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First-order differential calculi over multi-braided quantum groups | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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First-order differential calculi over multi-braided quantum groups

Micho ÐURÐEVICH*

Instituto de Matemáticas, UNAM, Area de la Investigacion Científica, Circuito Exterior, Ciudad Universitaria, México DF, CP 04510, Mexico

*Corresponding Author:
Micho ÐURÐEVICH
Instituto de Matemáticas,
UNAM, Area de la Investigacion Científica,
Circuito Exterior, Ciudad Universitaria,
México DF, CP 04510, Mexico
E-mail: [email protected]

Received date: May 16, 2008 Accepted Date: December 07, 2008

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Abstract

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 equation is interpretable as a morphism in a braided category generated by the basic algebra A and the associated braiding equation In our generalized framework two standard pentagonal diagrams expressing compatibility between  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 equation Furthermore, two braid operators generate in a natural manner a generally infinite system of braid operators equation 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 Gcl-bundles (over the same manifold) holds. Here Gcl 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 σn 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 equation 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 equation Similarly, right σ-covariance requires extendability of σ to a flip-over operator equation 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 σn-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 equation Similarly, right covariance will be characterized by a possibility of a right action equation 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 σn-covariance, for each n ∈ Z. The corresponding flip over operators equation naturally describe twisting properties of the left action equation. Besides the study of properties of maps equation and equation , 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

equation

where Γinv 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(ε).

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 A0 obtained from A by an appropriate change of the product. The lineal R should also be left-invariant with respect to the action of  .

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 A0 obtained from A by an appropriate change of the product. The lineal R should also be left-invariant with respect to the action of  .

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 σn-covariance, for each n ∈ Z.The study of bicovariant differential structures is the topic of Section 4. In the bicovariant case the action maps equation and equation, as well as the flip-over maps equation are mutually compatible, in a natural manner.

We shall characterize bicovariance in terms of the corresponding right A0-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 k: A → A. Such structures will be called k-covariant. As we shall see, a symbiosis of k 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 A0-ideal R. It turns out that a left-covariant calculus is *-covariant iff equation 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 equation 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

equation

and such that equation is surjective. Here, equation are the left and the right A-module structures of Γ.

Let us observe that d(1) = 0, and that the surjectivity of equation is equivalent to the surjectivity of equation, which is given by equation

Now, let us assume that X is a braided quantum space. In other words, we have in addition a bijective braid operator equation such that the following identities hold:

equation

The operator σ naturally induces a structure of an associative algebra on equation with the unit element equation. Explicitly, the product is given by

equation

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 equation satisfying

equation

Similarly, we say that Γ is right σ-covariant iff there exists a linear operatorequation such that

equation

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 σl and σr, if they exist, are uniquely determined by (2.4) and (2.5) respectively.

Requirement (2.4) can be replaced by the equivalent

equation

Similarly, the operator σr appearing in the context of the right σ-covariance can be characterized by

equation

In the following proposition, the most important general properties of σ-covariant structures are collected.

Proposition 2.2. (i) If Γ is a left σ-covariant calculus, then

equation

The map σl is surjective. Its kernel is an A-subbimodule of equation. Furthermore, we have

equation

(ii) Similarly, if Γ is right σ-covariant then

equation

The map σr is surjective, its kernel is an A-subbimodule ofequation and the following identities hold

equation

(iii) Finally, if Γ is bi-σ-covariant then

equation

Proof. Let us assume that Γ is left σ-covariant. Identities (2.8) are obvious. Let us check (2.10)–(2.12). Using (2.4) we obtain

equation

Furthermore,

equation

Similarly, we find

equation

The first term on the right-hand side of the above equality is equal to

equation

while the second term reads

equation

Combining the last two expressions we conclude

equation

We prove (2.9). Direct transformations give

equation

To prove the surjectivity of σl, it is sufficient to check that the elements of the form equation belong to im(σl). Let us define

equation

Using (2.8) and (2.10) we obtain

equation

The fact that ker(σl) is an A-subbimodule of equation 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

equation

It is possible to construct ‘pathological’ examples in which maps σl or σr 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 σl or σr. Modulo this assumption, left σ-covariance and right σ-1-covariance are equivalent properties. In other words,

equation

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.

