LTCC LECTURES ON NONCOMMUTATIVE DIFFERENTIALmajid/LTCCnotes2016.pdf · Noncommutative geometry is...

LTCC LECTURES ON NONCOMMUTATIVE DIFFERENTIAL

GEOMETRY

SHAHN MAJID

Abstract. Noncommutative geometry is the idea that when geometry is done

in terms of coordinate algebras, one does not really need the algebra to be

commutative. We provide an introduction to the relevant mathematics from aconstructive ‘differential algebra’ point of view that works over general fields

and includes the noncommutative geometry of quantum groups as well as of

finite groups. We also mention applications to models of quantum spacetime.

Contents

1. Converting geometry to algebra 12. Quantum groups 53. Noncommutative differential forms 93.1. Differentials on algebras 93.2. Differentials on quantum groups 124. Noncommutative vector bundles 154.1. Projective modules and K-theory 154.2. K-theory and cyclic cohomology 185. Noncommutative Riemannian geometry 225.1. Quantum principal bundles and framing 225.2. Bimodule connections 26Exercises 27Further Reading 28

28

1. Converting geometry to algebra

Noncommutative geometry of any flavour entails replacing a space and geomet-ric structures on it by an algebra with structures on that, inspired by a precisedictionary such as the one shown in the table below. The dictionary is a crutchwhich we eventually have to discard as we extend the structures on the algebra soas to make sense even when our algebra is noncommutative. The result is a moregeneral conception of geometry that can even be useful when our algebra is in factcommutative. For example, noncommutative differential structures on a finite setcorrespond to directed graphs with the given set as vertices. Differentials here donot necessarily commute with functions even though the latter commute amongst

Based on lectures at the London Taught Course Centre in 2011, revised January 2016. Thisis a preprint, now published in LTCC Lecture Notes Series, Vol 6 “Analysis and Mathematical

Physics” eds. S. Bulllet, T. Fearn and F. Smith. World Scientific, 2017. pp 139–176.

1

2 S. MAJID

Geometry X Commutative algebra Apolynomial subset ⊆ Cn reduced finitely generated algebracompact Hausdorff space unital (commutative) C∗ algebra

vector bundle finitely generated projective moduleexterior differential forms differential graded algebra

Dirac operator spectral triplegroup Hopf algebra coproduct ∆

principal bundle Hopf −Galois extensionDefined by the algebra Allow A noncommutative

themselves. In fact the only commutative calculus is the zero calculus, so the aboveis only possible within noncommutative geometry.

Of the different approaches, our quantum groups approach will be one where webuild up the different layers of geometry constructively, guided by group or quantumgroup symmetries. The latter already revolutionised knot theory and Lie theorythrough the constructions of knot invariants and canonical bases respectively, sowe also want to include these key examples in the same way as classical differentialgeometry includes Lie groups and their homogeneous spaces. This is in contrast tothe well-known operator algebras approach of Alain Connes[1] which comes out ofKO-homology and the Dirac operator as the starting point but does not fit too wellwith the geometry of quantum groups (in Connes approach the most famous ex-ample is the ‘noncommutative torus’ which turns out not to be a quantum group).Our approach is also motivated by potential applications to physics through thequantum spacetime hypothesis that our actual spacetime coordinates do not com-mute due to quantum gravity effects.

These lectures will assume an elementary knowledge of algebra. Suffice it to recallthat an algebra A over a field k means that A is a vector space over k equippedwith a linear map A⊗A→ A defining an associative product, and is unital if thereis an element 1 ∈ A which is the identity for the product. We also recall that anonzero element of an algebra is called nilpotent if some power of it vanishes. Analgebra is said to be reduced if it has no nilpotent elements. Next we recall thatone can think of many geometrical spaces as defined by equations. For example,we can think of a sphere as the set of solutions of x2

1 + x22 + x2

3 = 1 in R3. Suchsets are called ‘polynomial’ because they are defined as the common zero set of oneor more polynomials. We see that the geometry is encoded in an algebra with, inthis example, 3 generators and one relation. Such a conversion to algebra becomesa correspondence if we work over an algebraically closed field such as C.

Theorem 1.1. There is a 1-1 correspondence between reduced commutative alge-bras over C with n generators and polynomial subsets of Cn.

Proof. This is basically the starting point of algebraic geometry. Recall that anideal I ⊆ A is a linear subspace for which I.A,A.I ⊆ I, in which case A/I inheritsan algebra structure (a+I)(b+I) = ab+I. For any ideal I, its radical rad(I) = a ∈A | an ∈ I, for some n ∈ N ⊇ I is clearly an ideal obeying rad(rad(I)) = rad(I).It is easy to see that A/I is reduced iff I = rad(I). By definition, a commutativealgebra with generators x1, · · · , xn, say, means a quotient of the polynomial algebraC[x1, · · · , xn] by some ideal I (generated by polynomials in the xi that are set tozero in defining the relations of the algebra). So, reduced algebras of this type are

NONCOMMUTATIVE DIFFERENTIAL GEOMETRY 3

in correspondence with ideals I ⊆ C[x1, · · · , xn] such that I = rad(I). Next, anyideal J ⊆ C[x1, · · ·xn] has a zero set

Z(J) = x ∈ Cn | a(x) = 0, ∀a ∈ J.A polynomial subset X ⊆ Cn precisely means that X = Z(J) of some ideal J(necessarily generated by a finite collection of polynomials) and Hilbert’s ‘nullstel-lensatz’ says that

rad(J) = a ∈ C[x1, · · · , xn] | a vanishes on Z(J),which implies that Z(rad(J)) = x ∈ Cn| a(x) = 0, ∀a ∈ rad(J) ⊇ Z(J). Clearly,Z(rad(J)) ⊆ Z(J) from the definition of radical, so Z(rad(J)) = Z(J). We definethe corresponding reduced algebra as C[x1, · · · , xn]/rad(J) and conversely, givenan ideal I = rad(I) ⊆ C[x1, · · · , xn], we define a polynomial subset Z(I). Theseare now clearly inverse.

This correspondence is functorial in the sense that appropriately defined mapsbetween objects on either side correspond contravariantly. Motivated by the dictio-nary, we can now allow our algebras to be noncommutative. Thus we consider anyreduced finitely generated algebra over an algebraically closed field as some kind of‘noncommutative algebraic space’.

Example 1.2. Let q ∈ C∗. The algebra of 2 × 2 ‘quantum matrices’ is Cq[M2] =C〈a, b, c, d〉/I where I is the ideal generated by the relations

ba = qab, ca = qac, db = qbd, dc = qcd, da− ad = (q − q−1)bc, cb = bc

Its further quotient by ad−q−1bc = 1 is the ‘quantum group’ Cq[SL2]. When q = 1,these correspond to C4 and SL2(C) ⊂ C4 respectively.

The correspondence does not work over R because R is not algebraically closed.Our approach to this is to work with polynomial subsets of Cn as the correspondingalgebra over C and specify a ‘real form’ of the subset by means of an antilinearinvolution ∗ : A → A with ∗2 = id and (ab)∗ = b∗a∗ for all a, b ∈ A. A mapbetween such ∗-algebras is an algebra homomorphism respecting the ∗ on eachside. As an example, if S2

C is defined by the algebra C[x1, x2, x3] modulo relationsx2

1 + x22 + x2

3 = 1, the additional information x∗i = xi for i = 1, 2, 3 picks out the‘real sphere’ as obtained if we wished to solve the sphere equation in C with ∗interpreted as complex conjugation. However, we do not actually have to solve forany such points and when the algebra is noncommutative we take the choice of a ∗as the definition of a ‘real form’ of the otherwise complex noncommutative space.This is a relatively unexplored field of ‘∗-algebraic geometry’.

Example 1.3. When q is real we denote by Cq[SU2] the algebra Cq[SL2] in Ex-ample 1.2 equipped with the ∗-structure a∗ = d and b∗ = −q−1c. When q = 1 thesubset of SL2(C) ⊂ C4 for which a∗ = a, b∗ = b, c∗ = c, d∗ = d is the set SU2(C) of2× 2 unitary matrices of determinant 1.

More generally, a set X is a topological space if it is equipped with a collectionτ of subsets (called ‘open’) obeying some obvious axioms. The familiar case iswhen X is equipped with “distance” function” d(x, y) with non-negative real values,vanishing iff x = y, symmetric and obeying the triangle inequality. Then U ⊆ X isopen if for every x ∈ U one can find ε > 0 so that the entire ball of points of distanceless than ε from x lies inside U . The topology in this case is Hausdorff in the sense

4 S. MAJID

that any two distinct points can be placed in disjoint open sets, while the closure Xof X ⊂ Rn means all points y with the property that a ball of any arbitrarily smallsize about y intersects X. An open cover of X means a collection Uα of opensets such that X = ∪αUα, and a topological space is compact if every open coverhas a finite subcover. On the algebra side, a C∗-algebra means a ∗-algebra over Cequipped with a norm || || : A→ R≥0 (which defines a distance by d(a, b) = ||a− b||and hence a topology on A) subject to some axioms, notably

(1) ||a∗a|| = ||a||2, ∀ a ∈ A.

If X is a compact Hausdorff space then the algebra C(X) of complex valued contin-uous functions on X and pointwise operations is a commutative unital C∗-algebrawith

a∗(x) = a(x), ∀x ∈ X, ||a|| = supx∈X|a(x)|.

The 1st theorem of Gelfand and Naimark (1945) says that there is in fact a 1-1correspondence between unital commutative C∗-algebras and compact Hausdorfftopological spaces. This is the second line in our table. Now, according to ourphilosophy we could regard a noncommutative unital C∗-algebras as a noncom-mutative analog of a compact Hausdorff space. The key example is the algebraB(H) of bounded operators on a Hilbert space H, as a noncommutative C∗-algebrawith sum and composition of operators,

a∗ = a†, ||a|| = supv∈H, ||v||=1

||a.v||, ∀a ∈ B(H)

where † refers to the hermitian conjugate with respect to the Hilbert space innerproduct, i.e. (v, a†w) = (av, w) for all v, w ∈ H, and the stated norm is theoperator norm on B(H). The 2nd Gelfand-Naimark theorem (1954) tells is thatany C∗-algebra can be realised as a norm-closed ∗-subalgebra A ⊆ B(H). Just asa space in geometry often can be visualised as embedded in some flat space Rn,a ‘noncommutative space’ in the sense of a C∗-algebra can still be visualised asembedded in B(H). If, as in many treatments of quantum mechanics, one onlyworks with B(H), this would be like only working with Rn and missing all thenontrivial geometry from the embedding. As a corollary, if we already have a ∗-algebra A, we have only to find a faithful representation π : A → B(H) as a

∗-algebra map and can then extend A to a C∗-algebra A as the norm closure π(A).The (image of) A appears as a dense ∗-subalgebra and one can say loosely (or moreabstractly) that A = A, a C∗-algebra ‘completion’ of the ∗-algebra.

Example 1.4. Let θ ∈ [0, 2π) and let Aθ be the ∗-algebra generated by invertibleu, v with relations vu = eıθuv and u∗ = u−1, v∗ = v−1. Let

H = l2(Z) = (an ∈ C) |∑n

|an|2 <∞, (a, b) =∑n∈Z

anbn.

