geometry of quantum principal bundles iii structure …micho/papers/bundles3.pdf · 2 micho...

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES IIIStructure of Calculi and Around

MICHO DURDEVICH

Abstract. We present a general constructive approach to differential calculuson quantum principal bundles. This includes a complete structural analysis

of graded differential *-algebras describing horizontal forms, the calculus onthe base, and the complete algebra of connections and covariant derivatives.

A particular attention is given to purely quantum phenomena appearing in

the theory, which include the deviation of regularity operator and a residualquantum term of the curvature, representing the obstacle to multiplicativity

of the connection form. The concept of a universal horizontal envelope of

a first-order horizontal calculus is introduced and investigated. This can beviewed as a generalization of the universal differential envelope of a first-order

differential calculus. We obtain in a constructive way a graded *-algebra of

horizontal forms and the complete differential calculus on the base quantumspace. Applications of the formalism, and illustrative examples are discussed.

Contents

1. Introduction 2

2. General Geometro-Algebraic Framework 42.1. Preliminaries2.2. Connections & Covariant Derivatives

3. From Horizontal Calculus to Full Calculus 183.1. Abstract Horizontal Calculi3.2. Elementary Properties of Covariant Derivatives3.3. Beyond Horizontality

4. First-Order Horizontal Calculus 324.1. Naturality of Relations4.2. Horizontal Envelopes4.3. More About Envelopes

5. Concluding Observations & Examples 385.1. Affine Spaces of Connections5.2. Specifics of the Regular Case5.3. Quantum Frame Bundles5.4. Classical Finite Structure Groups

Appendix A. Graded Derivations 53

Appendix B. On Canonical Generating Elements 55

References 59

2 MICHO DURDEVICH

1. Introduction

In the realm of quantum geometry, which is a geometric incarnation of quantumphysics, the game is played by a new kind of spaces. They are based on unification,generalization and simplification, of things having the origins in mutually oppositeand at a first sight contradictory ideas form the old classical world. As part of thisunifying way of thinking, geometric intuition and algebraic manipulations go handby hand. Because quantum spaces do not in general possess parts nor points, astandard reductionistic approach to define and express objects and their propertiesin terms of coordinates and local formulae, is not feasible. However, a new kind oftechnical paradigm, involving calculations having a charm of their own, inviting itsproper way of thinking about quantum spaces, emerges naturally, at all the levelsbetween concrete examples and abstract and general considerations.

This treatise can be considered as a technical supplement to the main theory ofquantum principal bundles [2, 3, 11]. It contains a selection of detailed calculationsand its focus is on the general geometro-algebraic situations, where connections anddifferential structures are not restricted by any special condition, being far awayfrom interesting properties that are the foundation of some of the most importantexamples of quantum spaces, groups and bundles.

On the other hand, the general framework can be interpreted as a kind of ‘per-turbation’ of a more restricted closer-to-classical theory, by viewing general thingsin terms of new entities appearing in the game. This allows us to envision the gen-eral context in terms of an array of interesting quantum phenomena. Specifically,we shall spend a considerable amount of time in analyzing non-regular connections.The regular connections are most closely related to the classical idea of a connectionform–as the associated covariant derivative is a hermitian antiderivation. A veryinteresting thing is that although a given quantum principal bundle could featureno regular connections at all, we can measure a deviation of regularity of an ar-bitrary connection ω in terms of a simple algebraic commutator-like expression `ωwhich nicely twists ω and horizontal forms. The values of this twisted commutatorare always horizontal, a counter-intuitive fact taking into account the ‘verticality’of ω. Another similar phenomenon is that certain quadratic combinations of theconnection (the ones arising from the quadratic relations defining the left-invariantforms on the structure quantum groups) are horizontal instead of being inherentlyvertical. Surprisingly, these combinations can be understood as a residual ‘purelyquantum’ part of the curvature tensor. As we shall see, this residual part is a neces-sary ingredient in the complete set of algebraic relations defining a general calculuson the bundle. There are interesting physical implications, involving unificationbetween Yang-Mills fields and Higgs fields, and originating from this expression.

So all the formulae for regular case, as Leibniz rule, *-compatibility, Bianchiidentity, Structure Equation, remain valid in the general case, in a modified formthat includes such ‘purely quantum’ terms.

The paper is organized in the following way. In the next section, after brieflydescribing the main geometric setup, we shall delve into a detailed analysis ofalgebraic interrelations of various operators naturally associated to a connectionform on a quantum principal bundle P . Such as the covariant derivative Dω, theassociated curvature tensor rω and the non-regularity operator `ω.

All these objects have been introduced previously, here we shall focus on theirmore subtle interrelations involving ‘purely quantum ’ properties. In particular we

quantum principal bundles 3

shall analyze the modifications and diverse versions of a general Leibniz rule for Dω

and its conjugateD∗ω, as well as `ω and rω. It turns out that the established relationsform a complete set, in the sense that they fully determine the differential calculuson the bundle, together with a framework for the calculus on the structure quantumgroup, knowing the calculus on the base and the ‘halo’ algebra of horizontal forms.

This theme is fully developed in Section 3, where we start from an abstractgraded *-algebra horP , representing ‘horizontal differential forms’ equipped witha right quantum group action F : horP → horP ⊗ A and a ‘covariant derivative’operator D : horP → horP . It is always possible to construct such a covariantderivative, by extending the differential on the base dM : ΩM → ΩM to the wholehorP . Here ΩM is the F -fixed point subalgebra of horP . We then construct thewhole calculus Ω(P ) on P which enables us to reinterpret D as the true covariantderivative associated to a natural connection ω. In the process, we shall describea class of natural bicovariant *-covariant differential calculi on G, compatible in anatural way, with the calculus on the bundle. The residual part of the curvaturetensor, and the non-regularity operator play an essential role in defining the algebraof differential forms on the bundle P .

In Section 4 we go further by establishing a universality property of the con-structed algebras. We do so by explicitly deriving the defining relations of thecalculus, as a logical consequence of embedding the horizontal forms in a suitabledifferential algebra. This property implies independence of the calculus on the ini-tial choice of the covariant derivative. We also construct explicitly isomorphismsbetween calculi associated to different covariant derivatives.

Another important topic of Section 4 is the problematics of extending a first-order calculus to a higher-order calculus. Our starting point will be a *-bimoduleover the bundle, equipped with a quantum group symmetry and a differential act-ing on the base space algebra V. The horizontal envelope is a graded *-algebra ofall ‘quantum horizontal forms’ and in particular we obtain the complete differentialcalculus on the base quantum space M . The construction can be interpreted asa bundle-type refinement of the construction of universal differential envelopes offirst-order calculi [2]-Appendix B. However, in contrary to the universal differen-tial envelopes which are quadratic algebras, the calculi on the base associated tothese universal horizontal envelopes generally exhibit a hierarchy of quadratic andhigher-order relations, appearing as a compatibility requirement between differen-tial structure and bundle structural morphisms ρ : V → Md(V).

Some interesting examples and concluding constructions are collected in Sec-tion 5. We start by discussing properties emerging from the affine space of allcovariant derivatives/connections. We shall see that the displacements from oneconnection to another, generate automorphisms of the differential calculus, pro-viding us with an effective invariant description. We shall also see that in orderto accommodate all covariant derivatives, the structure quantum group G mustbe equipped with the maximal calculus–the universal one. In a sense this prop-erty can be interpreted as justification of all modes of possible differential calculion G. Even in the case when both bundle P and the group G are classical, westill have to admit ‘quantum’ connections, being quantum here measured by thedeviation of the global graded Leibniz rule. A very important special case of regularconnections is discussed separately, and the construction of horizontal envelopes isformulated independently. Here we shall apply systematically a kind of extended

4 MICHO DURDEVICH

bimodule technique in studying the corresponding horizontal envelope. In the clas-sical case, the affine space of regular connections generates precisely the classicalcalculus on the structure group, and all covariant derivatives/connections are clas-sical. Another important illustration of the general formalism is given by quantumframe bundles, where the entire calculus can be constructed on base of a covariantsystem of coordinate derivations. Finally, we shall dedicate some time to discussdifferential structures and associated objects, in the case of finite classical struc-ture groups. This context provides a rich class of purely quantum examples andcounterexamples.

The paper ends with two appendices having an independent conceptual interest.In the first one, some simple algebraic properties of general covariant left and rightderivatives are collected. We show that there exists a natural duality betweenthese two types of derivatives, in the sense that every left derivative determinesa right derivative and vice versa. Therefore, both of them represent one and thesame geometrical object–a quantum version of the classical covariant derivative. Inthe second Appendix, we present a complete construction of canonical multiplets ofgenerating elements for a quantum principal bundle. We shall be using C*-algebraicproperties of the algebra representing the quantum base space, in order to provethe existence of these canonical multiplets. They can be interpreted as pull-backsof the generators of the corresponding quantum classifying space.

2. General Geometro-Algebraic Framework

2.1. Preliminaries

In this section, we are going to introduce and recall the main ideas of thegeometro-algebraic setup for quantum principal bundles. We refer to [2, 3, 11]for a general theory. We are in a category of quantum spaces where spaces arerepresented by certain generally non-commutative complex *-algebras, and theirtransformations contravariantly represented by appropriate algebraic morphisms.When our algebras are commutative, we are in a restricted context equivalent tostandard differential geometry and topology. In such a quantum context, objectscorresponding to groups are represented by appropriate Hopf *-algebras, the Hopfalgebra structure being interpreted as the reflection of a group structure, in accor-dance with a general theory of quantum groups developed in [17, 18] (which in manyaspects applies far beyond its original framework devoted to compact structures).

Let us consider a quantum space M represented by a unital *-algebra V and aquantum space P represented by a unital *-algebra B. The idea of a simple fibrationof P over M is achieved by considering a unital *-monomorphism i : V → B. Thisfibration corresponds to a principal bundle, if there exists a quantum group Gacting freely by ‘symmetries’ of P , so that the invariant subalgebra of B coincideswith the image of V.

The quantum groupG will be represented by a Hopf *-algebraA. The elements ofA are interpreted as ‘polynomial functions’ on G. We shall denote by φ : A → A⊗Athe coproduct, while ε : A → C and κ : A → A are the counit and the antipode.We shall be using a standard Sweedler symbolic notation φn(a) = a(1) ⊗ · · · ⊗ a(n)

for the n-fold coproduct of a ∈ A so that in particular we have

ε(a)1 = κ(a(1))a(2) = a(1)κ(a(2)) a = ε(a(1))a(2) = a(1)ε(a(2))


Now we like so see G acting by symmetries of P . The corresponding algebraicsetup is given by a unital *-homomorphism F : B → B ⊗A modelling the idea of aright action P ×G P of G on P . In accordance with this

(id⊗ φ)F = (F ⊗ id)F (id⊗ ε)F = id

and as mentioned a minute ago, the subalgebra V is recovered as the F -fixed pointsubalgebra of B, that is V =

f ∈ B

∣∣ F (f) = f ⊗ 1

. The functions on thebase are those functions on the bundle, constant along the fibers. Equivalently,invariant under right translations by the structure group symmetries. We shalluse an extended symbolic notation, for the diverse right actions on algebras andmodules, appearing throughout this study–by using index 0 to label the componentof the transforming object belonging to the algebra or module, and positive indexesto label the components of A. So for example F (b) = b(0) ⊗ b(1) and the n-foldaction would read b(0) ⊗ · · · ⊗ b(n).

Finally, the group G is acting ‘freely’ on P in the sense that a map q⊗b 7→ qF (b)between B ⊗ B and B ⊗A is surjective. This is a straightforward generalization ofthe classical freeness condition expressed by injectivity of the map P ×G→ P ×Pgiven by (p, g) 7→ (p, pg). In such a way we arrive to the concept of a quantumprincipal G bundle over a quantum space M , these objects being represented bytriplets P = (B, i, F ). In fact a much more elaborated structure emerges, as theabove map can be considered ‘fiber by fiber’ and in such a way we project it downto B ⊗V B where it can be shown that it is always an isomorphism

(2.1) B ⊗V B! B ⊗A P ×M P ! P ×G

This enables us to consider the associated translation map τ : A → B⊗V B which isthe only non-trivial part of the above isomorphism, so that τ(a)↔ 1⊗a. Classicallythe translation map transforms (p, pg) 7→ g it is just the second coordinate projec-tion of the natural product identification. Following [9] we shall write symbolicallyτ(a) = [a]1 ⊗ [a]2. It is worth recalling the following elementary identities

[a(1)]1 ⊗ [a(1)]2 ⊗ a(2) = [a]1 ⊗ F [a]2

a(1) ⊗ [a(2)]1 ⊗ [a(2)]2 = F?[a]1 ⊗ [a]2[a]1[a]2 = ε(a)1

f [a]1 ⊗ [a]2 = [a]1 ⊗ [a]2f

[a]∗2 ⊗ [a]∗1 = [κ(a)∗]1 ⊗ [κ(a)∗]2where f ∈ V and F? : B → A ⊗ B is a left action associated to F , explicitly givenby F?(b) = κ−1(b(1))⊗ b(0). In addition a kind of multiplicativity equation holds

[c]1[a]1 ⊗ [a]2[c]2 = [ac]1 ⊗ [ac]2for every a, c ∈ A.

To make our geometrical setup complete, we shall also assume important ana-lytical properties for algebras V and B. Namely, we shall imagine B as embeddedas everywhere dense *-subalgebra of a C*-algebra B. The map F is assumed to becontinuously extendible to F : B → B ⊗ A where A = A is the C*-algebra of Gand the tensor product is the standard C*-algebraic one. In the algebra B we canconsider the closure of V. This C*-subalgebra V consists exactly of all F -invariantelements of B. In what follows we shall also assume that V is stable in V under

6 MICHO DURDEVICH

holomorphic functional calculus. This is a nice way to generalize the idea of asmooth function.

Remark 1. As far as the very idea of a quantum space is concerned, we fol-low a general conceptual framework established in [16]. At the topological level,the spaces are described by appropriate C*-algebras, intuitively interpretable, inthe spirit of classical Gelfand-Neimark theory, as ‘continuous functions’ vanish-ing at infinity, over the corresponding spaces. So unital algebras correspond tocompact spaces, and the idea of a locally compact non-compact quantum spaceis sculptured over non-unital C*-algebras. The ‘continuous maps’ between spacesX and Y , represented by C*-algebras A and B respectively, correspond to unital*-homomorphisms between multiplier algebras M(B) and M(A) which are in ad-dition continuous in the natural strict topology. In the classical ⇔ commutativecase, the multiplier algebras correspond to bounded continuous functions, and theirspectrums being identified with Stone-Cech compactifications of the original topo-logical spaces. More subtle geometrical structures, giving a finer-than-topologicaldescription, are given by appropriate dense *-subalgebras, as the ‘smooth’ functionsmentioned above. So this would be a sketch of our ‘universe’ of quantum spaces.Such a formulation is not equivalent to the one presented in [1] where the focus is oncyclic cohomology and K-theory, and in particular Morita equivalent C*-algebrasimply essentially the same geometrical contents. A very interesting and entirelydifferent way to quantize geometry has been developed systematically in [15] wherea quantum generalization is made via a change of the concept of point, into a sto-chastic point, a kind of a probabilistic entity possessing an operational meaning ofaverage values of coordinates.

Remark 2. In accordance with the interpretation of V as smooth functions on thebase, and A as polynomial functions on G the algebra B corresponds to smoothfunctions on the bundle which are polynomial along the fibers.

As a consequence of these analytical conditions, it can be demonstrated that forevery irreducible (matrix and unitary) representation u ∈ Mn(A) of G there existsa natural number d and a double-indexed multiplet of elements bαi ∈ B where1 ≤ α ≤ d and 1 ≤ i ≤ n such that

(2.2) F (bαj) =n∑i=1

bαi ⊗ uijd∑

α=1

b∗αibαj = δij

as discussed in more detail in Appendix A.

Remark 3. The lower bound for such numbers d has a geometrical interpretationof a topological complexity of the bundle P . The above relations are crucial forestablishing a general theory of quantum classifying spaces, as explained in [5].

2.2. Connections & Covariant Derivatives

In the previous subsection we have established the basic conceptual frameworkfor quantum principal bundles P = (B, i, F ). Let us recall the interpretation. The*-algebra B defines P as a quantum space, while the *-algebra V corresponds tothe base quantum space M . The natural inclusion i : V → B is interpreted as a‘fibration’ of P over M . There is a symmetry of P by a quantum group G via


F : B → B ⊗A and the base space algebra V consists precisely of all elements of Bthat are invariant under this action F .

The differential calculus on P essentially repeats this scheme, at the level ofgraded-differential *-algebras. As explained in detail in [3] such a calculus isspecified by a graded differential *-algebra Ω(P ) over B, its differential beingdP : Ω(P )→ Ω(P ), together with a differential morphism F : Ω(P )→ Ω(P )⊗Γ∧ ex-tending the group action F . Here Γ∧ is the enveloping differential algebra of a givenbicovariant *-covariant first-order calculus Γ over G, as explained and constructedin [2]–Appendix B. So a new kind of ‘supersymmetry’ emerges

(id⊗ ε)F = id (F ⊗ id)F = (id⊗ φ)F

where φ : Γ∧ → Γ∧ is the graded differential extension of the coproduct and thecounit ε is extended trivially. The geometrical interpretation of F and φ is thatthey are pullbacks, at the level of differential forms, of P ×G P and G×G G.

Let us recall that for a given such a differential calculus Ω(P ), the associatedalgebra of horizontal forms hor(P ) is defined as

hor(P ) =w ∈ Ω(P )

∣∣∣ F (w) ∈ Ω(P )⊗A

and the restriction of F to hor(P ) induces a map F∧ : hor(P ) → hor(P )⊗A. Thegraded differential *-algebra Ω(M) representing the calculus on the base quantumspace M is defined as the F∧-invariant elements of hor(P ) or the F -fixed-pointsubalgebra of Ω(P ). We shall denote by dM : Ω(M) → Ω(M) the correspondingrestricted differential.

Remark 4. We have Ω0(P ) = B = hor0(P ). On the other hand Ω0(M) = V. Asa differential algebra, it is assumed that Ω(P ) is generated by B. This means that

Ωn(P ) =∑

bdP (b1) . . . dP (bn) =∑

dP (b1) . . . dP (bn)b

however in general Ω(M) will not be generated by its 0-th part V although thereis an array of important special contexts where this property holds true (as hori-zontal envelopes–one of our principal topics for this study). The geometrical ideabehind the definition of horizontal forms is that they only exhibit trivial differentialproperties along the ‘vertical fibers’ of P .

Differential calculus on P extends canonical isomorphism (2.1) to the level ofdifferential algebras. Namely

(2.3) Ω(P ) ⊗Ω(M) Ω(P )! Ω(P ) ⊗ Γ∧

we also have the following natural decompositions of horizontal forms, as easilyfollows by playing with the translation map

(2.4)hor(P )⊗Ω(M) hor(P )! hor(P )⊗A

hor(P )! B ⊗V Ω(M)! Ω(M)⊗V B

and we can write a nice commutation formula

(2.5) [a]1 ⊗ [a]2ϕ = ϕ(0)[aϕ(1)]1 ⊗ [aϕ(1)]2for each ϕ ∈ hor(P ). In particular τ(a)w = wτ(a) for each a ∈ A and w ∈ Ω(M).

8 MICHO DURDEVICH

We shall now focus our attention to the algebraic structures in the horizontalforms realm, induced from a given complete differential calculus on a quantumprincipal bundle P , and connections.

Let ω : Γinv → Ω(P ) be a connection on P . This means [3] that ω is a linearone-forms valued map satisfying

ω(ϑ∗) = ω(ϑ)∗ Fω(ϑ) = (ω ⊗ id)ad(ϑ) + 1⊗ ϑthese conditions generalize the classical idea of a real lie(G)-valued one-form, whichis pseudotensorial and maps fundamental vertical vector fields back into their gen-erators. Here ad: Γinv → Γinv⊗A is the adjoint action of G on Γinv. It is explicitlygiven by

adπ(a) = π(a(2))⊗ κ(a(1))a(3)

where π : A → Γinv is the corresponding ‘quantum germs’ projection map π(a) =κ(a(1))d(a(2)). Let us recall the following interesting properties

π(ab) = ε(a)π(b) + π(a) bπ(a)∗ = −π[κ(a)∗]

dπ(a) = −π(a(1))π(a(2))

d(a) = a(1)π(a(2)) dκ(a) = −π(a(1))κ(a(2))

involving the quantum germs map and the canonical structure on the space Γinvof left-invariant elements of Γ.

Remark 5. The entire formalism of connections is directly inspired by classicalexposition of [13].