Definition 2.4. A first order calculus Γ over a T -braided quantum space X is called left/right/bi T -covariant iff it is left/right/bi y-covariant for each braiding γ ∈ T .

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 = αβa-1 γ . Explicitly, T + is the union of systems Tn, where T0 = T and Tn+1 is obtained from Tn 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.

(i) If Γ is left T -covariant then it is also left T +-covariant. We have

equation

for each α; β; γ ∈ T +.

(ii) Similarly, if Γ is right T -covariant then it is right T +-covariant, too. We have

equation

for each α; β; γ ∈ T +.

(iii) If Γ is bi-T -covariant, it is consequently also bi-T +-covariant and

equation

for each α; β; γ ∈ T +.

Proof. Let us assume that Γ is left T -covariant. Then,

equation

for each α; β; γ ∈ T . This means that Γ is left αβa-1 γ -covariant and (2.20) holds for the braidings from T . Now, we can proceed inductively and conclude that Γ is left Tn-covariant for each n ∈ N, and that (2.20) holds on T +.

Similarly, if Γ is right T -covariant then

equation

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 equation the coproduct map, and by ε : A → C and k: A → A the counit and the antipode map respectively. Let equation 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

equation

The operators fσ; g form a braid system, and the completion F = {σ; g}+ is consisting of maps of the form

equation

where n ∈ Z.

Proposition 2.6. (i) If Γ is left σ-covariant then it is also left F-covariant and

equation

Moreover, the following twisting properties hold

equation

for each n;m ∈ Z.

(ii) Similarly, if Γ is right σ-covariant then it is also right F-covariant and

equation

We also have

equation

for each n;m ∈ Z.

Proof. Let us assume left σ-covariance of Γ, and consider a map equation determined by the right hand side of (2.26). Direct transformations give

equation

Consequently, Γ is left τ-covariant and ξ = τl. According to Proposition 2.5 the calculus is left F-covariant.

Let us denote by a map determined by the right hand side of (2.27). We have then

equation

Consequently, Γ is right τ-1-covariant and ψ= (τ-1)r = ( l)τ-1.

Let us prove the twisting property (2.29). Using the standard braid relations we obtain

equation

Case (ii), when Γ is right σ-covariant, can be treated in a similar way.

Finally, let us describe twisting relations involving the antipode k and a σ-covariant first-order calculus Γ.

Proposition 2.7. If Γ is left σ-covariant then

equation

Similarly, if Γ is right F-covariant then

equation

for each n ∈ Z.

Proof. Let us assume that Γ is left σ-covariant. We have

equation

If the calculus is right F-covariant then

equation

The structure of left-covariant calculi

We pass to definitions of first order differential structures which are covariant with respect to the comultiplication map equation

Definition 3.1. A first-order calculus Γ over G is called right-covariant iff there exists a linear map equation such that

equation

The map equation is called the right action of G on Γ. It is uniquely determined by the above condition.

Definition 3.2. The calculus Γ is called left-covariant iff there exists a left action mapequation equation satisfying

equation

The map is uniquely determined by this condition.

Definition 3.3. We shall say that the calculus Γ is bicovariant, iff it is both left and rightcovarian

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..

Proposition 3.4. We have

equation

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:

equation

It is worth noticing that

equation

which also characterizes the map equation. The following proposition shows that equation gives a left comodule structure on Γ.

Proposition 3.5. We have

equation

Proof. Applying (3.2) and performing further elementary transormations with the counit we obtain

equation

Furthermore,

equation

which completes the proof. We have used the ‘octagonal’ compatibility property between  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.

Proposition 3.6. (i) The calculus Γ is, being left-covariant, also left F-covariant.

(ii) The diagram

equation

is commutative.

Proof. Let equationbe a map determined by equation

We shall prove that ξ satisfies a characteristic property for the flip-over operator σl. A direct computation gives

equation

where we have introduced equation

The last term in the above sequence of transformations can be further written as follows:

equation

Thus, Γ is left σ-covariant,  = σl and diagram (3.8) is commutative. According to (i) Proposition 2.6 the calculus Γ is automatically left F-covariant.