This is a Hilbert space on which Aθ is represented by

(π(u)a)n = aneınθ, (π(v)a)n = an+1

and the noncommutative torus Aθ is the resulting C∗-algebra completion of Aθ.Here (π(u)π(v)a)n = (π(v)a)ne

ınθ = an+1eıθn while (π(v)π(u)a)n = (π(u)a)n+1 =

eıθ(n+1)an+1, so we indeed have a representation of the algebra Aθ. Also, if θ = 0the C∗-algebra is commutative and hence isomorphic to C(X) for some compact


Hausdorff space X given by the set of ∗-algebra homomorphisms. These are nec-essarily of the form φ(u) = eıφ1 , φ(v) = eıφ2 for a pair of angles (φ1, φ2) so thatX = S1 × S1, a classical torus.

Examples 1.3 and 1.4 go back to Woronowcz and Connes/Rieffel respectively forthe operator algebras setting.

2. Quantum groups

So far we have not yet captured any geometry, only topology. The first ‘geo-metrical’ structure we look at is a group law on the ‘underlying space’ but in away that makes sense even when the latter does not exist. Given that our functorin the classical case between geometry and algebra was contravariant, this shouldappear on the algebra A as a coalgebra. Here we fix the field k over which we workand think of the algebra product as a linear map m : A⊗A → A and the unitelement 1A equivalently as a linear map η : k → A given by η(1) = 1A. A coalgebraconsists of the same data obeying the same axioms in an arrow reversed form whenwritten out as commutative diagrams. So, there is a coproduct ∆ : A→ A⊗A anda counit ε : A→ k obeying coassociativity and counity,

(∆⊗ id)∆ = (id⊗∆)∆, (ε⊗ id)∆ = (id⊗ ε)∆ = id.

If ∆ were to correspond to an actual product map on an underlying space and ε toan identity element then these would be (unital) algebra homomorphisms, whereA⊗A has the tensor product algebra. And if the space was actually a group thenthere would be an operation S : A → A induced by inversion. Thus a quantumgroup or Hopf algebra is an algebra A which is also a coalgebra, with ∆, ε unitalalgebra maps and an antipode S obeying

·(id⊗S)∆ = ·(S⊗ id)∆ = 1ε.

It can be shown that S is necessarily antimultiplicative. It should be clear thatthese axioms, when written as diagrams, are invariant under arrow reversal. Thatmeans that for every Hopf algebra construction built from the composition of thesestructure maps there is a dual ‘co’ construction with arrows reversed. We refer toour books [2, 3] for more details. We will also need a compact notation for thecoproduct, called the ‘Sweedler notation’ where we write ∆a = a(1)⊗ a(2) for anya ∈ A. The suffices here merely tell us which tensor factor of the output of ∆a werefer to, and there is a possible sum of such terms understood. We then extend thisnotation to include iterated coproducts so

(∆⊗ id)∆a = a(1)⊗ a(2)⊗ a(3) = (id⊗∆)∆a

where we may renumber provided we keep the order of the tensor factors.It was already observed by the topologist E. Hopf in the 1940s that none of

this requires A to be commutative, but it is only in the 1980s that many examplesemerged (from mathematical physics) which were truly beyond functions on actual(or algebraic) groups or their duals under arrow reversal, i.e. which were non-commutative and noncocommutative. These were (1) the quasitriangular quantumgroups of V.G. Drinfeld, of which the next example is a coordinate algebra ver-sion, and (2) the bicrossproduct quantum groups of my PhD thesis. Both includeexamples associated to every complex semisimple Lie algebra g.

6 S. MAJID

Example 2.1. Cq[SL2] in Example 1.2 is a Hopf algebra with

∆

(a bc d

)=

(a bc d

)⊗(a bc d

), ε

(a bc d

)=

(1 00 1

), S

(a bc d

)=

(d −qb

−q−1c a

)We used a matrix notation and more explicitly,

∆a = a⊗ a+ b⊗ c, ε(a) = 1, Sa = d

and so forth. Because we know that ∆, ε are algebra maps and S is an anti-algebramap, it suffices to define them on generators, provided these definitions respect therelations of the algebra, for example

∆(ba) = (a⊗ b+ b⊗ d)(a⊗ a+ b⊗ c) = a2⊗ ba+ ab⊗ bc+ ba⊗ da+ b2⊗ dc= qa2⊗ ab+ qab⊗ ad+ qab⊗(q − q−1)bc+ ab⊗ bc+ qb2⊗ cd= q(a⊗ a+ b⊗ c)(a⊗ b+ b⊗ d) = ∆(qab).

We define a ‘real form’ of a Hopf algebra over C or Hopf ∗-algebra as ∗-algebra Awhich is also a Hopf algebra with

∆ ∗ = (∗⊗∗)∆, ε ∗ = ε( ), (S ∗)2 = id

An example is Cq[SU2] as in Example 1.3 as a real form of Cq[SL2] when q isreal. When q ∈ (0, 1] this Hopf ∗-algebra has a natural completion to a C∗-algebraCq(SU2). Similarly C[u, u−1] with u∗ = u−1 and ∆u = u⊗u is a Hopf ∗-algebraC[S1] and completes to the C∗-algebra C(S1).

A left A-module or representation of an algebra A on a vector space V is a mapA⊗V → V (an ‘action’) obeying axioms which polarise the ones for multiplication.Reversing arrows, a left A-comodule for a coalgebra A means a vector space V anda ‘coaction’ ∆L : V → A⊗V obeying

(id⊗∆L) ∆L = (∆⊗ id) ∆L, (ε⊗ id)∆L = id.

We may use a shorthand notation ∆Lv = v ¯(1)⊗ v ¯(2). One similarly has the notion ofa right coaction ∆R : V → V ⊗A. A comodule algebra means an algebra B which isa comodule with coaction an algebra map, usually required to be a ∗-algebra mapin ∗-algebra setting.

Example 2.2. Let q ∈ C∗ and Cq[C2] = C〈x, y〉/I where I is the ideal generatedby the relations yx = qxy (the ‘quantum plane’). This is a Cq[SL2]-module algebrawith

∆L

(xy

)=

(a bc d

)⊗(xy

)Again, we use a matrix notation where ∆Lx = a⊗x + b⊗ y etc. We extend to

products as an algebra map and can check that this is well defined. When q = 1this coaction corresponds to the matrix action SL2(C)× C2 → C2.

Hopf algebras provide a cleaner and more logical way of doing many classicalconstructions, as well as generalising them to the noncommutative case. Here welook at integration. The reader may know that on a (locally) compact group onehas a unique translation-invariant integration.

Definition 2.3. Let A be a Hopf algebra over a field k. A left A-Hopf moduleis a vector space V which is both an A-module and an A-comodule and for which∆L : V → A⊗V is a left A-module map, i.e. ∆L(a.v) = (∆a).(∆Lv) for alla ∈ A, v ∈ V .


Here the dot denotes both the action of A on V and the product in A. Anexample is A itself with the (co)product supplying the left regular (co)action. Also,if V is a left A-comodule then its space of coinvariants is

AV = v ∈ V | ∆Lv = 1⊗ v.

Lemma 2.4. (Hopf module lemma) Let V be a left A-Hopf module. Then V∼=A⊗(AV )where the right hand side has the Hopf module structure of A.

Proof. In one direction H ⊗(AV ) → V the map is the left action a⊗ v 7→ a.v. Inthe other direction we provide the map

v 7→ F (v) = v ¯(1)(1)⊗Sv ¯(1)

(2).v¯(2).

Let us first verify that this in fact lands in A⊗(AV ), thus

(id⊗∆L)F (v) = v ¯(1)(1)⊗(Sv ¯(1)

(2))(1)v¯(2) ¯(1)⊗(Sv ¯(1)

(2))(2).v¯(2) ¯(2)

= v ¯(1)(1)⊗(Sv ¯(1)

(2)(2))v¯(2) ¯(1)⊗(Sv ¯(1)

(2)(1)).v¯(2) ¯(2)

= v ¯(1)(1)(1)⊗(Sv ¯(1)

(1)(2)(2))v¯(1)

(2)⊗(Sv ¯(1)(1)(2)(1)).v

¯(2)

= v ¯(1)(1)⊗(Sv ¯(1)

(3))v¯(1)

(4)⊗(Sv ¯(1)(2)).v

¯(2) = v ¯(1)(1)⊗ 1⊗Sv ¯(1)

(2).v¯(2)

so that the second factor of F (v) lives in AV . The first equality used the Hopfmodule property of V , the second anticomultiplicativity of S, the third that ∆L isa coaction. We then renumber coproducts in order to identify two consecutive onesto cancel by the antipode axiom. The map v 7→ F (v) is inverse to the first mapas v 7→ F (v) 7→ v ¯(1)

(1).(S¯(1)

(2).v¯(2)) = (v ¯(1)

(1)Sv¯(1)

(2)).v¯(2) = ε(v ¯(1))v ¯(2) = v. Similarly

the other way.

We recall that a Hopf algebra coacts on itself by ∆, say from the right.

Definition 2.5. A (right) invariant integral on a Hopf algebra A means a linearmap A→ k such that (

∫⊗ id)∆ =

∫⊗ 1.

We will also need the notion of dual Hopf algebra. Recall that if you have alinear map V → W then its adjoint is a linear map W ∗ → V ∗. Thus the algebraand coalgebra on A are adjoint to a coalgebra and algebra on H = A∗ when Ais finite-dimensional. From the arrow-reversal symmetry of the axioms of a Hopfalgebra, we obtain a Hopf algebra again. More generally, two Hopf algebras aredually paired if

〈ab, h〉 =∑〈a, h(1)〉〈b, h(2)〉, 〈a, hg〉 =

∑〈a(1), h〉〈a(2), g〉

etc., for all a, b ∈ A and h, g ∈ H.

Theorem 2.6. A finite-dimensional Hopf algebra has a unique right-invariant in-tegral up to normalisation.

Proof. We note that H = A∗ is canonically an A-Hopf module by

a.h = 〈h(1), Sa〉h(2), ∆Lh =∑i

ei⊗hf i

where ei is a basis of A with dual basis f i. Here ∆L defines by evaluationthe right multiplication of H on itself as 〈h, ei〉gf i = hg (summation understood),while the action is the left coaction of H on itself turned into a right action of A

8 S. MAJID

similarly by evaluation, turned into a left acton by the antipode S. We verify thatthe two together provide a Hopf module, i.e.,

〈h(1), Sa〉ei⊗h(2)fi = a(1)ea⊗〈(hf i)(1), Sa(2)〉(hf i)(2)

on inserting the definitions (the LHS is ∆L(a.h) etc.). To see this we evaluateagainst g ∈ H. Then the right hand side becomes

〈g(1), a(1)〉〈(hg(2))(1), Sa(2)〉(hg(2))(2) = 〈g(1), a(1)〉〈Sg(2)Sh(1), a(2)〉h(2)g(3)

= 〈g(1)Sg(2), a(1)〉〈Sh(1), a(2)〉h(2)g(3) = 〈h(1), Sa〉h(2)g

which is what the left hand side of the required identity becomes. Once we have aHopf module, Lemma 2.4 tells us that H∼=A⊗(AH) where

AH = h ∈ H | ei⊗hf i = 1⊗h = h ∈ H | ei〈hf i, a〉 = h(a), ∀a ∈ A= h ∈ H | h(a(1))a(2) = h(a), ∀a ∈ A

We see that this is the space of right-invariant integrals∫

: A → k. Hence thisspace is 1-dimensional as A,H have the same dimension.

