unammicho/papers/qpb-diff.pdf · 2003. 9. 20. · differential structures on quantum principal...

DIFFERENTIAL STRUCTURES ON QUANTUMPRINCIPAL BUNDLES

MICHO DURDEVICH

Abstract. A fully constructive approach to differential calculus on quantumprincipal bundles is presented. The calculus on the bundle is built in an in-trinsic way, starting from given graded (differential) *-algebras representing

horizontal forms on the bundle and differential forms on the quantum basespace, together with a family of antiderivations acting on horizontal forms,playing the role of covariant derivatives of certain special connections. Theseconnections are used as global counterparts of local trivializations. In thisconceptual framework, a natural differential calculus on the structure quan-tum group is described. Higher-order calculi on the structure quantum groupcoming from both universal envelopes and braided exterior algebras are con-sidered.

1. Introduction

The structure of a classical smooth manifold is completely expressible in termsof the associated *-algebra of complex-valued smooth functions. Accordingly, it ispossible to translate, at least in principle, all concepts and constructions of classicalgeometry into the language of algebra. This is the starting point of noncommutativedifferential geometry [2], where the classical concept of space is essentially enlargedby considering the appropriate non-commutative *-algebras. The elements of thesealgebras are then interpreted as ‘smooth functions’ over qualitatively new ‘quantum’underlying spaces.

In the noncommutative context we meet an interesting new phenomena con-sisting in the existence of various conceptually non-equivalent ways of generalizingbasic components of the classical formalism. Though, some fundamental [2] geomet-rical invariants (standard K-groups, cyclic homology and corresponding quantumde Rham cohomology) are uniquely and intrinsically associated to the quantumspace algebra.

The mentioned non-uniqueness reflects the fact that noncommutative geometryis a much deeper theory, compared with its classical counterpart. This becomesexplicitly visible in the foundations of quantum differential calculus.

For example, it is possible to start from the analogs of the vector fields anda natural formulation is to take derivations [3] of the quantum space algebra ascounterparts of the vector fields. Then the algebra representing differential formscan be extracted from the corresponding Chevalley complex. Such a construction isparticularly suitable [4, 5] for quantum spaces described by matrix algebras, or forbundles of such spaces over classical smooth manifolds. It is automatically covariantwith respect to all classical-type symmetries of the quantum space (described byautomorphisms of the quantum space algebra).

Reports in Mathematical Physics 41 (1) 91–115 (1998).

2 MICHO DURDEVICH

On the other hand, the derivation approach to quantum differential calculusis completely inappropriate for quantum groups and quantum spaces associatedto them. In the context of quantum groups, it is natural to consider differentialstructures covariant [17, 19] with respect to the left/right action of the quantumgroup. The same approach is most suitable for quantum spaces possessing someintrinsic quantum group symmetry [14]. However [19], there will be generally avariety of differential calculi satisfying postulated covariance properties (all of themhaving potential geometrical significance).

This naturally leads to a viewpoint that in noncommutative differential geometrythe choice of the most appropriate differential calculus should be context-dependent.

There exist some situations where the additional geometrical structure on thequantum space uniquely fixes the most appropriate differential calculus. An in-teresting class of quantum spaces with such a pre-existing structure is given byquantum principal bundles.

Quantum principal bundles are noncommutative-geometric analogs of standardprincipal bundles, where quantum groups play the role of structure groups. Thebase space and the bundle are quantum objects.

The aim of this study is to develop a fully constructive approach to differentialcalculus on quantum principal bundles. Our considerations are logically based ona general theory of quantum principal bundles, developed in papers [6, 7, 8].

As far as locally-trivial quantum principal bundles over classical smooth mani-folds are concerned [6], it is possible to construct in a natural manner a differentialcalculus on them. The algebra of differential forms on the bundle can be con-structed by combining standard differential forms on the base manifold with theappropriate differential calculus on the structure quantum group, such that everylocal trivialization of the bundle can be ‘extended’ to a local trivialization of thecorresponding calculus.

However, this local triviality property implies relatively strong constraints forthe algebra of differential forms on the structure quantum group. At the first-orderlevel, in the class of left-covariant differential structures (on the structure quantumgroup), there exists the unique minimal element satisfying mentioned constraints.This calculus is also *-covariant and bicovariant. If the higher-order differentialcalculus on the structure quantum group is described by the corresponding universalenvelope, then all compatibility conditions are resolved already at the first-orderlevel. The same situation holds if the higher-order calculus is described by thecorresponding braided exterior algebra [19].

In summary, quantum principal bundles over classical smooth manifolds are suf-ficiently ‘structuralized’ geometrical objects. This opens the possibility to constructthe whole differential calculus in an intrinsic manner, starting from the idea of localtriviality.

On the other hand, in the theory of general quantum principal bundles (overquantum spaces) we meet just the opposite situation. From the conceptual pointof view, the most natural approach to differential calculus is to start from appro-priate differential *-algebras representing differential calculi on the bundle and thestructure quantum group. Then quantum counterparts of all basic entities appear-ing in the classical formalism can be derived from these algebras. In particular,differential forms on the base manifold can be viewed as differential forms on thebundle, invariant under the ‘pull back’ induced by the right action map.

QUANTUM PRINCIPAL BUNDLES 3

However, a more detailed geometrical analysis [7] shows that there exist non-trivial correlations between a given differential calculus on the bundle, the calculuson the structure quantum group, and the algebra of quantum horizontal forms—asfar as the calculus on the bundle is chosen in a geometrically reasonable way.

In the general quantum context, it is not possible to speak about local trivial-izations (however there exist interesting special classes of quantum spaces and bun-dles [2, 6, 15] admitting the analogs of localization and local trivialization maps).It turns out that the role of local trivializations is effectively played by regularconnections [7], which are well-defined ‘global’ entities. With the help of regularconnections, we can incorporate the above mentioned constructive approach of [6]into the general quantum context.

In this paper we shall mainly consider bundles possessing compact matrix quan-tum structure groups. Let G be a compact matrix quantum group [18]. Let A bethe Hopf *-algebra of polynomial functions onG. We shall denote by φ : A → A⊗A,� : A → C and κ : A → A the coproduct, counit and the antipode respectively.

Let Γ be a bicovariant [19] first-order *-calculus over G, and let us assume that acomplete differential calculus on G is given by the corresponding [6]–Appenix B uni-versal differential envelope Γ∧ = Γ⊗/S∧. The quadratic ideal S∧ is, by definition,generated by the elements ψ of the form

ψ =∑

i

d(ai) ⊗ d(bi),∑

i

aid(bi) = 0,

where ai, bi ∈ A.Let M be a quantum space represented by a *-algebra V , and let us consider a

quantum principal G-bundle P = (B, i, F ) over M . Here B is a *-algebra represent-ing the quantum space P , while i : V → B is a *-monomorphism playing the roleof the dualized projection of P on M . The map F : B → B ⊗ A is a coassociativecounital *-homomorphism representing the dualized right action of G on P .

Let Ω(P ) be an arbitrary graded-differential *-algebra representing the com-plete differential calculus on the bundle. By definition [7] this means that Ω(P ) isgenerated by Ω0(P ) = B and that there exists a (necessarily unique) differential*-algebra homomorphism F̂ : Ω(P ) → Ω(P ) ⊗̂ Γ∧ extending the action F . Here⊗̂ is the graded tensor product. Then the horizontal forms can be defined as theelements of a graded *-subalgebra

hor(P ) = F̂−1{Ω(P ) ⊗A

}.