The decomposition (2.3) allows us to extend the translation map τ from A tothe whole calculus τ : Γ∧ → Ω(P )⊗Ω(M) Ω(P ). Explicitly

(2.6) τ(ϑ) = 1⊗ ω(ϑ)− ω(ϑ(0))[ϑ(1)]1 ⊗ [ϑ(1)]1for every connection ω where we have extended the indexes notation for the adjointaction so that ad(ϑ) = ϑ(0) ⊗ ϑ(1) = π(a(2)) ⊗ κ(a(1))a(3) for ϑ = π(a). Thepossibility to introduce translation maps (equivalent to the decompositions of theform (2.1) and (2.3)) ensures the existence of the the corresponding right modulestructure on commutants or graded commutants of algebras of invariants, like Vor Ω(M). By definition given a graded *-algebra Y with its graded *-subalgebraX then the graded commutant Z(X ,Y) is the set of elements ζ ∈ Y such thatζξ = (−)∂ξ∂ζξζ for each ξ ∈ X . Such a commutant is always a graded *-subalgebra.For example the elements of Z(V,B) or Z(Ω(M), hor(P )) carry a natural right A-module structure given by

(2.7) ψ a = [a]1ψ[a]2It is easy to verify that the following identities hold:

F∧(ϕ a) = (ϕ(0) a(2))⊗ κ(a(1))ϕ(1)a(3)

(ϕ a)∗ = ϕ∗ κ(a)∗

(ϕψ) a = (ϕ a(1))(ψ a(2))

We also have the following important twisted commutation rule

(2.8) ζϕ = (−)∂ζ∂ϕϕ(0)(ζ ϕ(1)

)


for ζ ∈ Z(Ω(M), hor(P )) and ϕ an arbitrary horizontal form. So we see that thegraded commutant objects, equipped with and the corresponding right action,are equivalent to a graded bicovariant *-algebra [18].

And the elements of Z(Ω(M),Ω(P )) in addition carry a right Γ∧ structure, con-structed with the help of the extended translation map:

(2.9) ζ γ =∑xy

(−)∂x∂ζxζy τ(γ) =∑xy

x⊗ y

and in particular we see that on left-invariant elements ϑ ∈ Γinv it reduces to

(2.10) ζ ϑ = ζω(ϑ)− (−)∂ζω(ϑ(0))(ζ ϑ(1))

The above constructions can be directly generalized to homogeneous additivemaps S : hor(P ) → hor(P ) satisfying S(wϕ) = (−)∂S∂wwS(ϕ) in other words weassume left superlinearity over Ω(M). If we define λS(a) = [a]1S[a]2 then it is easyto verify that

(2.11) S(ϕ) = (−)∂ϕ∂Sϕ(0)λS(ϕ(1))

The map λS is always valued in Z(Ω(M), hor(P )) so in particular we can composeit with the -structure. And the following equivalences hold:

F∧S = (S ⊗ id)F∧ ⇔ F∧λS(a) = λS(a(2))⊗ κ(a(1))a(3)

S(ϕ∗) = S(ϕ)∗ ⇔ λS(a)∗ = −λS [κ(a)∗]

S(ϕψ) = S(ϕ)ψ + ϕS(ψ) ⇔ λS(ab) = ε(a)λS(b) + λS(a) b

These properties will be find useful in various considerations of this study. Let usreturn to connections and focus to associated covariant derivatives. For a givenconnection ω its covariant derivative Dω : hor(P )→ hor(P ) is then defined by

(2.12) Dω(ϕ) = dP (ϕ)− (−)∂ϕϕ(0)ωπ(ϕ(1))

We can also define the sister derivative of Dω by moving the germs part to theleft of ϕ and keeping everything horizontal:

(2.13) D′ω(ϕ) = dP (ϕ) +ωπκ−1(ϕ(1))

ϕ(0)

It is a matter of a direct verification to see that

(2.14) D′ω(ϕ) = Dω(ϕ) + `ω(πκ−1(ϕ(1)), ϕ(0)

)where `ω : Γinv×hor(P )→ hor(P ) is the regularity deviation measure [11] a twistedcommutator between connections and horizontal forms, explicitly given by

(2.15) `ω(ϑ, ϕ) = ω(ϑ)ϕ− (−)∂ϕϕ(0)ω(ϑ ϕ(1))

A very interesting and somewhat surprising property is that the above expressionalways takes values from horizontal forms. It satisfies the following importantidentities

`ω(ϑ, ϕψ) = `ω(ϑ, ϕ)ψ + (−)∂ϕϕ(0)`ω(ϑ ϕ(1), ψ)(2.16)

−`ω(ϑ, ϕ)∗ = `ω(ϑ∗ κ(ϕ(1))∗, ϕ(0)∗)(2.17)

F∧`ω(ϑ, ϕ) = `ω(ϑ(0), ϕ(0))⊗ ϑ(1)ϕ(1)(2.18)

10 MICHO DURDEVICH

Remark 6. Equation (2.17) is telling us how to conjugate the non-regularity ob-stacle, while (2.18) shows that the entire expression is covariant. Property (2.16)is a kind of a twisted graded Leibniz rule. It allows us to link the action of `ω andthe -structure:

(2.19) `ω(ϑ a, ϕ) = [a]1`ω(ϑ, [a]2ϕ)− [a]1`ω(ϑ, [a]2)ϕ

Indeed after (2.16) and some elementary transformations

[a]1`ω(ϑ, [a]2ϕ) = [a]1`ω(ϑ, [a]2)ϕ+ [a]1[a](0)2 `ω(ϑ [a](1)

2 , ϕ)

= [a]1`ω(ϑ, [a]2)ϕ+ `ω(ϑ a, ϕ)

This property will play an important role in the next section, where we shall beapplying a variation on the same theme, in order to (re)construct a natural differ-ential calculus on G, in terms of geometrical entities operating on horizontal formsonly.

It worth emphasizing that, by construction, both operators Dω and D′ω extendthe differential dM : Ω(M)→ Ω(M) to hor(P ).

Proposition 1. (i) The following identities hold:

Dω(ϕψ) =Dω(ϕ)ψ + (−)∂ϕϕDω(ψ) + (−)∂ϕϕ(0)`ωπ(ϕ(1)), ψ

(2.20)

D′ω(ϕψ) =D′ω(ϕ)ψ + (−)∂ϕϕD′ω(ψ) + `ωπκ−1(ψ(1))κ−1(ϕ(1)), ϕ(0)

ψ(0)(2.21)

and in particular

Dω(wϕ) =dM (w)ϕ+ (−)∂wwDω(ϕ)(2.22)

D′ω(ϕw) =D′ω(ϕ)w + (−)∂ϕϕdM (w)(2.23)

for w ∈ Ω(M) and ϕ ∈ hor(P ).

(ii) We have

(2.24) Dω(ϕ)∗ = D′ω(ϕ∗)

for all horizontal forms.

Proof. We compute

Dω(ϕψ) = dP (ϕψ)− (−)∂ϕ+∂ψϕ(0)ψ(0)ωπ[ϕ(1)ψ(1)]

= dP (ϕ)ψ + (−)∂ϕϕdP (ψ)− (−)∂ϕ+∂ψϕ(0)ψ(0)ω[π(ϕ(1)) ψ(1)

]− (−)∂ϕ+∂ψϕψ(0)ωπ(ψ(1))

= Dω(ϕ)ψ + (−)∂ϕϕDω(ψ) + (−)∂ϕϕ(0)`ωπ(ϕ(1)), ψ


where we have used a Leibniz identity π(ab) = ε(a)π(b) + π(a) b for quantumgerms. Similarly,

D′ω(ϕψ) = dP (ϕ)ψ + (−)∂ϕϕdP (ψ) + ωπκ−1(ϕ(1)ψ(1))ϕ(0)ψ(0)

= dP (ϕ)ψ + (−)∂ϕϕdP (ψ) + ωπ[κ−1(ϕ(1))κ−1(ψ(1))]ϕ(0)ψ(0)

= d(ϕ)ψ + (−)∂ϕϕd(ψ) + ωπκ−1(ϕ(1))ϕ(0)ψ+

+ ω[πκ−1(ψ(1)) κ−1(ϕ(1))]ϕ(0)ψ(0)

= D′ω(ϕ)ψ + (−)∂ϕϕD′ω(ψ) + aω[πκ−1(ψ(1)) κ−1(ϕ(1))]−

− (−)∂ϕϕ[ωπκ−1(ψ(1))]ψ(0)

= D′ω(ϕ)ψ + (−)∂ϕϕD′ω(ψ) + `ωπκ−1(ψ(1)) κ−1(ϕ(1)), ϕ(0)

ψ(0)

The identity (2.24) is a direct consequence of the defining equations for Dω andD′ω and the reality properties for ω and dP : Ω(P ) → Ω(P ). Equations (2.20) and(2.21) are obtainable one from another by conjugating and applying (2.17).

Because of (2.24) we shall also write

(2.25) D′ω = D∗ω = ∗Dω∗

From the fact that B generates the whole calculus, and expressions for the covariantderivative maps, we conclude that

(2.26) hor1(P ) =∑

bDω(q)∣∣∣ b, q ∈ B =

∑D∗ω(b)q

∣∣∣ b, q ∈ Band later on this will be complemented by much more detailed information aboutgenerating elements of horizontals, in all orders.

There is another more subtle connection between Dω and D∗ω expressed in termsof the translation map. Let us observe that partial Leibniz rules (2.22) and (2.23)ensure the consistency of a map D2,ω : hor(P )⊗Ω(M)hor(P )→ hor(P )⊗Ω(M)hor(P )defined by

(2.27) D2,ω(ϕ⊗ ψ) = D∗ω(ϕ)⊗ ψ + (−)∂ϕϕ⊗Dω(ψ)

Indeed the expression factorizes through the tensor product over Ω(M) which alsoensures its consistency:

ϕw ⊗ ψ − ϕ⊗ wψ D∗ω(ϕw)⊗ ψ + (−)∂ϕ+∂wϕw ⊗Dω(ψ)

−D∗ω(ϕ)⊗ wψ − (−)∂ϕϕ⊗Dω(wψ)

= D∗ω(ϕ)w ⊗ ψ + (−)∂ϕϕdM (w)⊗ ψ + (−)∂ϕ+∂wϕw ⊗Dω(ψ)

−D∗ω(ϕ)⊗ wψ − (−)∂ϕ+∂wϕ⊗ wDω(ψ)− (−)∂ϕϕ⊗ dM (w)ψ = 0

Proposition 2. We have

(2.28) D2,ω

([a]1 ⊗ [a]2) = D∗ω

[a]1⊗ [a]2 + [a]1 ⊗Dω

[a]2

= 0

for all a ∈ A. In particular,

(2.29) D∗ω

[a]1

[a]2 + [a]1Dω

[a]2

= 0

12 MICHO DURDEVICH

Proof. Using explicit formulae for Dω and D∗ω, transformation properties of thetranslation map and the identity [a]1[a]2 = ε(a) we obtain

D∗ω

[a]1

[a]2 + [a]1Dω

[a]2

= d

[a]1

[a]2 + ωπκ−1(k(a(1))

)[a(2)]1[a(2)]2

− [a]1d

[a]2− [a(1)]1[a(1)]2ωπ(a(2)) = d

[a]1[a]2

+ ωπ(a(1))[a(2)]1[a(2)]2 − [a(1)]1[a(1)]2ωπ(a(2)) = ε(a)d(1) + ωπ(a)− ωπ(a) = 0

which proves (2.29). Now let us transform the middle term of (2.28) by the canon-ical bijection between hor(P )⊗Ω(M) hor(P ) and hor(P )⊗A. We obtain

D∗ω

[a]1⊗ [a]2 + [a]1 ⊗Dω

[a]2 D∗ω

[a(1)]1

[a(1)]2 ⊗ a

(2)

+ [a(1)]1Dω

[a(1)]2

⊗ a(2) = 0

according to (2.29). Therefore (2.28) holds.

A very interesting and important (and surprising perhaps) fact is that the oper-ator `ω is completely determined by the covariant derivative Dω. Indeed,

Proposition 3. If a ∈ ker(ε) then

(2.30) `ω(π(a), ϕ) = [a]1Dω

[a]2ϕ

− [a]1Dω

[a]2ϕ

for all horizontals.

Proof. First of all, let us observe that the above formula makes sense, because fora given ϕ a map

ψ 7→ Dω(ψϕ)−Dω(ψ)ϕ

supercommutes with the left multiplication by elements of Ω(M). A direct cal-culation, with the help of (2.20) and elementary identities [a]1[a]2 = ε(a)1 and[a]1[a](0)

2 ⊗ [a](1)2 = 1⊗ a involving the translation map, now gives

[a]1Dω

[a]2ϕ

− [a]1Dω

[a]2ϕ = [a]1Dω

[a]2ϕ+ [a]1[a]2D(ϕ)+

+ [a]1[a](0)2 `ω

(π([a](1)

2 ), ϕ)− [a]1Dω

[a]2ϕ = ε(a)D(ϕ) + `ω

(π(a), ϕ

)which completes the proof.

Proposition 4. The following conditions are equivalent:(i) A connection ω is regular, in other words `ω = 0;(ii) The standard graded Leibniz rule holds for Dω or D∗ω.

In this case we also have

(2.31) Dω = D∗ω

Proof. If ω is regular then by (2.14) we conclude that both covariant derivativescoincide. We also see that the standard graded Leibniz rule holds, as a directconsequence of (2.20). Now if Dω is an antiderivation, then

`ω(π(a), ϕ) = [a]1Dω

[a]2ϕ

− [a]1Dω

[a]2ϕ = [a]1[a]2D(ϕ) = 0

according to Proposition 3, and we see that ω is regular.


Remark 7. The coincidence of two covariant derivatives is equivalent to

(2.32) `ω(πκ−1(ϕ(1)), ϕ(0)

)= 0

as it follows from (2.14). However, in general this alone does not imply the regularityproperty. We can conclude that the graded Leibniz rule holds universally on hor(P )if the restricted map Dω : B → H = hor1(P ) is a standard derivation. But moreabout this later.

Another interesting special case, more general than regular connections, as wellas the coincidence of two covariant derivatives Dω and D∗ω, occurs when they satisfythe two-sided Leibniz rule over Ω(M). Important realization of this is the classicalcontext, where hor(P ) is supercommutative (with the complete calculus possiblyhighly ‘quantum’).

Proposition 5. (i) The following equations

(2.33)D∗ω(wϕ) = dM (w)ϕ+ (−)∂wwD∗ω(ϕ)

Dω(ϕw) = Dω(ϕ)w + (−)∂ϕϕdM (w)

are mutually equivalent and hold iff `ω(ϑ,w) = 0 for each ϑ ∈ Γinv and w ∈ Ω(M).In other words the connection form supercommutes with the calculus on the base.

(ii) If this holds, then we can act by a canonical structure on ω. Such aconnection ω will be regular iff

(2.34) ω(ϑ a) = ω(ϑ) a

in other words, regular connections intertwine the corresponding -products.

Proof. If Dω satisfies right Leibniz rule over Ω(M), equivalently to saying that aleft Leibniz rule over Ω(M) holds for D∗ω, then (2.20) implies that `ω vanishes onΩ(M). Now if ω is regular then ω(ϑ)ϕ = (−)∂ϕϕ(0)ω(ϑϕ(1)) but on the other handfor every ω with values in Z(Ω(M),Ω(P )) we have ω(ϑ)ϕ = (−)∂ϕϕ(0)

(ω(ϑ)ϕ(1)

).

Therefore regularity is equivalent to (2.34).

Let us now dedicate some time to see in more detail about the curvature of ageneral connection. It is intrinsically linked to the square of the covariant derivativeoperator. As it follows from the restricted Leibniz rules over Ω(M) the square ofDω/D∗ω is left/right Ω(M)-linear map.

Proposition 6. (i) We have

(2.35)D2ω(ϕ) = −ϕ(0)rω(ϕ(1))

rω(a) = −[a]1D2ω[a]2

with rω : A → hor(P ) expressible in a spirit of the classical structure equation

(2.36) rω(a) = dPωπ(a) + ωπ(a(1))ωπ(a(2))

(ii) The following identities hold

Dωrω(a) + `ωπ(a(1)), rω(a(2))

= 0(2.37)

rω(a)∗ = −rω[κ(a)∗](2.38)

F∧rω(a) = rω(a(2))⊗ κ(a(1))a(3)(2.39)

14 MICHO DURDEVICH

Proof. We compute

D2ω(ϕ) = Dω

(dP (ϕ)− (−)∂ϕϕ(0)ωπ(ϕ(1))

)= d2

P (ϕ)− (−)∂ϕdP (ϕ(0))ωπ(ϕ(1))

− ϕ(0)dP [ωπ(ϕ(1))] + (−)∂ϕ(dP (ϕ(0))ωπ(ϕ(1))− (−)∂ϕϕ(0)ωπ(ϕ(1))ωπ(ϕ(2))

)= −ϕ(0)dPωπ(ϕ(1))− ϕ(0)ωπ(ϕ(1))ωπ(ϕ(2)) = −ϕ(0)rω(ϕ(1))

Equations (2.35) are just a special case of a canonical representation of an ar-bitrary left Ω(M)-linear map S : hor(P ) → hor(P ). Such maps are in one-onecorrespondence with ‘tensors’ τS : A → hor(P ) via

(2.40)S(ϕ) = ϕ(0)τS(ϕ(1))

τS(a) = [a]1S[a]2The map S intertwines F∧ iff F∧τS = (τS ⊗ id)ad. Equation (2.39) is thereforeequivalent to the covariance of D2

ω. Equation (2.38) follows from (2.36) plus therules for *-structure coupled with elementary transformations:

rω(a)∗ =(dPωπ(a) + ωπ(a(1))ωπ(a(2))

)∗ = −dPωπ[κ(a)∗]

− ωπ[κ(a(2))∗]ωπ[κ(a(1))∗] = −rω[κ(a)∗]

Finally equation (2.37) is a quantum version of classical Bianchi identity:

Dωrω(a) = dP[dPωπ(a) + ωπ(a(1))ωπ(a(2))

]− dPωπ(a(2))ωπ[κ(a(1))a(3)]

− ωπ(a(2))ωπ(a(3))ωπ[κ(a(1))a(4)]

= rω(a(1))ωπ(a(2))− ωπ(a(1))rω(a(2))− rω(a(2))ωπ[κ(a(1))a(3)]

= rω(a(3))ωπ(a(1)) [κ(a(2))a(4)]

− ωπ(a(1))rω(a(2))

= −`ωπ(a(1)), rω(a(2))

and this completes our proof.

Remark 8. In general, the map rω will not project down to germs, the casewhich would naturally allow us to introduce a direct analog of the curvature ten-sor Rω : Γinv → hor(P ), via rω = Rωπ. This purely quantum phenomenon canbe interpreted as a kind of inadequacy of the calculus Γ for the bundle P . Theprojectability holds iff

rω(R) = 0where R = ker(ε) ∩ ker(π) is the right A-ideal that defines the calculus Γ in accor-dance with [18]. And according to (2.36) this is equivalent to

ωπ(a(1))ωπ(a(2)) = 0 ∀a ∈ Rin other words the connection must be multiplicative. An alternative way we han-dled the general connection tensor Rω in [3] was to introduce an embedded differ-ential map δ : Γinv → Γinv ⊗ Γinv as an additional geometrical component. Thiswould imply that the difference rω −Rωπ vanishes on selected elements of A only.Another way to express this phenomenon is to say that there exist horizontal qua-dratic expressions involving the vertical connection form only. These are exactlythe above expressions

rω(a) = ωπ(a(1))ωπ(a(2)) a ∈ R


a ‘residual part’ of the curvature tensor, understood as obstacle to multiplicativity.

Remark 9. The square of the dual covariant derivative is represented as follows

(2.41) D∗2ω (ϕ) = −rωκ−1(ϕ(1))ϕ(0)

Let us consider the square of the combined D2,ω. A direct computation gives

(2.42) D22,ω(ϕ⊗ ψ) = D∗2ω (ϕ)⊗ ψ + ϕ⊗D2

ω(ψ)

and in particular we have

D∗2ω

[a]1⊗ [a]2 + [a]1 ⊗D

2ω

[a]2

= 0(2.43)

D∗2ω

[a]1

[a]2 + [a]1D2ω

[a]2

= 0(2.44)

for all a ∈ A.

Let us now derive some useful identities that have an interest of their own, andwhich also will help us establish further algebraic properties of the curvature tensor.

Lemma 7. The following identity holds:

(2.45) ωπ(a(1))ωπ(a(2))ϕ− ϕ(0)ω[π(a(1)) ϕ(1)]ω[π(a(2)) ϕ(2)]

= `ω[π(a(1)), `ω(π(a(2)), ϕ)

]+ (−)∂ϕ`ω(ϑ(0), ϕ(0))ω[π(ϑ(1)) ϕ(1)]

where ϑ = π(a) and a ∈ ker(ε) while ϕ ∈ hor(P ). In particular, for a ∈ R itsimplifies to

(2.46) ωπ(a(1))ωπ(a(2))ϕ− ϕ(0)ω[π(a(1)ϕ(1))]ω[π(a(2)ϕ(2))]

= `ω[π(a(1)), `ω(π(a(2)), ϕ)

]telling us how to twist between horizontal forms and obstacles to multiplicativity.