In the infinite-dimensional case one can show similarly that the space of right-invariant integrals has dimension at most 1, i.e. an integral may not exist but if it

does, it is unique. If A is a Hopf ∗-algebra, we require∫

(a∗) =∫a for all a ∈ A.

Example 2.7. On Cq[SL2] the ‘Haar integral’ takes the form∫(bc)n =

(−1)nqn

[n+ 1]q2, [m]q =

1− qm

1− q.

and zero on other monomials in a, b, c or in b, c, d. We suppose that q is generic inthe sense that all the denominators are invertible. For example,

(

∫⊗ id)∆(bc) = (

∫⊗ id)((a⊗ b+ b⊗ d)(c⊗ a+ d⊗ c))

=

∫ac⊗ ba+

∫ad⊗ bc+

∫bc⊗ da+

∫bd⊗ dc

=

∫(1 + q−1bc+ qbc)⊗ bc+

∫bc⊗ 1 =

∫bc⊗ 1

since∫bc = −1/(q + q−1).

There are many applications of this theory. Here we mention only one: For anyfinite dimensional Hopf algebra A with dual H one has Fourier transform

F : A→ H, F(a) =∑i

(

∫eia)f i, F−1(h) =

1∫ ∗(∫

)

∑i

S−1ei

∫ ∗hf i.

Here∫

is a right-invariant integral, which we know exists by the above, and∫ ∗

is similarly one on H. The 2π normalisation factor in usual Fourier theory be-comes

∫ ∗(∫

). One can show that both this and S are invertible. Usually, Fouriertransform is limited to Abelian groups but using Hopf algebra technology it appliesmore generally. For example, the quantum group Cq[SL2] has a finite-dimensionalquotient cq[SL2] when q is a primitive r-th root of 1, namely by the additionalrelations br = cr = 0 and ar = dr = 1. Its dual is the Hopf algebra uq(sl2) and soF : cq[SL2]→ uq(sl2).


3. Noncommutative differential forms

In differential geometry one equips a topological space with the structure ofa differentiable manifold. This means that in each open set of a cover, one has‘local coordinates’ xi identifying the open set with a region of Rn (some fixed nwhich is the dimension of the manifold) and with the transition functions betweendifferent such ‘patches’ being differentiable. One also defines the tangent bundlewith sections spanned by vector fields of the form

∑i vi(x) ∂

∂xiin each local patch.

The cotangent bundle is dual to this and the space of ‘1-forms’ Ω1 is spanned byelements of the form

∑ωi(x)dxi in each local patch. Here dxi are a dual basis to

∂∂xi

at each point. One also has an abstract map d which turns a function f into a

differential 1-form df =∑i∂f∂xi

dxi. We turn all this around and define Ω1 by itsdesired properties as a ‘noncommutative differentiable structure’.

3.1. Differentials on algebras. Let A be a unital algebra over a field k. A vectorspace is a bimodule over A if it is both a left and a right A-module and if the twoactions commute.

Definition 3.1. A first order ‘differential calculus’ (Ω1,d) over A means1. Ω1 an A-bimodule.2. A linear map d : A→ Ω1 (the exterior derivative) such that

d(ab) = (da)b+ adb, ∀a, b ∈ A.3. Ω1 = spanadb | a, b ∈ A.4. (Optional connectedness condition) ker d = k.1.

This is more or less the minimum that one could require for an abstract notionof ‘differentials’ – one should be able to multiply them from the left and right byelements of A and have a Leibniz rule with respect to this. In usual algebraicgeometry one would assume that the left and right modules coincide, i.e. thatadb = db.a for all a, b ∈ A, but this is not reasonable to impose when our algebrasare noncommutative. For if we did, we would have d(ab− ba) = 0 so that d wouldbe far from connected. Our definition is necessarily more general and interestingeven for commutative algebras. In some cases, for example if Ω1 is free as a left orright A-module, we have a well-defined left or right cotangent dimension.

Proposition 3.2. Every A has a universal differential calculus given by1. Ω1

univ = kerm ⊆ A⊗A (the kernel of the product map).2. d : A→ Ω1

univ is given by da = 1⊗ a− a⊗ 1 for a ∈ A.The calculus is connected. Any other differential calculus on A is a quotient Ω1 =Ω1univ/N , for some subbimodule N ⊆ Ω1

univ. Its exterior derivative is that of Ω1univ

followed by the projection to the quotient.

Proof. It is elementary to check that Ω1univ is indeed a differential calculus. To see

that it obeys axiom 4 (is connected), suppose that A′ is a chosen complement tok1 so that A = k1⊕ A′. If a = λ1 + b ∈ k1⊕ A′ then da = 1⊗ b− b⊗ 1. Hence ifda = 0 we see that 1⊗ b = b⊗ 1. By projecting the first factor onto A′ we concludethat b = 0 as required. The universal property follows from the surjectivity axiom3.

Note that universal here means with the obvious notion of morphisms betweencalculi, namely bimodule maps that form a commutative triangle with the exteriorderivatives.

10 S. MAJID

Example 3.3. Let X be a finite set and A = C(X) the algebra of functions onit. (Connected) Ω1 on C(X) are in 1-1 correspondence with (connected) directedgraphs Γ with vertices X. We write x → y for the edges, then Ω1 = spanCex→ywith

df =∑x→y

(f(y)− f(x))ex→y, f.ex→y = f(x)ex,y, ex→y.f = ex→yf(y).

We can deduce that ex→y = δxdδy for all x → y, where δx is the Kronecker delta-function. Note that A⊗A has basis δx⊗ δy of which Ω1

univ is the subspace withbasis restricted to x 6= y, and dunivδx = 1⊗ δx − δx⊗ 1 =

∑y 6=x δy ⊗ δx − δx⊗ δy

which takes the form stated with the maximal graph x → y for all x 6= y. Hereδxdunivδy = δx⊗ δy when x 6= y. Now suppose that we have some other calcu-lus defined by a sub-bimodule N . If n =

∑x 6=y nx,yδx⊗ δy ∈ N then δxnδy =

nx,yδx⊗ δy ∈ N . Hence either nx,y = 0 for all elements n or δx⊗ δy ∈ N . HenceN = δx⊗ δy | (x, y) ∈ E for some subset E ⊆ (X×X)\diagonal. The quotient ofthe universal calculus by N can therefore be identified with the subspace spanned byδx⊗ δy for (x, y) ∈ E where E is the complement of E in (X ×X)\diagonal. SuchE are the edges of our digraph. Clearly ker d consists of those functions for whichf(y) = f(x) for all x → y, which are a multiple of 1 iff the graph is connected. Ifthe graph is regular in the sense that out of every vertex there are a fixed numbern of edges then X has left contangent dimension n and is, moreover, parallelizablein the sense Ω1∼=C(X)⊗Cn.

Definition 3.4. An exterior algebra on A or ‘differential graded algebra’ means1. A graded algebra Ω = ⊕nΩn with Ω0 = A2. d : Ωn → Ωn+1 such that d2 = 0 and

d(ωρ) = (dω)ρ+ (−1)nωdρ, ∀ω, ρ ∈ Ω, ω ∈ Ωn.

3. A,Ω1 generate Ω.Its noncommutative de Rham cohomology is defined to be

Hn(A) = ker(d|Ωn)/image(d|Ωn−1).

In the last definition we understand dΩn = 0 if n < 0. The volume dimension (ifit exists) is the top degree and can be different from the cotangent dimension, if itthat exists. In practice one can just construct Ω up to and including any degree ofinterest and ‘fill in’ all higher degrees automatically. For example, every first ordercalculus Ω1 on A has a maximal prolongation to an exterior algebra generated bythese with the minimal further relations contained in the definition of an exterioralgebra.

Theorem 3.5. Let A be an algebra. The maximal prolongation of its universal firstorder calculus is its universal exterior algebra Ωuniv =

⊕n Ωnuniv where Ωnuniv ⊂

A⊗(n+1) is the joint kernel of all the product maps between adjacent copies of A inthe tensor product. The product and differential are

(a0⊗ · · ·⊗ an)(b0⊗ · · ·⊗ bm) = (a0⊗ · · ·⊗ anb0⊗ · · ·⊗ bm)

d(a0⊗ · · ·⊗ an) =

n+1∑i=0

(−1)ia0⊗ · · ·⊗ ai−1⊗ 1⊗ ai⊗ · · ·⊗ an.

Moreover, Hi(A) = k if i = 0 and is zero otherwise (the complex is acyclic).


Proof. We leave the reader to verify that this is the maximal prolongation and focuson the cohomology. Consider

ω =∑α

aα0 daα1 · · · daαn =∑α

λαdaα1 · · · daαn +∑α

bαdaα1 · · · daαn

where we replace aα0 = λα1 ⊕ bα according to A = k1 ⊕ A′ (as before). Thendω =

∑α dbαdaα1 · · · daαn as d1 = 0 and d2 = 0. Hence if ω is closed, and writing

out dbα, we see that

1⊗∑α

bαdaα1 · · · daαn =∑α

bα⊗daα1 · · · daαn

Now projecting the first factor toA′ we conclude that in this case∑α b

αdaα1 · · · daαn =0 and hence

ω =∑α

λαdaα1 · · · daαn = d(∑α

λαaα1 daα2 · · · daαn)

so that ω is exact. The last step requires n > 1. The n = 0 case was already dealtwith and we computed ker d = k1 in this case.

We see that the universal calculus is both too big and has no nontrivial ‘topology’in view of this theorem. For a more non-trivial example of the same dimensions asthe classical geometry, one has:

Example 3.6. On the algebraic noncommutative torus Aθ we let Ω1 = Aθdu,dvwith

du.u = u.du, dv.v = v.dv, dv.u = eıθu.dv, du.v = e−ıθv.du.