The map F̂ is reduced to a *-homomorphism F∧ : hor(P ) → hor(P ) ⊗ A. Thecalculus on the base is given by the F∧-fixed point graded *-subalgebra Ω(M) ofhor(P ). It is a differential subalgebra of Ω(P ).

Following classical geometry, it is possible to develop the formalism of connec-tions on quantum principal bundles [6, 7]. Connections are defined as hermitianfirst-order linear maps ω : Γinv → Ω(P ) satisfying

F̂ω(ϑ) = (ω ⊗ id)$(ϑ) + 1 ⊗ ϑ,where Γinv is the left-invariant part of Γ and $ : Γinv → Γinv ⊗A is the associatedadjoint action. We have Γinv = ker(�)/R, where R ⊆ ker(�) is the right A-idealcanonically corresponding [19] to Γ. Let π : A → Γinv be the natural projectionmap, explicitly given by

π(a) = κ(a(1))d(a(2)).

4 MICHO DURDEVICH

The map $ can be explicitly defined by

$π = (π ⊗ id)ad,where ad: A → A⊗A is the adjoint action of G on itself, defined by

ad(a) = a(2) ⊗ κ(a(1))a(3).

Every connection ω induces the covariant derivative map Dω : hor(P ) → hor(P ),via the formula

Dω(ϕ) = d(ϕ) − (−1)∂ϕ∑

k

ϕkωπ(ck),(1.1)

where∑

kϕk ⊗ ck = F∧(ϕ). This map intertwines the right action F∧, and extends

the corresponding differential Md : Ω(M) → Ω(M). A particularly important role isplayed by regular connections [7], that are characterized by the identity

ω(ϑ)ϕ = (−1)∂ϕ∑

k

ϕkω(ϑ ◦ ck),

where ϕ ∈ hor(P ) and ϑ ∈ Γinv, and ◦ is the canonical right A-module structure onΓinv, explicitly given by ϑ ◦ a = κ(a(1))ϑa(2). If ω is regular then Dω is a hermitianantiderivation.

The left-invariant part Γ∧inv ⊆ Γ∧ is given by

Γ∧inv = Γ⊗inv/S

∧inv S

∧inv = gen

{S∧2inv

}S∧2inv =

{π(a(1)) ⊗ π(a(2))

∣∣∣ a ∈ R}.A connection ω is called multiplicative [7] iff it extends (necessarily uniquely) to

a unital *-homomorphism ω∧ : Γ∧inv → Ω(P ). As shown in [7], if the calculus admitsregular connections, then it is always possible to assume that all regular connectionsare multiplicative (passing if necessary to an appropriate factor-calculus).

Starting from a given regular connection ω we can construct a cross-productisomorphism

mω : hor(P ) ⊗ Γ∧inv ↔ Ω(P ) mω(ϕ⊗ ϑ) = ϕω

∧(ϑ),(1.2)

and this allows us to re-express the structure of the whole calculus Ω(P ) in termsof

{hor(P ), F∧, Dω,Γ

∧inv

}.

The starting point for considerations of this paper consists of a graded *-algebrahorP (the elements of which are interpretable as horizontal forms on P ), a *-homo-morphism F∧ : horP → horP ⊗ A (playing the role of the induced action of Gon horizontal forms) and the graded subalgebra ΩM ⊆ horP consisting of F∧-invariant elements (representing differential forms on M) endowed with a differ-ential Md : ΩM → ΩM . Then the abstract counterparts of covariant derivatives(of regular connections) can be defined as hermitian first-order antiderivationsD : horP → horP , intertwining F∧ and extending the map

Md. These maps form

a real affine space der(P ). Their elementary properties are studied in Section 2.Starting from the space der(P ) it is possible to construct, in a natural manner,

a graded-differential *-algebra ΩP imaginable as consisting of differential forms onP , together with an appropriate bicovariant first-order *-calculus Φ on G and theassociated pull back map F̂ : ΩP → ΩP ⊗̂Φ∧. These constructions will be presentedin Section 3. The constructed algebra ΩP contains horP as its graded *-subalgebra.Actually horP coincides with the graded *-subalgebra representing horizontal forms


in the general theory. The map F∧ : horP → horP ⊗A coincides with the reductionof F̂ , as in the framework of the general theory.

Furthermore, there exists a natural correspondence between regular connections(relative to the constructed calculus) and derivatives D ∈ der(P ). This correspon-dence is interpretable as labeling regular connections by the corresponding covariantderivative operators. In such a way, the circle will be closed. Finally, in Section 4some concluding remarks are made.

The paper ends with two technical appendices. In the first one, the constructionof the calculus on the bundle is sketched in the case when the higher-order differ-ential calculus on the structure quantum group is described by the correspondingbicovariant braided exterior algebra [19]. The calculus can be constructed essen-tially in the same way as in the universal envelope case, the only additional point isthat we should check the compatibility with extra relations from braided exterioralgebra. The second appendix discusses the extension of the formalism to arbitrary(non-compact) structure quantum groups.

2. Abstract Covariant Derivative Operators

Let G be a compact matrix quantum group and let P = (B, i, F ) be a quantumprincipal G-bundle over a quantum space M ↔ V . The image of the map i : V → Bcoincides with the F -fixed point subalgebra of B. The action F : B → B⊗A is free,in the sense that for each a ∈ A there exist elements bi, qi ∈ B such that∑

i

qiF (bi) = 1 ⊗ a.(2.1)

Let us consider a graded *-algebra

horP =∑⊕k≥0

horkP

such that B = hor0P and let F∧ : horP → horP ⊗ A be the grade-preserving *-homomorphism extending F and satisfying

(id ⊗ φ)F∧ = (F∧ ⊗ id)F∧(2.2)(id ⊗ �)F∧ = id.(2.3)

Geometrically, F∧ determines a (left) action of G by ‘automorphisms’ of horP .The elements of horP will be interpreted as horizontal differential forms on P .

Let ΩM ⊆ horP be the graded *-subalgebra consisting of F∧-invariant elements. Theelements of ΩM will be interpreted as differential forms on the base ‘manifold’ M .(These geometrical interpretations will be completely justified after constructingthe calculus on the bundle P ).

Proposition 2.1. Let us assume that a linear map ∆: horP → horP is given suchthat

F∧∆ = (∆ ⊗ id)F∧(2.4)∆(ΩM ) = {0}.(2.5)

(i) If ∆ is an (odd) antiderivation then there exists a unique µ = µ∆ : A → horPsuch that

∆(ϕ) = (−1)∂ϕ∑

k

ϕkµ(ck)(2.6)

6 MICHO DURDEVICH

for each ϕ ∈ horP , where∑

kϕk ⊗ ck = F∧(ϕ). The following identity holds

µ(a)ϕ = (−1)∂ϕ∑

k

ϕkµ(ack),(2.7)

where a ∈ ker(�).(ii) Similarly, if ∆ is an (even) derivation on horP then there exists a unique

linear map µ = µ∆ : A → horP such that

∆(ϕ) =∑

k

ϕkµ(ck).(2.8)

The following identities hold

∆µ = 0(2.9)

µ(a)ϕ =∑

k

ϕkµ(ack)(2.10)

where a ∈ ker(�).(iii) In both cases

µ(1) = 0(2.11)

(µ⊗ id)ad = F∧µ.(2.12)The map ∆ is hermitian iff

µ(κ(a)∗) = −µ(a)∗(2.13)for each a ∈ A.