Proof. By a direct calculation

ωπ(a(1))ωπ(a(2))ϕ = ωπ(a(1))`ω(π(a(2)), ϕ

)+ (−)∂ϕωπ(a(1))ϕ(0)ω[π(a(2)) ϕ(1)]

= ϕ(0)ω[π(a(1)) ϕ(1)]ω[π(a(2)) ϕ(2)] + ωπ(a(1))`ω(π(a(2)), ϕ

)+ (−)∂ϕ`ω(π(a(1)), ϕ(0))ω[π(a(2)) ϕ(1)]

= ϕ(0)ω[π(a(1)) ϕ(1)]ω[π(a(2)) ϕ(2)] + (−)∂ϕ`ω(π(a(1)), ϕ(0))ω[π(a(2)) ϕ(1)]

+ `ω[π(a(1)), `ω

(π(a(2)), ϕ

)]− (−)∂ϕ`ω(π(a(3)), ϕ(0))ω

[π(a(1)) κ(a(2))a(4)ϕ(1)

]= ϕ(0)ω[π(a(1)) ϕ(1)]ω[π(a(2)) ϕ(2)] + `ω

[π(a(1)), `ω(π(a(2)), ϕ)

]+ (−)∂ϕ`ω(ϑ(0), ϕ(0))ω[π(ϑ(1)) ϕ(1)]

where ϑ(0) ⊗ ϑ(1) = π(a(2))⊗ κ(a(1))a(3) = ad(ϑ).

Remark 10. An equivalent, but more expanded way to write equation (2.45) is

(2.47) ωπ(a(1))ωπ(a(2))ϕ− ϕ(0)ωπ(a(1)ϕ(1))ωπ(a(2)ϕ(2))+

+ ϕ(0)ω(ϑ ϕ(1))ωπ(ϕ(2)) + ϕ(0)ωπ(ϕ(1))ω(ϑ ϕ(2))

= `ω[π(a(1)), `ω(π(a(2)), ϕ)

]+ (−)∂ϕ`ω(ϑ(0), ϕ(0))ω[π(ϑ(1)) ϕ(1)]

where a ∈ ker(ε) and π(a) = ϑ.

16 MICHO DURDEVICH

And now on to calculate the twisting rules involving the differential of the con-nection. This, together with the above expressions, will allow us to write an explicitformula for the covariant derivative of the `ω, interpretable as twisting rules for thecomplete curvature tensor as well. Differentiating the expression for `ω and using(2.47) and (2.36) as well as the definition of the covariant derivative Dω we obtain

d`ω(ϑ, ϕ) = [dω(ϑ)]ϕ− ω(ϑ)d(ϕ)− (−)∂ϕd(ϕ(0))ωπ(aϕ(1))− ϕ(0)dωπ(aϕ(1))

= rω(a)ϕ− ωπ(a(1))ωπ(a(2))ϕ− (−)∂ϕDω(ϕ(0))ωπ(aϕ(1))

− ϕ(0)ωπ(ϕ(1))ωπ(aϕ(2))− ω(ϑ)Dω(ϕ)− (−)∂ϕω(ϑ)ϕ(0)ωπ(ϕ(1))

− ϕ(0)rω(aϕ(1)) + ϕ(0)ωπ(a(1)ϕ(1))ωπ(a(2)ϕ(2))

= rω(a)ϕ− ϕ(0)rω(aϕ(1))− `ω(ϑ,Dω(ϕ))− (−)∂ϕ`ω(ϑ, ϕ(0))ωπ(ϕ(1))

− ϕ(0)ωπ(aϕ(1))ωπ(ϕ(2))− ϕ(0)ωπ(ϕ(1))ωπ(aϕ(2))

− ωπ(a(1))ωπ(a(2))ϕ+ ϕ(0)ωπ(a(1)ϕ(1))ωπ(a(2)ϕ(2))

Continuing in the same way of elementary manipulations and regrouping, our cal-culations lead us into

rω(a)ϕ− ϕ(0)rω(aϕ(1))− `ω(ϑ,Dω(ϕ))− (−)∂ϕ`ω(ϑ, ϕ(0))ωπ(ϕ(1))

− `ω[π(a(1)), `ω(π(a(2)), ϕ)

]− (−)∂ϕ`ω(ϑ(0), ϕ(0))ω[π(ϑ(1)) ϕ(1)]

= rω(a)ϕ− ϕ(0)rω(aϕ(1))− `ω(ϑ,Dω(ϕ))− `ω[π(a(1)), `ω(π(a(2)), ϕ)

]− (−)∂ϕ`ω(ϑ(0), ϕ(0))ω[π(ϑ(1)ϕ(1))]

Or to put it in terms of the covariant derivative:


(2.48) Dω`ω(ϑ, ϕ) = rω(a)ϕ− ϕ(0)rω(aϕ(1))− `ω(ϑ,Dω(ϕ))

− `ω[π(a(1)), `ω

(π(a(2)), ϕ

)]where ϑ = π(a) and a ∈ ker(ε).

Remark 11. We see that property (2.46) follows from (2.48) as a special case whena ∈ R and hence ϑ = 0.

With the help of the operators `ω and rω and their properties we have derivedso far, we gain more insight into the general structure of horizontal forms.

Proposition 9. (i) The algebra of horizontal forms is generated by elements: b ∈B = hor0(P ), first order elements Dω(b), second order elements rω(a) with a ∈ Rand finally iterative applications of `ω(ϑ, ∗) on elements Dω(b) and rω(a). Themodule H = hor1(P ) of first-order horizontal forms is spanned by elements of theform bDω(q) where b, q ∈ B.

(ii) Let us assume that the module H = hor1(P ) of the first-order horizontalforms is spanned by elements of the form bdM (f) where b ∈ B and f ∈ V. Equiv-alently we can say that dM (f)b span the module H. This is further equivalent tosaying that N = Ω1(M) is spanned by fdM (g). Then for each n ≥ 2 we have

(2.49) horn(P ) =∑

bdM (f1) . . . dM (fn)

=∑

dM (f1) . . . dM (fn)b

and in particular the calculus Ω(M) on the base is generated by V = Ω0(M).


Proof. The module G = Ω1(P ) is spanned by elements of the form bdP (q) whereb, q ∈ B. By taking the left horizontal projection, we conclude that H is spannedby bDω(q). In creating a horizontal-vertical decomposition

Ω(P )! hor(P )⊗ Γ∧invfor the calculus Ω(P ) as discussed in [3], we have to know how to ‘exchange’ hor-izontal forms from H and connection ω, as well as to identify the residual part ofthe curvature rω(a) where a ∈ R. The horizontal-vertical decomposition transformsproducts of elements from H and im(ω) into sums of special elements of the formϕω∧(λ) where ϕ ∈ hor(P ) and λ ∈ Γ∧inv.

Due to the twisted Leibniz rule for `ω during such rearrangements there wouldappear, besides expressions involving H the elements of the form rω(a) and iteratedactions by `ω on Dω(b) and rω(a). Finally if

H = BdM (V) ⇔ dM (V)B = Hthen Dω is reduced in the subalgebra of hor(P ) generated by H. But then it impliesthat all the values of `ω and rω are from this subalgebra and hence (2.49) holds.

In the classical theory, the square of Dω, as a square of a graded derivation, is astandard derivation of order 2. The following result is a general quantum versionof this important property.


(2.50) D2ω(ϕψ) = D2

ω(ϕ)ψ + ϕD2ω(ψ) + ϕ(0)`ω

[π(ϕ(1)), `ω

(π(ϕ(2)), ψ

)]+ ϕ(0)`ω

(π(ϕ(1)), Dω(ψ)

)+ ϕ(0)Dω`ω(π(ϕ(1)), ψ)

Proof. Basically, as in the classical theory, just apply twicely the quantum Leibnizrule (2.20) elementary properties of Dω and in addition use (2.48):

D2ω(ϕψ) = Dω

Dω(ϕ)ψ + (−)∂ϕϕDω(ψ) + (−)∂ϕϕ(0)`ω(π(ϕ(1)), ψ)

= D2

ω(ϕ)ψ − (−)∂ϕDω(ϕ)Dω(ψ)− (−)∂ϕDω(ϕ(0))`ω(π(ϕ(1)), ψ

)+ (−)∂ϕDω(ϕ)Dω(ψ) + ϕD2

ω(ψ) + ϕ(0)`ω(π(ϕ(1)), Dω(ψ)

)+ (−)∂ϕDω(ϕ(0))`ω(π(ϕ(1)), ψ) + ϕ(0)Dω`ω(π(ϕ(1)), ψ)

+ ϕ(0)`ω[π(ϕ(1)), `ω

(π(ϕ(2)), ψ

)]= D2

ω(ϕ)ψ + ϕD2ω(ψ) + ϕ(0)`ω

(π(ϕ(1)), Dω(ψ)

)+ ϕ(0)Dω`ω(π(ϕ(1)), ψ) + ϕ(0)`ω

[π(ϕ(1)), `ω

(π(ϕ(2)), ψ

)]It is interesting also to observe that if we now apply (2.48) the derived formulatransforms into a tautology.

The curvature tensor rω satisfies its own twisted version of a ‘germ’ Leibniz rule.It can be obtained directly from (2.48) and applying elementary transformations ofthe translation map.

Lemma 11. We have

(2.51) rω(ab) = ε(a)rω(b) + [b]1rω(a)[b]2 − [b]1`ω[π(a(1)), `ω

(π(a(2)), [b]2

)]− [b]1Dω`ω

(π(a), [b]2

)− [b]1`ω

(π(a), Dω[b]2

)

18 MICHO DURDEVICH

for all a, b ∈ A.

We shall conclude this section by further generalizing our quantum Bianchi iden-tity, to covariant derivatives of appropriate higher order expressions of the curvaturetensor. This is particularly helpful in calculations involving quantum characteristicclasses and Chern character. Let us define a new operator

(2.52) ıω(a, ϕ) = rω(a(1))`ω(π(a(2)), ϕ

)− `ω

(π(a(1)), rω(a(2))ϕ

)It is worth observing that ıω(a, 1) = −`ω

(π(a(1)), rω(a(2))

). It is also useful to write

down an alternative expression

(2.53) ıω(a, ϕ) = rω(a(2))`ω[π(κ(a(1))a(3)

), ϕ]− `ω

(π(a(1)), rω(a(2))

)ϕ

as it follows from the Leibniz rule (2.16) for `ω. We can now describe the action ofDω on products of values of the curvature tensor.

(2.54) Dω

[rω(a1) . . . rω(an)

]=

n∑k=1

rω(a1) . . . rω(ak−1)ıω(ak, rω(ak+1) . . . rω(an)

)where a1, . . . , an ∈ A. This formula can be derived inductively, from the basicquantum Bianchi identity (2.37) and applying the Leibniz rule (2.20) for Dω.

Proposition 12. As a special case of the above property, we find that the followinggeneralized Bianchi identity holds:

(2.55) Dω

rω(a(1)) . . . rω(a(n))

+ `ω

(π(a(1)), rω(a(2)) . . . rω(a(n+1))

)= 0

Proof. Inserting∑

a1 ⊗ · · · ⊗ an = a(1) ⊗ · · · ⊗ a(n) in formula (2.54) we see thatof all 2n+ 1 terms on the right all but the last mutually cancel.

3. From Horizontal Calculus to Full Calculus

3.1. Abstract Horizontal Calculi

In the previous section we have derived a set of formulae establishing a completehorizontal differential calculus, in terms of the covariant derivative Dω, its curva-ture tensor rω and the non-regularity measure `ω. The operators rω and `ω arecompletely determined by Dω and the principal bundle structure on P .

Now we shall revert this, repeating the highlights of the previous section in thisabstract context: Our starting point will be an abstract graded *-algebra horPequipped with a right action F : horP → horP ⊗A and a ‘covariant derivative’ mapD : horP → horP extending a given differential structure on the subalgebra ΩM ofF -invariants. Our goal is to construct a complete calculus on the bundle, includingthe appropriate calculus on the structure group G so that the initial algebra horPbecomes the horizontal forms algebra of this complete calculus, and the map Dbecomes the covariant derivative associated to a naturally appearing connection ω.In the process, we will construct various entities intrinsically associated to initialdata (horP , F,ΩM , D) and they will be given an old geometric interpretation interms of the complete calculus on the bundle. The constructions of this section canbe understood as a simple and logical generalization of the results presented in [6]where the focus was on regular connections only.

So, we assume that horP is a graded *-algebra such that

horP = ⊕k≥0horkP hor0P = B


and that F : horP → horP ⊗ A is a grade-preserving *-homomorphism extendingF : B → B ⊗A and maintaining the coassociativity and counitality.

We also assume that the F -invariant part ΩM of horP is equipped with a differ-ential structure dM : ΩM → ΩM so that we have a graded-differential *-algebra andΩ0M = V. This algebra will represent the calculus on the base quantum space M .

Definition 12. We shall call such a triplet (horP , F,ΩM ) a horizontal calculus overa quantum principal bundle P .

For every representation u : Cu → Cu⊗A of G on a vector space Cu let Fu be thegraded ΩM -bimodule of the intertwiner maps between u and F : horP → horP ⊗A.In other words we consider maps ζ : Cu → horP such that the diagram

(3.1)

Cuu−−−−→ Cu ⊗A

ζy yζ ⊗ id

horP −−−−→F

horP ⊗A

commutes. These intertwiner bimodules correspond to associated vector bundles.As explained in [4, 11] the following natural bimodular decomposition holds

(3.2) horP =⊕u∈TFu ⊗ Cu ζ(v)↔ (ζ, v)

where T is a given complete set of irreducible representations of G. In terms of thisdecomposition, the action F reduces to id ⊗ u in each component. The bimoduleF∅ associated to the trivial representation ∅ correspond to the base algebra ΩM .

Now we shall consider the question of the existence of maps D : horP → horP ⊗Aextending the differential dM and such that

(3.3)

D(horkP ) ⊆ hork+1P

FD = (D ⊗ id)F

D(wϕ) = dM (w)ϕ+ (−)∂wwD(ϕ)

where w ∈ ΩM and ϕ ∈ horP . If such a D is given, then taking compositions withintertwiners from Fu we obtain a collection of maps Du : Fu → Fu. In particular

(3.4) D∅ = dM

Every Du satisfies

(3.5)D(wζ) = dM (w)ζ + (−)∂ζwDu(ζ) ∀w ∈ ΩM ζ ∈ Fu

Du(Fku ) ⊆ Fk+1u

Now conversely, consider a given non-trivial u ∈ R(G). As explained in [4] thespace Fu is finite and projective as left and right ΩM module. In particular itadmits a ‘covariant derivative’ map Du : Fu → Fu satisfying the above conditions(3.5). Now let us chose one such a map Du for every nontrivial irreducible u ∈ Tand use (3.4) for the trivial representation. Finally, define D to be the direct sumof dM and maps of the form Du⊗ id acting within the decomposition (3.2). In sucha way we obtain a map D : horP → horP satisfying properties (3.3). In summary

20 MICHO DURDEVICH

Proposition 13. There exists a natural correspondence between covariant mapsD : horP → horP and families (Du)u∈T \∅ of maps Du : Fu → Fu. In termsof this D will satisfy left/right graded Leibniz rule over ΩM iff the correspondingfamily members satisfy the same rule. The map D is homogeneous iff all the familymembers are homogeneous and of the same grade.

3.2. Elementary Properties of Covariant Derivatives

We shall assume that an extension D : horP → horP of dM is given so that (3.3)holds. Let us consider the operator D∗ = ∗D∗. It is also F -covariant, extends dMand satisfies the right Leibniz rule over ΩM :

(3.6) D∗(ϕw) = D∗(ϕ)w + (−)∂ϕϕdM (w)

Given the maps D and D∗ we can combine them naturally

(3.7) D2(ϕ⊗ ψ) = D∗(ϕ)⊗ ψ + (−)∂ϕϕ⊗D(ψ)

to a transformation D2 : horP ⊗M horP → horP ⊗M horP an the tensor product isover ΩM . The left and right Leibniz rules for D and D∗ ensure the consistencyof this definition. The map D2 is covariant, and commutes with the natural *-structure on horP ⊗M horP in the sense of

(3.8) D2

ϕ⊗ ψ

∗ = (−)∂ϕ∂ψD2

ψ∗ ⊗ ϕ∗

Lemma 14. The following identities are equivalent:

(3.9)D2

[a]1 ⊗ [a]2

= D∗

[a]1⊗ [a]2 + [a]1 ⊗D

[a]2

= 0

D∗

[a]1

[a]2 + [a]1D

[a]2

= 0

for a ∈ A.

In what follows we shall assume that (3.9) holds.

Definition 13. Every such an operator D is called a covariant derivative for thehorizontal calculus (horP , F,ΩM ).

Remark 14. Given D the above property (3.9) uniquely and consistently deter-mines a right derivative D∗, and vice versa, the maps D and D∗ are in a naturalduality besides being conjugate one of another. So in reality, our condition is thatthe right form of the left ΩM -derivative D coincides with the conjugate D∗. Moreabout this is collected in the first Appendix.

Because of the Leibniz rules over ΩM the squares D2 and D∗2 act as left/rightΩM -linear transformations on horP respectively. In particular, we have

(3.10) D2(ϕ) = −ϕ(0)%(ϕ(1)) %(a) = −[a]1D2[a]2

Since D extends dM its square vanishes on ΩM , so %(1) = 0. Taking the square ofD2, we get

(3.11) D22(ϕ⊗ ψ) = D∗2(ϕ)⊗ ψ + ϕ⊗D2(ψ)

and we see that

(3.12)D∗2

[a]1⊗ [a]2 + [a]1 ⊗D

2

[a]2

= 0

D∗2

[a]1

[a]2 + [a]1D2

[a]2

= 0

We are going to derive two elementary algebraic properties of the curvature tensor.


Lemma 15. The following identities hold:

F%(a) = %(a(2))⊗ κ(a(1))a(3)(3.13)

%(a)∗ = −%[κ(a)∗](3.14)

for every a ∈ A.

Proof. This is quite a general feature on maps acting on horizontal forms thatgraded-commute with forms on the base, as explained in the previous section. Letus give an explicit proof here. The covariance of D and hence D2 coupled with thetransformation of the translation map, gives

F%(a) = −F

[a]1D2[a]2

= [a(2)]1D

2[a(2)]2 ⊗ κ(a(1))a(3) = %(a(2))⊗ κ(a(1))a(3)

which proves (3.13). By conjugating the definition of % we obtain

D∗2(ϕ) = −%∗(ϕ1)ϕ(0)

with %∗ = ∗ρ∗. On the other hand (3.12) reads

%∗κ(a(1))[a(2)]1 ⊗ [a(2)]2 = −[a(1)]1 ⊗ [a(1)]2%(a(2))

and by applying the product of horP and the identity [a]1[a]2 = ε(a) we concludethat %∗κ = −% in other words (3.14) holds.

Let us now consider a map ` : A× horP → horP defined by

(3.15) `(a, ϕ) = [a]1D

[a]2ϕ− [a]1D

[a]2ϕ

for a ∈ ker(ε) and `(1, ϕ) = 0. The above formula makes sense, due to the fact thata map

(3.16) S : ϕ 7→ D(ϕψ)−D(ϕ)ψ

supercommutes with the left multiplication by elements of ΩM . Let us also observethat `(a, 1) = 0. An equivalent way to define ` is

(3.17) `(a, ϕ) = •D2([a]1 ⊗ [a]2ϕ)

as it directly follows from the definition of D2 and property (3.9). It is also clearthat ` is covariant in the sense of

(3.18)F`(a, ϕ) = `(a(2), ϕ(0))⊗ κ(a(1))a(3)ϕ(1)

F`(a, ϕ(0))κ(ϕ(1)) = `(a(2), ϕ)⊗ κ(a(1))a(3)

as it directly follows from covariance equations for D and the translation map.

Lemma 16. The following modified Leibniz rule holds:

(3.19) D(ϕψ) = D(ϕ)ψ + (−)∂ϕϕD(ψ) + (−)∂ϕϕ(0)`(ϕ(1), ψ)

Proof. Let us fix ψ then consider a slightly modified map (3.16) in the form

ϕ 7→ S′(ϕ) = S(ϕ)− (−)∂ϕϕD(ψ) = (−)∂ϕϕ(0)λ(ϕ(1))

where λ(a) = [a]1S′[a]2. Expanding this we arrive at

D(ϕψ)−D(ϕ)ψ − (−)∂ϕϕD(ψ) = (−)∂ϕϕ(0)`(ϕ(0), ψ)

with an equivalent description taking care of scalars

(3.20) `(a, ψ) = [a]1D

[a]2ψ− [a]1D

[a]2ψ − ε(a)D(ψ)

where a ∈ A is arbitrary.

22 MICHO DURDEVICH

The map ` itself exhibits a kind of a twisted commutator Leibniz rule.

Lemma 17. The following identities hold:

`(a, ϕψ) = `(a, ϕ)ψ + (−)∂ϕϕ(0)`(aϕ(1), ψ)(3.21)

`(ab, ϕ) = [b]1`(a, [b]2ϕ)− [b]1`(a, [b]2)ϕ(3.22)

where we assume that a ∈ ker(ε) and b, ϕ and ψ are arbitrary.