Its maximal prolongation has Ω2 = Aθdudv and Ωi = 0 for all i > 2. The noncom-mutative de Rham cohomology is the same as classically, namely

H0 = C, H1 = C2, H2 = C

Here u, u−1 generate a classical circle C[S1] and Ω1 restricts to this as classically.Similarly for v, v−1. Of interest is how the two interact. From d applied to therelations we need d(vu) = dv.u + v.du = eıθd(uv) = eıθdu.v + eıθu.dv, whichis provided by the relations shown. These relations also ensure that the calculusbecomes the usual one on S1×S1 when θ = 0. We thus define Ω1 as a free moduleon the left (i.e. just by the product of Aθ) and use the relations between 1-formsand the algebra generators to define the right module structure. Next, apply d tothese relations to find

(du)2 = 0, (dv)2 = 0, dvdu+ eıθdudv = 0.

which tells is that Ω2 is 1-dimensional over Aθ. That the cohomology is the same asclassically is a calculation along the same lines as when θ = 0. One still has a basisumvn : m,n ∈ Z of Aθ and there are extra eıθ factors in the computations butthe steps are the same. The natural basis for computations is given by e1 = u−1duand e2 = v−1dv with de1 = −u−1du.u−1du = 0 from the Leibniz rule and thecommutation relations above, as classically. Similarly for de2. Now suppose that

λe1 + µe2 = d(∑

amnumvn) =

∑amn(mum−1du.vn + umnvn−1dv).

as d on powers of u alone or v alone behaves as classically. Putting du to the rightintroduces a facfor e−nıθ but we still need amn = 0 for all m 6= 0 or n 6= 0 for thedu and dv coefficients to match. Hence only a0,0 can contribute and this does not

12 S. MAJID

as d1 = 0. Hence all λe1 + µe2 are not exact. In fact, H0 = C.1, H1 = Ce1 ⊕ Ce2

and H2 = Ce1e2.

Definition 3.7. If A is a ∗-algebra we ask the calculus or exterior algebra to be∗-compatible in the sense that this extends to Ω a graded-∗-algebra (i.e. with anextra sign factor according to the degrees) with [∗,d] = 0.

Examples are the universal calculus on any ∗-algebra and the above calculus onAθ, where e∗i = −e∗i in keeping with the classical picture u = eıφ1 , v = eıφ2 whereei = ıdφi.

3.2. Differentials on quantum groups. In the classical case of a Lie group onehas a unique translation-covariant differential calculus. In the quantum group weagain have relatively few such calculi but usually without uniqueness.

Definition 3.8. Ω1 on a Hopf algebra A is left covariant if:(1) There is a left coaction ∆L : Ω1 → A⊗Ω1.(2) Ω1 becomes a left Hopf module.(3) d : A→ Ω1 is a comodule map, where A coacts on itself by ∆.

Note that since Ω1 is spanned by elements of the form adb, conditions (2),(3)imply that ∆L is a bimodule map and defined by the formula

∆L(adb) =∑

a(1)b(1)⊗ a(2)db(2).

Conversely, if ∆L is well-defined by such a formula then it is easy to verify that(1),(2),(3) hold. So this is all we need for a calculus to be left-covariant. Theleft-invariant 1-forms are the coinvariants Λ1 = AΩ1.

Theorem 3.9. Let A be a Hopf algebra. Left-covariant Ω1 on A have the formΩ1∼=A⊗Λ1 where Λ1∼=A+/I for some right ideal I ⊆ A+ = ker ε.

Proof. The first part is the Hopf module lemma, Lemma 2.4. Next, consider the‘Maurer-Cartan form’ ω : A+ → Λ1 defined by

ω(a) = Sa(1)da(2), ∀a ∈ A+.

It’s image is invariant since we have ∆Lω(a) = (Sa(1))(1)a(2)(1)⊗(Sa(1))(2)da(2)(2) =(Sa(1)(2))a(2)(1)⊗Sa(1)(1)da(2)(2) = 1⊗ω(a). Moreover, if α = aidbi is left-invariantthen ai(1)bi(1)⊗ ai(2)dbi(2) = 1⊗α. Applying S to the first factor and multiplyingtells us that α = ω(ε(ai)bi). Hence ω is surjective and Λ1∼=A+/I where I = kerω.The Leibniz rule requires I a right ideal.

The converse is also true, so left-covariant differential calculi (Ω1,d) are in 1-1correspondence with right ideals I ⊆ A+. Given an ideal I we have Λ1 = A+/I asa right H-module by right multiplication in A. The left (co)action on Ω1 is justthe (co)action on A as in the Hopf module lemma. The rest of the structure is justtracing through the above isomorphism. Thus

a.(b⊗ v) = ab⊗ v, ∆L(a⊗ v) = (∆⊗ id)(a⊗ v)

(a⊗ v).b =∑

ab(1)⊗ v.a(2)

where the last of these follows from ω(a).b =∑a(1)ω(ba(2)). Similarly,

da = (id⊗π)(∆a− a⊗ 1), π : A+ → Λ1

where π is the canonical surjection. I = 0 gives the universal calculus.


There is a similar notion of right-covariance and a calculus is called bicovariantif it is both left and right covariant. The two coactions will necessarily commute(so that Ω1 is a bicomodule).

Corollary 3.10. [4] A left-covariant Ω1 is bicovariant iff AdR(I) ⊆ I ⊗A whereAdR(a) =

∑a(2)⊗Sa(1)a(3). If so then Λ1 is a right A-comodule from AdR.

Proof. If ∆R is well-defined, we compute

∆R(aω(b)) = a(1)(Sb(1))(1)db(2)(1)⊗ a(2)(Sb(1))(2)b(2)(2)

= a(1)Sb(2)(1)db(2)(2)⊗h(2)(Sb(1))b(3) = a(1)ω(b(2))⊗ a(2)(Sb(1))b(3).

We recognise the right adjoint coaction AdR(b) which at the classical level corre-sponds to conjugation in the group, and require that AdR(I) ⊆ I ⊗A. Conversely,in this case we define ∆R as stated. Equivalently, we define the right coaction onA⊗Λ1 as the tensor product of ∆ and AdR.

An advantage of the bicovariant case is that there is a natural prolongation insome sense compatible with Poincare duality. We define

Ψ : Λ1⊗Λ1 → Λ1⊗Λ1, Ψ(ω(a)⊗ω(b)) = ω(b(2))⊗ω(a(Sb(1))b(3))

obeying the Yang-Baxter or braid relations. Any permutation σ ∈ Sn can bewritten in reduced form σ = σi1 · · ·σil(σ) where l(σ) is the length and σi = (i, i+ 1)

and we define Ψσ = Ψi1 · · ·Ψil(σ) on (Λ1)⊗n, where Ψi denotes the braiding in thei, i+ 1 tensor factors, and ‘antisymmetrize’ by

An =∑σ

(−1)l(σ)Ψσ, Λn =(Λ1)⊗n

kerAn, Ωn = A⊗Λn.

This is the original definition of Woronowicz[4] while another approach is due tomyself based on braided-integers and Λ as a superbraided Hopf algebra canonicallyassociated to Λ1. We now focus on bicovariant Ω1. One says that a calculus isirreducible if it has no proper quotients.

Example 3.11. For A = C[x] with ∆x = x⊗ 1 + 1⊗x (the affine line) the irre-ducible bicovariant Ω1 are parametrixed by λ ∈ C and take the form

Ω1 = C[x].dx, dx.f(x) = f(x+ λ)dx, df =f(x+ λ)− f(x)

λdx.

Only the Newton-Leibniz calculus at λ = 0 has [dx, f ] = 0. Here AdR is trivial soleft covariant calculi are the same as bicovariant ones and C[x] is a principal idealdomain so A+ = 〈x〉 (the ideal generated by x) and I = 〈xm(x)〉 for some monicirreducible polynomial. Hence calculi are in 1-1 correspondence with such m. OverC, the only possible m are m(x) = x−λ for λ ∈ C. It remains only to work out thestructure of the calculus in this case. We have Λ1 = 〈x〉/〈x(x−λ)〉 is 1-dimensionaland

dx = 1⊗π(x), dx.x = x(1)⊗π(xx(2)) = x⊗π(x) + 1⊗π(x2) = xdx+ λdx

as x2 = x(x − λ) + λx. The bimodule relations with general f(x) follow. Alsodxn = dx.xn−1 + xdxn−1 = (x + λ)n−1dx + xdxn−1 gives the formula for d onmonomials by induction.

The first part of the analysis applies to k[x] for any field k; calculi are againclassified by monic irreducibles m(x) and Ω1 as a left k[x]-module is identified withkλ[x], where λ generates the field extension defined by m(λ) = 0.

14 S. MAJID

Example 3.12. For A = C[t, t−1] with ∆t = t⊗ t (the complex circle) the irre-ducible bicovariant Ω1 are parametrixed by q ∈ C∗ and take the form

Ω1 = C[t, t−1].dt, dt.f(t) = f(qt)dt, df =f(qt)− f(t)

q(t− 1)dt.

Only the Newton-Leibniz calculus at q = 1 has [dt, f ] = 0. Here A+ = 〈t − 1〉,with the elements given by tn − 1 = [n]t(t − 1) in terms of the q-integer [n]t =1 + t+ · · · tn−1. As before, an ideal in A+ takes the form I = 〈(t− 1)(t− q)〉. Then

dt = t⊗π(t− 1), dt.t = t2⊗π((t− 1)t) = qt2⊗π(t− 1) = qtdt

where (t− 1)t = (t− 1)(t− q) + q(t− 1). We see that Ω1 = A.e1 where e1 = t−1dtis the more natural left-invariant generator. We then obtain the bimodule relationswith general f(t). Finally,

dtn = dt.tn−1 + tdt.tn−2 + · · ·+ tn−1dt = tn−1(qn−1 + · · ·+ q + 1)dt = [n]qtn−1dt

to give the result stated. It is instructive to compute the cohomology. We assume qis not a nontrivial root of unity, which is equivalent to [n]q 6= 0 for all n 6= 0. Thend(∑ant

n) =∑tn−1[n]qandt = 0 implies that an = 0 for all n 6= 0, i.e. H0 = C.1

and our calculus is connected. Also, Ωi = 0 for i > 1 since in any prolongation wehave 0 = d(dt.t − qtdt) = −(dt)2(1 + q) = −[2]q(dt)

2, and hence (dt)2 = 0. Thenwe similarly have ∑

antndt = d

(∑ antn+1

[n+ 1]q

)iff a−1 = 0, and hence H1 = Ct−1dt. This changes if q is a root of unity.

Example 3.13. For A = k(G) on a finite group G, the irreducible bicovariant Ω1

are in 1-1 correspondence with nontrivial conjugacy classes C ⊂ G. The 1-formsea =

∑g∈G δgdδga for a ∈ C form a basis of Λ1 and

ea.f = Ra(f)ea, df =∑a∈C

(Ra(f)− f)ea.

The corresponding graph is the Cayley graph of G with respect to C. Here Ra(f) =f(( )a) is right-translation. For Λ1 we set to zero all delta-functions except δaa∈C .These project to our basis ea, which we identify in terms of d. The calculus isconnected iff C is a generating set.

We see that any generating set of a finite group gives a left-covariant connectedcalculus while ad-stable such sets give a bicovariant one. In the second case wehave an exterior algebra Ω(G). Its volume dimension carries deep information(conjecturally related to Lusztig’s canonical basis in the case where G is the Weylgroup of a semisimple Lie algebra). Here G = S3 with C = (12), (13), (23) hasdimensions dim(Ωi) = 1 : 3 : 4 : 3 : 1, so volume dimension 4 but cotangentdimension 3.