Proof. We shall prove the statements assuming that ∆ is a derivation. The casewhen ∆ is an antiderivation can be treated similarly.

At first, it is clear that µ is unique, if exists. Indeed, property (2.8) implies

µ(a) =∑

i

qi∆(bi),

where qi, bi ∈ B are such that (2.1) holds.Conversely, the above formula can be taken as the starting point for a definition

of µ. However, such a definition will be consistent iff{∑iqiF (bi) = 0

}=⇒

{∑iqi∆(bi) = 0

}.

Here, the map µ will be constructed in a slightly different way, without explicitlyproving this consistency condition. The notation introduced in [7]–Appendix Bwill be followed. Let T be the set of equivalence classes of irreducible unitaryrepresentations of G.

For each class α ∈ T let us consider an irreducible representation uα ∈ α (actingin Cnα , with matrix elements uαij) and choose elements b

αki ∈ Bα (where Bα is the

multiple irreducible V-submodule of B, corresponding to α) such that

F∧(bαki) =

∑j

bαkj ⊗ uαji∑

k

bα∗ki bαkj = δij1.


In fact to ensure the second equation some additional ‘positivity’ assumptions arenecessary, but this does not influence the final result. Let µ : A → horP be a linearmap specified by µ(1) = 0 and

µ(uαij) =∑

k

bα∗ki ∆(bαkj).(2.14)

Let us assume that elements ξ1, . . . , ξnα ∈ horP form one uα-multiplet. We have

then ∑i

ξiµ(uαij) =

∑ki

ξibα∗ki ∆(b

αkj) =

∑ki

∆(ξibα∗ki b

αkj) =

∑i

δij∆(ξi) = ∆(ξj),

because of (2.5) and the F∧-invariance of∑

iξib

α∗ki . Consequently, property (2.8)

holds, because irreducible α-multiplets span Bα, and

B =∑⊕α∈T

Bα.

Let us check (2.12). Applying (2.4) and (2.14) we obtain

F∧µ(uαij) =

∑k

F (bα∗ki )(∆ ⊗ id)F (bαkj) =

∑kmn

(bα∗km ⊗ uα∗mi)

(∆(bαkn) ⊗ u

αnj

)=

∑mn

µ(uαmn) ⊗ uα∗miuαnj = (µ⊗ id)ad(uαij).

Acting by (∆ ⊗ id) on the identity∑k

b∗αki F (bαkj) = 1 ⊗ uαij(2.15)

we find ∑k

(∆(bα∗ki )F (b

αkj) + b

α∗ki (∆ ⊗ id)F (bαkj)

)= 0.

Identity (2.8) together with the above equality and (2.4) gives

0 =∑

k

(∆(bα∗ki )∆(b

αkj) + b

α∗ki ∆

2(bαkj))

= ∆(∑

k

bα∗ki ∆(bαkj)

)= ∆µ(uαij).

Hence, (2.9) holds. Multiplying (2.15) by F∧(ϕ) on the right and using (2.8) weobtain ∑

k

bα∗ki ∆(bαkjϕ) =

∑kml

bα∗ki bαkmϕlµ(u

αmjcl) =

∑l

ϕlµ(uαijcl).

On the other hand

µ(uαij)ϕ =∑

k

bα∗ki ∆(bαkj)ϕ =

∑k

bα∗ki ∆(bαkjϕ) − δij∆(ϕ).

Consequently,

µ(uαij)ϕ+ δij∆(ϕ) =∑

l

ϕlµ(uαijcl),

which proves (2.10).

8 MICHO DURDEVICH

Finally, let us assume that ∆ is hermitian. This implies

µ(uαij)∗ =

[∑k

bα∗ki ∆(bαkj)

]∗=

∑k

∆(bα∗kj )bαki

= −∑

k

bα∗kj ∆(bαki) = −µ(uαji) = −µ(κ(uαij)∗).

Hence, (2.13) holds. Conversely, if (2.13) holds then applying (2.8), (2.10) and(2.11) we obtain

∆(ϕ)∗ =∑

k

(ϕkµ(ck)

)∗ = ∑k

µ(ck)∗ϕ∗k

= −∑

k

µ(κ(ck)∗)ϕ∗k =

∑k

ϕ∗kµ(c∗k) = ∆(ϕ

∗)

which completes the proof.

It is worth noticing that degrees of the maps µ and ∆ are the same. Further, ∆is completely fixed by its restriction on B, because µ is expressible in terms of thisrestriction.

It is possible to ‘reverse’ the above construction of µ. If a linear homogeneousmap µ : A → horP is given such that

µ(a)ϕ = (−1)∂µ∂ϕ∑

k

ϕkµ(ack)(2.16)

where a ∈ ker(�), then (2.6)/(2.8) determines a map ∆: horP → horP , which is aneven/odd (anti)derivation, depending on the parity of µ.

In appendix B we shall give an alternative proof of the above proposition, suitablefor non-compact quantum structure groups. Before going further, let us observethat the introduced multiplets bαki have a very subtle geometrical meaning. Theyare pull backs of the canonical generators of the universal G-bundle QEG overthe quantum classifying space QBG, under the classifying maps [12] for P . Theexistence of such multiplets is essentially based on the compactness of G.

Let us now assume that ΩM is endowed with a differential *-algebra structure,specified by a first-order differential map Md : ΩM → ΩM . In other words, the fol-lowing properties hold

Md(Ω∗M ) ⊆ Ω

∗+1M(2.17)

Md(ϕψ) = Md(ϕ)ψ + (−1)∂ϕϕMd(ψ)(2.18)

Md2 = 0(2.19)

Md(ϕ∗) = Md(ϕ)∗.(2.20)

Definition 1. A bundle derivative for P with respect to{horP , F∧,ΩM

}is a linear

map D : horP → horP satisfyingD(hor∗P ) ⊆ hor

∗+1P(2.21)

F∧D = (D ⊗ id)F∧(2.22)(D�ΩM ) =

Md(2.23)

D(ϕψ) = D(ϕ)ψ + (−1)∂ϕϕD(ψ)(2.24)D(ϕ∗) = D(ϕ)∗.(2.25)


In other words, bundle derivatives are hermitian right-covariant first-order an-tiderivations on horP , which extend

Md.

Let us observe that every linear map D acting in horP and satisfying (2.22) isreduced in the space ΩM . If a map D : horP → horP satisfies (2.21)–(2.24) then∗D∗ possesses the same property, and hence (∗D ∗ +D)/2 is an element of der(P ).

The set der(P ) of all bundle derivatives on P is a real affine space, in a naturalmanner. We shall assume that der(P ) 6= ∅. The corresponding vector space −→der(P )consists of hermitian first-order right-covariant antiderivations E on horP satisfying

E(ΩM ) = {0}.(2.26)

Lemma 2.2. (i) For each E ∈ −→der(P ) there exists a unique χE : A → horP suchthat

E(ϕ) = −(−1)∂ϕ∑

k

ϕkχE(ck),(2.27)

for each ϕ ∈ horP . We haveF∧χE(a) = (χE ⊗ id)ad(a)(2.28)χE(κ(a)

∗) = −χE(a)∗(2.29)

χE(a)ϕ = (−1)∂ϕ∑

k

ϕkχE(ack)(2.30)

for each a ∈ ker(�) and ϕ ∈ horP .(ii) Similarly, for each D ∈ der(P ) there exists a unique %D : A → horP satisfying

D2(ϕ) = −∑

k

ϕk%D(ck),(2.31)

for each ϕ ∈ horP . We haveF∧%D(a) = (%D ⊗ id)ad(a)(2.32)

D%D(a) = 0(2.33)

%D(κ(a)∗) = −%D(a)

∗(2.34)

%D(a)ϕ =∑

k

ϕk%D(ack)(2.35)

for each a ∈ ker(�).