Proof. A direct calculation gives

`(a, ϕψ) = [a]1D

[a]2ϕψ− [a]1D

[a]2ϕψ = [a]1D

[a]2ϕψ

− [a]1D

[a]2ϕ

ψ

+ [a]1D

[a]2ϕψ − [a]1D

[a]2ϕψ

= (−)∂ϕϕ(0)[aϕ(1)]1D

[aϕ(1)]2ψ− (−)∂ϕϕ(0)[aϕ(1)]1D

[aϕ(1)]2

ψ

+

[a]1D

[a]2ϕ− [a]1D

[a]2ϕψ

= `(a, ϕ)ψ + (−)∂ϕϕ(0)`(aϕ(1), ψ)

where we have applied [a]1 ⊗ [a]2ϕ = ϕ(0)[aϕ(1)]1 ⊗ [aϕ(2)]2. Furthermore,

`(ab, ϕ) = [ab]1D

[ab]2ϕ− [ab]1D

[ab]2

ϕ = [b]1[a]1D

[a]2[b]2ϕ

− [b]1[a]1D

[a]2[b]2

ϕ = [b]1[a]1D

[a]2[b]2ϕ

− [b]1[a]1D

[a]2

[b]2ϕ

+ [b]1[a]1D

[a]2

[b]2ϕ− [b]1[a]1D

[a]2[b]2ϕ

= [b]1`(a, [b]2ϕ

)− [b]1`(a, [b]2)ϕ

and we have used the multiplicativity property [ab]1 ⊗ [ab]2 = [b]1[a]1 ⊗ [a]2[b]2 forthe translation map.

Here is a compatibility property between ` and the *-structure.

Lemma 18. If a ∈ ker(ε) then

(3.23)`(a, ϕ)∗ = `

(κ(a)∗k(ϕ(1))∗, ϕ(0)∗)

`(aκ−1(ϕ(1)), ϕ(0)

)∗ = `(κ(a)∗, ϕ∗

)Proof. Conjugating the defining formula (3.17) for ` we obtain

`(a, ϕ)∗ = •D2

[a]1 ⊗ [a]2ϕ

= •D2

ϕ∗[κ(a)∗]1 ⊗ [κ(a)∗]2

= •D2

[κ(a)∗κ−1(ϕ(1)∗)]1 ⊗ [κ(a)∗κ−1(ϕ(1)∗)]2ϕ

(0)∗ = `(κ(a)∗κ(ϕ(1))∗, ϕ(0)∗)

and we have also used the conjugation identity [a]∗2 ⊗ [a]∗1 = [κ(a)∗]1 ⊗ [κ(a)∗]2 forthe translation map.

And not surprisingly, as we could guess from the general theory of connections,the map ` links the derivative D and its conjugate derivative D∗.

Lemma 19. We have

(3.24) D∗(ϕ) = D(ϕ) + `(κ−1(ϕ(1)), ϕ(0)

)


Proof. We compute

`(κ−1(ϕ(1)), ϕ(0)

)= [κ−1(ϕ(1))]1D

[k−1(ϕ(1))]2ϕ

− εκ−1(ϕ(1))D(ϕ(0))

− [κ−1(ϕ(1))]1D

[k−1(ϕ(1))]2ϕ

= •D2

[κ−1(ϕ(1))]1 ⊗ [κ−1(ϕ(1))]2ϕ

(0)−D(ϕ) = D∗(ϕ)−D(ϕ)

Here we have used identity [κ−1(ϕ(1))]1 ⊗ [κ−1(ϕ(1))]2ϕ(0) = ϕ⊗ 1 and (3.9).

In classical geometry, as well as in quantum context with regular connections,the derivative the curvature tensor vanishes. As we have seen in the previoussection, for general connections this will no longer be the case, however a simplemodification of the standard form of the identity holds. In our abstract contextof horizontal calculi, we can derive a similar quantum version of classical Bianchiidentity.


(3.25) D%(a(1)) . . . %(a(n))

+ `(a(1), %(a(2)) . . . %(a(n+1))

)= 0

for each a ∈ A and n ∈ N. In particular,

(3.26) D%(a) + `(a(1), ρ(a(2))

)= 0

Proof. The base Bianchi formula is a straightforward derivation, by expanding thepurely quantum term of the Bianchi identity,

`(a(1), %(a(2))

)= [a(1)]1D

[a(1)]2%(a(2))

− [a(1)]1D

[a(1)]2

%(a(2))

− ε(a(1))D%(a(2)) = −[a]1DD2[a]2

+ [a]1D

2D

[a]2−D%(a) = −D%(a)

and we have used (3.20) & the right covariance of the translation map. The mainpart follows from a more general expression, describing the action of D on arbitraryproducts of %. If we define

(3.27)ı(a, ϕ) = %(a(1))`

(a(2), ϕ

)− `(a(1), %(a(2))ϕ

)= %(a(2))`

(κ(a(1))a(3), ϕ

)− `(a(1), %(a(2))

)ϕ

then by inductively applying the generalized Leibniz rule (3.19) for the covariantderivative map we arrive at

(3.28) D[%(a1) . . . %(an)

]=

n∑k=1

%(a1) . . . %(ak−1)ı(ak, %(ak+1) . . . %(an)

)where a1, . . . , an ∈ A. And now the full Bianchi identity derives easily by inserting∑

a1 ⊗ · · · ⊗ an = a(1) ⊗ · · · ⊗ a(n) and observing a pairwise cancellation of 2n (allbut the last) terms in the right-hand side of the above expression.

So we know how to calculate the action of D on algebraic expressions involvingthe curvature tensor. A related question is to compute the action of the covariantderivative on `.

Lemma 21. (i) For a ∈ ker(ε) and arbitrary ϕ we have

(3.29) D`(a, ϕ) = %(a)ϕ− ϕ(0)%(aϕ(1))− `(a,D(ϕ)

)− `(a(1), `(a(2), ϕ)

)

24 MICHO DURDEVICH

(ii) Equivalently, to put it in terms of the translation map,

(3.30) %(ab) = ε(a)%(b) + [b]1%(a)[b]2 − [b]1`[a(1), `

(a(2), [b]2

)]− [b]1D`

(a, [b]2

)− [b]1`

(a,D[b]2

)for every a, b ∈ A.

Proof. Let us first observe that

(3.31) D2

[a]1 ⊗ [a]2ϕ

− [a]1 ⊗ [a]2D(ϕ) = [a(1)]1 ⊗ [a(1)]2`(a

(2), ϕ)

for arbitrary a and ϕ. Indeed,

[a(1)]1 ⊗ [a(1)]2`(a(2), ϕ) = [a(1)]1 ⊗ [a(1)]2[a(2)]1D

[a(2)]2ϕ

− [a(1)]1 ⊗ [a(1)]2[a(2)]1D

[a(2)]2

ϕ− [a(1)]1 ⊗ [a(1)]2ε(a

(2))D(ϕ)

= [a]1 ⊗D

[a]2ϕ− [a]1 ⊗D

[a]2ϕ− [a]1 ⊗ [a]2D(ϕ)

= D2

[a]1 ⊗ [a]2ϕ

− [a]1 ⊗ [a]2D(ϕ)

where we have played with [a(1)]1 ⊗ [a(1)]2[a(2)]1 ⊗ [a(2)]2 = [a]1 ⊗ 1 ⊗ [a]2. Now,proceeding with the main computation

%(a)ϕ− ϕ(0)%(aϕ(1))− `(a,D(ϕ)

)− `(a(1), `(a(2), ϕ)

)= −[a]1D

2

[a]2ϕ+ ϕ(0)[aϕ(1)]1D

2

[aϕ(1)]2− [a]1D

[a]2D(ϕ)

+ [a]1D

[a]2D(ϕ)− [a(1)]1D

[a(1)]2`(a

(2), ϕ)

+ [a(1)]1D

[a(1)]2`(a(2), ϕ) + ε(a(1))D`(a(2), ϕ)

= −[a]1D2

[a]2ϕ+ [a]1D

2

[a]2ϕ− •D2

[a]1 ⊗ [a]2D(ϕ)

− •D2

[a(1)]1 ⊗ [a(1)]2`(a

(2), ϕ)

+D`(a, ϕ)

= D`(a, ϕ) + •D22

[a]1 ⊗ [a]2ϕ

− •D2

[a]1 ⊗ [a]2D(ϕ)

− •D2

2

[a]1 ⊗ [a]2ϕ

+ •D2

[a]1 ⊗ [a]2D(ϕ)

= D`(a, ϕ)

which establishes our first formula. The second one gives an equivalent expressionin terms of the translation map.

We shall now use the derived properties of ` and % to provide an interestingalternative and simplifying insight into the Bianchi identities. Let us consider thespace Ξ of intertwiners λ : A → horP between the adjoint action ad on A and F .The cuvature tensor and its convolution iterations belong to this space, and as wehave already explained the maps λ = λT are in one-to-one correspondence with leftsuper ΩM -linear covariant transformations T = Tλ : horP → horP . The compositionof such transformations is reflected in Ξ by the corresponding convolution product.Precisely,

(3.32) λXY (a) = (−)∂X∂Y λX(a(1))λY (a(2))

Let D : Ξ→ Ξ be an operator defined by

(3.33) Dλ(a) = Dλ(a) + `(a(1), λ(a(2))

)so that in particular the quantum Bianchi identity reads D% = 0.


Lemma 22. In terms of the introduced correspondence, we have

(3.34) Dλ! DTλ − (−)∂λTλD

in other words the action of D is represented as the graded commutator with D.

Proof. The right hand side of (3.34) acts as follows(DTλ − (−)∂λTλD

)(ϕ) = (−)∂λ∂ϕD

(ϕ(0)λ(ϕ(1))

)− (−)∂λ(−)∂λ∂ϕ+∂λD(ϕ(0))λ(ϕ(1)) = (−)∂λ∂ϕ+∂ϕϕ(0)Dλ(ϕ(1))

+ (−)∂λ∂ϕ+∂ϕϕ(0)`(ϕ(1), λ(ϕ(2))

)= (−)∂λ∂ϕ+∂ϕϕ(0)Dλ(ϕ(1))

It is worth observing that the graded commutator with D is an antiderivation inΞ, if we interpret it as left super ΩM -linear maps on horP .

Now taking the square of (3.34) we find

(3.35) D2λ! D2Tλ − TλD2

and representing D2 as −%, the operator product as the convolution product (3.32)we further simplify this into

(3.36) D2λ(a) = %(a(1))λ(a(2))− λ(a(1))%(a(2))

It is instructive to derive this formula for the square of D by doing a direct calcu-lation, with the help of the established properties of % and `. So we compute

D2λ(a) = D2λ(a) +D`(a(1), λ(a(2))

)+ `(a(1), Dλ(a(2))

)+ `[a(1), `

(a(2), λ(a(3))

)]= −λ(a(2))%

(κ(a(1))a(3)

)+ %(a(1))λ(a(2))− λ(a(3))%

[(a(1) − ε(a(1))

)κ(a(2))a(4)

]= %(a(1))λ(a(2))− λ(a(1))%(a(2))

Taking into account that

(3.37) λ(a) = (−)n%(a(1)) . . . %(an) λ! D2n

we see that the quantum Bianchi identities are interpretable as a simple statementthat D2 and its powers commute with D. Another way to express the same thingis to say that

(3.38) Dϕ(0)%(ϕ(1)) . . . %(ϕ(n))

= D(ϕ(0))%(ϕ(1)) . . . %(ϕ(n))

for every ϕ ∈ horP and n ∈ N.It is worth defining as a separate entity these twisted graded commutators with

maps from Ξ. We shall write

(3.39) [λ | a |ϕ] = λ(a)ϕ− (−)∂λ∂ϕϕ(0)λ(aϕ(1))

for a ∈ ker(ε) and ϕ an arbitrary horizontal form. We shall assume that [λ | 1 |ϕ] =0. For example in formula (3.29) there appear a kind of twisted graded commutatorwith the curvature tensor and later on we shall see how the change of the covariantderivative reflects in the change of the corresponding ` expressed in terms of thesame braided commutator with the connection displacement. We shall also use asimplified notation

(3.40) ℘(a, ϕ) = %(a)ϕ− ϕ(0)%(aϕ(1)) = [% | a |ϕ]

for twisted commutators with the curvature tensor.

26 MICHO DURDEVICH

Lemma 23. The following identity holds

(3.41) D℘(a, ϕ) = ℘(a,D(ϕ)

)+D2`(a, ϕ)− `

(a,D2(ϕ)

)+ ℘

(a(1), `(a(2), ϕ)

)− `(a(1), ℘(a(2), ϕ)

)where a ∈ ker(ε) and ϕ ∈ horP .

Proof. A direct computation, with the help of the Bianchi identity and the Leibnizrule for D, gives

D℘(a, ϕ) = D(%(a)ϕ− ϕ(0)%(aϕ(1))

)= D%(a)ϕ+ %(a)D(ϕ)−D(ϕ(0))%(aϕ(1))

− (−)∂ϕϕ(0)D%(aϕ(1))− (−)∂ϕϕ(0)`(ϕ(1), %(aϕ(2))

)− `(a(1), %(a(2))

)+ %(a(2))`

(κ(a(1))a(3), ϕ

)= ℘

(a,D(ϕ)

)+ (−)∂ϕϕ(0)`

(a(1)ϕ(1), %(a(2)ϕ(2))

)+ %(a(2))`

(κ(a(1))a(3), ϕ

)− `(a(1), %(a(2))

)ϕ− (−)∂ϕϕ(0)`

(ϕ(1), %(aϕ(2))

)In accordance with (3.27) let us recall that

`(a(1), %(a(2))ϕ

)= `(a(1), %(a(2))

)ϕ+ %(a(1))`(a(2), ϕ)− %(a(2))`

(κ(a(1))a(3), ϕ

)and applying this we further simplify our expressions into

℘(a,D(ϕ)

)− `(a(1), %(a(2))ϕ

)+ (−)∂ϕϕ(0)`

[(a(1) − ε(a(1))

)ϕ(1), %(a(2)ϕ(2))

]+ %(a(1))`

(a(2), ϕ) = ℘

(a,D(ϕ)

)− `(a(1), %(a(2))ϕ

)+ %(a(1))`

(a(2), ϕ)

+ `[a(1), ϕ(0)%(a(2)ϕ(1))

]− `(a(1), ϕ(0)

)%(a(2)ϕ(1))

= ℘(a,D(ϕ)

)− `(a(2), ϕ(0))%

[κ(a(1))a(3)ϕ(1)

]− `(a(1), ℘(a(2), ϕ)

)− `(a,D2(ϕ)

)+ ℘

(a(1), `(a(2), ϕ)

)= ℘

(a,D(ϕ)

)+D2`(a, ϕ)− `

(a,D2(ϕ)

)+ ℘

(a(1), `(a(2), ϕ)

)− `(a(1), ℘(a(2), ϕ)

)and we have also applied the covariance property for `.

In fact, the above property is a special case of a more general rule for takingcovariant derivatives of braided commutators.

Proposition 24. The following identity holds

(3.42)D[λ | a |ϕ] + `

a(1), [λ | a(2) |ϕ]

− (−)∂λ[λ | a(1) | `(a(2), ϕ)] =

= [D(λ) | a |ϕ] + (−)∂λ[λ | a |D(ϕ)] + `(a, Tλ(ϕ)

)− (−)∂λTλ`(a, ϕ)

where a ∈ ker(ε) while ϕ ∈ horP and λ ∈ Ξ.

3.3. Beyond Horizontality

Now we are ready to ‘complete the circle’ by including the algebra horP in astandard differential calculus framework for P , within which it can be interpretedas the associated horizontal forms algebra, and D as the covariant derivative of adistinguished connection ω. We shall also construct a natural differential calculuson the structure quantum group G.

Let R be the set of all elements a ∈ ker(ε) for which

(3.43) `(a, ϕ) = 0 ∀ϕ ∈ horP


This space is a right ideal in A, according to (3.22). Furthermore, property (3.23)implies that

(3.44) κ(R)∗ = R

while (3.18) ensures that

(3.45) ad(R) ⊆ R ⊗A

Hence, the ideal R defines [18] a left-covariant first order differential calculus Γwhich is *-covariant and right-covariant. In other words, a first order bicovariant*-calculus over G.

In what follows, we shall base the calculus on G on appropriate right subidealsR ⊆ R which are ad-covariant and ∗κ-invariant. That is, a kind of *-covariantand bicovariant refinement of Γ. For a moment let us consider an arbitrary such acalculus Γ↔ R, and later on we shall find a more specific condition for R.

Let us extend the graded *-algebra horP by a vector space Γinv of first-ordergenerators. Let us assume that relations between generators ϑ ∈ Γinv and elementsϕ ∈ horP have the form:

(3.46) ϑϕ = (−)∂ϕϕ(0)(ϑ ϕ(1)) + `(a, ϕ) π(a) = ϑ

These relations are compatible with the *-structure on horP and Γinv as it followsfrom (3.23). Indeed ϑ∗ = −π[κ(a)∗] and

ϑϕ− (−)∂ϕϕ(0)(ϑ ϕ(1))∗ = (−)∂ϕϕ∗ϑ∗ − (ϑ∗ κ(ϕ(1)∗))ϕ(0)∗

= `(a, ϕ)∗ = `(κ(a)∗κ(ϕ(1)∗), ϕ(0)∗)

The relations are also transparent to the product in horP , as we can see from(3.21). Explicitly,

ϑ(ϕψ)− (−)∂ϕ∂ψ(ϕ(0)ψ(0))ϑ (ϕ(1)ψ(1)) = `(a, ϕψ) =ϑϕ− (−)∂ϕϕ(0)(ϑ ϕ(1))

ψ + (−)∂ϕϕ(0)

(ϑ ϕ(1))ψ − (−)∂ψψ(0)(ϑ (ϕ(1)ψ(1))

= `(a, ϕ)ψ + (−)∂ϕϕ(0)`(aϕ(1), ψ)

Definition 15. The above relations (3.46) will be called primary relations fordifferential calculus associated to (horP , F,ΩM , D).

In such a way we have constructed a graded *-algebra W(P ) containing horPand Γ⊗inv as its subalgebras, and products between the elements of these subalgebrasinduce natural decompositions

(3.47) W(P )! horP ⊗ Γ⊗inv! Γ⊗inv ⊗ horP

So W(P ) is given by a crossed-product construction and the product in W(P ) isencoded in a flip-over operator Σ: Γ⊗inv ⊗ horP → horP ⊗ Γ⊗inv satisfying

(3.48) Σ(ϑ⊗ ϕ) = `(a, ϕ)⊗ 1 + (−)∂ϕϕ(0) ⊗ (ϑ ϕ(1))

for generators ϑ ∈ Γinv. There are two naturally appearing pentagonal identities

Σ(id⊗ •P ) = (•P ⊗ id)(id⊗ Σ)(Σ⊗ id)(3.49)

Σ(•⊗ ⊗ id) = (id⊗ •⊗)(Σ⊗ id)(id⊗ Σ)(3.50)

28 MICHO DURDEVICH

the first extends the product to the whole tensor algebra and the second expressesthe compatibility of the relations with the product on horP . As well as the standard*-compatibility relation

(3.51) ∗Σ∗ = Σ−1

ensuring the consistency of the *-structure.Let us now assume that the refining calculus Γ possesses the following additional

consistency property:

(3.52) π(a(1))⊗ π(a(2)) = 0 ⇒ %(a) = 0 ∀a ∈ R

Remark 16. When in doubt, we can always find a refuge in the universal calculuswhere R = 0 and Γinv ↔ ker(ε). Another nice solution is given by considering allthe elements a ∈ R satisfying in addition %(a) = 0. According to the correspondingelementary properties of % established in the previous subsection, the kernel of % inker(ε) is a ∗κ-invariant and ad-covariant right ideal.

