morphism in a braided category generated by the basic algebra A and the associated braiding σ : A ⊗ A → A ⊗ A. In our generalized framework two standard pentagonal diagrams expressing compatibility between φ and σ are replaced by a single more general octagonal diagram. We refer to [4] for a fully diagrammatic generalized categorical formulation of multi-braided quantum groups.
It turns out that the lack of the functoriality of the coproduct map is 'measurable' by a second braid operator τ : A ⊗ A → A ⊗ A. Furthermore, two braid operators generate in a natural manner a generally infinite system of braid operators σ n : A ⊗ A → A ⊗ A, where n ∈ Z, which elegantly express twisting properties of all the maps appearing in the game. This explains our attribute multi-braided, for the structures we are dealing with.
Multi-braided quantum groups include various completely 'pointless' structures, overcoming in such a way an inherent geometrical inhomogeneity of standard quantum groups and braided quantum groups. This inhomogeneity is explicitly visible in geometrical situations in which 'diffeomorphisms' of quantum spaces appear. For example, in the theory of locally trivial quantum principal bundles over classical smooth manifolds [2] a natural correspondence between quantum G-bundles (where G is a standard compact quantum group) and ordinary G cl -bundles (over the same manifold) holds. Here G cl is the classical part of G.
The multi-braided formalism reduces to the standard braided quantum groups iff σ = τ , which means that all the operators σ n coincide with σ. This is also equivalent to the multiplicativity of the counit map.
In the formalization of the concept of a first-order calculus, we shall follow [9]: If the algebra A represents a quantum space X, then every first-order calculus over X will be represented by an A-bimodule Γ, playing the role of the 1-forms on X, together with a standard derivation d : A → Γ playing the role of the differential. Such a formalization reflects noncommutativegeometric [1] philosophy, according to which the concept of a differential form should be the starting point for a foundation of the quantum differential calculus.
The paper is organized as follows. In Section 2 we first study differential calculi over a quantum space X, compatible in the appropriate sense with a single braid operator σ : A ⊗ A → A ⊗ A. In this context, left/right, and bi-σ-covariant differential structures are distinguished. The notion of left σ-covariance requires a natural extendability of σ to a flip-over operator σ l : Γ ⊗ A → A ⊗ Γ. Similarly, right σ-covariance requires extendability of σ to a flip-over operator σ r : A⊗Γ → Γ⊗A. Finally, the concept of bi-σ-covariance is simply a symbiosis of the previous two.
We shall then briefly analyze general situations in which the calculus is covariant relative to a given braid system T operating in A.
At the end of Section 2 we begin the study of differential structures over multi-braided quantum groups. We shall prove that if A and σ are associated to a multi-braided quantum group G then the left/right σ-covariance implies the left/right σ n -covariance, for each n ∈ Z. We shall also analyze interrelations between all possible flip-over operators and maps determining the group structure.
All considerations with braid operators can be performed at the language of braid and tangle diagrams, as in the framework of braided categories. The unique additional moment is that crossings of diagrams should be appropriately labeled, since we are in a multi-braided situation. At the diagramatic level, many of the proofs become very simple. However, in this study the considerations will be performed in the standard-algebraic way. For the reasons of completeness, all the proofs are included in the paper. Through sections 3-5 we shall exclusively deal with a given multi-braided quantum group G. In Section 3 we begin with formulations of braided counterparts of concepts of the left, right and bi-covariance [9]. All these notions are intrinsically related to the problematics of generalizing the concept of Lie algebra. As in the standard theory [9], the notion of left covariance will be formulated by requiring a possibility of defining a left action Γ : Γ → A ⊗ Γ of G on Γ. Similarly, right covariance will be characterized by a possibility of a right action Γ : Γ → Γ ⊗ A. The notion of bicovariance is a symbiosis of the previous two. If we interpret the left-invariant elements of a left-covariant calculus as the dual of the associated quantum Lie algebra, then the bicovariance essentially adds the requirement of the existence of the adjont action of the group on the corresponding Lie algebra.