Proof. Let us consider a bundle derivative D. Properties (2.21)–(2.25) imply thatD2 is a second-order right-covariant hermitian derivation on horP satisfying

D2(ΩM ) = {0}Applying Proposition 2.1 (ii) to the case ∆ = D2 we conclude that there exists aunique map %D : A → horP such that (2.31) holds. Identities (2.32)–(2.35) followfrom statements (ii)–(iii) in Proposition 2.1. The statement (i) follows by a similarreasoning.

We have χE(1) = %D(1) = 0. Introduced maps are mutually correlated, as wesee from the next lemma.

Lemma 2.3. The following identity holds

%D+E(a) = %D(a) +DχE(a) + χE(a(1))χE(a

(2)).(2.36)

10 MICHO DURDEVICH

Proof. For a given a ∈ A let us choose elements qi, bi ∈ B such that (2.1) holds,and let us assume that F (bi) =

∑kbki ⊗ cki.

A direct computation gives

−%D+E(a) =∑

i

qi(D + E)2(bi)

= −∑ki

qibki%D(cki) −∑ki

qibkiχE(c(1)ki )χE(c

(2)ki )

−∑ki

qiD(bkiχE(cki)

)+

∑ki

qi(Dbki)χE(cki)

= −%D(a) − χE(a(1))χE(a(2)) −DχE(a).

For eachD ∈ der(P ) and E ∈ −→der(P ) let RD,PE ⊆ ker(�) be subspaces consistingof elements anihilated by %D and χE respectively.

Lemma 2.4. The spaces RD and PE are right A-ideals. Moreover,ad(PE) ⊆ PE ⊗A(2.37)κ(PE)∗ = PE(2.38)κ(RD)∗ = RD(2.39)

ad(RD) ⊆ RD ⊗A.(2.40)

Proof. For a given a ∈ A let us choose elements bi, qi ∈ B such that (2.1) holds.Applying (2.10) we find∑

i

qi%D(b)bi =∑ki

qibki%D(bcki) = %D(ba),

for each b ∈ ker(�), where∑

kbki ⊗ cki = F (bi). In particular, if b ∈ RD then

ba ∈ RD, too. Similarly, it follows that PE are right A-ideals. Finally, (2.37)–(2.40) directly follow from properties (2.28), (2.29), (2.32) and (2.34).

3. Constructions of Differential Structures

In this section two constructions will be presented. At first, starting from thesystem der(P ) of all bundle derivatives we shall construct a canonical differentialcalculus on the structure quantum group G. Secondly, a canonical differentialstructure on the bundle P will be constructed, by combining this calculus on Gwith the algebra horP . As we shall see, there exists a natural correspondencebetween bundle derivatives and regular connections on P . Bundle derivatives areinterpretable as covariant derivatives associated to regular connections.

Let R̂ be the intersection of all ideals RD and PE . According to Lemma 2.3

R̂ = RD⋂{⋂

EPE},(3.1)

for an arbitrary D ∈ der(P ). Indeed,

χE(a(1))χE(a

(2)) =12χE(a

(2))χE[κ(a(1))a(3)

],(3.2)

as follows from (2.28) and (2.30). In particular if a belongs to the right-hand sideof (3.1) then %D+E(a) = 0, for each E ∈

−→der(P ).


Let Φ be the left-covariant first-order differential calculus on G canonically cor-responding to R̂ (in the sense of [19]). Lemma 2.4 implies

ad(R̂) ⊆ R̂ ⊗A(3.3)κ(R̂)∗ = R̂.(3.4)

In other words [19], Φ is a bicovariant *-calculus. Let us consider a derivativeD ∈ der(P ), and let us factorize the map %D through R̂. The factorized map willbe denoted by the same symbol %D : Φinv → horP .

Definition 2. The map %D is called the curvature of D.

Similarly, for each E ∈ −→der(P ) we have the factorized map χE : Φinv → horP . Inother words, we can write

%Dπ = %D,(3.5)

χEπ = χE .(3.6)

where π : A → Φinv = ker(�)/R̂ is the canonical projection map.The following identities summarize results of the previous section:

%D+E = %D +DχE− < χE , χE > D%D = 0F∧χE = (χE ⊗ id)$ χE(ϑ∗) = χE(ϑ)∗

F∧%D = (%D ⊗ id)$ %D(ϑ∗) = %D(ϑ)∗

−D2(ϕ) =∑

k

ϕk%Dπ(ck)

%D(ϑ)ϕ =∑

k

ϕk%D(ϑ◦ck)

−E(ϕ) =(−1)∂ϕ∑

k

ϕkχEπ(ck)

χE(ϑ)ϕ =(−1)∂ϕ∑

k

ϕkχE(ϑ◦ck)

where are the brackets associated to an arbitrary embedded differential mapδ : Φinv → Φinv⊗Φinv (a hermitian covariant lifting of the differential d : Φinv → Φ∧2invalong the factor projection). The ◦ denotes the natural right A-module structure.

We shall assume that the complete differential calculus on G is based on thecorresponding universal envelope Φ∧. Let us now consider a *-algebra vhP repre-senting ‘vertically-horizontally’ decomposed forms [7]. At the level of graded vectorspaces

vhP = horP ⊗ Φ∧inv,and the *-algebra structure on vhP is specified by

(ϕ⊗ ϑ)∗ =∑

k

ϕ∗k ⊗ (ϑ∗◦c∗k)(3.7)

(ψ ⊗ η)(ϕ ⊗ ϑ) = (−1)∂η∂ϕ∑

k

ψϕk ⊗ (η◦ck)ϑ.(3.8)

By construction horP and Φ∧inv are interpretable as *-subalgebras of vhP .

The formulas

∂D(ϕ) = D(ϕ) + (−1)∂ϕ∑

k

ϕkπ(ck)(3.9)

∂D(ϑ) = %D(ϑ) + d(ϑ),(3.10)

12 MICHO DURDEVICH

where ϕ ∈ horP and ϑ ∈ Φinv, while d : Φ∧inv → Φ∧inv is the corresponding differ-ential, determine (via the graded Leibniz rule) a hermitian first-order differential∂D : vhP → vhP , for each D ∈ der(P ).

Differential *-algebras (vhP , ∂D) are mutually naturally isomorphic.

Proposition 3.1. (i) For each E ∈ −→der(P ) there exists a unique homomorphismhE : vhP → vhP such that

hE(ϕ) = ϕ(3.11)

hE(ϑ) = ϑ− χE(ϑ),(3.12)

for each ϕ ∈ horP and ϑ ∈ Φinv.(ii) The maps hE are hermitian and bijective.

(iii) We have

hEhW = hE+W(3.13)

for each E,W ∈ −→der(P ).(iv) The map hE is an isomorphism between differential structures (vhP , ∂D)

and (vhP , ∂D+E), for each E ∈−→der(P ) and D ∈ der(P ).