Example 3.14. [4] Cq[SL2] has a left-covariant Ω1 (compatible with ∗) with

e− = ddb− qbdd, e+ = q−1adc− q−2cda, e0 = dda− qbdc

e±f = q|f |fe±, e0f = q2|f |fe0

d

(ac

)=

(ac

)e0 + q

(bd

)e+, d

(bd

)=

(ac

)e− − q−2

(bd

)e0.


Here |f | of a monomial is the number of a, c minus the number of b, d. As with thenoncommutative torus, for generic q the maximal prolongation gives an exterioralgebra of classical dimensions, e.g. a unique top form e+e− up to normalisation.

For all standard quantum groups Cq[G] associated to simple groups, there donot exist bicovariant calculi of classical dimensions. For generic q and a q → 1limiting requirement, one has one calculus for every irreducible representation V ofthe group, of cotangent dimension (dim(V ))2. For Cq[SL2] the smallest bicovariantcalculus is therefore 4-dimensional with dimension pattern dim(Ωi) = 1 : 4 : 6 : 4 : 1like a 4-manifold. Interestingly, the noncommutative de Rham cohomology is thesame for Cq[SL2] (or Cq[SU2] as a real form) as for the above calculus on S3, namely

H0 = C, H1 = C, H2 = 0, H3 = C, H4 = C.

4. Noncommutative vector bundles

Cotangent bundles are special cases of vector bundles. On a manifold X, a vectorbundle of rank n means a smooth assignment of a vector space Ex at each point xwith each Ex isomorphic to Cn. More precisely, E is itself a manifold and there isa surjection π : E → X such that in an open neighborhood of any point we haveπ−1(U)∼=U × Cn compatible with π and compatible with a vector space structureon each Ex = π−1(x). The main thing we will need is that every vector bundle Eis the direct summand of a trivial one: there is some bundle E′ → X such thatE ⊕ E′∼=X × CN for some sufficiently large N . We also need the space of sections

Γ(E) = smooth maps s : X → E | π s = id

which is a module over the algebra of functions on X by pointwise multiplicationif we think of sections as ‘functions’ with values s(x) ∈ π−1(x) = Ex. In termsof sections, Γ(E)⊕ Γ(E′)∼=C∞(X)⊗CN and hence Γ(E) = image(e) for some e ∈MN (C∞(X)) a projection matrix e2 = e. In this way (made precise in the ‘Serre-Swan theorem’) vector bundles become equivalent to finitely generated projectivemodules over the algebra of functions.

4.1. Projective modules and K-theory. Let A be an algebra over a field k. Inview of the above,

Definition 4.1. A ‘vector bundle’ over A is defined as a finitely generated projectiveE module over A.

If Ω1 is a differential calculus on A and E a vector bundle on A, we define aconnection on E as a linear map such that

D : E → Ω1⊗AE , D(as) = da⊗

As+ aDs, ∀a ∈ A, s ∈ E .

If higher forms are defined we extend the definition to ‘form valued sections’

D : Ωn⊗AE → Ωn+1⊗

AE , D(ω⊗

As) = dω⊗

As+ (−1)nωDs, ∀ω ∈ Ωn.

We can then define the curvature FD = D2 : E → Ω2⊗A E .

Proposition 4.2. Let E be a vector bundle over A with projector e, so E = AN .efor some number N copies of A and e ∈MN (A). Then

D(v.e) = d(v.e).e = dv.e+ v.de.e, v ∈ AN

16 S. MAJID

is a connection, the ‘Grassmann connection’ on E, with curvature given by

FD = −de.de.e.

Proof. We write a element v ∈ E as the image of a row vector v = (v1, · · · , vN ).The first expression for D shows that it is well-defined and depends only on v.The second follows by the Leibniz rule applied component-wise and e2 = e. Theconnection rule follows from the first form as

D(av.e) = (d(av.e)).e = (da.v.e+ ad(v.e)).e = da.v.e+ aD(v.e),

for all a ∈ A, v.e ∈ E . For the curvature, we compute D2(v.e) as

D(dv.e+ v.de.e) = (d(dv.e)).e+ (d(v.de.e)).e = −dv.de.e+ dv.de.e− v.de.de.e

where we treat dv and v.de as Ω1-valued row vectors (a form-valued section). Weused the Leibniz rule and d2 = 0. Also, e2 = e implies e.de.de = (de2)de−de.e.de =de.de2 − de.e.de = de.de.e so that

FD(v.e) = −v.de.de.e = −v.e.de.de = −v.e.de.de.e

acts simply as the MN (Ω2)-valued matrix as stated on E .

The converse is also true in the case of the universal calculus on suitable A;in this case a finitely generated module E is projective iff it admits a connectionwith the universal calculus (Cuntz-Quillen theorem). As a nontrivial example wetreat CP 1 first classically (but using our algebraic methods) and a noncommutativeversion.

Example 4.3. CP 1 means the set of lines in C2 and has a tautological bundle Ewhere Ex at each x ∈ CP 1 is the line itself in C2. Each Ex∼=C and we have a rank1 vector bundle. Now, lines in C2 are in 1-1 correspondence with certain matrices

CP 1 ↔ (eij) ∈M2(C) | e2 = e, Tr(e) = 1, e∗ij = eji.

Here any such hermitian e has real eignevalues and since e2 = e these are each 0 or1. Since Tr(e) = 1, exactly one eigenvalue is 1 and the corresponding eigenvectordefines a 1-dimensional subspace. Using the dictionary in Theorem 1.1 the complexpolynomial subset of C4 defined by the relations corresponds to an algebra C[CP 1]defined as the ∗-algebra with generators eij, i, j = 1, 2, relations e2 = e, Tre = 1 and∗-structure e∗ij = eji. The sections of the tautological vector bundle is the projective

module E = C[CP 1]2.e where e ∈ M2(C[CP 1]) is the matrix of generators. TheGrassmann connection recovers us the monopole connection in CP 1∼=S2. Note thatC[CP 1] is a ‘complexified CP 1’ or S2

C and is an affine variety while the ∗-algebrastructure remembers the ‘real form’ CP 1 or S2. One can chose more conventionalcoordinates by parametrizing

e =

(1− x zz∗ x

)for self-adjoint generator x and generator z. The form of e solves the trace andhermitian conditions and the remaining projector relations become

zx = xz, zz∗ = z∗z = x(1− x)


which indeed describes a sphere of radius 1/2 if we write z = x1+ıx2 and x = x3+ 12 .

For the Grassmann connection note that dz.z∗ + zdz∗ = (1− 2x)dx from d appliedto the quadratic relation. Hence in one patch

de.de =

(−dx dzdz∗ dx

)2

=

(dzdz∗ −2dxdz

2dxdz∗ −dzdz∗

)=

dzdz∗

1− 2x

(1− 2x 2z

2z∗ −(1− 2x)

)=

dzdz∗

1− 2x(2e− 1)

From which

FD = − dzdz∗

1− 2xe = −ıdx1dx2

x3e

which is a constant multiple of the volume form (the top form) on S2 as it shouldbe for the monopole.

Notice in this example that we did not impose a priori that the ∗-algebra wascommutative, this came out of the projector relations! Hence we can discovernoncommutative versions just as easily.

Example 4.4. The ‘fuzzy sphere’ Cλ[CP 1] or Cλ[S2] is defined exactly as in Ex-ample 4.3 but with Tr(e) = 1 + λ where λ ∈ R. The result is a ∗-algebra

[x, z] = λz, [z, z∗] = 2λ(x− 1 + λ

2), z∗z = x(1− x)

and the Grassmann connection defines the ‘fuzzy monopole’ with respect to anydifferential calculus. As before, we write

e =

(1 + λ− x z

z∗ x

)which solves the deformed trace condition. The remaining projector relations comeout as stated. The first two relations are in fact the enveloping algebra U(su2) as‘fuzzy R3’ and the last relation sets the Casimir equal to a constant, i.e. this is thestandard quantisation of S2 as a coadjoint orbit in su∗2 as mentioned at the end ofSection 2.

Example 4.5. The q-sphere Cq[CP 1] or Cq[S2] is defined exactly as in Example 4.3but with a modified ‘q-trace’ Trq(e) = e11 + q2e22 where q ∈ R∗. The result is the∗-algebra

zx = q2xz, zz∗ = q4z∗z + q2(1− q2)x, z∗z = x(1− x)

and the Grassmann connection is the q-monopole with respect to any calculus. Herewe solve the Trq(e) = 1 condition with

e =

(1− q2x zz∗ x

).

The remaining projector relations come out as stated. The origin of the q-trace isthat this is Cq[SU2]-invariant and as a result Cq[S2] necessarily has a (left) coactionof Cq[SU2] corresponding to the classical picture. One can well ask what happens ifone combines this with the previous example and asks for Trq(e) = 1+λ. The resultis of course the ‘q-fuzzy sphere’ with two parameters q, λ. It turns out for q 6= 1 tobe isomorphic to a ‘constant time slice’ (depending on λ) of the unit hyperboloid inq-Minkowski space.

18 S. MAJID

For the noncommutative torus, we work with a Schwarz space version

Aθ = ∑

amnumvn | (amn) ∈ S(Z2)

where the Schwarz space means functions on Z2 decaying faster than any power ofn,m (more precisely (|m| + |n|)N |amn| is bounded on Z2 for all N ∈ N). This islarger than the algebraic version but still workable. One defines S(R) similarly asfunctions on R decaying faster than any power.

Example 4.6. For irrational θ, Aθ has finitely generated projective modules Ep,qwhere p, q ∈ Z. As a vector space Ep,q = S(R,Cq) and

(u.s)(x) = u0s(x− θ +p

q), (v.s)(x) = e2πıxv0s(x)

where u0, v0 are fixed unitary matrices on Cq obeying v0u0 = e2πı pq u0v0 and s ∈S(R,Cq) = S(R)⊗Cq. One may verify that this is a module much as we did inconstructing Aθ in Section 1. The associated ‘Powers projector’ ep,q is a bit beyondour scope but see [1] and related papers.

4.2. K-theory and cyclic cohomology. Vector bundles are one of the tools usedin geometry to obtain topological invariants. From the operation ⊕ among vectorbundles one can construct K0(X) as an abelian group made, loosely speaking, outof stable equivalence classes. Two vector bundles are stably equivalent if theyare isomorphic after direct sum with some third vector bundle. More precisely,isomorphism classes of vector bundles form a semigroup under direct sum. Givenany semigroup S with operation ⊕ (say) its associated group consists of equivalenceclasses of S × S where

(E,F ) ∼ (E′F ′) iff ∃G ∈ S s.t. E ⊕ F ′ ⊕G = E′ ⊕ F ⊕G

(Grothendieck’s construction). One can sloppily think of (E, 0) as the ‘positive ele-ments’ and (0, E) as their adjoined negatives. In algebraic terms, we define K0(A)similarly as direct sum of isomorphism classes of finitely generated projective mod-ules made into a group. Elements of S(A) are equivalence classes [e] of projectors,where

1. Extension by zero:

(e 00 0

)∈MN+k(A)

2. Conjugation: ueu−1 for any u ∈ GLN (A)define the same projective module up to isomorphism. Likewise, if the extensionof one projector is “conjugate” in a suitable sense to an extension of another,we consider them equivalent members of S(A). In a ∗-algebra setting we requireprojectors to be hermitian and the u above to be unitary. For example, over C,one has K0(S2) = Z × Z generated by tensor products of the monopole bundleand direct sums. We have the same picture for generic q for K0(Cq[S2]) with thenontrivial generator being the [e] for e in Example 4.5.