Our attention will be then confined to the left-covariant structures. As we shall see, left covariance implies left σ-covariance and, consequently, left σ n -covariance, for each n ∈ Z. The corresponding flip over operators σ l n naturally describe twisting properties of the left action Γ . Besides the study of properties of maps Γ and σ l n , and their interrelations, we shall also analyze the internal structure of left-covariant calculi. It turns out that the situation is more or less the same as in the standard theory [9]. As a left/right A-module, every left-covariant Γ is free and can be invariantly decomposed where Γ inv is the space of left-invariant elements of Γ (as mentioned above, the dual of the quantum Lie algebra). We shall also prove a braided generalization of the structure theorem [9] by establishing a natural correspondence between (classes of isomorphic) left-covariant Γ and certain lineals R ⊆ ker( ).
However, a full analogy with [9] breaks, because R is generally not a right ideal in A, but a right ideal in a simplified [3] algebra A 0 obtained from A by an appropriate change of the product. The lineal R should also be left-invariant with respect to the action of τ .
Concerning the concept of the right covariance, it is in some sense symmetric to that of the left covariance. For this reason, we shall not repeat completely analogous considerations for right-covariant calculi. The most important properties of them are collected, without proofs, in Appendix A. In particular, right covariance implies right σ n -covariance, for each n ∈ Z.
The study of bicovariant differential structures is the topic of Section 4. In the bicovariant case the action maps Γ and Γ , as well as the flip-over maps σ l n , σ r n are mutually compatible, in a natural manner.
We shall characterize bicovariance in terms of the corresponding right A 0 -ideals R. It turns out that the calculus is bicovariant if and only if R satisfies two additional conditions. The first one correspond to the adjoint invariance in the standard theory [9]. In its formulation, a braided analogue of the adjoint action of G on itself appears naturally. For this reason, the most important properties of this map are collected in Appendix B. The second additional condition for R, trivial in the standard theory, consists in its right τ -invariance.
In Section 5, we shall analyze differential structures which are covariant with respect to the antipodal map κ : A → A. Such structures will be called κ-covariant. As we shall see, a symbiosis of κ and left covariance is equivalent to bicovariance.
In Section 6 we shall first introduce the concept of a *-structure on a multi-braided quantum group. Then, we pass to the study of *-covariant calculi, in the context of multi-braided quantum groups.
Besides other results we shall obtain a characterization of *-covariant left covariant structures Γ, in terms of the corresponding right A 0 -ideal R. It turns out that a left-covariant calculus is *-covariant iff κ(R) * ⊆ R, which is identical as in the standard theory [9].
In this paper only the abstract theory will be presented. Concrete examples will be included in the next part of the study, after developing a higher-order differential calculus. This will include differential structures over already considered groups, as well as new examples of 'differential' multi-braided quantum groups coming from the developed theory. Finally, let us mention that we shall assume here trivial braiding properties of the differential d : A → Γ. Our philosophy is that the non-trivial braidings involving the differential map should be interpreted as an extra structure given over the whole differential calculus.

The concept of braided covariance
Let A be a complex unital associative algebra. Let us denote by m : A⊗A → A the multiplication in A. The algebra A will be interpreted as consisting of 'smooth functions' over some quantum space X.
By definition, a first order differential calculus over X is a unital A-bimodule Γ, equipped with a linear map d : A → Γ satisfying the Leibniz rule and such that ι l Γ = m l Γ (id⊗d) : A⊗A → Γ is surjective. Here, m l Γ : A⊗Γ → Γ and m r Γ : Γ⊗A → Γ are the left and the right A-module structures of Γ.
Let us observe that d(1) = 0, and that the surjectivity of ι l Γ is equivalent to the surjectivity of ι r Γ : A ⊗ A → Γ, which is given by ι r Γ = m r Γ (d ⊗ id). Now, let us assume that X is a braided quantum space. In other words, we have in addition a bijective braid operator σ : A ⊗ A → A ⊗ A such that the following identities hold: The operator σ naturally induces a structure of an associative algebra on A ⊗ A with the unit element 1 ⊗ 1 ∈ A ⊗ A. Explicitly, the product is given by We are going to analyze natural compatibility conditions between Γ and σ.
Definition 2.1. A first-order differential calculus Γ over X is called left σ-covariant iff there exists a linear operator σ l : Γ ⊗ A → A ⊗ Γ satisfying Similarly, we say that Γ is right σ-covariant iff there exists a linear operator σ r : Finally, Γ is called bi-σ-covariant iff it is both right and left σ-covariant.
The idea beyond this definition is that 'twistings' between elements from A and Γ are performable 'term by term' such that twistings between the symbol d and elements from A are trivial.
It is easy to see that maps σ l and σ r , if they exist, are uniquely determined by (2.4) and (2.5) respectively.