The operator σl figures in the right multiplicativity law for the left action map.

Proposition 3.7. The diagram

equation

is commutative.

Proof. According to Proposition 3.5 and diagram (3.8),

equation

Our next proposition describes twisting properties of the left action map, with respect to the braid system F.

Proposition 3.8. We have

equation

for each n;m ∈ Z. In particular, it follows that

equation

Proof. Using (3.2) and the main properties of F we obtain

equation

We pass to the study of the internal structure of left-covariant calculi. For a given Γ, let Γinv be the space of left-invariant elements of Γ. In other words

equation

Let P : Γ → Γ be a linear map defined by

equation

We are going to show that P projects Γ onto Γinv. Evidently, the elements of Γinv are P-invariant.

Lemma 3.9. We have

equation

Proof. Applying (3.2)–(3.3), (3.13) and performing standard braided transformations we obtain

equation

Now, it follows that P(Γ) ⊆ Γinv. Indeed, according to the previous lemma, it is sufficient to check that Pd(A) ⊆ Γinv. We compute

equation

Consequently, P projects Γ onto Γinv and the composition

equation

is surjective.

It is easy to see, by the use of (3.10), that the flip-over operators equation map equation onto equation. Moreover, the corresponding restrictions mutually coincide.

Lemma 3.10. We have

equation

for each n ∈ Z.

Proof. Applying the appropriate twisting properties we obtain

equation

We are going to prove that the space Γ is naturally isomorphic to equationv, as a left A-module.

Proposition 3.11. The equation is an isomorphism of left A-modules. Its inverse is given by equation

Proof. The map equation is evidently a left A-module homomorphism. Let us check thatequation are mutually inverse maps. Using (3.6)–(3.7) and (3.13) we obtain

equation

On the other hand P(aϑ) = ε(a)ϑ for all a ∈ A and ϑ 2 Γinv. Using this and (3.4) we obtain

equation

Consequently, the two maps are mutually inverse left A-module isomorphisms.

The above proposition allows us to identify equation In terms of this identification, the following correspondences hold

equation

The following technical lemma will be useful in some further computations.

Lemma 3.12. We have

equation

Proof. We compute

equation

According to (3.15), this is further viewable as

equation

The first term in the above difference is transformed further

equation

Concerning the second term,

equation

Let R be the intersection of spaces ker(π) and ker(ε). As follows directly from the previous lemma, the space R is a right ideal in the algebra A0, which coincides as a vector space with A, but which is endowed with the product m0 = mτ-1σ, as discussed in [3]-Appendix. According to (3.15), we have

equation

The map π induces the isomorphism

equation

It is easy to see that the map equation given by

equation

defines a right A0-module structure on the space Γinv. In the above formula it is assumed that a ∈ ker(ε), while b is arbitrary.

terms of the identification equation the right A-module structure is given by

equation

where equation is the common left-invariant part of all operators equation We shall now prove that Γ is trivial as a right A-module.

Proposition 3.13. The multiplication map

equation

is an isomorphism of right A-modules. Its inverse is given by

equation

Proof. Clearly, (3.24) is a right A-module homomorphism. A direct computation gives

equation

Furthermore,

equation

The above computations are performed in the spaces equation respectively.

In the framework of the identification equation the following correspondences hold:

equation

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 A0-ideal which satisfies (3.20) naturally gives rise to a first-order left-covariant calculus.

Proposition 3.14. Let R ⊆ ker(ε) be an arbitrary  -invariant right A0-ideal. Let us define spaces Γinv and Γ, together with mapsequationequation by the equalities

equation

Finally, define the maps

equation

by equalities (3.17)–(3.18) and (3.23), respectively.

Then, equation and equation 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 equation respectively.

Proof. It is clear that equation determines a left A-module structure on Γ. Let us prove thatequation determines a right A-module structure. We have

equation

Here, we have used (3.22)–(3.23), and identities

equation

which follow from (3.15) and (3.22).

The maps equation and equation commute, because

equation

It is easy to see that the bimodule Γ is unital.

According to Lemma 3.1∈ And equation (3.22),

equation

Using this, equations (3.16) and (3.18) and (3.23) we obtain

equation

To complete the proof, let us observe that (3.16) implies that equation 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 equation and the left action equation are mutually compatible.