Let Υ be a subalgebra of Γ⊗inv generated by elements of the form π(a(1))⊗π(a(2))where a ∈ R. Clearly,

(3.53) Υ↔π(a(1))⊗ π(a(2)) | a ∈ R

⊗There exists a natural Υ-bimodule structure on horP defined by

(3.54)

(π(a(1))⊗ π(a(2))

)· ϕ = %(a)ϕ

ϕ ·(π(a(1))⊗ π(a(2))

)= ϕ%(a)

The above definition is consistent, due to the definition of the ideal R. We have infact a Υ-*-bimodule, because of[(

π(a(1))⊗ π(a(2)))· ϕ]∗ = ϕ∗%(a)∗ = −ϕ∗ρ[κ(a)∗] = ϕ∗ ·

(π(a(1))⊗ π(a(2))

)∗Let us take Γ∧—the universal differential envelope of Γ to represent the complete

calculus on the structure group G. We have

Γ∧ ↔ A⊗ Γ∧inv ↔ Γ∧inv ⊗A

at the level of left/right A-modules, and the algebra Γ∧inv is given by quadraticrelations

(3.55) Γ∧inv = Γ⊗inv/π(a(1))⊗ π(a(2))

∣∣ a ∈ RProposition 25. (i) There exists a unique homomorphism F : W(P )→W(P )⊗Γ∧

extending F : horP → horP ⊗A and satisfying

(3.56) F (ϑ) = ad(ϑ) + 1⊗ ϑ ∀ϑ ∈ Γinv

(ii) The map F is grade-preserving, intertwines the *-structures and

(3.57) (F ⊗ id)F = (id⊗ φ)F

Proof. The uniqueness of F follows from the fact that Γinv together with horP ,generates the wholeW(P ). To check the consistency of the definition of F we must


verify that the defining formula (3.56) passes through the generating relations forW(P ). We have

F(ϑϕ− (−)∂ϕϕ(0)(ϑ ϕ(1))

)= (−)∂ϕϑ(0) ⊗ (ϑϕ(1))− (−)∂ϕϕ(0) ⊗ ϕ(1)(ϑ ϕ(2))

+ ϑ(0)ϕ(0) ⊗ (ϑ(1)ϕ(1))− (−)∂ϕϕ(0)(ϑ(0) ϕ(1))⊗ (ϑ(1)ϕ(2))

= ϑ(0)ϕ(0) ⊗ (ϑ(1)ϕ(1))− (−)∂ϕϕ(0)(ϑ(0) ϕ(1))⊗ (ϑ(1)ϕ(2)) = F`(a, ϕ)

and hence F exists, and by construction is grade-preserving. Both sides of (3.57)are homomorphisms from W(P ) to W(P ) ⊗ Γ∧ ⊗ Γ∧. Therefore it is sufficient tocheck that the two homomorphisms coincide on horP and Γinv. In the same spirit Fintertwines the *-structures, when acted on horP and Γinv and hence it is globally*-preserving.

Let us now consider an ideal I(P ) in W(P ) generated by

(3.58)%(a)− π(a(1))⊗ π(a(2))

∣∣ a ∈ Rand let Ω(P ) be the corresponding factoralgebra.

Definition 17. We shall call these relations the secondary relations for the calculusassociated to (horP , F,ΩM , D).

Proposition 26. (i) The ideal I(P ) is *-invariant and F (I(P )) ⊆ I(P ) ⊗ Γ∧.Hence the map F projects to a *-homomorphism F : Ω(P ) → Ω(P ) ⊗ Γ∧ which isgrade-preserving and

(3.59) (F ⊗ id)F = (id⊗ φ)F

(ii) There exists a unique antiderivation dP : Ω(P )→ Ω(P ) satisfying

dP (ϕ) = D(ϕ) + (−)∂ϕϕ(0)π(ϕ(0))(3.60)

dP [π(a)] = %(a)− π(a(1))π(a(2))(3.61)

where ϕ ∈ horP and a ∈ A.(iii) The map dP commutes with ∗ and satisfies

(3.62) d2P = 0

in other words Ω(P ) is a graded-differential *-algebra.

(iv) The map F intertwines the differentials on Ω(P ) and Ω(P ) ⊗ Γ∧.

Proof. Let us start by verifying the * and F invariance of the defining relations forI(P ). We have

F%(a)−π(a(1))⊗π(a(2))

= %(a(2))⊗κ(a(1))a(3)−

(π(a(2))⊗π(a(3))

)⊗κ(a(1))a(4)

− 1⊗ π(a(1))π(a(2)) + π(a(2))⊗ π(a(1))κ(a2)a(4) − π(a(2))⊗ κ(a(1))a(3)π(a(4))

= q(a(2))⊗ κ(a(1))a(3) − π(a(2))⊗ d(κ(a(1))a(3)

)= (q ⊗ id)ad(a)

where we have used ad-invariance of R as well as identities dκ(a) = −π(a(1))κ(a(2))and π(a(1))π(a(2)) = −dπ(a) defining

q(a) = %(a)− π(a(1))⊗ π(a(2))

30 MICHO DURDEVICH

Furthermore,

q(a)∗ = %(a)∗−(π(a(1))⊗π(a(2))

)∗ = −%[κ(a)∗]+π[κ(a(2)∗)]⊗π[κ(a(1)∗] = −q[κ(a)∗]

and this proves our first assertion. Let us proceed with the construction of dP .Since its action on generating spaces of Ω(P ) is fixed, we only have to check thecompatibility of these formulae with the defining relations of W(P ) and Ω(P ). Letus start by verifying the graded Leibniz rule on horP . We compute

dP (ϕ)ψ + (−)∂ϕϕdP (ψ) = D(ϕ)ψ + (−)∂ϕϕ(0)π(ϕ(1))ψ + (−)∂ϕϕD(ψ)

+ (−)∂ϕ+∂ψϕψ(0)π(ψ(1)) = D(ϕψ)− (−)∂ϕϕ(0)`(ϕ(1), ψ)

+ (−)∂ϕϕ(0)`(ϕ(1), ψ) + ϕ(0)ψ(0)(π(ϕ(1)) ψ(1)) + (−)∂ϕ+∂ψϕψ(0)π(ψ(1))

= D(ϕψ) + (−)∂ϕ+∂ψϕ(0)ψ(0)π(ϕ(1)ψ(1)) = dP (ϕψ)

and hence indeed it holds on horizontals. Furthermore, the primary relations trans-form as follows

dPϑϕ− (−)∂ϕϕ(0)(ϑ ϕ(1))

= dP (ϑ)ϕ− ϑdP (ϕ)

− (−)∂ϕd(ϕ(0))(ϑ ϕ(1))− ϕ(0)dP (ϑ ϕ(1))

= %(a)ϕ− π(a(1))π(a(2))ϕ− π(a)D(ϕ)− (−)∂ϕπ(a)ϕ(0)π(ϕ(1))

− (−)∂ϕD(ϕ(0))π(aϕ(1))− ϕ(0)π(ϕ(1))π(aϕ(2))

− ϕ(0)%(aϕ(1)) + ϕ(0)π(a(1)ϕ(1))π(a(2)ϕ(2))

= %(a)ϕ− ϕ0%(aϕ(1))− `(a,D(ϕ))− π(a(1))π(a(2))ϕ− (−)∂ϕ`(a, ϕ0)π(ϕ(1))

− ϕ(0)π(aϕ(1))π(ϕ(2))− ϕ(0)π(ϕ(1))π(aϕ(2)) + ϕ(0)π(a(1)ϕ(1))π(a(2)ϕ(2))

Applying identity

π(a(1))π(a(2))ϕ = ϕ(0)π(a(1)ϕ(1))π(a(2)ϕ(2)) + (−)∂ϕ`(ϑ(0), ϕ(0))(π(ϑ(1)) ϕ(1)

)+ `(a(1), `(a(2), ϕ)

)− ϕ(0)π(aϕ(1))π(ϕ(2))− ϕ(0)π(ϕ(1))π(aϕ(2))

and using property (3.29) this further transforms into

%(a)ϕ− ϕ0%(aϕ(1))− `(a,D(ϕ))− `(a(1), `(a(2), ϕ)

)− (−)∂ϕ`(ϑ(0), ϕ(0))

(π(ϑ(1)) ϕ(1)

)− (−)∂ϕ`(a, ϕ0)π(ϕ(1))

= D`(a, ϕ)− (−)∂ϕ`(a(2), ϕ(0))π[κ(a(1))a(3)ϕ(1)] = dP `(a, ϕ)

where a ∈ ker(ε) and π(a) = ϑ. As for the secondary relations we compute

dP(%(a)− π(a(1))π(a(2))

)= D%(a) + %(a(2))π[κ(a(1))a(3)]− %(a(1))π(a(2))

+ π(a(1))%(a(2)) + π(a(1))π(a(2))π(a(3))− π(a(1))π(a(2))π(a(3))

= D%(a) + `(a(1), %(a(2))) = 0

where a ∈ A is arbitrary and we have applied the quantum Bianchi identity. So thisimplies at first, that the secondary relations are compatible with the definition ofdP , and hence we conclude that dP exists as an antiderivation on Ω(P ). The secondimportant conclusion from the above lines is that d2

P vanishes on the elements π(a).


In order to prove that the square of dP vanishes globally, it is sufficient to checkthat it vanishes on horP (the square of an antiderivation is a derivation). We have

d2P (ϕ) = dP

D(ϕ) + (−)∂ϕϕ(0)π(ϕ(1))

= −ϕ(0)%(ϕ(1))− (−)∂ϕD(ϕ(0))π(ϕ(1))

+(−)∂ϕD(ϕ(0))π(ϕ(1))+ϕ(0)π(ϕ(1))π(ϕ(2))+ϕ(0)%(ϕ(1))−ϕ(0)π(ϕ(1))π(ϕ(2)) = 0

Let us now check that dP commutes with the *-structure. Because of the gradedLeibniz rule, it is sufficient to verify this property on the generating spaces horPand Γinv. Having in mind that dPπ(a) = [q(a)] we have already checked that dPand d∗P = ∗dP ∗ coincide on Γinv. For horizontals, we have

d∗P (ϕ) = D∗(ϕ) + (−)∂ϕ(ϕ(0)∗π(ϕ(1)∗)

)∗ = D∗(ϕ)− π[κ−1(ϕ(1))]ϕ(0)

= D∗(ϕ)− `(κ−1(ϕ(1)), ϕ(0)

)− (−)∂ϕϕ(0)

[π(κ−1(ϕ(2))

) ϕ(1)

]= D∗(ϕ)− `

(κ−1(ϕ(1)), ϕ(0)

)+ (−)∂ϕϕ(0)π(ϕ(1))

= D(ϕ) + (−)∂ϕϕ(0)π(ϕ(1)) = dP (ϕ)

in accordance with (3.24). Finally, to check property (iv) it is sufficient to verify iton horP and Γinv. We have already checked out the action on F on special elementsq(a). For horizontals, we have

F dP (ϕ) = D(ϕ(0))⊗ ϕ(1) + (−)∂ϕ(ϕ(0) ⊗ ϕ(1))(π(ϕ(3))⊗ κ(ϕ(2))ϕ(4))

+(−)∂ϕϕ(0)⊗ϕ(1)π(ϕ(2)) = D(ϕ(0))⊗ϕ(1)+(−)∂ϕϕ(0)π(ϕ(1))⊗ϕ(2)+ϕ(0)⊗d(ϕ(1))

= dP (ϕ(0))⊗ ϕ(1) + (−)∂ϕϕ(0) ⊗ d(ϕ(1))

By multiplicativity and the graded Leibniz rule, this property extends to the wholealgebra Ω(P ).

Proposition 27. (i) Within Ω(P ) the algebra horP is characterized as the graded*-subalgebra of all elements ϕ satisfying F (ϕ) ∈ Ω(P ) ⊗ A. In other words, it isinterpretable as the horizontal forms associated to the calculus (Ω(P ), dP , F ).

(ii) The formula

ω(ϑ) = 1⊗ ϑ/I(P )

defines a natural connection ω : Γinv → Ω(P ) ! horP ⊗Υ Γ⊗inv. The followingidentifications hold

D = Dω `(a, ϕ) = `ω(π(a), ϕ) % = rω ı = ıω

Proof. Let us first observe that Ω(P ) naturally decomposes as

(3.63) Ω(P )! horP ⊗Υ Γ⊗inv! Γ⊗inv ⊗Υ horP

We basically have to synthesize all the structural elements developed in this section,and close the circle.

Remark 18. The algebra ΩM can now be viewed as the F -fixed point subalgebraof Ω(P ) and the differential on ΩM is the restriction of dP .

32 MICHO DURDEVICH

4. First-Order Horizontal Calculus

4.1. Naturality of Relations

Having established the structure of a natural differential calculus Ω(P ) associatedto a quadruplet (horP , F,ΩM , D) we can now turn to the problematics to whatextent this calculus is defined by its first-order part, which is a certain bimoduleover B. We shall see that the secondary relations are a logical consequence of thefirst-order relations, so that the calculus coincides with the universal differentialenvelope of its first-order part. We follow the identifications established in theprevious section.

Let us start by showing that the graded Leibniz rule for the differential dP ,together with its action on horizontal forms completely determines the action onΩ(P ) including its secondary relations.

From the definition of the action of dP on horP it follows that

(4.1) ωπ(a) = [a]1(dP −D)[a]2

for each a ∈ A. Assuming the graded Leibniz rule for dP and performing elementarycalculations we obtain

dPωπ(a) = dP

[a]1

(dP −D)[a]2 − [a]1dPD[a]2=D[a(1)]1

[a(1)]2ωπ(a(2)) + [a(2)]1ωπ

κ−1(a(1))

[a(2)]2ωπ(a(3))

− [a]1D2[a]2 + [a(1)]1D[a(1)]2ωπ(a(2)) = %(a)− ωπ(a(1))ωπ(a(2))

and we have applied (3.9) together with (3.24) and

(4.2) [a(2)]1`(κ(a(1)), [a(2)]2

)= `(a(1), [a(2)]1

)[a(2)]2

which is a direct consequence of (3.21) and [a]1[a]2 = ε(a). Therefore we have re-covered the formula for differentiating the canonical connection, and in a particularcase of a ∈ R we conclude that the secondary relations

%(a) = ωπ(a(1))ωπ(a(2))

must hold.Now we shall explain how primary relations naturally emerge from the require-

ment of a simple extendibility of horizontal forms to a complete calculus over P .Let Λ(P ) be a differential *-algebra including horP as its subalgebra, so that thedifferential dΛ : Λ(P )→ Λ(P ) extends the differential on the base dM : ΩM → ΩM .Therefore the map dΛ−D supercommutes with the left multiplication by elementsof ΩM . Let ς : A → Λ(P ) be a map given by

(4.3) ς(a) = [a]1(dΛ −D)[a]2

From this definition it follows that ς(C) = 0 and

(4.4) dΛ(ϕ) = D(ϕ) + (−)∂ϕϕ(0)ς(ϕ(1)) ∀ϕ ∈ horP

We also have

(4.5) ς(a)∗ = −ς[κ(a)∗] ∀a ∈ A


Indeed a direct application of (3.9) gives,

ς(a)∗ =

(dΛ −D)[a]2∗[a]∗1 = (dΛ −D

∗)[κ(a)∗]1[κ(a)∗]2= −[κ(a)∗]1(dΛ −D)[κ(a)∗]2 = −ς[κ(a)∗]

The map ς satisfies the following variant of Leibniz rule

(4.6) ς(ab) = ε(a)ς(b) + [b]1ς(a)[b]2 − [b]1`(a, [b]2)

as it follows by elementary computations

ς(ab) =D∗[ab]1

[ab]2 + [ab]1dΛ[ab]2 = D∗

[b]1[a]1

[a]2[b]2

+ [b]1[a]1dΛ[a]2[b]2 + [b]1[a]1[a]2dΛ[b]2 = [b]1]

[a]1(dΛ −D)[a]2

[b]2+ ε(a)[b]1(dΛ −D)[b]2 − [b]1[a(1)]1[a(1)]2`(a

(2), [b]2)

= ε(a)ς(b) + [b]1ς(a)[b]2 − [b]1`(a, [b]2)

Now let us define R to be the set of elements a ∈ ker ε in which ς(a) = 0 and`(a, ϕ) = 0 for every ϕ ∈ horP . It is easy to see that R is a right-ideal in ker(ε)and κ(R)∗ = R. Therefore it determines a *-covariant differential calculus Γ overG. Let $ : Γinv → Λ(P ) be a ‘connection form’ map defined by

(4.7) $π = ς

As a consequence of (4.5) we conclude that

(4.8) $(ϑ)∗ = $(ϑ∗)

We have the following commutation relation between the elements $(ϑ) and hori-zontal forms:

(4.9) $(ϑ)ϕ− (−)∂ϕϕ(0)$(ϑ ϕ(1)) = `(a, ϕ) π(a) = ϑ

Indeed,

0 = ϕdΛ(ψ) + (−)∂ϕdΛ(ϕ)ψ − (−)∂ϕdΛ(ϕψ) = ϕ(0)ς(ϕ(0))ψ + (−)∂ψϕψ(0)ς(ψ(1))

− (−)∂ψϕ(0)ψ(0)ς(ϕ(1)ψ(1))− ϕ(0)`(ϕ(1), ψ)

and property (4.9) easily follows.Thus, our primary relations are also a logical consequence of the existence of a

differential algebra extension of horP and the algebra Ω(P ) constructed in previoussection exhibits a kind of universality property.

Proposition 28. (i) There exists a unique homomorphism Φ: Ω(P ) → Λ(P )intertwining differentials and extending identity on horP . This map is grade/*-preserving and

(4.10) Φω = $

(ii) Let us assume that there exists a FΛ : Λ(P ) → Λ(P ) ⊗ Γ∧ which is a dif-ferential algebra morphism extending F : horP → horP ⊗ A. Then $ satisfies acharacteristic connection property

(4.11) FΛ$(ϑ) = ($ ⊗ id)ad(ϑ) + 1⊗ ϑ

and the map Φ is a monomorphism between graded-differential *-algebras intertwin-ing F and FΛ.

34 MICHO DURDEVICH

Proof. To complete the proof, let us demonstrate (4.11). By formula (4.3) andsome elementary transformations we obtain

FΛς(a) = FΛ

[a]1(dΛ −D)[a]2

= [a(2)]1(dΛ −D)[a(2)]2 ⊗ κ(a(1))a(3)

+ [a(2)]1[a(2)]2 ⊗ κ(a(1))d(a(3)) = ς(a(2))⊗ κ(1)a(3) + 1⊗ π(a)

Therefore $ is a connection form for the calculus Λ(P ) and Φ is indeed a calculimonomorphism.

4.2. Horizontal Envelopes

As explained in [5] a fundamental element of the structure of a quantum principalbundle is a *-morphism ρ : V → Md(V) induced by the structural relations (2.2).Explicitly, this map is given by

(4.12) ραβ =n∑k=1

bαkb∗βk

Here we shall assume that the generating representation u is unimodular in the senseof conjugate representation u being realizable as a subrepresentation of a poweru× · · · × u. In this case, the structural relations together with the specification ofG-invariant combinations of generators bαi enable us to reconstruct the bundle, viaan appropriate cross-product involving generalized Cuntz algebras. More precisely,

(4.13) B! V ⊗µ On,d! On,d ⊗µ V

where On,d is a *-algebra generated by elements ψαi together with the relations

(4.14)d∑

α=1

ψ∗αiψαj = δij Fn,d(ψαi) =n∑j=1

ψαj ⊗ uji

defining also a symmetry action Fn,d : On,d → On,d ⊗ A of G. Basically, we con-structed an abstract quantum principal bundle, starting from fundamental relations(2.2). There is a natural *-homomorphism µ : O0

n,d → V where O0n,d is the invari-

ant *-subalgebra of Fn,d. This map is a restriction of a covariant bundle mapµ : On,d → B defined by µ : ψαi 7→ bαi. The algebras On,d are building blocks ofthe quantum classifying space BG together with its universal bundle EG. Equa-tion (4.13) features a natural flip-over operator which we shall use extended onhorizontal forms σµρ : On,d ⊗µ ΩM ↔ ΩM ⊗µ On,d given by

(4.15) σµρ(ψαi ⊗ w) =d∑

β=1

ραβ(w)ψβi

and extended on products of generators by standard pentagonal properties

(4.16)(σµρ ⊗ id)(id⊗ •) = (• ⊗ id)(id⊗ σµρ)(σµρ ⊗ id)

(id⊗ σµρ)(• ⊗ id) = (id⊗ •)(σµρ ⊗ id)(id⊗ σµρ)

So we can formulate another decomposition of horizontal forms

(4.17) horP ! ΩM ⊗µ On,d! On,d ⊗µ ΩMWe shall also use the same formulae (4.12) and (4.15) in the context of bimodulesover B and why dont we now proceed to define a very important concept of a


horizontal first-order calculus. Let us assume that a *-B-bimodule H is given,together with a *-derivation dM : V → H satisfying

(4.18) H = BdM (V) ⇔ dM (V)B = H

In other words we assume that the elements of H can be written as sums∑

bdM (f)

or equivalently∑

dM (f)b for some b ∈ B and f ∈ V.

We shall also assume that H carries a natural right action of G in the form of amap F∼ : H → H⊗A satisfying

(4.19) F∼(dM (f)b

)= dM (f)F (b)

for each f ∈ V and b ∈ B. Because of property (4.18) the action F∼ is unique, if itexists. We also conclude that

(4.20) F∼(ψb) = F∼(ψ)F (b) ∀ψ ∈ H, b ∈ B

for each b ∈ B and ψ ∈ H. Another direct consequence of the above properties(4.18) and (4.19) is that

(id⊗ φ)F∼ = (F∼ ⊗ id)F∼(4.21)

(id⊗ ε)F∼ = id(4.22)

hold. It is also worth observing that the *-conjugation on H is unique, if it exists.Indeed dM (f)b 7→ b∗dM (f∗) and this fixes it completely.

Let us observe that the elements of the form

N = VdM (V) = dM (V)V

are precisely those elements of H which are F∼-invariant. Indeed ψ 7→ ψ(0)h(ψ(1))where h : A → C is the Haar measure for G, is the canonical projection on thecorresponding invariant subspace and dM (f)b 7→ dM (f)b(0)h(b(1)). In the bimoduleH we can naturally ‘act’ by morphisms ραβ . We shall assume that these morphismspreserve the invariant part N , in other words

(4.23) ραβ(N ) ⊆ N

so that we have a bimodule morphism ρ : N → Md(N ).