Requirement (2.4) can be replaced by the equivalent Similarly, the operator σ r appearing in the context of the right σ-covariance can be characterized by In the following proposition, the most important general properties of σ-covariant structures are collected.
The map σ l is surjective. Its kernel is an A-subbimodule of Γ ⊗ A. Furthermore, we have 14) The map σ r is surjective, its kernel is an A-subbimodule of A ⊗ Γ and the following identities hold Similarly, we find The first term on the right-hand side of the above equality is equal to while the second term reads Combining the last two expressions we conclude We prove (2.9). Direct transformations give To prove the surjectivity of σ l , it is sufficient to check that the elements of the form a ⊗ bd(q) belong to im(σ l ). Let us define Using (2.8) and (2.10) we obtain The fact that ker(σ l ) is an A-subbimodule of Γ ⊗ A directly follows from equalities (2.10) and (2.12).
In such a way we have shown (i). The right σ-covariance case can be treated in a similar manner. Finally, if Γ is bi-σ-covariant then It is possible to construct 'pathological' examples in which maps σ l or σ r are not injective. However, besides certain technical complications such a possibility gives nothing essentially new. For this reason, we shall assume from this moment that every left/right σ-covariant calculus we are dealing with possesses bijective flip-over operator σ l or σ r . Modulo this assumption, left σ-covariance and right σ −1 -covariance are equivalent properties. In other words, Now, we shall generalize the previous consideration to situations in which, instead of one, a system of mutually compatible braided quantum space [3] structures on X appears. Definition 2.3. Let us assume that A is equipped with a braid system T . Then we shall say that X is a T -braided quantum space.
As explained in [3]-Appendix, every braid system T can be naturally completed. The completed system T * is defined as the minimal extension of T , invariant under ternar operations of the form δ = αβ −1 γ. Explicitly, T * is the union of systems T n , where T 0 = T and T n+1 is obtained from T n by applying the above mentioned operations.
Proposition 2.5. Let X be a T -braided quantum space and Γ a first-order calculus over X.
Proof. Let us assume that Γ is left T -covariant. Then, for each α, β, γ ∈ T . This means that Γ is left αβ −1 γ-covariant and (2.20) holds for the braidings from T . Now, we can proceed inductively and conclude that Γ is left T n -covariant for each n ∈ N, for each α, β, γ ∈ T . This implies that Γ is right T * -covariant and that (2.22) holds for each α, β, γ ∈ T * . Identities (2.21) and (2.23)-(2.24) can be derived in essentially the same manner as it is done in the proof of Proposition 2.2, in the case of a single flip-over operator.
From this moment, as well as through the next three sections we shall deal exclusively with braided quantum groups, in the sense of [3]. Let G be such a group, represented by A. We shall denote by φ : A → A ⊗ A the coproduct map, and by : A → C and κ : A → A the counit and the antipode map respectively. Let σ : A ⊗ A → A ⊗ A be the intrinsic braid operator.
As explained in [3], twisting properties of the coproduct and the antipode are not properly expressible in terms a single braid operator σ. This is the place where a 'secondary' braid operator naturally enters the game. Explicitly, it is given by The operators {σ, τ } form a braid system, and the completion F = {σ, τ } * is consisting of maps of the form Moreover, the following twisting properties hold for each n, m ∈ Z.
(ii) Similarly, if Γ is right σ-covariant then it is also right F-covariant and We also have for each n, m ∈ Z.
Proof. Let us assume left σ-covariance of Γ, and consider a map ξ : Γ ⊗ A → A ⊗ Γ determined by the right hand side of (2.26). Direct transformations give Consequently, Γ is left τ -covariant and ξ = τ l . According to Proposition 2.5 the calculus is left F-covariant. Let us denote by ψ a map determined by the right hand side of (2.27). We have then Let us prove the twisting property (2.29). Using the standard braid relations we obtain Case (ii), when Γ is right σ-covariant, can be treated in a similar way.
Finally, let us describe twisting relations involving the antipode κ and a σ-covariant first-order calculus Γ.
Proof. Let us assume that Γ is left σ-covariant. We have If the calculus is right F-covariant then The structure of left-covariant calculi We pass to definitions of first order differential structures which are covariant with respect to the comultiplication map φ : A → A ⊗ A.
The map Γ is called the right action of G on Γ. It is uniquely determined by the above condition.