Proposition 4.1. The diagram

equation

is commutative.

Proof. Applying (3.2) and (A.1) we obtain

equation

As a simple consequence of (4.1) we find that the spaces Γinv and invΓ are right/left-invariant respectively. The following proposition characterizes the corresponding restrictions of equation and equation. Let ad: equation be the adjoint action of G on itself, as defined in Appendix B.

Proposition 4.2. The following identities hold

equation

Proof. We compute

equation

Completely similarly,

equation

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

equation

Proof. A direct computation gives

equation

Similarly we obtain

equation

As a simple consequence of the previous proposition we find

equation

The following proposition describes the corresponding restriction twistings.

Proposition 4.4. The following identities hold

equation

Proof. Using standard twisting transformations we obtain

equation

The second identity can be derived in a similar manner.

Let R ⊆ ker(ε) be the right A0-ideal which canonically corresponds to Γ. In the following proposition we have characterized bicovariance in terms of R.

Proposition 4.5. (i) We have

equation

(ii) Conversely, if R ⊆ ker(ε) corresponding to a left-covariant calculus Γ is ad-invariant, then the calculus Γ is bicovariant. Moreover, in terms of the identificationequation the right action equation is given by

equation

where the map equation is given by

equation

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 equation as well as equalities (3.26) and (4.13) we obtain

equation

Furthermore, (3.27) implies

equation

Consequently, Γ is bicovariant andequation

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 k-covariant iff the following equivalence holds

equation

Let us assume that Γ is k-covariant. Then the formula

equation

consistently and uniquely determines a bijective map μ: Γ → Γ. It follows that

equation

Let us analyze properties of Γ, in the case when it is also σ-covariant.

Proposition 5.2. (i) If Γ is left σ-covariant (and accordingly, left F-covariant) then

equation

(ii) If Γ is right F-covariant then

equation

Proof. Assume that Γ is left F-covariant. A direct computation gives

equation

Furthermore,

equation

Symmetrically, assuming the right F-covariance of Γ we get

equation

Finally,

equation

Now, we shall analyze interrelations between k-covariance and bicovariance.

Proposition 5.3. A left-covariant calculus Γ is k-covariant if and only if it is bicovariant. In this case the following identities hold:

equation

Moreover, the diagram

equation

is commutative. Here, the vertical arrows are the corresponding double-sided actions and products.

Proof. Let us assume that Γ is left-covariant and k-covariant, and let us consider a map ξ : Γ →equation defined by

equation

It turns out that ξ is the right action for Γ. Indeed,

equation

Consequently Γ is right-covariant withequation and (5.8) holds.

Similarly, if Γ is k-covariant and right-covariant then a mapequation given by

equation

satisfies

equation

This implies that Γ is also left-covariant withequation and equality (5.9) holds.

Finally, let us assume that Γ is bicovariant and consider a map μ: Γ → Γ defined by diagram (5.10). Then a straightforward computation shows that equality (5.1) holds, and that { is bijective. In other words, Γ is k-covariant and (5.10) holds by construction.

Let us assume that Γ is bicovariant. Then the quadrupletsequation corresponding to the left/right-covariant structure on Γ are naturally related via the antipodal maps. According to (5.8)–(5.9)

equation

Proposition 5.4. The following identities hold:

equation

where equation is the antipode associated to A0.

Proof. A direct computation gives

equation

Similarly,

equation

Relations (5.15) immediately follow from (5.12), definition of spaces R and K and the fact that equation

Let us check (5.13)–(5.14). On the space equation the following equalities hold

equation

Similarly, in the framework of the space equation we can write

equation

In terms of the bimodule structuralizations equation the operator μ has a particularly simple form.

Proposition 5.5. The following identities hold:

equation

Proof. We compute

equation

Similarly,

equation

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

equation

for each α γ ∈ T . Here, equation is the standard transposition. It is easy to see that then (6.1) holds for every α γ ∈ T +. It is worth noticing that the operators

equation

also form a braid system over A.