Definition 19. The triplet (H, F∼, dM ) is called a first-order horizontal calculusover a quantum principal bundle P .

Remark 20. There is always a trivial way to extend a given first-order horizontalcalculus to a complete horizontal calculus (horP , F∧,ΩM ) over P , by postulatinghor1

P = H and horkP = 0 for k ≥ 2. Then ΩM = V ⊕N and F∧ = F ⊕ F∼.

Lemma 29. (i) The map F∼ is also left-multiplicative, in the sense

(4.24) F∼(bψ) = F (b)F∼(ψ)

and a *-intertwiner, so that

(4.25) (∗ ⊗ ∗)F∼ = F∼∗

(ii) The module multiplication induces the following natural isomorphisms

(4.26) H! B ⊗V N ! N ⊗V B

36 MICHO DURDEVICH

Proof. Let us consider the canonical multiplet generators bαi. For every f ∈ V andq ∈ B we have

F∼(bαjdM (f)q

)=∑β

F∼(ραβ [dM (f)]bβjq

)=∑βi

ραβ [dM (f)]bβiq(0) ⊗ uijq

(1)

=∑i

bαidM (f)q(0) ⊗ uijq(1) = F (bαj)F∼

(dM (f)q

)so we see that (4.24) holds for the elements of the form b = bαj . It is also clear fromthe very definition of F∼, that (4.24) holds for all b ∈ V. On he other hand, theset of all such elements b ∈ B for which this left multiplicativity property holds isclearly a subalgebra of B and we conclude that the mysterious subalgebra is nothingbut B itself. Now the *-intertwining property easily follows

F∼[(dM (f)b)∗] = F∼[b∗dM (f∗)] = b(0)∗dM (f∗)⊗ b(1)∗

= [dM (f)b(0)]∗ ⊗ b(1)∗ = (∗ ⊗ ∗)F∼[dM (f)b]

The decomposition property is now a corollary of considering already mentionedtrivial horizontal calculus based on B⊕H, where there are no higher-order horizontalforms.

Remark 21. The left multiplicativity property (4.24) is actually equivalent to theinvariance of N under the pull-back morphisms ραβ .

Let us construct the tensor algebra

(4.27) N⊗ =⊕k≥0

N⊗k N⊗0 = V N⊗k = N ⊗V · · · ⊗V N︸︷︷︸k

over the invariant part N . We need to find a minimal system of relations whichare compatible with the initial differential dM : V → N as well as with the matrixmorphism ρ. Let N∧ be the universal differential envelope of the calculus (N , dM ).Let us recall that N∧ = N⊗/J0 where J0 is a graded ideal generated by a V-bimodule of quadratic relations

J 20 =

∑dM (f)⊗ dM (g)

∣∣∣∑ fdM (g) = 0

This is the minimal set of relations necessary to get a consistent differentialextension of (N , dM ). So up to now we have a graded-differential algebra N∧ wherein general we can not consistently apply the morphism ρ and the tensor algebra N⊗where, again in general, we can not act by dM . There exist a natural non-decreasingsequence of ideals

(Jk)k≥0

inductively defined as follows. For each k let us firstintroduce auxiliary ideal Jk,ρ as the minimal ρ-invariant ideal containing Jk. Everygraded ideal X projects down to a graded ideal X∧ in N∧. Then X∧ + dM (X∧)is the minimal dM -invariant space in N∧ containing X∧, and due to the gradedLeibniz rule it is an ideal, too. Finally, we define Jk+1 to be the pull-back ofJ ∧k,ρ + dM (J ∧k,ρ). So that we have

Jk ⊆ Jk,ρ ⊆ Jk+1 dM (J ∧k ) ⊆ J ∧k+1

Let us defineJ =

⋃k≥0

Jk so that J ∧ =⋃k≥0

J ∧k


Now by construction J is a ρ-invariant *-ideal, such that J ∧ is dM -invariant. Thusthe factor-*-algebra

(4.28) ΩM = N∧/J ∧

is a higher-order calculus for (N , dM ) we shall use the same symbol dM : ΩM → ΩMfor the projected differential. Moreover, the morphism ρ naturally projects to agrade-preserving *-homomorphism ρ : ΩM → Md(ΩM ).

Remark 22. So the procedure starts with J0. The ideal J0,ρ is generated by matrixelements of all the morphisms ρm : N⊗ → Mdm(N⊗) where m ≥ 0, restricted toinitial generators J 2

0 . Projecting them down to N∧ differentiating, and pullingback to N⊗ gives the generators of the ideal J1, which are therefore quadratic andcubic. And so on—in general the ideal Jk will contain the generating elementsfrom order 2 up to k + 2 and Jk,ρ is generated by the matrix elements of all ρm

restricted on the generators of Jk.

We can now complete the construction of the full algebra of horizontal forms,using the cross-product and the classifying map, as described at the beginning ofthis subsection. We shall combine the map ρ : ΩM → Md(ΩM ) together with thehomomorphism µ : O0

n,d → V to build the flip-over operator σµρ : On,d ⊗µ ΩM ↔ΩM ⊗µ On,d satisfying (4.15) and (4.16), and finally define

(4.29) H∧! ΩM ⊗µ On,d! On,d ⊗µ ΩM

where the algebra structure on the tensor product is defined with the help of σµρ.The action F∧ : H∧ → H∧ ⊗A is induced from Fn,d : On,d → On,d ⊗A.

Proposition 30. The constructed triplet (H∧, F∧,ΩM ) is a horizontal calculus ex-tending the first-order calculus (H, F∼, dM ).

Remark 23. Nothing impends a possible disastrous outcome of our beautiful pro-cedure, in which the ideals Jk just eat up everything they can. But even in thiscase, we are still left with an intact first-order calculus :) There would be no secondor higher order differential forms in such a situation—the already mentioned triv-ial extension. Another ‘extreme’ occurs when J = J0 in other words, the initialideal J0 is ρ-invariant. Universal first-order calculi are such an example, becauseJ0 = 0 in this case. Another example–this will always be the case when M is aclassical smooth manifold equipped with a classical first-order calculus. A nice classproviding ‘intermediary’ examples is given by carefully chosen non-standard calculiover classical spaces, finite classical groups are discussed in this light, later on.

Proposition 31. Consider a graded *-algebra D. Assume that L : D → D⊗A is a*-homomorphism and action of G, such that its invariant graded *-subalgebra DMis equipped with a differential structure δM : DM → DM . Let us also assume thatcovariant identifications B ↔ D0 and H ↔ D1 are given so that

V D0M

dM

y yδMN D1

M

38 MICHO DURDEVICH

In other words (D, L,DM ) is a horizontal calculus extending the first ordercalculus (H, F∼, dM ). Under such assumptions, there exists a unique homomor-phism ν : H∧ → D extending the above identifications. This maps is grade/*-preserving and intertwines the corresponding actions of G. When restricted on theG-symmetric parts, it gives a homomorphism between graded differential *-algebrasΩM and DM .

Proof. We can extend, in a unique and covariant/* way the initial identificationsB ↔ D0 and H ↔ D1 to a homomorphism ν : H⊗ → D. Because of the covariancewe have ν(N⊗) ⊆ DM . Let us prove that ν(Jk) = 0 for each k ≥ 0. For k = 0 itfollows from the intertwining differentials property and universality of the envelopeN∧. Let ν2 : N∧ → DM be the induced homomorphism of graded differential *-algebras. Now if ξ ∈ H⊗ is such that ν(ξ) = 0 then ν[ραβ(ξ)] = 0 because of (4.12).And if ζ ∈ N∧ is such that ν2(ζ) = 0 then ν2dM (ζ) = δMν2(ζ) = 0. These twosimple properties imply that if ν(Jk) = 0 then ν(Jk+1) = 0. Hence ν(J ) = 0and the map ν factorizes to ν∧ : H∧ → D.

4.3. More About Envelopes

So far we have elaborated 2 ways to reconstruct a complete calculus given some‘partial’ information. The first was to build the full calculus on the bundle, startingfrom graded *-algebra of horizontal forms and the calculus on the base. The secondconstruction was to build the full algebra of horizontal forms, given a *-bimodule offirst-order horizontal forms together with a first-order differential calculus over M .As a simple addition to this collection of natural extensions, let us now study therelationship between the algebra Ω(P ) and its first-order part G = Ω1(P ). Weare going to prove that Ω(P ) can be viewed as the universal differential envelopeG∧ of G.

Proposition 32. Let us assume that the algebra hor(P ) is given by the universalhorizontal envelope of its first-order part. Then G is spanned by sums of elementsbdP (q) where b, q ∈ B and the corresponding universal map G∧ Ω(P ) is anisomorphism of graded differential *-algebras.

Proof. If the horizontal forms are given by the horizontal envelope construction,then the entire hierarchy of the relations in hor(P )! H∧ is in fact induced fromthe quadratic relations of the universal envelope of N plus the requirement thatΩM be embedded in a larger horizontal forms algebra–implying a second compati-bility mode with the pull-back morphism ρ. As we already explained, the primaryand secondary relations, which allow us to go from hor(P ) to Ω(P ) are a logicalconsequence of the existence of an abstract differential structure. Therefore, allthese relations are fulfilled in G∧ and the universal envelope G∧ itself provides avalid solution for a complete calculus on the bundle.

5. Concluding Observations & Examples

5.1. Affine Spaces of Connections

The set of all covariant derivatives D possesses a natural structure of a real(infinite-dimensional, in general) affine space d(horP , F,ΩM ). In order to adoptmore covariant derivatives, we may ‘refine’ the calculus Γ to be compatible with


all covariant derivatives from a given family. If Γ is compatible with 2 covariantderivatives say K and L then it is compatible with all of the derivatives belongingto the line tE + (1− t)F | t ∈ R passing through K and L. Let us elaborate thisin more detail.

Proposition 33. The vector space−→d (horP , F,ΩM ) associated to d(horP , F,ΩM )

is represented as real left-ΩM graded-linear first-order maps E : horP → horP , orequivalently as first-order linear maps λE : A → horP satisfying

(5.1)

FλE(a) = λ(a(2))⊗ κ(a(1))a(3)

λE(1) = 0

λE(a)∗ = −λE [κ(a)∗]

The correspondence E ↔ λE being given by

E(ϕ) = −(−)∂ϕϕ(0)λE(ϕ(1))

λE(a) = −[a]1E[a]2

By removing the third reality condition for λE we get the representation of thecomplex vector space

−→dC(horP , F,ΩM ).

Let us now explicitly calculate the regularity obstacle and the curvature ten-sor of the displaced connection D + E, in terms of those associated to the initialconnection.

Lemma 34. For a ∈ ker(ε) and ϕ ∈ horP we have

`D+E(a, ϕ) = `D(a, ϕ) + λE(a)ϕ− (−)∂ϕϕ(0)λE(aϕ)(5.2)

%D+E(a) = %D(a) + λE(a(1))λE(a(2)) +DλE(a) + `D(a(1), λE(a(2))

)(5.3)

Proof. Let us first compute the anticommutator between D and E. Applying thedefinition of E and the Leibniz rule for D we find

(DE + ED)(ϕ) = (−)∂ϕD(ϕ(0))λE(ϕ(1))− (−)∂ϕD[ϕ(0)λE(ϕ(1))

]= (−)∂ϕD(ϕ(0))λE(ϕ(1))− ϕ(0)DλE(ϕ(1))− ϕ(0)`D

(ϕ(1), λE(ϕ(2))

)− (−)∂ϕD(ϕ(0))λE(ϕ(1)) = −ϕ(0)

DλE(ϕ(1)) + `D

(ϕ(1), λE(ϕ(2))

)We now have

%D+E(a) = −[a]1(D + E)2[a]2 = −[a]1D2[a]2 − [a]1(DE + ED)[a]2 − [a]1E

2[a]2= %D(a) +DλE(a) + `D

(a(1), λE(a(2))

)+ λE(a(1))λE(a(2))

and finally

`D+E(a, ϕ) = [a]1(D + E)[a]2ϕ − [a]1(D + E)[a]2ϕ = `D(a, ϕ)

− (−)∂ϕ[a(1)]1[a(1)]2ϕ(0)λE(a(2)ϕ(1)) + [a(1)]1[a(1)]2λE(a(2))ϕ

= `D(a, ϕ) + λE(a)ϕ− (−)∂ϕϕ(0)λE(aϕ(1))

which proves the regularity obstacle transformation rule.

40 MICHO DURDEVICH

It also worth to remember the following charming identity:

(5.4) `D+E

(a(1), λE(a(2))

)= `D

(a(1), λE(a(2))

)+ 2λE(a(1))λE(a(2))

− λE(a(2))λEκ(a(1))a(3)

If a suitable calculus Γ is given, then the set of all Γ-compatible covariant derivativesis a (real) affine subspace of d(horP , F,ΩM ).

Proposition 35. Requiring that this space be the whole d(horP , F,ΩM ), that is, thecompatibility with all possible covariant derivatives, leads to the universal calculuson G where Γinv ↔ ker(ε) and R = 0.

Proof. If a first-order calculus Γ is compatible with covariant derivatives D andD + E then λE(a) = 0 for each a ∈ R. We shall prove that for each non-zeroa ∈ ker(ε) there exists E ∈ −→dC(horP , F,ΩM ) such that λE(a) 6= 0. This would implythat the only calculus compatible with all covariant derivatives is the universalcalculus. Let us consider an irreducible multiplet a1, . . . , am for the adjoint actionad: ker(ε) → ker(ε) ⊗ A, corresponding to some irreducible representation v ∈Mm(A). Let us pick up a similar irreducible multiplet ψ1, . . . , ψm in hor1

P . Fromthe decomposition into multiple irreducible modules, and their basic properties, weknow that such multiplets exist. So we have

ad(ai) =m∑j=1

aj ⊗ vji F (ψi) =m∑j=1

ψj ⊗ vji

Finally, let C be an arbitrary ad-invariant complement of the multiplet a1, ..., am inker(ε). Let us define a map λ : A → horP by λ(ai) = ψi λ(1) = 0 and λ(C) = 0and declare λE = λ. Such an E is a generic example of an elementary complexcovariant derivative ‘fluctuation’. Now for an arbitrary a ∈ ker(ε) let us find asplitting

ker(ε)! C ⊕ lina1, . . . , amsuch that its multiplet projection is non-zero.

Proposition 36. (i) For given D and E there exists a unique homomorphism\D,D+E : Ω(P,D)→ Ω(P,D + E) satisfying

(5.5) \D,D+Eπ(a) = π(a)− λE(a) ∀a ∈ A

and extending the identity on horP .(ii) This map is an isomorphism of graded differential *-algebras and the diagram

(5.6)

Ω(P,K) F−−−−→ Ω(P,K) ⊗ Γ∧

\K,L

y y\K,L ⊗ id

Ω(P,L) −−−−→F

Ω(P,L) ⊗ Γ∧

is commutative for every K,L ∈ d(horP , F,ΩM ).(iii) We have

(5.7) \D,D = id \D,K = \−1K,D \D,K\K,L = \D,L


Proof. Let us check that the defining formula (5.5) is compatible with primary andsecondary relations characterizing the complete calculus. We start at Ω(P,D) andend in Ω(P,D + E). At first the primary relations:

π(a)ϕ−(−)∂ϕϕ(0)π(aϕ(1)) (π(a)−λE(a))ϕ−(−)∂ϕϕ(0)(π(aϕ(1))−λE(aϕ(1))

)= `D+E(a, ϕ)− λE(a)ϕ+ (−)∂ϕϕ(0)λE(aϕ(1)) = `D(a, ϕ)

according to (5.2). For the secondary relations let us take a ∈ R and apply (5.3)and (5.4). Also, take into account that λE vanishes on ad-invariant R. We compute

π(a(1))π(a(2)) (π(a(1))− λE(a(1))

)(π(a(2))− λE(a(2))

)= π(a(1))π(a(2))

− λE(a(2))π(a(1))− π(a(1))λE(a(2)) + λE(a(1))λE(a(2))

= %D+E(a)− `E+D

(a(1), λE(a(2))

)+ λE(a(1))λE(a(2))

= %D+E(a)− `D(a(1), λE(a(2))

)− λE(a(1))λE(a(2)) = %D(a)

This proves the compatibility with the relations, and hence \D,D+E exists and isunique.

Remark 24. We see that the only additional constraint to the calculus is broughtby the requirement that λE vanishes on R. Such a calculus Γ would be compatiblewith all covariant derivatives from the line D + tE | t ∈ R.

Remark 25. According to Proposition 13 which describes the complex versionof this affine space, it is quite a ‘big’ object: The direct product of spaces ofall derivatives for bimodules Fu over non-trivial irreducible representations u ∈T \ ∅. The regular connections form a generally tiny (and possibly even empty)affine subspace d∗(horP , F,ΩM ).

5.2. Specifics of the Regular Case

The theory of connections and covariant derivatives assumes an especially elegantand simple form, when we restrict our context to regular connections, where thecovariant derivative globally satisfies the graded Leibniz rule. Here we shall presenta relatively self-contained exposition of the formalism, and it turns out we canmake several shortcuts, by taking the advantage of regularity. In particular, theconstruction of horizontal envelopes can be performed by adopting an extendedbimodule technique [18].

So our starting point will be a unital B-*-bimodule H equipped with a rightaction F∼ : H → H⊗A of G and a *-derivation D : B → H covariant in the sensethat the diagram

(5.8)

B F−−−−→ B ⊗A

Dy yD ⊗ id

H −−−−→F∼

H⊗A

is commutative. The invariant part N is a V *-submodule and covariance impliesD(V) ⊆ N . We shall also require that

(5.9) BD(V) = H ⇔ H = D(V)B

42 MICHO DURDEVICH

which in particular implies N = VD(V) = D(V)V. We also conclude that Hnaturally decomposes as

(5.10) H! B ⊗V N ! N ⊗V B

Remark 26. So we are studying some special first-order horizontal calculi overP such that there exists a covariant derivative satisfying the graded Leibniz ruleintertwining *-conjugations on B and H. The standard notion of a first-order differ-ential calculus, as introduced in [18] and focusing on quantum groups, is certainlya special case of the above definition, where V = B and we are interested in spacesand not groups, so the ‘structure group’ is trivial, or when B = A and the groupsis understood as a trivial bundle over the one-point set V = C.

Our main task is to extend the calculus of one-forms to the complete calculus,by constructing a graded *-algebra of all horizontal forms H∧ with its extendedcovariant derivative D : H∧ → H∧ and taking the advantage of the special propertyof D being an antiderivation.

Let us consider an extended bimodule H. Its simple construction goes as follows.Take a symbol X and define, at the level of left B-modules

(5.11) H = BX ⊕H

while the right B-module structure is specified by

(5.12) Xb = bX +D(b) ∀b ∈ B

together with an obvious requirement that on H it coincides with the given B-module structure on H. It is easy to see that the above rules define a B-bimodulestructure on H, extending the bimodule H. There is also a natural extension ofthe *-structure from H to H determined by requiring X∗ = −X. By postulatingF∼(bX) = b(0)X ⊗ b(1) we define an extended action F∼ : H → H ⊗ A.

We also have a natural short exact sequence of bimodules

(5.13) 0→ H −→ H −→ B → 0

where H projects to B via Xb 7→ b.Our construction enables us to view the derivation D as an inner derivation of

the extended bimodule H, a key property in constructing a higher-order horizontalcalculus.

Remark 27. Actually the problematics of classifying exact sequences (5.13) givenH and B is equivalent to the problem of studying derivations D : B → H. Twoderivations will give equivalent extensions, iff they differ by an inner derivation.

The invariant part of H is the extended V-*-bimodule N and for its naturaldecomposition we get

(5.14) H! B ⊗V N ! N ⊗V B

Let us consider the tensor algebra built over the extended bimodule H. In otherwords,

H⊗ =∞∑k=0

⊕ H⊗k H⊗k =

k︷︸︸︷H ⊗B · · · ⊗B H


for k ≥ 2 and H⊗0,1 = B, H. In the same spirit, we can consider

H⊗ =∞∑k=0

⊕H⊗k H⊗k = H⊗B · · · ⊗B H︸︷︷︸k

a smaller tensor algebra associated to H. There are natural *-bimodule inclusionsof H⊗k into H⊗k so that the algebra H⊗ contains H⊗ as its graded *-subalgebra.

Remark 28. Actually it is not that obvious, we need to verify explicitly that theinclusion of H into H induces inclusions of H⊗k into H⊗k for all k ≥ 2. Thisproperty follows from interpreting the tensor algebra H⊗ as a deformation of thetensor algebra associated to the direct sum H+ = H⊕B and constructing a natural(C-linear) identification

(5.15) H⊗! H⊗+

In this picture D is interpretable as a ‘deformation parameter’. The symbolicgenerator X introduces a natural filtration in H⊗. The graded algebra associatedto this filtration is H⊗+. The above identification can be constructed by taking adecomposition into submodules

(5.16) H⊗n+ =⊕

S⊆1,...,n

H⊗|S|

and moving each summand into H⊗n by replacing the units 1 labeling the sub-set 1, . . . , n \ S in the tensor product expansion of elements from H⊗|S| by thesymbol X.