The map Γ is uniquely determined by this condition.
Definition 3.3. We shall say that the calculus Γ is bicovariant, iff it is both left and rightcovariant.
The above definitions naturally formulate braided generalizations of standard concepts of right/left and bi-covariance in the standard theory [9]. Throughout the rest of the section, we shall consider left-covariant differential structures.
Proof. Identity (3.3) is a direct consequence of (3.2). To prove (3.4), we start from (3.2) and apply elementary properties of the product and the coproduct maps: It is worth noticing that which also characterizes the map Γ . The following proposition shows that Γ gives a left Acomodule structure on Γ.
Proposition 3.5. We have Proof. Applying (3.2) and performing further elementary transormations with the counit we obtain which completes the proof. We have used the 'octagonal' compatibility property between φ and σ.
As we shall now see, every left-covariant differential calculus Γ is left σ-covariant. According to Proposition 2.6, this means that Γ is left F-covariant, too.
We shall prove that ξ satisfies a characteristic property for the flip-over operator σ l . A direct computation gives The last term in the above sequence of transformations can be further written as follows: Thus, Γ is left σ-covariant, χ = σ l and diagram (3.8) is commutative. According to (i)-Proposition 2.6 the calculus Γ is automatically left F-covariant.
The operator σ l figures in the right multiplicativity law for the left action map.
is commutative.
Proof. According to Proposition 3.5 and diagram (3.8), Our next proposition describes twisting properties of the left action map, with respect to the braid system F.
for each n, m ∈ Z. In particular, it follows that Proof. Using (3.2) and the main properties of F we obtain We pass to the study of the internal structure of left-covariant calculi. For a given Γ, let Γ inv be the space of left-invariant elements of Γ. In other words Let P : Γ → Γ be a linear map defined by We are going to show that P projects Γ onto Γ inv . Evidently, the elements of Γ inv are P -invariant.
Lemma 3.9. We have P ι l Γ = ( ⊗ P d)σ −1 τ (3.14) Proof. Applying (3.2)-(3.3), (3.13) and performing standard braided transformations we obtain Now, it follows that P (Γ) ⊆ Γ inv . Indeed, according to the previous lemma, it is sufficient to check that P d(A) ⊆ Γ inv . We compute It is easy to see, by the use of (3.10), that the flip-over operators σ l n map Γ inv ⊗ A onto A ⊗ Γ inv . Moreover, the corresponding restrictions mutually coincide. Proof. Applying the appropriate twisting properties we obtain We are going to prove that the space Γ is naturally isomorphic to A⊗Γ inv , as a left A-module.
On the other hand P (aϑ) = (a)ϑ for all a ∈ A and ϑ ∈ Γ inv . Using this and (3.4) we obtain Consequently, the two maps are mutually inverse left A-module isomorphisms.
The above proposition allows us to identify Γ ↔ A ⊗ Γ inv . In terms of this identification, the following correspondences hold The following technical lemma will be useful in some further computations.
Lemma 3.12. We have According to (3.15), this is further viewable as The first term in the above difference is transformed further Concerning the second term, Let R be the intersection of spaces ker(π) and ker( ). As follows directly from the previous lemma, the space R is a right ideal in the algebra A 0 , which coincides as a vector space with A, but which is endowed with the product m 0 = mτ −1 σ, as discussed in [3]-Appendix. According to (3.15), we have (3.20) The map π induces the isomorphism It is easy to see that the map • : Γ inv ⊗ A → Γ inv given by defines a right A 0 -module structure on the space Γ inv . In the above formula it is assumed that a ∈ ker( ), while b is arbitrary. In terms of the identification Γ ↔ A ⊗ Γ inv the right A-module structure is given by where σ * : Γ inv ⊗ A → A ⊗ Γ inv is the common left-invariant part of all operators σ l n . We shall now prove that Γ is trivial as a right A-module.
Proposition 3.13. The multiplication map is an isomorphism of right A-modules. Its inverse is given by

Proof. Clearly, (3.24) is a right A-module homomorphism. A direct computation gives
The above computations are performed in the spaces A ⊗ ker( ) and ker( ) ⊗ A respectively.
In the framework of the identification Γ ↔ Γ inv ⊗ A, the following correspondences hold: These correspondences follow from (3.24)-(3.25), performing simple algebraic transformations. We are ready to present a braided counterpart of the reconstruction theorem [9] of the standard theory. As we have seen, every left-covariant calculus Γ is completely determined by the corresponding R. The following proposition shows that conversely, every right A 0 -ideal which satisfies (3.20) naturally gives rise to a first-order left-covariant calculus.