Proposition 6.1. Let Γ over X be a *-covariant calculus. Then

(i) If Γ is left T -covariant then it is also left Tc-covariant and

equation

(ii) Similarly, if Γ is right T -covariant then it is right Tc-covariant, with

equation

Now switch to multi-braided quantum groups G. Let us assume that the *-structure on A satisfies

equation

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.

This implies a number of further compatibility relations between*and maps appearing at the group level. At first, we have

equation

The above equality implies

equation

Furthermore, as in the classical theory we have

equation

Really,

equation

and consequently

equation

Furthermore, let us examine interrelations between *, and braid operators  and σ.

Proposition 6.3. The following identities hold:

equation

Proof. Direct transformations give

equation

where equation This proves (6.8). Furthermore, applying (6.8) and the definition of  we obtain

equation

Condition (6.4) says that the comultiplication Φ is a hermitian map, if equation s endowed with the *-structure induced by σ and *: A → A.

Proposition 6.4. Consider a *-covariant differential calculus Γ over G.

(i) Assume that Γ is, in addition, left-covariant. Then

equation

(ii) Similarly, if Γ is right-covariant then

equation

(iii) If Γ is k-covariant then

equation

For the end of this section, let us characterize *-covariance of a left-covariant calculus in terms of the corresponding right A0-ideal. It turns out that the characterization is the same as in the standard theory.

Proposition 6.5. Let Γ be an arbitrary left-covariant calculus over G and R ⊆ ker(ε) the associated right A0-ideal. Then the calculus Γ is *-covariant if and only if R is +k-invariant.

Proof. If Γ is *-covariant then (6.11), together with the definition of R, implies that R is +k-invariant.

Conversely, let us assume that R is*k-invariant. Then the formula (6.11) consistently defines an antilinear involution*: Γinv → Γinv

According to Proposition 3.11 the formula (aϑ)+ = ϑ+a+, where a ∈ A and ϑ 2 Γinv consistently defines an antilinear extension +: Γ → Γ. Applying the elementary transformations with d and π we obtain

equation

Consequently, Γ is *-covariant.

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) = equation. 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(ε)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 equation can be also characterized by

equation

The right action map satisfies equalities

equation

Every right-covariant calculus is automatically right F-covariant. In particular, the flip-over operator equation is determined by the diagram

equation

The operator σr expresses the left multiplicativity of equation via the diagram

equation

The following twisting properties hold:

equation

Let invΓ be the set of all right-invariant elements of Γ. Then the map Q: Γ → Γ defined by

equation

projects Γ onto invΓ. Moreover,

equation

The composition

equation

is surjective. All flip-over operators equation mapequation Their restrictions on this space are given by

equation

for each n ∈ Z.

As a right A-module, the space Γ is naturally identificable with equation The isomorphism is induced by the multiplication map equation Moreover,

equation

In terms of the structuralization equation the following correspondences hold

equation

Here, equation is the restriction of the operatorsequation and the mapequation is given by

equation

This map determines a left A0-module structure on invΓ. We have also

equation

The space Γ is also trivial as a left A-module. The corresponding isomorphism equation is induced by the product map, and explicitly

equation

In terms of the structuralization equation the following correspondences hold:

equation

The structure of every right-covariant calculus Γ is completely determined by the space K =ker(equation) ∩ ker(ε). This space is a left A0-ideal satisfying

equation

Conversely, let K ⊆ ker(ε) be a left A0-ideal such that equality (A.23) holds. The space invΓ and maps equation and ε can be recovered as

equation

The whole right-covariant calculus Γ is then constructed with the help of the above established correspondences.

B Elementary properties of the adjoint action

By definition, the adjoint action of G onto itself is a linear map ad: equation defined by

equation

Lemma B.1. The following identities hold

equation

In other words, ad is a counital and coassociative map.

Proof. We compute

equation

Computation of the left-hand side of (B.3) gives

equation

Further useful identities are

Lemma B.2. We have

equation

Proof. Let us check the second identity. A direct computation gives

equation

Finally, let us study the twisting properties of the adjoint action.

Lemma B.3. The following identities hold:

equation

Proof. We compute

equation

Furthermore, we have

equation

Acknowledgment

I would like to thank Professor Zbigniew Oziewicz for carefully reading a preliminary version of the paper, and for numerous valuable remarks and observations.

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