More novel is another ‘topological invariant’ which is more like homology of aspace. As usual we let A be a unital algebra over k.

Definition 4.7. The cyclic cochain complex of A is

Cnλ (A) = φ : A⊗(n+1) → k | φ(a1, · · · , an, a0) = (−1)nφ(a0, a1, · · · , an)


(bφ)(a0, · · · , an+1) =

n∑j=0

(−1)jφ(a0, · · · , ajaj+1, · · · , an+1)

+(−1)n+1φ(an+1a0, a1, · · · , an).

where b : Cnλ → Cn+1λ obeys b2 = 0. The cyclic cohomology HCnλ (A) is the kernel

of b on degree n modulo the image of b on degree n− 1.

A cyclic n-cochain is called unital if it vanishes when any of its arguments is 1and n ≥ 1. When n = 0 the condition is taken to be empty. To bring out the‘geometric meaning’ we suppose that A has an exterior algebra Ω of sufficientlyhigh top degree and that

∫: Ωn → k is a linear map such that∫

dω = 0,

∫ωρ = (−1)|ω||ρ|

∫ρω

on forms of appropriate homogeneous degree. Such an ‘n-cycle’ is like integrationon a closed submanifold of dimension n.

Proposition 4.8. [1] Let A be equipped with an exterior algebra and∫

an n-cycle.Then

φ(a0, a1, · · · , an) =

∫a0da1 · · · dan

is a unital cyclic n-cocycle.

Proof. We first check cyclicity

φ(a1, · · · an, a0) =

∫a1da2 · · · da0 = (−1)n−1

∫a1d(da2 · · · dan.a0)

= (−1)n−1(

∫d(a1da2 · · · dan.a0)−

∫da1 · · · dan.a0)

= (−1)n∫

da1 · · · dan.a0 = (−1)nφ(a0, a1, · · · , an)

using the graded Leibniz rule and the first property of∫

. Next,

(bφ)(a0, · · · , an+1) =

∫a0a1da2 · · · dan+1

+

n∑j=1

(−1)j∫a0da1 · · · d(ajaj+1) · · · dan+1 + (−1)n+1

∫an+1a0da1 · · · dan

and we expand the summed terms by the Leibniz rule as

j=n∑j=1

(−1)j∫a0da1 · · · daj .aj+1daj+2 · · · dan+1

+

j=n∑j=1

(−1)j∫a0da1 · · · daj−1.ajdaj+1 · · · dan+1.

Now the interior of the first sum at j cancels with the interior of the second at j+1.What remains is the boundary j = 1 of the second sum, which cancels with thefirst term of bφ above, and j = n of the first sum giving (−1)n

∫a0da1 · · · dan.an+1.

This cancels with the last term of bφ above due to the second requirement of∫

ann-cycle. Note that we only really need this in the form

∫aω =

∫ωa for all a ∈ A

and ω ∈ Ωn but this implies and is therefore equivalent to the graded version for

20 S. MAJID

general ρ ∈ Ω (a short proof by induction on degree of ρ). Clearly φ is unital sinced1 = 0.

The converse is also true in the case of the universal calculus on A; in this casethe two notions are equivalent. Starting with a unital n-cocycle φ we define

∫on

degree n by the same formula as above but read the other way. In degree 0 anelement of HC0(A) and a 0-cycle both mean a ‘trace’, i.e. a map φ : A → k suchthat φ(ab) = φ(ba). In degree 1 a unital 2-cocycle is a linear map φ : A⊗ 2 → kwith

φ(1, a) = 0, φ(a0a1, a2)− φ(a0, a1a2) + φ(a2a0, a1) = 0

(which implies that φ is antisymmetric). Now∫a0da1 = φ(a0, a1) is well-defined

because if∑aα0 daα1 = 0 then

∑aα0 ⊗ aα1 =

∑aα0 a

α1 ⊗ 1. Writing aα1 = λα +

bα ∈ k1⊕ A′ (choosing a complement of k1) we find∑aα0 ⊗ bα =

∑aα0 b

α⊗ 1 andconclude that

∑aα0 ⊗ bα = 0 and hence

∑aα0 ⊗ aα1 ∈ A⊗ 1. Hence in this case∑

φ(aα0 , aα1 ) = 0 as required. Once we know that

∫is well-defined, we just push

the proof of the proposition in reverse; by definition∫

da = φ(1, a) = 0 while thecocycle condition amounts

∫a0da1.a2 =

∫a2a0da1 as required.

Example 4.9.∫ ∑

amnumvn = a00 defines a 0-cycle on the noncommutative torus

Aθ. Remembering that the amn are at θ = 0 the Fourier coefficients of a function onS1×S1, this becomes the Haar integral on the torus. For the calculus in Example 3.6we define a 2-cycle by

∫ae1e2 =

∫a for a ∈ Aθ. This defines a unital 2-cocycle on

Aθ. To see what this looks like, we use as basic 1-forms e1 = u−1du and e2 = v−1dvwhich commute with Aθ and anticommute as classically among themselves. In thatcase da = (∂ua)e1 +(∂va)e2 defines two derivations ∂u, ∂v : Aθ → Aθ. They look asclassically ∂u = u−1 ∂

∂u and similarly for ∂v provided we understand all expressionsas ‘normally ordered’ with u to the left of v. We have

φ(a, b, c) =

∫a(∂ube1 + ∂vbe2)(∂uce1 + ∂vce2) =

∫a(∂ub∂vc− ∂vb∂uc)e1e2

=

∫a(∂ub∂vc− ∂vb∂uc).

Finally, we can put together these ideas.

Theorem 4.10. [1] Suppose that k has characteristic 0. There is a ‘Chern-Connes’pairing between K0(A) and HC2m(A) given by

〈[e], [φ]〉 =1

m!

∑i0,··· ,i2m

φ(ei0i1 , ei1e2 , · · · , ei2mi0) =1

m!Trφ(e, e, · · · , e)

Proof. We sketch why this well-defined on both sides. If we change e by extensionby zero then clearly the right hand side does not change. The right hand sideis also unchanged if we conjugate e ∈ MN (A). In degree 0 this is φ(uijejku

−1ki ) =

φ(u−1ki uijejk) = φ(eii) (summations understood) by the cocycle (trace) requirement.

For higher degree we similarly need to use the cocycle condition. For example, fordegree 2 this is

φ(a0a1, a2, a3)− φ(a0, a1a2, a3) + φ(a0, a1, a2a3)− φ(a3a0, a1, a2) = 0.

Suppose for the moment that N = 1 so e, u ∈ A. Then using e2 = e,

φ(ueu−1, ueu−1, ueu−1) = φ(ue.eu−1, ueu−1, ueu−1) = φ(ue, eu−1, ueu−1)


where we applied the cocycle condition with a0 = ue, a1 = eu−1, a2 = a3 = ueu−1.Siimilarly, the right-hand side is equal to φ(ue, eu−1, ue.eu−1) = 2φ(ue, e, eu−1) −φ(ue, eu−1, ue.eu−1), where we use that φ is invariant under cyclic rotations. Hence,as 2 is invertible, we obtain φ(ue, e, eu−1) = φ(ue, e.e, eu−1) = 2φ(ue, e, eu−1) −φ(e, e, e). Hence we obtain φ(e, e, e). Now when N > 1 we have matrix indices onthe u, e being matrix multiplied. As Tr on a product of matrices is invariant undercyclic rotations used in some of the above steps, the same steps still work. Theproof for general even degree is best done by more sophisticated arguments.

On the other side, if φ = bψ we have

Tr(

2m−1∑j=0

(−1)jψ(e, e, · · · , e) + (−1)2mψ(e, e, · · · , e)) = Trψ(e, e, · · · , e)

since e2 = e and there are an odd number of alternating such terms. But ψ is oddand cyclic, hence changes sign under cyclic rotation of its arguments while the tracedoes not change under the cyclic rotation of the corresponding matrix products.Hence Trψ(e, e, · · · , e) = 0.

The interpretation of this is best seen if φ is given by a 2m-cycle. Then [1]

〈[e], [φ]〉 =1

m!Tr

∫ede · · · de =

(−1)m

m!Tr

∫FmD = Tr

∫e−FD

for D the Grassmann connection. Here FD = −edede = −dede.e as we notedbefore, so we may move all the e’s to the left. Also, as in differential geometry, weconsider

∫to be zero on forms of the wrong degree. Here e−FD it is the analogue

of the classical Chern character ch : K0(X) → HDR(X). The classical de Rhamcohomology pairs with homology in X represented by cycles C ⊆ X and the pairingis∫C

on forms of correct degree.

Example 4.11. We consider Cq[S2] with the tautological or monopole bundle inExample 4.5. For q 6= 1 this has a certain 0-cocycle φ defined by

φ(zm) = φ(z∗m) = 0, φ(zmxn) = φ(z∗mxn) =δm,0

1− q2n, n > 0.

Then

〈[e], [φ]〉 = φ(Tr

(1− q2x zz∗ x

)) = φ(1 + (1− q2)x) = 1.

This shows that the q-monopole bundle is non-trivial. The 1 here is the topologicalcharge of the q-monopole. We use basis zmxn | m > 0, n ≥ 0 ∪ z∗mxn | m >0, n ≥ 0 ∪ xn | n ≥ 0 for Cq[S2] in view of its relations. Note that φ has nolimit as q → 1 and although broadly similar to it, is not the restriction of the Haarintegral on Cq[SU2] discussed in Section 2. The latter obeys

∫ba =

∫σ(a)b for all

a, b, where σ is a certain ‘twisting automorphism’. Thus we have to use somethingdifferent to fit the standard axioms of cyclic cohomology or we can use

∫but have

to use a q-deformed or ‘twisted’ cyclic cohomology to properly accommodate suchq-deformed geometries in a way that has a classical limit as q → 1.

This example goes back to work of P. Hajac and the author.

22 S. MAJID

5. Noncommutative Riemannian geometry

We now take a deeper point of view on both vector bundles and Riemanniangeometry, starting with the notion of principal G-bundle P over a manifold X. Thisis defined exactly like a vector bundle with a surjection π : P → X but each fibrePx = π−1(x) now has the structure of a fixed group G. This is achieved smoothlyby starting with a free right action of G on the manifold P such that X = P/G.A connection on P is defined concretely as an equivariant complement in Ω1(P ) tothe ‘horizontal forms’ (those pulled back from Ω1(X)). This is, however, equivalentto ω ∈ Ω1(P )⊗ g with certain properties. Here g is the Lie algebra of G. Giventhis data, there is an associated vector bundle E = P ×G V and a connection D onit, for any representation V of G.