Proof. Uniqueness of maps hE follows from the fact that horP and Φinv generatevhP . To establish their existence, it is sufficient to check that conditions (3.11)–(3.12) are compatible with the product rule for ϑϕ, and with the quadratic con-straint defining the algebra Φ∧inv. We have

ϑϕ = (−1)∂ϕ∑

k

ϕk(ϑ◦ck) −→ (−1)∂ϕ∑

k

(ϕk(ϑ◦ck) − ϕkχE(ϑ◦ck)

)=

[ϑ− χE(ϑ)

]ϕ,

for each ϑ ∈ Φinv and ϕ ∈ horP . Further, if a ∈ R̂ then

0 = π(a(1))π(a(2)) −→ π(a(1))π(a(2)) + χEπ(a(1))χEπ(a(2))−

[χEπ(a

(1))]π(a(2)) − π(a(1))χEπ(a(2)) = 0,

because of (3.2) and

π(a(1))χEπ(a(2)) = −χEπ(a(3))

[π(a(1))◦

(κ(a(2))a(4)

)]= −

[χEπ(a

(1))]π(a(2)) + χEπ(a

(2))π[κ(a(1))a(3)

]= −

[χEπ(a

(1))]π(a(2)).

Hence, hE exists. In order to prove the hermicity of hE it is sufficient to check thatrestrictions of hE on Φinv and horP are hermitian maps. This imediately followsfrom the hermicity of χE . Similarly, it is sufficient to check that (3.13) holds onhorP and Φinv. This trivially follows from (3.11)–(3.12). Now (3.13) implies thathE are bijective maps.


Let us prove (iv). Because of the graded Leibniz rule, it is sufficient to checkthat hE∂D = ∂D+EhE holds on Φinv and horP . We have

hE∂D(ϕ) = D(ϕ) + (−1)∂ϕ∑

k

ϕk[π(ck) − χEπ(ck)

]= (D + E)(ϕ) + (−1)∂ϕ

∑k

ϕkπ(ck) = ∂D+E(ϕ).

Furthermore, using the identity dπ(a) = −π(a(1))π(a(2)) [6]-Appendix B, formulas(2.36) and (3.2), and performing elementary transformations we obtain

hE∂Dπ(a) = −hE[π(a(1))π(a(2))

]+ %Dπ(a)

= −π(a(1))π(a(2)) − χEπ(a(1))χEπ(a(2))+

[χEπ(a

(1))]π(a(2)) + π(a(1))χEπ(a

(2)) + %Dπ(a)

= −π(a(1))π(a(2)) − χEπ(a(1))χEπ(a(2))+ χEπ(a

(2))π[κ(a(1))a(3)

]+ %Dπ(a)

= %Dπ(a) +DχEπ(a) + χEπ(a(1))χEπ(a

(2))

−DχEπ(a) − π(a(1))π(a(2))+ χEπ(a

(2))π[κ(a(1))a(3)

]− χEπ(a

(2))χEπ[κ(a(1))a(3)

]= %D+Eπ(a) −DχEπ(a) − π(a(1))π(a(2))

+ χEπ(a(2))π

[κ(a(1))a(3)

]− χEπ(a(2))χEπ

[κ(a(1))a(3)

]= ∂D+Eπ(a) −DχEπ(a) − EχEπ(a) + χEπ(a(2))π

[κ(a(1))a(3)

]= ∂D+Eπ(a) − ∂D+EχEπ(a) = ∂D+EhEπ(a).

Now, a manifestly invariant differential calculus on P can be constructed by‘gluing’ algebras (vhP , ∂D), with the help of isomorphisms hE. Let ΩP be a graded-differential *-algebra obtained in this way.

More precisely, let us first consider a direct external sum of graded-differential*-algebras

ΣP =⊕D

(vhP , ∂D)

over all bundle derivatives D. Let us define ΩP to be a graded-differential *-subalgebra of ΣP , consisting of the elements w satisfying

πD+E(w) = hEπD(w),

for each D ∈ der(P ) and E ∈ −→der(P ). Here πD are coordinate projection maps. Byconstruction, it is clear that every such projection gives a differential *-isomorphismbetween ΩP and (vhP , ∂D), and in what follows we shall restrict the domain of mapsπD to the algebra ΩP . Let

Pd be the differential on ΩP .

The map F is naturally extendible to a homomorphism F̂ : ΩP → ΩP ⊗̂ Φ∧ ofgraded-differential *-algebras. Explicitly,

(πDF̂ π−1D )(ϕ⊗ ϑ) = F∧(ϕ)$̂(ϑ)

where $̂ : Φ∧inv → Φ∧inv ⊗̂ Φ∧ is the graded-differential *-homomorphism extendingthe (co)adjoint action $ : Φinv → Φinv ⊗A.

14 MICHO DURDEVICH

The following equality holds

horP = F̂−1

{ΩP ⊗A

}.

This justifies the interpretation of horP , as the algebra of horizontal forms. Also,the above equality justifies the interpretation of ΩM , as consisting of differentialforms on the base manifold M . In particular, ΩM is

Pd-invariant, and Pd�ΩM =

Md.

Proposition 3.2. There exists a natural affine isomorphism between der(P ) andthe space of regular connections on P (relative to the calculus ΩP ). More precisely,for each D ∈ der(P ), the connection ω = ωD given by

ω(ϑ) = π−1D (1 ⊗ ϑ)(3.14)is regular. Moreover

D = Dω, π−1D = mω(3.15)

where mω : horP ⊗ Φ∧inv → ΩP is the associated decomposition map.Conversely, if ω is an arbitrary regular connection on P then there exists a

unique D ∈ der(P ) such that (3.14) holds.

Proof. The first part of the proposition directly follows from the definition of ω,and formulas (1.1), (1.2) and (3.9).

Let ω : Φinv → ΩP be an arbitrary regular connection. Let D = Dω : horP →horP be the covariant derivative associated to ω. By construction, Dω ∈ der(P )and the second equality in (3.15) holds. This further implies that (3.14) holds. Onthe other hand, if (3.14) holds then automatically D = Dω.

It is worth noticing that the above introduced connection ω satisfies

%D = Rω = dω − 〈ω, ω〉,where Rω is the curvature form associated to ω, in the framework of the generaltheory [7].

4. Concluding Remarks and Examples

In the presented construction we have assumed that the horizontal algebra horPis given. The natural interpretation is that horP reflects some geometrical structureexisting on the base space M . As a concrete illustration, let us mention quantumframe bundles [9, 10]. In this context, we first fix a bicovariant *-bimodule Ψ overG. The space V = Ψinv is equipped with the induced *-structure, the adjoint actionκ : V → V⊗A and the natural right-module structure ◦ : V⊗A → V. These mapscan be naturally extended to the exterior algebra V∧, given by

V∧ = V⊗/gen{im(I + τ)}

where τ : V ⊗ V → V ⊗ V is the associated [19] braid operator. Horizontal formsare given by

horP = B ⊗ V∧

equipped with the appropriate cross-product structure. The right action F∧ is givenby the product of F and κ : V∧ → V∧ ⊗A. The elements of C1 ⊗ V ↔ V play therole of the canonical coordinate first-order forms.

Another general method for constructing quantum horizontal algebras horP is tore-express the structure of the bundle P in terms of the associated vector bundles[13], and to built horP with the help of these bundles and a given differential


calculus ΩM on the base. To every representation u : Hu → Hu ⊗ A of G in afinite-dimensional Hilbert space Hu we can associate the intertwiner V-bimoduleEu = Mor(u, F ). These spaces are analogs of the associated vector bundles.