Proposition 3.14. Let R ⊆ ker( ) be an arbitrary τ -invariant right A 0 -ideal. Let us define spaces Γ inv and Γ, together with maps • : Γ inv ⊗ A → Γ inv and π : A → Γ inv , as well as σ * : Γ inv ⊗ A → A ⊗ Γ inv by the equalities Then, m l Γ and m r Γ determine a structure of a unital A-bimodule on Γ. Moreover, Γ is a left-covariant first-order differential calculus over G, with the differential and the left action coinciding with the introduced d and Γ respectively.
Proof. It is clear that m l Γ determines a left A-module structure on Γ. Let us prove that m r Γ determines a right A-module structure. We have Here, we have used (3.22)-(3.23), and identities which follow from (3.15) and (3.22). The maps m l Γ and m r Γ commute, because It is easy to see that the bimodule Γ is unital. According to Lemma 3.12 and equation (3.22), Using this, equations (3.16) and (3.18) and (3.23) we obtain To complete the proof, let us observe that (3.16) implies that Γ given by (3.17) is indeed the left action.

Bicovariant calculi
In this section we shall study bicovariant differential calculi Γ over G. As in the standard theory [9] the right action Γ and the left action Γ are mutually compatible.
is commutative.

Proof. Applying (3.2) and (A.1) we obtain
As a simple consequence of (4.1) we find that the spaces Γ inv and inv Γ are right/left-invariant respectively. The following proposition characterizes the corresponding restrictions of Γ and Γ . Let ad : A → A ⊗ A be the adjoint action of G on itself, as defined in Appendix B.

Proposition 4.2. The following identities hold
Proof. We compute Completely similarly, We pass to the the analysis of the specific twisting properties of the left and the right action maps.

Proposition 4.3. The following equalities hold
As a simple consequence of the previous proposition we find The following proposition describes the corresponding restriction twistings.
Proposition 4.4. The following identities hold Proof. Using standard twisting transformations we obtain The second identity can be derived in a similar manner.
Let R ⊆ ker( ) be the right A 0 -ideal which canonically corresponds to Γ. In the following proposition we have characterized bicovariance in terms of R. (ii) Conversely, if R ⊆ ker( ) corresponding to a left-covariant calculus Γ is ad-invariant, then the calculus Γ is bicovariant. Moreover, in terms of the identification Γ = Γ inv ⊗ A, the right action Γ : Γ → Γ ⊗ A is given by where the map : Γ inv → Γ inv ⊗ A is given by Proof. The first statement of the proposition is a direct consequence of (4.2) and (4.8). Concerning the second part, it is sufficient to check that the map ξ given by the right-hand side of (4.12) satisfies (A.2)-(A.3). Using the structuralization Γ = Γ inv ⊗ A as well as equalities (3.26) and (4.13) we obtain Consequently, Γ is bicovariant and ξ = Γ .

Antipodally covariant calculi
In this Section we shall consider differential structures covariant relative to the antipode map.
Definition 5.1. A first-order calculus Γ is called κ-covariant iff the following equivalence holds Let us assume that Γ is κ-covariant. Then the formula consistently and uniquely determines a bijective map κ : Γ → Γ. It follows that Let us analyze properties of Γ, in the case when it is also σ-covariant.
Symmetrically, assuming the right F-covariance of Γ we get Finally, Now, we shall analyze interrelations between κ-covariance and bicovariance.
Moreover, the diagram is commutative. Here, the vertical arrows are the corresponding double-sided actions and products.
Proof. Let us assume that Γ is left-covariant and κ-covariant, and let us consider a map ξ : Γ → Γ ⊗ A defined by It turns out that ξ is the right action for Γ. Indeed, Consequently Γ is right-covariant with ξ = Γ and (5.8) holds. Similarly, if Γ is κ-covariant and right-covariant then a map ξ : Γ → A ⊗ Γ given by This implies that Γ is also left-covariant with ξ = Γ , and equality (5.9) holds. Finally, let us assume that Γ is bicovariant and consider a map κ : Γ → Γ defined by diagram (5.10). Then a straightforward computation shows that equality (5.1) holds, and that κ is bijective. In other words, Γ is κ-covariant and (5.10) holds by construction.