The action map F∼ extends to a *-preserving action F∼ : H⊗ → H⊗⊗A and weobviously have F∼(H⊗) ⊆ H⊗ ⊗A. Furthermore, (5.10) and (5.14) imply that thefollowing natural decompositions hold

(5.17)H⊗n! B ⊗V N

⊗n! N⊗n ⊗V B N⊗n =

n︷︸︸︷N ⊗V · · · ⊗V N

H⊗n! B ⊗V N⊗n! N⊗n ⊗V B N⊗n = N ⊗V · · · ⊗V N︸︷︷︸

n

The derivation map D admits a natural extension to the entire tensor algebraH⊗ by simply defining

(5.18) D(T ) = XT − (−)∂TTX

It is evident by construction, that such an extended D is a hermitian antiderivationof grade one. Its square is a hermitian derivation of grade two. In particular, wehave

(5.19) D2(T ) = X2T − TX2

It is trivial but worth mentioning

(5.20) D(X) = 2X2 D2(X) = 0

44 MICHO DURDEVICH

and the diagram

(5.21)

H⊗F∼−−−−→ H⊗ ⊗A

Dy yD ⊗ id

H⊗ −−−−→F∼

H⊗ ⊗A

is commutative.

Remark 29. As is well known, antiderivations and derivations on a graded algebra,constitute a Lie superalgebra the bosonic part of which is given by derivations andthe fermionic part of which is given by antiderivations. The super-Lie bracket isgiven by graded commutators.

Now let J be a quadratic ideal in H⊗ generated by all the elements of the form

(5.22) X2f − fX2

In other words, we consider the image D2[V] as being the generating space for thisideal.

Proposition 37. The ideal J is *-invariant and D-invariant. Moreover we haveF∼(J ) ⊆ J ⊗A.

Proof. The hermicity property of J is a direct consequence of the hermicity of Dand the fact that V is a *-subalgebra in B. In order to prove the D-invariance,it is sufficient to check, due to the graded Leibniz rule for D, that the generatingelements are transformed by D into the elements of J . And indeed, we have

D3(f) = DD2(f) = X[X2, f ]− [X2, f ]X f ∈ V

which belongs to J . Finally the F∼-invariance is a consequence of the fact that thegenerating elements for J are within N⊗2 and thus are F∼-symmetric.

Having constructed such a beautiful ideal J , nothing prevents us to considereven more beautiful factor-algebra

(5.23) H∧ = H⊗/J

By construction, it is a graded *-algebra, naturally containing B and H as its gradezero and one components respectively. Moreover, the map D according to the aboveproposition, projects (denoted by the same symbol) to a hermitian antiderivationD : H∧ → H∧. The map F∼ projects down to a *-homomorphism F∧ : H∧ → H∧⊗Awhich follows tradition of its predecessors in being an action:

(id⊗ φ)F∧ = (F∧ ⊗ id)F∧(5.24)

(id⊗ ε)F∧ = id.(5.25)

We can pick up a couple of important subalgebras of H∧. At first, let H∧ be thesubalgebra generated by B and H. It is the projection of H⊗. As such it is a graded*-subalgebra and F∧(H∧) ⊆ H∧ ⊗A. Let Ω be the graded *-subalgebra consistingof invariants of H∧ and let similarly Ω be the invariant part of H∧. These algebras


are factor-projections of N⊗ and N⊗ and the following natural decompositionshold

H∧! B ⊗V Ω! Ω⊗V B(5.26)

H∧! B ⊗V Ω! Ω⊗V B(5.27)

From the covariance property for D it follows that Ω is D-invariant in other wordsD(Ω) ⊆ Ω. Let dM : Ω→ Ω be the corresponding restriction map.

Proposition 38. (i) The derivation D2 : H∧ → H∧ is Ω-linear on both sides. Inparticular the square of dM vanishes:

(5.28) d2M = 0

In other words Ω equipped with dM is a graded differential *-algebra.

(ii) The algebra Ω is the minimal differential subalgebra of Ω containing V inother words

(5.29) Ωm =∑

fdM (f1) . . . dM (fm)

=∑

dM (f1) . . . dM (fn)f

The algebra H∧ is D-invariant and

(5.30) Hm∧ =∑

bdM (f1) . . . dM (fm)

=∑

dM (f1) . . . dM (fn)b

Proof. The module N gives a standard first-order calculus over M in other wordsN = VdM (V) = dM (V)V. Since the defining relations of H∧ are quadratic, the firstorder part is intact so H1

∧ = H and H1∧ = H, in particular N = Ω1 and N = Ω1.

As we already mentioned, higher order parts of Ω are projections of blocks N⊗k.Therefore, applying the graded Leibniz rule for dM we conclude that decompositions(5.29) hold. This, together with (5.26) implies (5.30). Now, our quadratic relationsare expressible in H∧ as d2

M (f) = 0 for each f ∈ V. Taking into account (5.29) and(5.20) as well as the Leibniz rule for D2 we conclude that D2 is Ω linear on bothsides, equivalently the square of dM vanishes. We now conclude that dM (Ω) ⊆ Ω.From (5.26) and graded Leibniz rule for D, it follows that H∧ is D-invariant.

Remark 30. The fact that the square of dM vanishes can be rephrased as thecommutativity of X2 and the elements of Ω in H∧. As we can see by comparing theextended bimodule techniques presented here and in [2, 18], our construction of thehorizontal envelope of H includes, as a very special case V = B, the construction ofthe universal differential envelope of the standard first-order differential calculus.

The subalgebra Ω of H∧ is generated by X and Ω. It can be described purely interms of Ω in the following way. At the level of vector spaces, we have

Ω = Ω⊗ Pol[X]

while the algebra structure is defined using a flip-over operator

σ : Pol[X]⊗ Ω→ Ω⊗ Pol[X]

given by σ(X2n+1⊗w) = (−)∂ww⊗X2n+1 +dM (w)⊗X2n for odd degrees of X andσ(X2n⊗w) = w⊗X2n for even ones. In a more general fashion, a straightforwardcalculation shows that

H∧! H∧ ⊗ Pol[X]

46 MICHO DURDEVICH

with the flip-over map σ : Pol[X]⊗H∧ → H∧ ⊗ Pol[X] given by

σ(X2n ⊗ ψ) =n∑k=0

(n

k

)D2k(ψ)⊗X2n−2k

σ(X2n+1 ⊗ ψ) =n∑k=0

(n

k

)D2k+1(ψ)⊗X2n−2k + (−)∂ψD2k(ψ)⊗X2n+1−2k

The flip-over operator satisfies the characteristic pentagonal identities

σ(id⊗ •) = (• ⊗ id)(σ ⊗ id)(id⊗ σ)

σ(• ⊗ id) = (id⊗ •)(id⊗ σ)(σ ⊗ id)

which expresses the product in H∧⊗Pol[X] in terms of the factors H∧ and Pol[X].

Remark 31. The above can be viewed as a special case of a cross-product construc-tion, involving an action of a Lie superalgebra g by antiderivations on a superalgebraΦ. In this case, the algebra structure on Φ⊗ U [g] is introduced with the help of aflip-over operator σ : U [g]⊗ Φ→ Φ⊗ U [g] which is defined by

σ(γ ⊗ ψ) = (−)∂γ∂ψψ ⊗ γ + [γ · ψ]⊗ 1 γ ∈ g ψ ∈ Φ

and extended to the tensor products with the full universal envelope U [g] by thepentagonal multiplicativity. The above is equivalent to introducing an extendedalgebra Φ consisting of Φ plus the vector space g of generators obeying the universalenvelope relations with

γψ = (−)∂γ∂ψψγ + [γ · ψ]

Our extended bimodule construction is recovered as a special case, when g is the2-dimensional Lie superalgebra generated by a single fermionic element X. In thiscase U [g]↔ Pol[X].

Remark 32. The flip-over operator allows us to ‘switch’ between the elements ofH∧ and polynomial expressions on X. So we can ‘move’ symbols X on the leftor right, in any algebraic combination within H∧. This flexibility is made possiblethanks to factorizing through a simple ideal J and gained D-invariance propertyof ‘true horizontals’ H∧ in H∧. The tensor algebra H⊗ is ‘too rigid’ to allow this.

Let us now consider situations when something supercommutes with the invari-ant elements: which leads us to the graded commutants. We have already explainedtheir special significance and relations with the translation maps.

Remark 33. For example, the elements of Z(Pol[X], H∧) are solutions of the equa-tion D(ϕ) = 0 in H∧. If we expand ϕ =

∑k≥0

Xkϕk with ϕk ∈ H∧ then

(5.31) D(ϕ) =∑k≥0

(−)kXkD(ϕk) + 2∞∑n=1

X2nϕ2n−1

and hence D(ϕ) = 0 is equivalent to a system of equations

ϕ2n−1 = −12D(ϕ2n) D(ϕ2n−1) = 0


Let us consider Z(Ω,H∧) and Z(Ω, H∧). Obviously the former is included in thelatter and X2 ∈ Z(Ω, H∧). Since we have X2n+1w− (−)∂wwX2n+1 = dM (w)X2n ageneral homogeneous polynomial ϕ =

∑k≥0

Xkϕk belongs to Z(Ω, H∧) iff

(5.32) ϕ2nw − (−)∂w∂ϕwϕ2n − (−)∂ϕϕ2n+1dM (w) = 0 ϕ2n+1 ∈ Z(Ω,H∧)

Both subalgebras are D-invariant and F∧-invariant

D(Z(Ω,H∧)

)⊆ Z(Ω,H∧) F∧

(Z(Ω,H∧)

)⊆ Z(Ω,H∧)⊗A

D(Z(Ω, H∧)

)⊆ Z(Ω, H∧) F∧

(Z(Ω, H∧)

)⊆ Z(Ω, H∧)⊗A

and let us recall that the expression

(5.33) ϕ a = [a]1ϕ[a]2defines a natural right A-module structure on them.

Formulae that involve left/right counterparts of general covariant derivativessimplify in our context, and it is instructive to derive them independently. Thisis because D is ‘self-dual’ thanks to the global graded Leibniz rule. So the mapD : H∧ → H∧ naturally induces a module derivation D2 : H∧ ⊗ΩH∧ → H∧ ⊗ΩH∧via D2(ϕ⊗ψ) = D(ϕ)⊗ψ+ (−)∂ϕϕ⊗D(ψ). The definition is consistent with thetensor product over Ω because of

D2(ϕw ⊗ ψ) = D(ϕw)⊗ ψ + (−)∂ϕ+∂wϕw ⊗D(ψ)

= D(ϕ)w ⊗ ψ + (−)∂ϕϕd(w)⊗ ψ + (−)∂ϕ+∂wϕw ⊗D(ψ)

= D(ϕ)⊗ wψ + (−)∂ϕϕ⊗ d(w)ψ + (−)∂ϕ+∂wϕ⊗ wD(ψ)

= D(ϕ)⊗ wψ + (−)∂ϕϕ⊗D(wψ) = D2(ϕ⊗ wψ)

The same can be said for extended structures Ω and H∧.

Proposition 39. (i) The following diagram

(5.34)

H∧ ⊗Ω H∧ −−−−→ H∧ ⊗A

D2

y yD ⊗ id

H∧ ⊗Ω H∧ −−−−→ H∧ ⊗Ais commutative, where the horizontal arrows are the Hopf-Galois identification. Inparticular,

(5.35) D2τ(a) = 0 ∀a ∈ A

(ii) Recall that Z(Ω,H∧) is D-invariant. We have

(5.36) D(ϕ a) = D(ϕ) a

for all ϕ ∈ Z(Ω, H∧) and a ∈ A.

Proof. The diagram is a direct consequence of the graded Leibniz rule and thecovariance property, for D. From the definition of τ and the fact that D(1) = 0it follows that (5.35) holds. The second part of the proposition now follows from(5.35) and the definition of the -structure:

D(ϕ a) = D([a]1ϕ[a]2

)= D[a]1ϕ[a]2 + [a]1D(ϕ)[a]2 + (−)∂ϕ[a]1ϕD[a]2

= [a]1D(ϕ)[a]2 = D(ϕ) a

48 MICHO DURDEVICH

where we used the fact that ϕ graded-commutes with the elements of Ω so it canbe ‘inserted’ in the middle of the tensor product over Ω.

Let us recall that the curvature tensor of D is a map %D : A → H∧ defined by%D(a) = −[a]1D

2[a]2. The map is well-defined, since the square of D is a Ω-linearmap. We have D2(ϕ) = −ϕ(0)%D(ϕ(1)) for every ϕ ∈ H∧. Compared with thegeneral theory, our context is very special. In particular D2 is a Ω-linear derivationand the curvature tensor satisfies 2 distinguished properties.

Proposition 40. Values of %D commute with elements of Ω and we have D%D = 0.

Proof. It is instructive to explicitly demonstrate commutativity

w%D(a) = −w[a]1D2[a]2 = −w•

([a]1 ⊗D

2[a]2)

= −•(w[a]1 ⊗D

2[a]2)

= −•(id⊗D2)(w[a]1 ⊗ [a]2

)= −•(id⊗D2)

([a]1 ⊗ [a]2w

)=

− •([a]1 ⊗D

2[[a]2w])

= −•([a]1 ⊗ [D2[a]2]w)

)= −

([a]1D

2[a]2)w = %D(a)w

and the simplified Bianchi identity

D%D(a) = D([a]1D

2[a]2)

= D[a]1D2[a]2 + [a]1D

3[a]2 = •(id⊗D2

)D2τ(a) = 0

completes our proof.

As a conclusion to this mini-topic, we shall derive a nice formula connecting the-operator acting on even powers of X with the curvature tensor.

Proposition 41. Even powers of X belong to Z(Ω, H∧) and we have

(5.37) X2n a =n∑k=0

(−)k(n

k

)X2k

[%D(a(1)) · · · %D(a(k))

]Proof. The formula for the square of D and definition of %D give

%D(a) = −[a]1D2[a]2 = [a]1

[a]2X

2 −X2[a]2

= [a]1[a]2X2 − [a]1X

2[a]2= ε(a)X2 −X2 a

Applying identity X2n a = (X2 a(1)) · · · (X2 a(n)) and performing elementarytransformations leads to the desired formula.

5.3. Quantum Frame Bundles

The structure of a quantum frame bundle is characterized by the existence ofspecial one-forms, commuting with the calculus on the base. These forms quantizethe classical concept of a local coordinate system. Another equivalent way to for-mulate this is via a covariant system of derivations acting on the base and takingtheir values in the functions on the bundle, together with a -structure on associ-ated coordinate symbols (the latter being trivial in the classical case). A generaltheory of quantum frame bundles is presented in [7, 8]. Here we shall focus ona natural appearance of a first order horizontal calculus, in this context of framebundles.


So let us fix a unitary and hermitian representation u ∈ Mn(A) for G. Theconcept of frame structure consists in two components. The first one is given by asystem of derivations ∂1, . . . , ∂n : V → B transforming covariantly

(5.38) F∂j(f) =n∑i=1

∂i(f)⊗ uij

for all f ∈ V and moreover satisfying the following completeness property. Thereexist elements qαi ∈ B and fα ∈ V obeying

(5.39)∑α

qαi∂j(fα) = δij

Remark 34. Such a completeness condition in particular ensures non-triviality ofthe system of derivations. We can always chose elements qαi in such a way thatF (qαi) =

∑n

j=1qαj ⊗ κ(uij).

Furthermore, let us assume that a homomorphism ν : B → Mn(B) is given whichextends the diagonal map

(5.40) V 3 f

f · · · 0...

. . ....

0 · · · f

∈ Mn(V)

and satisfies the following two *-conditionsn∑k=1

νkiν∗jk(b)

= δijb(5.41)

∂j(f∗) =

n∑i=1

νij∂i(f)∗

(5.42)

as well as the following covariance condition

(5.43) Fνij(b)

=

n∑k,l=1

νkl(b(0))⊗ κ(uik)b(1)ulj

for all b ∈ B. The above formulation is equivalent to a simpler and more ele-gant picture involving appropriate first-order horizontal calculi. Namely, that of afirst-order horizontal calculus (H, F∼, dM ) over P equipped with a hermitian baseθ1, . . . , θn ∈ H of the free left B-module structure, and such that

F∼(θi) =n∑j=1

θj ⊗ κ(uij)(5.44)

fθi = θif ∀f ∈ V(5.45)

Then derivations ∂1, . . . , ∂n and homomorphism ν are recovered as

θib =∑j

νij(b)θj(5.46)

dM (f) =n∑i=1

∂i(f)θi(5.47)

50 MICHO DURDEVICH

Remark 35. It is worth observing that derivations ∂1, . . . , ∂n and homomorphismν are only connected via the second *-condition (5.42) which ensures hermicity ofdM . Condition (5.43) in general prohibits the trivial extensions of the diagonalmap (5.40) due to the noncommutativity of A. And the base space compatibil-ity condition (5.45) ensures that we can ‘act’ on the coordinate forms θi by thecorresponding right A-module structure where θi a = [a]1θi[a]2.

The multiplet θ1, . . . , θn is also a basis for the right B-module structure on H,with the connection between free right and left module representation given byθib = b(0)(θi b(1)). In other words we can write

(5.48) ν(b) = b(0)(b(1)) : A → Mn

[Z(V,B)

]with representing the structure on coordinate one-forms

θi a =n∑j=1

ij(a)θj

so that we have

ij(ab) =n∑k=1

[ik(a) b(1)]kj(b(2))(5.49)

ijκ(a)∗

= νkj

ik(a)∗

(5.50)

Fij(a) =n∑

k,l=1

kl(a(2))⊗ κ(a(1))κ(uik)a(3)ulj(5.51)

We can apply our general theory and construct the associated horizontal envelopecalculus (H∧, F∧,Ω) and consider covariant derivatives D ∈ d(H∧, F∧,Ω). Let usobserve that the relations of the horizontal envelope are equivalent to the followingsystem

(5.52)∑α

Qα(f)dM (fα) = 0 ∀f ∈ V

where Qα : V → V are first order differential operators given by

(5.53) Qα(f) =n∑i=1

∂i(f)qαi

so that we have a kind of ‘coordinate system’ representation

(5.54) dM (f) =∑α

Qα(f)dM (fα)

of the differential dM : Ω→ Ω. By definition, the torsion tensor of D is the multipletof 2-forms

(D(θ1), . . . , D(θn)

).

Amongst the most interesting geometrical situations are those with vanishingtorsion tensor. Let us observe that if D is torsion-free, and in addition satisfiesboth left and right graded Leibniz rule over Ω, then initial coordinate one-forms θinecessarily belong to Z(Ω,H∧). Indeed, for f ∈ V we have

D(f)θi = D(f)θi + fD(θi) = D(fθi) = D(θif) = −θiD(f) +D(θi)f = −θiD(f)

and it follows that θi graded-commute with all the elements of Ω. Another richgeometrical context is provided by quantum spin bundles, with associated Diracoperators being interpretable in purely diagrammatic terms [14].


To conclude this discussion, let us observe that in a quantum Riemannian geom-etry context our frame structures assume a particularly elegant form–we can adjustthe generating functions so that

(5.55) q∗αi = ∂i(fα) = bαi

which provides a link between classifying maps, quantum Riemannian metrics, andembeddings into flat quantum Euclidean spaces.

5.4. Classical Finite Structure Groups

A very interesting and completely quantum phenomena brought by the generaltheory of differential calculi on quantum groups, is a possibility to apply it in aclassical context of finite groups, where from the viewpoint of standard geometryand the idea of infinitesimals, there should not exist any differential calculus at all.

So let us describe in more detail irreducible bicovariant *-covariant calculi overthe group G. The algebra A is commutative and finite-dimensional, and naturallyspanned by the elements of the group g ∈ G identifying them with the characteristicfunctions of the one-point sets g. We thus have

(5.56)∑h∈G

h = 1 q · g =

q when q = g

0 otherwise

here we multiplied elements of G as functions. Here we use the symbol · to denotethe product of such functions. The group structure is given by

(5.57) φ(g) =∑h∈G

h⊗ (h−1g) κ(g) = g−1

and the adjoint action is specified by

(5.58) ad(g) =∑h∈G

(hgh−1)⊗ h

The ideals R = RS in A are naturally labeled by subsets S = SR of G. Everyideal is trivially bilateral, and also *-invariant. So here we deal with a very simpleapplication of Gelfand-Naimark theory:

RS =f ∈ A

∣∣∣ f(x) = 0 ∀x ∈ SR

SR =x ∈ G

∣∣∣ f(x) = 0 ∀f ∈ RS

Remark 36. Let us recall that in a C*-algebra a closed two-sided ideal is nec-essarily *-invariant, that is, it is a C*-ideal—as revealed by a simple play withapproximate units.