The conjugate representation operation induces antilinear anti-isomorphisms∗u : Eu → Eū, and every intertwiner f : Hu → Hv between u, v ∈ R(G) inducesa V-bimodule homomorphism f? : Ev → Eu. It turns out that the system of objects{Eu, f?, ∗u

}completely determines the structure of the initial quantum principal

bundle P .In a similar way we can introduce the spaces Fu of ‘vector bundle-valued forms’

on M , starting from a given algebra horP . These spaces are graded ΩM -bimodulesand we have the following natural decompositions

Fu ↔ Eu ⊗V ΩM ↔ ΩM ⊗V Eu.In particular, we can introduce the flip-over operators

σu : Eu ⊗V ΩM → ΩM ⊗V Eu.The operators σu satisfy a number of compatibility conditions [13] with the systemof objects

{Eu, f?, ∗u

}. When these conditions are satisfied we can reverse the whole

analysis and construct the algebra horP .It is worth noticing that the presented construction of the calculus on the bundle

works for an arbitrary bicovariant first-order *-calculus Γ based on a right A-idealR satisfying R ⊆ R̂. It is also possible to perform the construction of differentialstructures on G and P , dealing with a restricted set of bundle derivatives, formingan appropriate affine subspace L ⊆ der(P ). Covariant derivatives of regular connec-tions (with respect to the associated differential structures) form an affine subspaceL̂ (containing L) of der(P ). Particularly interesting are subspaces L satisfying thestability property L = L̂.

Let us now consider an arbitrary quantum principal bundle P = (B, i, F ) , andlet us assume that the calculus on P is based on a graded-differential *-algebraΩ(P ). Let us assume that the calculus on G is based on a bicovariant *-calculusΓ. Finally, let us assume that the calculus Ω(P ) admits regular multiplicativeconnections.

Then for every regular connection ω the corresponding covariant derivative mapDω : hor(P ) → hor(P ) is a bundle derivative—relative to

{hor(P ), F∧,Ω(M)

}.

However, the converse is generally not true. In general only an affine subspaceof der(P ) will be induced by regular connections. On the other hand, starting fromhor(P ) and der(P ) and applying constructions presented in this study we obtainthe bicovariant *-calculus Φ on G and the graded-differential *-algebra ΩP . Ingeneral algebras Ω(P ) and ΩP are not mutually naturally related. However, if allbundle derivatives are interpretable as covariant derivatives of regular connectionsthen R ⊆ R̂, so that Γ factorizes to Φ and the algebra Ω(P ) is includable into thepresented formalism.

As we have already mentioned, presented constructions of differential structuresgeneralize the corresponding constructions of the theory [6] of locally-trivial quan-tum principal bundles over standard smooth manifolds.

Let us assume that P is a locally-trivial quantum principal G-bundle over acompact smooth manifold M . The algebra hor(P ) representing horizontal formscan be constructed by combining standard differential forms on M with the algebraB of ‘smooth functions’ on P , independently of the specifications of the complete

16 MICHO DURDEVICH

calculus on the bundle and the quantum structure group. If the bundle is locallytrivialized over some open set, then hor(P ) will be locally trivialized in a naturalmanner, too.

It turns out that there exists a natural bijection between the elements of der(P ),and standard connections on the classical part Pcl of P . The right A-ideal R̂consists precisely of those elements a ∈ ker(�) satisfying

(X ⊗ id)ad(a) = 0,for each X ∈ lie(Gcl). Here the elements of lie(Gcl) are understood as hermitianfunctionals X : A → C satisfying X(ab) = �(a)X(b) +X(a)�(b).

Hence, R̂ determines the minimal admissible (bicovariant *-) calculus on G,in the terminology of [6]. In other words, Φ is the minimal left-covariant first-order calculus on G compatible, in a natural manner, with all local ‘transitionfunctions’ associated to the bundle P . Let Ω(P ) be the graded-differential *-algebraconstructed from Φ and P , with the help of G-cocycles [6]. Then the identity mapon B extends to the graded-differential *-isomorphism between ΩP and Ω(P ).

The mentioned local trivializability condition does not respect the classical limit.If G is a classical group then Φ will be a classical calculus. However, if we considerthe quantum SU(2) group [17], the calculus Φ will be infinite-dimensional (for anynon-trivial quantum principal G-bundle P ), with Φinv naturally isomorphic [6] tothe space of polynomial functions over a quantum 2-sphere [14].

Now, we shall illustrate the presented general formalism on the example of thequantum Hopf fibration. This is a quantum U(1) bundle over a quantum 2-sphere[14]. The bundle space P is the quantum SU(2) group [17]. We follow here theexample presented in Section 6 of [7], in a slightly modified way.

By definition [17], the *-algebra B is generated by elements α and γ and thefollowing relations

αα∗ + µ2γγ∗ = 1 α∗α+ γ∗γ = 1

αγ = µγα αγ∗ = µγ∗α γγ∗ = γ∗γ.

Next, the fundamental representation is given by

u = (u†)−1 =(α −µγ∗γ α∗

)where µ ∈ (−1, 1) \ {0}.

The structure group G = U(1) is the classical part of the quantum SU(2), andthere exists a canonical *-Hopf-algebra epimorphism j : B → A, specified by j(α) =z and j(γ) = 0 where z is the canonical unitary generator of U(1).

Within this example, primed quantum group entities refer to the quantum SU(2)group (the total space of our bundle). The right action F : B → B ⊗ A is given byF = (id ⊗ j)φ′.

Let Υ be the 3D first-order calculus over P constructed in [17]. By definition,this calculus is based on a right B-ideal

R′ = gen{γ2, γγ∗, γ∗2, αγ − γ, αγ∗ − γ∗, µ2α+ α∗ − (1 + µ2)1

}.

The space Υinv is 3-dimensional. It is spanned by elements

η3 = π′(α− α∗) η+ = π′(γ) η− = π′(γ∗).


The associated right B-module structure ◦ is specified byµ2η3 ◦ α = η3µη± ◦ α = η±

η3 ◦ α∗ = µ2η3η± ◦ α∗ = µη±

with Υinv ◦ γ = Υinv ◦ γ∗ = {0}. This right B-module structure is naturallyprojectable to the right A-module structure on Υinv. Furthermore, it is possible tointroduce, in a natural manner, a right adjoint action κ : Υinv → Υinv ⊗A of G onthis calculus. Explicitly

κ(η3) = η3 ⊗ 1, κ(η−) = η− ⊗ z2, κ(η+) = η+ ⊗ z−2.Let horP be a graded *-algebra generated by the elements η±, the algebra B and

the following relations:

η±b = K(b)η±,(4.1)

η+η− = −µ2η−η+,(4.2)η2+ = η

2− = 0,(4.3)

where b ∈ B, and K : B → B is an automorphism given by K = (id ⊗ �µ)F , while�µ : A → C is the character specified by

�µ(z) = µ−1 �µ(z

∗) = µ.

The elements η± are here assumed to be of the grade one, and obviously thenontrivial components of horP are

hor0P = B, hor2P = Bη+η−,

hor1P = Bη+ ⊕ Bη−.The *-structure is induced from B and Υinv (in other words η∗+ = µη− and η∗3 =−η3). The maps κ and F admit a unique common extension F∧ : horP → horP ⊗A,which is a graded *-algebra homomorphism.

Let us also observe that η± ◦ a = �µ(a)η± and hence relations (4.1) can berewritten in the form

η±b =∑

k

bk(η± ◦ ck),

where F (b) =∑

kbk ⊗ ck.