As an example, the frame bundle on a manifold of dimension n is a certain princi-pal SOn-bundle. In noncommutative geometry we would not know which quantumgroup version of this to take or even what n would be if there was no cotangentdimension, so we are forced to generalise the notion even in classical differentialgeometry to that of a G-framing. This means a general principal G-bundle P , arepresentation V and θ ∈ Ω1(P, V ∗) that is G-equivariant and horizontal such thatthe induced map E → Ω1(X) on the associated bundle sections given by multipli-cation by θ pointwise (by contracting the V ∗ of θ with the V -value of a section)is an isomorphism. The entries of θ in local coordinates are the ‘n-bein’. In thiscontext, the framing isomorphism turns the the covariant derivative D induced bya ‘spin connection’ ω on the principal bundle into a connection ∇ on Ω1. Thisbeing torsion free amounts to Dθ = 0 when θ is itself viewed as a section of anassociated bundle. In classical geometry we would also want to choose ω such that∇ is metric compatible, hence the Levi-Civita connection for the metric. However,for a general G-framing, and also in the quantum case, this may not be possible andwe are forced to a weaker concept of Riemannian geometry. One such is the notionof both a G-framing and a G-coframing. Given the former, the latter just means θ∗

so that (V ∗, θ∗) is also a framing, which is equivalent to Ω1 now being isomorphicto E∗, i.e. to Ω1 being isomorphic to its own dual, i.e. to a possibly non-symmetricmetric g = 〈θ∗, θ〉 ∈ Ω1⊗C∞(X) Ω1 where the angular brackets denote evaluation

of V ∗, V and the result lies in the tensor square of Ω1(X) due to equivariance andhorizontality of θ, θ∗. In this context we can ask that ω is torsion free with respectto θ∗, i.e. Dθ∗ = 0. We call this condition ‘cotorsion free’ and the remarkablething is that if the connection is already torsion free then cotorsion free amounts to(∧⊗ id)∇g = 0, a weaker classical notion of metric compatibility but which tendsto work in the noncommutative case and which one can solve for directly.

5.1. Quantum principal bundles and framing. We now turn to the noncom-mutative formulation on an algebra A with structure quantum group H.

Definition 5.1. A ‘quantum principal bundle’ over A means:(1) P a right H-comodule algebra via ∆R : P → P ⊗H with A = PH .(2) Compatible differential structures with Ω1(H) = H.Λ1

H bicovariant, Ω1(P )right H-covariant and Ω1(A) = A(dA)A ⊆ Ω1(P ).

(3) 0→ PΩ1(A)P → Ω1(P )→ P ⊗Λ1H → 0 is exact.

Here the map ver : Ω1(P ) → P ⊗Λ1H is the analogue of the ‘vertical vector

fields’ generated by the Lie algebra of the fibre group acting on the total space ofthe bundle. Here Λ1

H = H+/IH is like the dual of the Lie algebra so ver is like a


coaction version of the vertical vector fields. It is defined at the level of universalcalculi ver : Ω1

univ(P ) → P ⊗H+ by ver(u⊗ v) = u∆Rv. Compatibility of thecalculi means

∆RNP ⊆ NP ⊗H, ver(NP ) ⊆ P ⊗ IH , NA = NP ∩ Ω1univ(A)

where we recall that calculi on algebras are defined by subbimodules N , and ensuresthat ver descends to the non-universal calculi. If we already have a bundle with theuniversal calculi then the further content of (3) is that we have in fact ver(NP ) =P ⊗ IH .

Surjectivity of the condition (3) corresponds classically to freeness of the action(i.e. only the identity element fixes all points), while exactness in the middle saysthat the horizontal forms PΩ1(A)P are exactly those killed by the ‘vertical vectorfields’. This algebraic and global condition in noncommutative geometry replacesthe concept of a local trivialisation, which would normally be used to prove suchthings in classical differential geometry. Following the classical geometry, a con-nection on a quantum bundle means a left P -module and right H-comodule map

Π : Ω1(P )→ Ω1(P ), Π2 = Π, ker Π = PΩ1(A)P

i.e. an equivariant complement to the horizontal forms. A connection is calledstrong if (id − Π)dP ⊆ Ω1(A)P , which is automatic if Ω1(A) commutes with allelements of P . We recall from Section 3.2 that H coacts on Λ1

H by AdR in thebicovariant case. Definition 5.1 and basic results and examples such as the followinggo back to work of T. Brzezinski and the author.

Proposition 5.2. Connections on a quantum principal bundle P correspond toright comodule maps ω : Λ1

H → Ω1(P ) such that ver ω = 1⊗ id.

Proof. One can check that ver is equivariant where P ⊗Λ1H has the tensor product

of ∆R on P and AdR in Corollary 3.10 on Λ1H . Then given ω, set

Π = ·(id⊗ω) ver

which is then equivariant, and clearly a projection due to ver ω = 1⊗ id. Con-versely, given Π, and exactness of our sequence in Definition 5.1 we define

ω(v) = Π ver−1(1⊗ v), ∀v ∈ Λ1H ,

meaning we choose any element mapping onto 1⊗ v under ver, then apply Π. Thisis well-defined just because Π and ver have the same kernel.

Nontrivial examples are provided by quantum homogeneous spaces. Here the‘total space’ algebra P is itself a quantum group with left-covariant calculus andthere is a Hopf algebra surjection π : P → H (so in the classical case a subgroup).There is then a canonical coaction ∆R = (id⊗π)∆ and we suppose that this givesa quantum bundle with the universal calculus. The condition

(id⊗π)AdR(IP ) ⊆ IP ⊗H, π(IP ) = IH

then gives a quantum bundle over A = PH with our given calculi. We also have aleft coaction ∆L = (π⊗ id)∆ and if i : H → P is a linear map with

π i = id, ∆R i = (i⊗ id)∆, ∆L i = (id⊗ i)∆, i(IH) ⊆ IPthen ω(h) =

∑Si(h)(1)di(h)(2) for any h ∈ H+ is a strong connection.

24 S. MAJID

Example 5.3. Over C, let P = Cq[SL2] with its 3D left-covariant calculus andH = Cq2 [t, t−1] with calculus dt.t = q2tdt, as in Examples 3.12, 3.14. Let

π : Cq[SL2]→ C[t, t−1], π

(a bc d

)=

(t 00 t−1

)which induces a coaction ∆Rf = f ⊗ t|f | where |f | is the degree on a monomial fas in Example 3.14. This is clear from

∆R

(a bc d

)=

(a⊗ t b⊗ t−1

c⊗ t d⊗ t−1

).

The coinvariants subalgebra PH = Cq[S2C] therefore means the degree 0 elements,

generated by x = −q−1bc, z = cd and w = −qab. We identify them with Cq[S2C] in

Example 4.5 with wx = q−2xw for the relations of w = z∗ there. The calculus onCq[SL2] corresponds to

IP = 〈a+ q2d− (1 + q2), b2, c2, bc, (a− 1)b, (d− 1)c〉

and one may verify that π(IP ) = IH = 〈(t−1)(t−q2)〉, as well as the other technicalconditions for a quantum bundle. The map

i(tn) = an, i(t−n) = dn, ∀n ≥ 0

has the desired bicovariance properties and gives

ω(tn − 1) = [n]q2e0, Fω(tn − 1) = q3[n]q2e+ ∧ e−.

For brevity we take ω(1) = 0. Then ω(t) = (Sa(1))da(2) = dda − qbdc = e0 in the3D calculus, so the claim holds for n = 1. When n ≥ 2,

ω(tn) = (San(1)dan

(2)) = S(a(1)a′(1)a′′

(1) · · · )d(a(2)a′(2)a′′

(2) · · · )= S(a′(1)a

′′(1) · · · )Sa(1) ((da(2))a

′(2)a′′

(2) · · ·+ a(2)d(a′(2)a′′

(2) · · · ))= ω(tn−1) + S(a′(1)a

′′(1) · · · )ω(t)a′(2)a

′′(2) · · ·

= ω(tn−1) + S(a′(1)a′′

(1) · · · )e0a′(2)a′′

(2) · · · = ω(tn−1) + q2(n−1)e0

where an = aa′a′′ · · · is the product of n copies of the generator a ∈ Cq[SL2] (theprimes are to keep the instances apart). We used the antimultiplicativity of theantipode S and the Leibniz rule for the third equality. For the last equality we usedthat a′(2)a

′′(2) · · · has degree n− 1 and hence its commutation relations with e0 give

a factor q2(n−1) after which we cancel using the antipode axioms and ε(a) = 1. Thecomputation for negative n is similar and the curvature computation is an exercise.

To complete the theory, let V be a finite-dimensional right H-comodule. Wehave an ‘associated bundle’ E = (P ⊗V )H which is now a vector bundle as inSection 4.1 at least if there is a connection at the universal calculus level (so thatit is projective). A strong connection ω, under some technical assumptions on thebundle, defines

D : E → Ω1(A)⊗AE , D = (id−Π)d

as the associated covariant derivative or vector bundle connection on E .

Example 5.4. For the quantum principal bundle in Example 5.3, let V = C.v be1-dimensional and ∆Rv = v⊗ tn. The associated bundle En = Cq[SL2]−n (the


degree −n part) defines the q-monopole of charge n. If u ∈ En then ver(du) =u ¯(1)⊗u ¯(2)−u⊗ 1 = u⊗(t−n− 1) projected to Λ1

H . Hence Π(du) = uω(t−n− 1) and

Du = (id−Π)du = du− uω(t−n − 1) = du+ uq−2n[n]q2e0

for the q-monopole connection in Example 5.3.

Next we define a framing of A as a quantum principal bundle and θ : V →PΩ1(A) such that the induced left A-module map

sθ : E → Ω1(A), p⊗ v 7→ pθ(v)

is an isomorphism.

Theorem 5.5. Let π : P → H be a quantum homogeneous bundle as above. ThenA = PH is framed by the bundle and

V = (P+ ∩A)/(IP ∩A), ∆Rv = v(2)⊗Sπ(v(1)), θ(v) = Sv(1)dv(2)

where v is a representative of v in P+ ∩ A. Hence every quantum homogeneousspace is a ‘quantum manifold’ in the framed sense.

Proof. First observe that v ∈ A means v(1)⊗π(v(2)) = v⊗ 1. Moreover, if v ∈ Athen v(1)⊗ v(2) ∈ P ⊗A because v(1)⊗ v(2)(1)⊗π(v(2)(2)) = v(1)(1)⊗ v(1)(2)⊗π(v(2))= v(1)⊗ v(2)⊗ 1, and if v ∈ P+∩A then ε(v(2))π(Sv(1)) = π(Sv) = Sπ(v(2))ε(v(1)) =1ε(v) = 0 so that ∆R restricts to P+ ∩A. Similarly,

∆Rv = v(1)⊗π(Sv(1)) = v(1)(2)⊗π(Sv(1)(1))π(v(2)) = v(2)⊗π(Sv(1)v(3))

which is the projected adjoint coaction. IP is stable under this, hence if v ∈ IP ∩Athen ∆Rv ∈ IP ∩ A⊗H and ∆R descends to V . Meanwhile, if v ∈ IP thenSv(1)⊗ v(2) ∈ NP and hence θ(v) = 0 in Ω1(P ), so this is well-defined. Moreover,if v ∈ A is a representative of v ∈ V then θ(v) = Sv(1)dv(2) ∈ PΩ1(A) as required.Hence all maps are defined as required and we have sθ : (P ⊗V )H → Ω1(A). Weprovide its inverse by quotienting the inverse in the universal calculus case, namely

s−1θ (adb) = [ab(1)⊗ b(2) − ab⊗ 1], ∀a, b ∈ A

where the expression in square brackets lies in P ⊗P+ ∩ A and [ ] denotes theequivalence class modulo IP ∩ A. That the result actually lies in (P ⊗V )H andgives the inverse of sθ is then a direct verification.