The inclusion R ⊆ ker(ε) is equivalent to ε /∈ SR. The space Γinv is spanned bythe canonical basic elements

π(g) = [g]

∣∣ g ∈ SR. The structure is given by

(5.59) [g] h =

[g] if g = h;0 otherwise

and the right A-module structure is defined by

(5.60) [g]h = (hg−1)[g]

The *-covariance property κ(R)∗ = R is equivalent to S−1R = SR and the bicovari-

ance property ad(R) ⊆ R⊗A is equivalent to gSRg−1 = SR for all g ∈ G, that is

to say, the set SR must be constituted of entire conjugation classes.

52 MICHO DURDEVICH

Left-covariant endomorphisms T of Γ are isomorphic to the algebra of functionsλ : SR → C, the correspondence is given by T [g] = λ(g)[g]. The *-preservingmorphisms are characterized by λ(g−1) = −λ(g)∗ and the bicovariant morphismscorrespond to λ that are constant along the conjugation classes in SR.

Therefore, irreducible *-covariant bicovariant calculi are given by either a sin-gle conjugate class coinciding with its inverse class, or by a union of two distinctmutually inverse conjugation classes.

The canonical braid-operator σ : Γ⊗2inv → Γ⊗2

inv can be calculated by the followingexpression

(5.61) σ([h]⊗ [g]

)= [hgh−1]⊗ [h]

and in particular we see that if gh = hg then σ acts as the standard flip, theelements [q]⊗ [q] are always σ-symmetric.

The universal differential envelope Γ∧ of Γ is described as follows. The set ofquadratic relations defining Γ∧inv is given by sums

(5.62)∑gq=h

[g]⊗ [q] ε 6= h 6∈ SR q, g ∈ SR

and therefore the dimension of the quadratic relations space is the same as thecardinality of the set of non-unit elements of G\SR expressible as a product of twoelements from SR.

Let us illustrate all this in the simplest non-Abelian case, of G being the groupof permutations of 3 symbols, or equivalently the group of isometries of EquilateralTriangle. So

G = ε, r, s, x, y, z x2 = y2 = z2 = ε = rs = sr

xy = yz = zx = r yx = xz = zy = s

and let us chose reflections x, y, z as the self-dual conjugate class for the calculus.Then we see that the space of quadratic relations is 2-dimensional and spanned by

[x]⊗ [y] + [y]⊗ [z] + [z]⊗ [x]

[y]⊗ [x] + [z]⊗ [y] + [x]⊗ [z]

and on the other hand the canonical braid operator σ is specified by

[x]⊗ [y] 7→ [y]⊗ [z] [y]⊗ [z] 7→ [z]⊗ [x] [z]⊗ [x] 7→ [x]⊗ [y]

[y]⊗ [x] 7→ [x]⊗ [z] [z]⊗ [y] 7→ [y]⊗ [x] [x]⊗ [z] 7→ [z]⊗ [y]

and we see that it satisfies σ3 = id. In accordance with the general theory, we candirectly verify that the elements of the quadratic relations are σ-invariant. Besidesthe symmetric elements, there are two doublets corresponding to eigenvalues eπi/3

and e−2πi/3 each:

[x]⊗ [y] + e2πi/3[y]⊗ [z] + e−2πi/3[z]⊗ [x]

[y]⊗ [x] + e2πi/3[z]⊗ [y] + e−2πi/3[x]⊗ [z]

as well as[x]⊗ [y] + e−2πi/3[y]⊗ [z] + e2πi/3[z]⊗ [x]

[y]⊗ [x] + e−2πi/3[z]⊗ [y] + e2πi/3[x]⊗ [z]


Furthermore, we see that the differential structure is very simple:

ε 7→ x[x] + y[y] + z[z] r 7→ x[y] + y[z] + z[x] s 7→ y[x] + x[z] + z[x]x 7→ ε[x] + r[y] + s[z] y 7→ ε[y] + r[z] + s[x] z 7→ ε[z] + r[x] + s[y]

with the above assignments defining −d.

Remark 37. Now it is easy to verify that, in accordance with what has beenmentioned in our discussion about horizontal envelopes, this setup provides us withexamples of a horizontal universal envelope such that the calculus on the baseinherits additional relations from the bundle.

Appendix A. Graded Derivations

In this Appendix we have collected, for the reasons of completeness, some impor-tant yet elementary algebraic properties of graded derivation maps. The conceptualcontext is that of Section 3. We shall consider a graded *-algebra horP equippedwith a free right action F : horP → horP ⊗ A of G, so that the fixed-point *-subalgebra ΩM is equipped with a differential structure dM : ΩM → ΩM . We areinterested in properties of general F -covariant maps from horP to horP extendingdM and satisfying left or right graded Leibniz rule over ΩM .

So let D : horP → horP be such a left derivation. In other words,

D(wϕ) = dM (w)ϕ+ (−)∂wwD(ϕ)

holds for ϕ ∈ horP and w ∈ ΩM .Let us consider a map D′ : horP → horP defined by

(A.1) D′(ϕ) = [κ−1(ϕ(1))]1D

[κ−1(ϕ(1))]2ϕ(0)− [κ−1(ϕ(1))]1D

[κ−1(ϕ(1))]2

ϕ(0)

The above graded left Leibniz rule for D ensures the consistency of this definition.

Lemma 42. The map D′ is F -covariant, extends dM and satisfies the graded rightLeibniz rule over ΩM .

Proof. The covariance of D′ is a direct consequence of the covariance of D and thecovariance of the transformation ϕ 7→ [k−1(ϕ(1))]1 ⊗ [k−1(ϕ(1))]2 ⊗ ϕ(0). Now forw ∈ ΩM we have F (w) = w⊗ 1 and therefore D′(w) = D(w) = dM (w). Let us nowcheck the right Leibniz rule. We have

D′(ϕw) = [κ−1(ϕ(1))]1D

[κ−1(ϕ(1))]2ϕ(0)w

−[κ−1(ϕ(1))]1D

[κ−1(ϕ(1))]2

ϕ(0)w

= [κ−1(ϕ(1))]1D

[κ−1(ϕ(1))]2ϕ(0)w

− [κ−1(ϕ(1))]1D

[κ−1(ϕ(1))]2ϕ

(0)w

+ [κ−1(ϕ(1))]1D

[κ−1(ϕ(1))]2ϕ(0)w − [κ−1(ϕ(1))]1D

[κ−1(ϕ(1))]2

ϕ(0)w

= D′(ϕ)w+[κ−1(ϕ(1))]1D

[κ−1(ϕ(1))]2ϕ(0)w

−[κ−1(ϕ(1))]1D

[κ−1(ϕ(1))]2ϕ

(0)w

= D′(ϕ)w + (−)∂ϕϕD′(w)

where we have applied identity [κ−1(ϕ(1))]1 ⊗ [κ−1(ϕ(1))]2ϕ(0) = ϕ ⊗ 1 factorizing

the remaining terms through the product over ΩM .

Now, we can combine D and D′ in order to construct a map D2 acting onhorP ⊗M horP , as follows:

D2(ϕ⊗ ψ) = D′(ϕ)⊗ ψ + (−)∂ϕϕ⊗D(ψ)

54 MICHO DURDEVICH

Lemma 43. We have

(A.2)D′

[a]1⊗ [a]2 + [a]1 ⊗D

[a]2

= 0

D′

[a]1

[a]2 + [a]1D

[a]2

= 0

for every a ∈ A.

Proof. The above two equalities are in fact equivalent. Let us prove the second.Consider an arbitrary unitary irreducible representation u ∈ Mn(A) of G and as-sociated functions bαj where α ∈ 1, . . . , d and j ∈ 1, . . . , n. A more detailedanalysis of this is given in Appendix B. So we have

d∑α=1

b∗αibαj = δij F (bαj) =n∑i=1

bαi ⊗ uij

As we have already mentioned, because of the above equations it is possible to write[uij ]1 ⊗ [uij ]2 =

∑αb∗αi ⊗ bαj . The definition of D′ gives

(A.3) D′(b∗αi) =∑jβ

b∗βiD(bβjb

∗αj

)− b∗βiD

(bβj)b∗αj

where we have used the fact that κ−1(u∗ij) = uji. Therefore, for a = uij we get

D′

[a]1

[a]2 + [a]1D

[a]2

=∑α

D′(b∗αi)bαj +∑α

b∗αiD(bαj)

=∑αβk

b∗βiD

(bβkb

∗αk

)bαj − b

∗βiD

(bβk)b∗αkbαj

+∑α

b∗αiD(bαj)

Using the fact that∑

kbβkb

∗αk = ρβα(1) belongs to V and applying the left Leibniz

rule for D this further transforms as∑αβk

b∗βiD

(bβkb

∗αkbαj

)− b∗βibβkb

∗αkD(bαj)

−∑βk

b∗βiD(bβk)δkj +∑α

b∗αiD(bαj)

=∑βk

b∗βiD(bβk)δkj −∑αk

δikb∗αkD(bαj)−

∑β

b∗βiD(bβj) +∑α

b∗αiD(bαj)

=∑β

b∗βiD(bβj)−∑α

b∗αiD(bαj) = 0

In other words the desired reciprocity holds.

The reciprocity formula (A.2) completely fixes one of the operators D and D′

when its companion is given. Here is one way to see it. If we denote by S a possiblevariation of one of them when the other is fixed, then

[a]1S([a]2) = 0 or S([a]1)[a]2 = 0

with S being a graded left linear or right ΩM -linear, depending on the variation ofwhich operator we consider. This, together with identities

1⊗ ϕ = ϕ(0)[ϕ(1)]1 ⊗ [ϕ(1)]2 ϕ⊗ 1 = [κ−1(ϕ(1))]1 ⊗ [κ−1(ϕ(1))]2ϕ(0)

implies that S must always be zero. The relation between D and D′ is completelysymmetrical. Given D′ then D is recovered as

(A.4) D(ϕ) = D′ϕ(0)[ϕ(1)]1

[ϕ(1)]2 − (−)∂ϕϕ(0)D′

[ϕ(1)]1

[ϕ(1)]2


Indeed inserting (A.1) into the above expression and considering special elementsϕ = wbαj where w ∈ ΩM , as well as the fact that both D and D′ extend dM , weobtain

D′ϕ(0)[ϕ(1)]1

[ϕ(2)]2 − (−)∂ϕϕ(0)D′

[ϕ(1)]1

[ϕ(1)]2

=∑βi

D′(wbαib∗βi)bβj − (−)∂ww

∑βi

bαiD′(b∗βi)bβj =

∑βi

dM (wbαib∗βi)bβj

+ (−)∂ww∑γβik

bαib

∗γiD(bγk)b∗βkbβj − bαib

∗γiD(bγkb

∗βk)bβj

here we have also applied (A.3). The standard graded Leibniz rule for dM the leftone for D over ΩM allow us to further simplify this into∑

βi

dM (wbαib∗βi)bβj − (−)∂ww

∑γβik

dM[bαib

∗γibγkb

∗βk

]bβj

+ (−)∂ww∑γβik

dM (bαib∗γi)bγkb

∗βkbβj + (−)∂ww

∑γik

bαib∗γiD(bγk)δjk

=∑βi

dM (wbαib∗βi)bβj + (−)∂ww

∑γi

bαib∗γiD(bγj)

− (−)∂ww∑βi

dM (bαib∗βi)bβj + (−)∂ww

∑γi

dM (bαib∗γi)bγj

=∑γi

D(wbαib∗γibγj) =

∑i

D(wbαi)δij = D(wbαj) = D(ϕ)

Since every horizontal form is a sum of such special horizontals ϕ, we conclude that(A.4) holds.

Because of this we can consider a pair (D′, D) as a unified entity representinga single object operating within horP in ‘dual mode’. All possible such matchedpairs form, in a natural way, a (generally infinite-dimensional) complex vector spaced(horP , F,ΩM ).

Lemma 44. The formula (D′, D)∗ = (D∗, D′∗) defines a natural *-involution onthe affine space d(horP , F,ΩM ).

Proof. It is clear that *-conjugation interchanges left and right Leibniz rule proper-ties. Because of the covariance of * and reality of d : ΩM → ΩM the maps D∗ andD′∗ possess all desired properties. What remains is to verify that * of a matchedpair remains a matched pair. This can be done explicitly, by conjugating (A.4),using the identity [k(a)∗]1⊗ [κ(a)∗]2 = [a]∗2⊗ [a]∗1 and arriving to (A.1) with D∗ ex-pressed in terms of D′∗. Another, even simpler way, is to conjugate the connectingidentities (A.2).

Appendix B. On Canonical Generating Elements

Let us consider a quantum principal G-bundle P = (B, i, F ) over a quantumspace M ! V. The main purpose of this Appendix is to explain why, for everyirreducible unitary representation u ∈ Mn(A) of G, there exists a natural numberd and a d× n matrix B with the coefficients from B satisfying

(B.1) B†B = I (id⊗ F )[B] = B©⊥u

56 MICHO DURDEVICH

in other words, the entries of B obey equationsd∑

α=1

b∗αibαj = δij F (bαi) =n∑j=1

bαj ⊗ uji

which are relations we have used several times in the main text. In particular thetranslation map for u is expressible via by know quite a familiar expression:

[uij ]1 ⊗ [uij ]2 =d∑

α=1

b∗αi ⊗ bαj

Let L ⊆ B⊗B be the set of all elements l =∑

q⊗b for which∑

qF (b) ∈ 1⊗A.In other words, we are considering a pull back in B⊗B of the image of the translationmap τ : A → B⊗VB. For every irreducible unitary u ∈ R(G) let Lu be the subspaceof elements of L projecting to the linear span of elements 1 ⊗ uij . We have thefollowing natural decompositions

(B.2)Lu! Mor(u, uL)⊗ Cu Mor(u, uL) → B ⊗Mor(u, F )

Au = spanuij! Cu ⊕ · · · ⊕ Cu︸︷︷︸n

= Cn ⊗ Cu

Because of the covariance of the projection Lu to Au it effectively acts betweenthe intertwiner spaces, and we can consider a splitting which gives us elementslij ∈ L

∣∣ i, j = 1, . . . , n

such that

(B.3) (id⊗ F )[lij ] =n∑k=1

lik ⊗ ukj •[lij ] = δij

and moreover lij =∑

αrαi ⊗ sαj with

(B.4) F (sαj) =n∑k=1

sαk ⊗ ukj

so that in particular

(B.5)∑α

rαisαj = δij

and acting by F on both sides of the above equality leads us to

(B.6)∑α

F (rαisαj) =∑αk

F (rαi)(sαk ⊗ ukj) = δij ⊗ 1

and now multiplying on right by 1⊗ u∗jl and summing over j we find

(B.7) 1⊗ u∗il =∑α

F (rαi)(sαl ⊗ 1)

From this we see that we can assume that the transformation properties of rαiinvolve only the conjugate representation u in other words rαi ∈ Bu and furthermorewe can project down to the components of rαi corresponding to u. Let qαipq be thecoefficient associated to u∗pq in other words

(B.8) F (rαi) =n∑

p,q=1

qαipq ⊗ u∗pq


Applying the coaction property we conclude that

(B.9) F (qαirq) =n∑p=1

qαipq ⊗ u∗pr

And on the other hand from (B.7) we get

(B.10)∑α

n∑p,q=1

qαipqsαl ⊗ u∗pq = 1⊗ u∗il

which reads, due to linear independence of matrix entries of irreducible representa-tions, as

(B.11)∑α

qαipqsαl = δipδql

Let us define new elements

(B.12) qαi =n∑r=1

qαrir

so that we have at first

(B.13) F (qαj) =n∑i=1

qαi ⊗ u∗ij

as it follows from (B.9) and also

(B.14)∑α

qαisαj = δij

according to (B.11). So we arrive at a particular form of expressing the translationmap, in terms of multiplets qαi and sαj . Now we shall use the C*-algebraic structureto prove that we can find a more specific representation, by a partial isometrymatrix. Let us recall a standard lemma from the theory of C*-algebra idempotents.

Lemma 45. (i) Let C be a unital C*-algebra and X, Y ∈ C elements satisfying

XYX = X YXY = Y

Then q = XY and p = Y X are idempotents in C satisfying qX = Xp = X andpY = Y q = Y .

(ii) Let p⊥, q⊥ ∈ C be the orthogonal projections associated to p and q and defineX = Xp⊥ and Y = Y q⊥. Then

XYX = X YXY = Y XY = q⊥ YX = p⊥

(iii) The elements X∗X and Y Y ∗ are strictly positive and mutually inverse inp⊥Cp⊥ while Y ∗Y and XX∗ are strictly positive and mutually inverse in q⊥Cq⊥.

(iv) If we define

(B.15) U = X 1√

X∗X

= 1√

XX∗

X V = Y

1√Y ∗Y

= 1√

Y Y ∗

Y

where the objects in braces refer to corresponding localized algebras, then

U = V ∗ UV = q⊥ V U = p⊥

in other words U and V are mutually conjugate partial isometries establishing equiv-alence between p⊥ and q⊥.

58 MICHO DURDEVICH

Proof. Let us first observe that spectra of positive elements pp∗ and p∗p coincide,both contain number 0 in non-trivial scenarios, and their strictly positive parts areinside the interval [1,∞). Moreover if we consider C faithfully realized in a Hilbertspace H then pp∗ is reduced in K = im(p) and the restriction pp∗ : K → K isinvertible: if we take ψ ∈ K then by Pythagoras theorem

(pp∗ψ,ψ) = ‖p∗ψ‖2 = ‖ψ‖2 + ‖ψ − p∗ψ‖2

The idempotent p has its naturally associated orthogonal projection p⊥ which isthe result of acting by any continuous positive function on R mapping 0 7→ 0 and[1,∞) into 1, we can also write p⊥ = limk fk(pp∗) where fk are appropriatelychosen polynomials mapping 0 in 0. The projector p⊥ is characterized by

pp⊥ = p⊥ p⊥p = p

and we have

(B.16) p⊥Cp⊥ =a ∈ C

∣∣ pa = a = ap∗

We can say the same for the idempotent q its orthogonal projector associate q⊥and L = im(q) with

(B.17) q⊥Cq⊥ =a ∈ C

∣∣ qa = a = aq∗

and let us also note that we can choose the same sequence fk for both p and q.Now let us observe that X provides a bijection between K and L while the

inverse of this bijection is Y : L→ K. The ‘normalized’ elements X and Y satisfyXY = Xp⊥Y q⊥ = XY q⊥ = qq⊥ = q⊥ and similarly YX = pp⊥ = p⊥. We alsohave XYX = q⊥X = X and similarly YXY = p⊥Y = Y . From these relations weconclude that the adjoints X∗ and Y ∗ map L onto K and K onto L as mutuallyinverse transformations. Therefore X∗X : K → K is invertible and its inverse isgiven by Y Y ∗ : K → K and similarly XX∗ : L → L is invertible and its inverse isgiven by Y ∗Y . From the definition (B.15) it is clear that U and V are mutuallyinverse viewed as transformations between K and L. We also compute

U∗U = 1√

X∗X

X∗X

1√X∗X

= p⊥

and similarly

UU∗ = 1√

XX∗

XX∗

1√XX∗

= q⊥

and therefore U is a partial isometry establishing equivalence between p⊥ and q⊥,we also see that V = U∗.

Remark 38. The spectra of these invertibles XX∗ and X∗X as well as theirinverses Y ∗Y and Y Y ∗ coincide. It is worth recalling that

(λ− ab)(λ+ a

1λ− ba

b)

=(λ+ a

1λ− ba

b)

(λ− ab) = λ2

so quite generally, the spectra σ(ab) and σ(ba) always coincide modulo number 0.

Let us return to our main context, and continue with exploring generating rela-tions for a quantum principal bundle P . So we have a d × n matrix Q and n × dmatrix S satisfying QS = 1n. We can now consider these 2 matrices as elements


of a C*-algebra C = K ⊗B where K are compact operators in the Hilbert spacel2(N), and apply Lemma 45. This will in particular transform

S B = 1√

SS∗

S

and the n× d matrix B would satisfy a couple of interesting properties.

Proposition 46. (i) Under the assumption of holomorphic stability of V, the en-tries bαi of the matrix B belong to B.

(ii) The following identities hold

F (bαj) =n∑i=1

bαi ⊗ uij(B.18)

d∑α=1

b∗αibαj = δij(B.19)

Proof. Let us observe that a d × d matrix SS∗ has its coefficients from V. Thematrix u is unitary and so

F( n∑i=1

sαis∗βi

)=∑ikl

sαks∗βl ⊗ ukiu

∗li =

∑kl

sαksβl ⊗ δkl =( n∑i=1

sαis∗βi

)⊗ 1

Now the holomorphic stability of V guarantees that the entries tαβ of the matrix(SS∗)−1/2

interpreted here as the element of q⊥Cq⊥ still belong to V. So we now

have

(B.20) bαi =d∑

β=1

tαβsβi

and the transformation property (B.18) directly follows from (B.4). Finally (B.19)expresses that B∗B = 1n = p⊥ the partial isometry property.

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60 MICHO DURDEVICH

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Institute of Mathematics, Universidad Nacional Autonoma de Mexico, Area de la

Investigacion Cientıfica, Circuito Exterior, Ciudad Universitaria, Mexico City, CP

04510, MEXICO

E-mail address: [email protected]

http://www.matem.unam.mx/~micho

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