It is easy to see that the formulas

D(b) =∑

k

(bkπ+(ck) + bkπ−(ck)

),

D(η+) = D(η−) = 0

consistently and uniquely define a hermitian first-order antiderivation D : horP →horP . This map is also F

∧-covariant, and in particular it reduces in the correspond-ing fixed-point subalgebra ΩM . Let us denote by

Md : ΩM → ΩM this restriction.

It folows by a direct computation that Md2 = 0. Hence, we can apply the generalconstruction presented in this paper to the quadruplet (horP , F

∧,ΩM ,Md), con-

sidering the single constructed covariant derivative map D (in other words, a 0-dimensional affine space L = {D}).

Next, it follows that the associated calculus overG is 1-dimensional, based on theright A-ideal R̂ = j(R′). The space Φinv is spanned by the element ζ = π(z − z∗).Furthermore we have Φ∧k = {0} for k ≥ 2. The calculus based on Φ∧ differs from

18 MICHO DURDEVICH

the classical differential calculus, because the right A-module structure on Φinv isgiven by

µ2ζ ◦ z = ζζ ◦ z∗ = µ2ζ.

Now applying the construction of the associated differential calculus on the bun-dle, it follows that the algebra ΩP is generated by horP and ζ, together with thefollowing relations:

µ4ζη+ = −η+ζζη− = −µ4η−ζ

ζ2 = 0.

If we identify the element ζ with the element η3 ∈ Υinv, we see that the wholegraded *-algebra ΩP can be naturally identified with the universal envelope Υ

∧,which is explicitly constructed in [17]. Moreover, this identification intertwines thecorresponding differentials Pd : B → Ω1P and d : B → Υ. Hence, by the universalityof Υ∧ it follows that graded differential *-algebras ΩP and Υ

∧ are isomorphic.The curvature of the canonical connection ω is given by

Rω(ζ) = dη3 = µ(1 + µ2)η−η+.

The presented example fits into the context of general frame structures [10],mentioned at the beginning of this section. The space V is spanned by η+ and η−.

Let us finally turn back to the context of general quantum principal bundles. Thecalculus on G given by the presented construction may be very complicated, even ifG is a classical group. This can be interpreted as a reflection of a topological non-triviality of the bundle. Actually, Φ is the minimal invariant calculus addopted tothe construction of characteristic classes [8] via the quantum Weil homomorphismW : Σ(G,Φ) → H(ΩM ). Here, Σ(G,Φ) is the G-invariant part of the braided-symmetric algebra built over Φinv. From this point of view it is not surprisingthat Φinv will be infinite-dimensional in sufficiently ‘irregular’ cases. For example,it is possible to construct [8] a quantum line bundle P over a compact smoothmanifold M , by taking the appropriate ‘cross-product’ between a standard linebundle P0 and a diffeomorphism γ : M →M . The operators D will be induced bystandard connections. However, the quantum curvature operators %D will be givenby expressions involving the action of arbitrary polynomials of γ on the standardcurvature forms R ∈ Ω2M . The resulting calculus Φ over G = U(1) will be generallyinfinite-dimensional.

Appendix A. Differential Structures From BraidedExterior Algebras

Let σ : Φ⊗AΦ → Φ⊗AΦ be the canonical flip-over operator [19]. This map isa bicovariant bimodule automorphism. Its ‘left-invariant’ part σ : Φ⊗2inv → Φ⊗2inv isgiven by

σ(η ⊗ ϑ) =∑

k

ϑk ⊗ (η◦ck)(A.1)

where∑

kϑk ⊗ ck = $(ϑ). Let Φ∨ be the corresponding braided exterior algebra

[19]. By definition, Φ∨ can be obtained by factorizing the ‘tensor bundle’ algebra


Φ⊗ through the (bicovariant *-) ideal S∨ = ker(A) where

A =∑n≥0

⊕An

is the corresponding ‘total antisymetrizer’. The maps An : Φ⊗n → Φ⊗n are given

byAn =

∑π∈S

n

(−1)πσπ.

Here σπ : Φ⊗n → Φ⊗n are obtained by replacing transpositions in a minimal de-

composition of π with the corresponding σ-twists.The space horP ⊗Φ∨inv = vh

∨P possesses a natural *-algebra structure (expressed

by the same formulas (3.7)–(3.8)). As shown in [7]–Appendix A the formulas (3.9)–(3.10) consistently determine a first-order hermitian differential ∂D on vh

∨P , for each

D ∈ der(P ). Now we shall prove that differential algebras (vh∨P , ∂D) are naturallyisomorphic.

Let us consider the *-algebra qP = horP ⊗ Φ⊗inv (the *-structure is specified

by the same formulas as for vhP ). For each E ∈−→der(P ) there exists a unique

*-automorphism hE : qP → qP satisfyinghE(ϕ) = ϕ

hE(ϑ) = ϑ− χE(ϑ)for each ϕ ∈ horP and ϑ ∈ Φinv. Moreover

hE+W = hEhW

for each E,W ∈ −→der(P ). We shall prove thathE(horP ⊗ S∨inv) = horP ⊗ S∨inv

for each E ∈ −→der(P ). Applying (A.1) and elementary properties of maps χE we find

h−E(ϑ) =∑

k+l=n

(χ⊗kE ⊗ idl)Akl(ϑ) =

∑k+l=n

1k!

(χ⊗kE Ak ⊗ idl)Akl(ϑ)(A.2)

for each ϑ ∈ Φ⊗ninv , where χ⊗E : Φ⊗inv → horP is the corresponding unital multiplicative

extension, andAkl =

∑π∈S

kl

(−1)πσπ−1 .

Here Skl ⊆ Sk+l is consisting of permutations preserving the order of sets {1, . . . , k}and {k + 1, . . . , k + l}, (and σπ are restricted to Φ

⊗inv).

In particular if ϑ ∈ S∨inv then hE(ϑ) ∈ horP ⊗ S∨inv, because ofAk+l = (Ak ⊗Al)Akl.

Thus, the maps hE can be ‘factorized’ through S∨inv. In such a way we obtain

*-automorphisms hE : vh∨P → vh

∨P . We have

hE∂D = ∂D+EhEfor each D ∈ der(P ). For simplicity, we have denoted by the same symbols basicmaps operating in different spaces.

As a consequence of the universality of Φ∧, we can write

Φ∨ = Φ∧/[S∨]∧,

20 MICHO DURDEVICH

where [ ]∧ denotes the image in Γ∧, under the factorization from Γ⊗ to Γ∧. Thealgebra vh∨P can be obtained by factorizing vhP through a graded *-ideal

IP = horP ⊗ [S∨inv]∧.

The ideal IP is hE-invariant, for each E ∈−→der(P ).

The rest of the construction of the corresponding invariant calculus on the bundleis the same as in the universal case. In such a way we obtain a graded-differential*-algebra ΛP . We have

ΛP = ΩP /ÎPwhere ÎP is a graded-differential *-ideal corresponding to IP .

Appendix B. Extension to Non-Compact Structure Quantum Groups

In this Appendix we shall see how the basic results of the paper can be incorpo-rated into the context of bundles with arbitrary (not necessarily compact) quantumstructure groups.

Let us assume that G is an arbitrary quantum group, represented by a Hopf*-algebra A. For a given quantum principal G-bundle P = (B, i, F ) let us considera map Ξ: B ⊗V B → B ⊗A, defined by

Ξ(b ⊗ q) = bF (q).One of the conditions forming the concept of a quantum principal bundle is thatthe map Ξ is surjective. This geometrically means that G acts freely on P .