Example 5.6. [5] At least for generic q, A = Cq[S2C] = Cq[SL2]0 in Exam-

ples 4.5,5.3 is framed, with

Ω1∼=E−2 ⊕ E2.

We identify the summands with the holomorphic and antiholomorphic quantumdifferentials Ω1,0, Ω0,1 in a double complex. There is also a coframing giving us thequantum metric on Cq[S2

C] as

g = qdw⊗A

dz + q−1dz⊗A

dw + q[2]q2dx⊗A

dx

and the q-monopole connection induces

∇dz = −q−1[2]q2zg, ∇dw = −q−1[2]q2wg, ∇dx = −q−1[2]q2(x− 1

[2]q2)g.

which is torsion free, cotorsion free and q-deforms the classical Levi-Civita connect-ion on the sphere. Here w = z∗ for the real form with q real.

26 S. MAJID

5.2. Bimodule connections. The above theory applies much more widely thanjust to q-deformations. For example, we can take A the Hopf algebra of functionon a finite group with differential structure as in Example 3.13, P = A⊗H with Ha second copy, ∆R = id⊗∆ and the canonical quantum metric g =

∑a ea⊗A ea,

say, and then solve for torsion free cotorsion free connections. For example, thepermutation group S3 with the Ω(S3) mentioned previously gets a noncommutativeRiemannian geometry of constant curvature. In fact this, the previous q-sphere andmany other examples can be characterised entirely at the level of Ω1(A) and a metricg ∈ Ω1⊗A Ω1 even without a full framing picture. Here torsion, cotorsion and theRiemann curvature appear as

T∇ := ∧∇− d : Ω1 → Ω2

coT∇ := (d⊗ id− (∧⊗ id)(id⊗∇))g ∈ Ω2⊗A

Ω1

R∇ := (d⊗ id− (∧⊗ id)(id⊗∇))∇ : Ω1 → Ω2⊗A

Ω1.

The property of being a connection is something we already covered for vectorbundles in Section 4.1 as

∇(aω) = a∇ω + da⊗A∇ω, ∀a ∈ A, ω ∈ Ω1.

Nondegeneracy of the metric, meanwhile, is existence of a bimodule map ( , ) :Ω1⊗A Ω1 → A such that

(id⊗( , ω))g = ω = ((ω, )⊗ id)g, ∀ω ∈ Ω1.

It turns out that the existence of ( , ) forces the metric g to be central, and wealso we typically require ‘quantum symmetry’ in the form g ∈ ker∧ so as to havea symmetric tensor in the classical limit. This too holds for Cq[S2] and C(S3). Infact, the connections in these and many other examples are bimodule connections,meaning there exists a bimodule map σ : Ω1⊗A Ω1 → Ω1⊗A Ω1 with

∇(ωa) = (∇ω)a+ σ(ω⊗da), ∀a ∈ A, ω ∈ Ω1.

If σ exists then it is uniquely determined, so this is a condition on the connection∇, not additional data. Bimodule connections go back to works of Dubois-Violette,Michor and Mourad. In this case the connection extends naturally to tensor prod-ucts, so full metric compatibility makes sense as

∇g := (∇⊗ id + (σ⊗ id)(id⊗∇))g = 0

although not the case for the mentioned examples which are merely cotorsion-free.The above then amounts to a self-contained formulation of ‘noncommutative Rie-mannian geometry’ at the level an algebra A equipped with a differential structureand data g,∇ defined relative to it. This misses the deeper structure needed, say,for the Dirac operator, but has the merit that you can take your favourite algebraA and directly solve for its moduli of quantum Riemannian geometries. In the∗-algebra case we have a notion of ‘real’ metric (namely invariant under (∗⊗∗)flip)and ‘real’ connection.

Example 5.7. [6] We take the ∗-algebra A with generators x, t and relations[x, t] = λx where x∗ = x, t∗ = t and λ is an imaginary parameter. This is aHopf algebra (the enveloping algebra of a nonAbelian Lie algebra) with additive ∆on the generators and its natural bicovariant calculus

[dx, x] = [dx, t] = 0, [x,dt] = λdx, [t, dt] = λdt


admits a unique 1-parameter form of ‘real’ quantum-symmetric metric up to nor-malisation, namely a λ-deformation of

g = bx2dt⊗A

dt+ (1 + bt2)dx⊗A

dx− bxt(dx⊗A

dt+ dt⊗A

dx)

where b is a nonzero real parameter. This in turn admits a unique ‘real’ torsionfree metric compatible bimodule connection having a classical limit as λ→ 0 (thereis also another ‘quantum Levi-Civita connection’ which blows up as λ → 0). Thegeometry forced out of the differential algebra is that of an expanding ‘big bang’type universe if b > 0 and a very strong gravitational source at x = 0 if b < 0,so strong that all timelike geodesics curve back in. This is not the only calculus;another choice forces one canonically to a λ-deformed Bertotti-Robinson metric anda quantum Levi-Civita bimodule connection for it.

Noncommutative geometry clearly has potential across mathematics. There isalso potential to explore the structure of actual quantum systems, for example inquantum information. Better explored to date in mathematical physics is its usein modelling quantum gravity corrections to spacetime, of which the above is anexample. Here it is well known that a measuring device cannot resolve distancessmaller than the wavelength used. However, particles of smaller wavelength alsohave higher energy in quantum theory. Hence to probe smaller and smaller distancesyou will need heavier and heavier particles until you reach the point where the probeparticles are so heavy that they curl up the geometry that you are trying to probe(form black holes in the extreme case). This means that distances less than thePlanck scale |λ| ∼ 10−33cm make no sense and continuum differential geometry doesnot apply. Instead, quantum fuzziness expressed in noncommutative spacetime mayappear at this scale.

Exercises

1. Prove that a commutative C∗-algebra is reduced.

2. Prove that the antipode of a Hopf algebra A is unique, antimultiplicative andanticomultiplicative in the sense ∆S = (S⊗S)flip∆.

3. Let u, v generate S3 with u2 = v2 = e and uvu = vuv (denoted w). Show thatΩ(S3) defined by u, v, w in Example 3.13 has dimensions 1 : 3 : 4 : 3 : 1.

4. Show that D for the charge 1 q-monopole in Example 5.4 is isomorphic to theGrassmann connection in Proposition 4.2 for the projector in Example 4.5.

5. In Ω(S3) with g =∑a ea⊗A ea, show that ∇eu = −eu⊗A eu − ev ⊗A ew −

ew ⊗A ev + 13θ⊗A θ is torsion free and cotorsion free. Here θ =

∑a ea.

Solutions

1. If x in the C∗-algebra is nilpotent then x∗x is also nilpotent and self-adjoint.So it is enough to show that there are no self-adjoint nilpotents. If y is self-adjointwith y2 = 0 then 0 = ||y2|| = ||y∗y|| = ||y||2 and hence y = 0. If x is self-adjointand xn = 0, xn−1 6= 0 and n > 2, let y = xn−1. Then y is self-adjoint and y2 = 0hence y = 0, which is a contradiction.

2. If S′ is another antipode then S′a = ε(a(1))S′a(2) = (Sa(1)(1))a(1)(2)S

′a(2) =(Sa(1))a(2)(1)(S

′a(2)(2)) = (Sa(1))ε(a(2)) = Sa. For S(ab) start by proving the identity(S(a(1)b(1)))a(2)b(2)⊗ b(3)⊗ a(3) = 1⊗ b⊗ a, then apply S in the middle and multiply

28 S. MAJID

the first two factors to obtain (S(a(1)b))a(2)⊗ a(3) = Sb⊗ a. Now apply S to thesecond factor and multiply. Deduce the last part by arrow-reversal of the precedingpart.

3. We let eu = ω(δu), etc., then Ψ(eu⊗ eu) = eu⊗ eu, Ψ(eu⊗ ev) = ew ⊗ eu etc.It follows that e2

u = 0 etc., and euev + evew + eweu = 0, eveu + euew + ewev = 0as these elements are Ψ-invariant in the tensor square so vanish under the wedgeproduct, and are a basis of such elements. Hence Λ2 is 9-5=4-dimensional. Usingthese relations one then finds three independent 3-forms in Λ3 and one top-formeueveuew, say. This is a finite-set calculus as in Example 3.3 for the Cayley graphof the generators.

4. We have Dd = dd+ q−1de0 = ce− and similarly Db = ae− using the formula inExample 5.4 and the relations of the 3D calculus in Example 3.14. Also from thesewe compute the differentials on A = Cq[S2

C] as

dz = c2e− + d2e+, dw = −q(a2e− + b2e+), dx = −ace− − bde+

and solve for Dd,Db in terms of these. Also factorise the projector e, so

e =

(d−b

)(a q−1c

), D

(d−b

)= de⊗

(d−b

)from which it is clear that u = fd − gb 7→ (f, g)e = (ua, uq−1c) for f, g ∈ A is therequired isomorphism E1 → A2e.

5. Here ∧∇eu = −evew − ewev = deu etc., by the Maurer-Cartan equations. Forthe cotorsion we apply wedge to the first two factors of (id⊗∇)(g) and comparewith (d⊗ id)g using the relations from Solution 3.

Further Reading

These are suggestions for further reading and not a bibliography; please seereferences therein and the extensive research literature.

[1] A. Connes, Noncommutative Geometry. Academic Press (1994).[2] S. Majid, Foundations of Quantum Group Theory. Camb. Univ. Press (1995).

[3] S. Majid, A Quantum Groups Primer. vol. 292, L.M.S. Lect. Notes, Camb. Univ. Press (2002).

[4] S. Woronowicz, Differential calculus on compact matrix pseudogroups (quantum groups), Com-mun. Math. Phys. 122, 125–170 (1989).

[5] S. Majid, Noncommutative riemannian and spin geometry of the standard q-sphere, Commun.

Math. Phys. 256, 255–285 (2005).[6] E. Beggs and S. Majid, Gravity induced from quantum spacetime, Class. Quantum. Grav. 31,

035020 (39pp) (2014).

School of Mathematical Sciences, Queen Mary University of London, 327 Mile EndRd, London E1 4NS, s [email protected]

LTCC LECTURES ON NONCOMMUTATIVE DIFFERENTIALmajid/LTCCnotes2016.pdf · Noncommutative geometry is...

Documents

Transcript of LTCC LECTURES ON NONCOMMUTATIVE DIFFERENTIALmajid/LTCCnotes2016.pdf · Noncommutative geometry is...