Proof. A direct computation gives
15) immediately follow from (5.12), definition of spaces R and K and the fact that κ 0 = .
Let us check (5.13)-(5.14). On the space ker( ) ⊗ A the following equalities hold Similarly, in the framework of the space A ⊗ ker( ) we can write In terms of the bimodule structuralizations Γ ↔ A ⊗ Γ inv and inv Γ ⊗ A ↔ Γ the operator κ has a particularly simple form.
Proposition 5.5. The following identities hold: Proof. We compute On * -covariant differential structures Let us consider a quantum space X, represented by a unital algebra A and assume that X is T -braided. Let us also assume that A is equipped with a *-structure such that ( * ⊗ * )α = ψα −1 ψ( * ⊗ * ) (6.1) for each α ∈ T . Here, ψ : A ⊗ A → A ⊗ A is the standard transposition. It is easy to see that then (6.1) holds for every α ∈ T * . It is worth noticing that the operators also form a braid system over A.
Now switch to multi-braided quantum groups G. Let us assume that the *-structure on A satisfies φ * = ( * ⊗ * )ψσ −1 φ (6.4) Definition 6.2. We shall say that the antimultiplicative *-involution on A satisfying the above equality is a *-structure on a braided quantum group G.
Proof. If Γ is *-covariant then (6.11), together with the definition of R, implies that R is * κ-invariant.
Let us observe that the above proof is the same as in the standard theory [9] (braidings are not included). A similar characterization of *-covariance holds for right-covariant structures.
As already mentioned at the beginning of this study, the whole theory of multibraided structures admits a natural axiomatic formulation, via some simple and elegant diagrammatic language [4]. New interesting algebraic structures are naturally includable in such a diagrammatic frameword, among others are Clifford algebras, spinors, Dirac operators, and their braided generalizations [5,8,7].
The quantum analog of the Lie algebra is recovered as lie(G, Γ) = Γ * inv . The whole analysis can be performed in terms of this dual object. However, we find the calculus and differential forms picture more suitable for the quantum context, as in the standard (non-braided) formulation [9]. It is important to emphasize a contextual nature of the notion of the associated Lie algebra, as it depends on the bimodule Γ chosen to represent the calculus. The phenomenon appears already in the classical (commutative) contexts, where we can chose non-classical R for example the jet bundle ideals at the neutral element R = ker( ) k for k ≥ 3. Such structures will be included, with other examples, in a sequel to this paper.

A Right-covariant calculi
Let Γ be a right-covariant first order differential calculus over a braided quantum group G. The corresponding right action Γ : Γ → Γ ⊗ A can be also characterized by The right action map satisfies equalities Every right-covariant calculus is automatically right F-covariant. In particular, the flip-over operator σ r : A ⊗ Γ → Γ ⊗ A is determined by the diagram The operator σ r expresses the left multiplicativity of Γ , via the diagram The following twisting properties hold: Let inv Γ be the set of all right-invariant elements of Γ. Then the map Q : Γ → Γ defined by projects Γ onto inv Γ. Moreover, The composition is surjective. All flip-over operators σ r n map A ⊗ inv Γ onto inv Γ ⊗ A. Their restrictions on this space are given by σ r n (id ⊗ ς) = (ς ⊗ id)τ (A.12) for each n ∈ Z. As a right A-module, the space Γ is naturally identificable with inv Γ ⊗ A. The isomorphism is induced by the multiplication map m r Γ . Moreover, In terms of the structuralization Γ ↔ inv Γ ⊗ A, the following correspondences hold Here, * σ : A⊗ inv Γ → inv Γ⊗A is the restriction of the operators σ r n , and the map • : A⊗ inv Γ → inv Γ is given by The space Γ is also trivial as a left A-module. The corresponding isomorphism Γ ↔ A ⊗ inv Γ is induced by the product map, and explicitly In terms of the structuralization Γ = A ⊗ inv Γ the following correspondences hold: The structure of every right-covariant calculus Γ is completely determined by the space K = ker(ς) ∩ ker( ). This space is a left A 0 -ideal satisfying In other words, ad is a counital and coassociative map.
Proof. We compute Computation of the left-hand side of (B.3) gives Further useful identities are Lemma B.2. We have Proof. Let us check the second identity. A direct computation gives Finally, let us study the twisting properties of the adjoint action.