If Ξ is in addition injective, then we are in the framework of Hopf-Galois exten-sions [16]. If the quantum structure group G is compact then the map Ξ is alwaysbijective [11]. On the other hand, if G is non-compact then generally Ξ fails to beinjective.

It turns out that bijectivity of Ξ is essential for generalizing in a simple way thepresented formalism to the level of non-compact quantum groups.

Let P = (B, i, F ) be a quantum principal G-bundle such that Ξ is bijective.Then we can introduce the translation [1] map τ : A → B ⊗V B, by the formula

τ(a) = Ξ−1(1 ⊗ a).

In our main context of compact matrix quantum structure groups, the translationmap is explicitly given by

τ(uαij) =∑

k

bα∗ki ⊗ bαkj .(B.1)

The translation map possesses the following elementary algebraic properties:

τ(a)∗ = τ [κ(a)∗] [a]1[a]2 = �(a)1(B.2)

(id ⊗ F )τ(a) = τ(a(1)) ⊗ a(2)(B.3)τ(ab) = [b]1[a]1 ⊗ [a]2[b]2(B.4)

(Fop ⊗ id)τ(a) = a(1) ⊗ τ(a(2))(B.5)[a(2)]1[κ(a

(1))]1 ⊗ [κ(a(1))]2 ⊗ [a(2)]2 = 1 ⊗ [a]1 ⊗ [a]2(B.6)where we have used the symbolic notation τ(a) = [a]1 ⊗ [a]2, and Fop : B → A⊗ Bis the opposite action.


Let us also assume that P is equipped with a horizontal *-algebra horP and theappropriate differential Md : ΩM → ΩM , and finally let us assume that a naturalextension Ξ: horP ⊗Ω

MhorP → horP ⊗A is bijective, too.

The following commutation property holds:

τ(a)w = wτ(a) ∀w ∈ ΩM .(B.7)

Let zhP ⊆ horP be the graded *-subalgebra consisting of the elements whichgraded-commute with ΩM . In other words,

zhP ={ϕ ∈ horP

∣∣∣ ϕw = (−1)∂w∂ϕwϕ, ∀w ∈ ΩM}.(B.8)The algebra zhP is F∧-invariant.

Proposition B.1. (i) The formula

ϕ ◦ a = [a]1ϕ[a]2(B.9)consistently determines a right A-module structure in the space zhP .

(ii) The following identities hold

F∧(ϕ ◦ a) =∑

k

(ϕk ◦ a(2)) ⊗ κ(a(1))cka(3)(B.10)

(ϕψ) ◦ a = (ϕ ◦ a(1))(ψ ◦ a(2))(B.11)(ϕ ◦ a)∗ = ϕ∗ ◦ κ(a)∗,(B.12)

where∑

kϕk ⊗ ck = F∧(ϕ). In other words, the quadruplet (zhP , F∧, ◦, ∗) canoni-

cally determines a *-algebra Θ which is bicovariant over G. In particular Θinv =zhP and Θ ↔ A⊗ zhP at the level of vector spaces.

Proof. It is clear that formula (B.9) is consistent, because the elements of zhP canbe inserted in tensor products over ΩM . Equality (B.7) gives

(ϕ ◦ a)w = [a]1ϕ[a]2w = (−1)∂ϕ∂ww[a]1ϕ[a]2 = (−1)∂ϕ∂ww(ϕ ◦ a),in other words we have a map ◦ : zhP ⊗A → zhP . This is a right A-module structurebecause

ϕ ◦ (ab) = [ab]1ϕ[ab]2 = [b]1[a]1ϕ[a]2[b]2 = [b]1(ϕ ◦ a)[b]2 = (ϕ ◦ a) ◦ b.Let us check properties (B.10)–(B.12). A direct calculation gives

F∧(ϕ ◦ a) =∑

k

F [a]1(ϕk ⊗ ck)F [a]2 =∑

k

[a(2)]1ϕk[a(2)]2 ⊗ κ(a

(1))cka(3)

=∑

k

(ϕk ◦ a(2)) ⊗ κ(a(1))cka(3).

Furthermore,

(ϕ ◦ a)∗ = [a]∗2ϕ∗[a]∗1 = [κ(a)

∗]1ϕ∗[κ(a)∗]2 = ϕ

∗ ◦ κ(a)∗.Finally, we have

(ϕψ) ◦ a = [a]1(ϕψ)[a]2 = [a(1)]1ϕ[a(1)]2[a(2)]1ψ[a(2)]2 = (ϕ ◦ a(1))(ψ ◦ a(2)),which completes the proof.

22 MICHO DURDEVICH

The introduced ◦-structure on zhP includes as a special case the right-modulestructure considered in [6], introduced with the help of local trivializations. Wehave the following commutation relation

ξϕ = (−1)∂ϕ∂ξ∑

k

ϕk(ξ ◦ ck)(B.13)

between the elements of zhP and arbitrary horizontal forms ϕ. The space zhP isD-invariant, for each D ∈ der(P ).

Proposition B.2. The maps %D, χE : A → horP are zhP -valued and satisfy theequalities

%D(ab) = �(a)%D(b) + %D(a) ◦ b(B.14)%D(a) = −[a]1D

2[a]2(B.15)

χE(a) = −[a]1E[a]2(B.16)χE(ab) = �(a)χE(b) + χE(a) ◦ b.(B.17)

Proof. It is instructive to perform the proof in a more general context. Let us firstconsider a homogeneous linear map ∆: horP → horP satisfying

∆(wϕ) = (−1)∂w∂∆w∆(ϕ)(B.18)for each w ∈ ΩM and ϕ ∈ horP . Let µ : A → horP be a linear map given by

µ(a) = [a]1∆[a]2.(B.19)

The map µ completely determines ∆, because of

∆(ϕ) = (−1)∂ϕ∂∆∑

k

ϕkµ(ck).(B.20)

We can now re-express various properties of ∆ in terms of µ. At first, ∆ inter-twines the action F∧ iff µ intertwines ad: A → A⊗A and F∧. Secondly, ∆ will beright ΩM -linear iff µ takes the values from zhP .

Let us assume that ∆ is also right ΩM -linear, and consider the graded Leibnizrule property

∆(ϕψ) = ∆(ϕ)ψ + (−1)∂ϕ∂∆ϕ∆(ψ).(B.21)The above equality will be satisfied iff

µ(ab) = �(a)µ(b) + µ(a) ◦ b.(B.22)

Indeed (B.21) together with (B.19) gives

µ(ab) = [b]1[a]1∆{[a]2[b]2} = [b]1µ(a)[b]2 + [b]1[a]1[a]2∆[b]2= �(a)µ(b) + µ(a) ◦ b.

On the other hand (B.22) together with (B.20) and commutation relation (B.13)implies (B.21).

To complete this analysis let us observe that a graded-derivation ∆ will be her-mitian iff µ(a)∗ = −µ[κ(a)∗], for each a ∈ A.

Equations (B.15)–(B.16) are definitions of maps %D and χE . Relations (B.14)and (B.17) tell us that after factorizing the maps %D and χE through the ideal R̂,we obtain right A-module homomorphisms between Φinv and zhP .


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Instituto de Matematicas, UNAM, Area de la Investigacion Cientifica, Circuito Ex-terior, México DF, cp 04510, MEXICO

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