Nonassociative Riemannian geometry by twisting

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Nonassociative Riemannian geometry by twistingTo cite this article Edwin Beggs and Shahn Majid 2010 J Phys Conf Ser 254 012002

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Edwin J Beggs and Shahn Majid-

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Nonassociative Riemannian geometry by twisting

Edwin Beggs1 and Shahn Majid2

1 Department of Mathematics Swansea UniversitySingleton Parc Swansea SA2 8PP UK2 School of Mathematical Sciences Queen Mary University of London327 Mile End Rd London E1 4NS

E-mail ejbeggsswanseaacuk smajidqmulacuk

Abstract Many quantum groups and quantum spaces of interest can be obtained by cochain(but not cocycle) twist from their corresponding classical object This failure of the cocycle con-dition implies a hidden nonassociativity in the noncommutative geometry already known to bevisible at the level of differential forms We extend the cochain twist framework to connectionsand Riemannian structures and provide examples including twist of the S7 coordinate algebrato a nonassociative hyperbolic geometry in the same category as that of the octonions

Received 17 November 2009

1 IntroductionThe idea of noncommutative geometry of course is to replace points in an actual space by acoordinate algebra These are generally generated by variables x y z say enjoying algebraicproperties for addition multiplication and other relations paralleling the numbers for which theycould be viewed as placeholders Thus solving x2 + y2 + z2 = 1 over R would bring one to theactual points of a sphere but one could also consider this equation more abstractly with x y zas generators of an algebra And if in place of the usual xy = yx etc we have some other non-commutation relations then clearly these generators could never be realised as numbers Overthe last two decades it has become obvious that the assumption of commutativity is a historicalaccident to do with the fact that classical mechanics was discovered before quantum mechanicsand there is no particular reason certainly from a mathematical perspective not to allownoncommutavity as a generalisation of usual geometry more applicable to the quantum worldThere are also physical reasons to think that plausibly the first quantum gravity corrections togeometry should be expressed as such noncommutativity [21] Accordingly a great deal of efforthas been put into developing such a geometry both in Connesrsquo lsquospectral triplersquo approach and ina lsquoquantum groupsrsquo approach modelled around quantum groups such as the deformations Cq(G)and their homogeneous spaces in the first instance Both approaches are by now somewhatmature and the quantum groups approach includes specific testable predictions for Planck scalephysics [4] Meanwhile the spectral triples approach includes specific predictions and massrelations for the standard model [11]

In this article we go one step further and consider geometry that is nonassociative in thesame algebraic sense While noncommutative geometry is motivated from quantum theory andhence somewhat familiar this is not so for nonassociativity If we simply drop associativity then

Quantum ndash a Festschrift for Tony Sudbery IOP PublishingJournal of Physics Conference Series 254 (2010) 012002 doi1010881742-65962541012002

ccopy 2010 IOP Publishing Ltd 1

most of what we do in algebra and geometry becomes vastly more complicated if not impossibleTherefore the first point is that we should only approach nonassociative geometry under duressHowever common to many approaches to noncommutative geometry is the idea of replacingdifferential structures by an algebraic formulation of lsquoexterior algebra of differential formsrsquo withgenerators dx dydz say formalized as an associative differential graded algebra (Ω d) overour possibly noncommutative lsquocoordinate algebrarsquo This is pretty much a prerequisite to doany kind of physical model building on noncommutative geometry wave equations Maxwellrsquosequations etc Unfortunately after about 20 years experience with model-building it turnedout to be not quite possible to do this while maintaining a strict correspondence with classicaldifferentials and classical symmetries When quantization fails in this respect one speaks of aquantum anomaly and in previous work we have similarly noted a fundamental anomaly orobstruction in the entire programme of noncommutative geometry

Theorem 11 [5] The standard quantum groups Cq(G) of simple Lie algebras g do not admit anassociative differential calculus Ω(Cq(G)) in deformation theory with left and right translation-covariance and classical dimensions

We showed that this obstruction like other anomalies in physics can be expressed at thesemiclassical level as curvature in our case of a certain Poisson preconnection and that for gsemisimple there is no flat connection of the required type The same applies to envelopingalgebras U(g) when g is semisimple viewed as quantisation of glowast with its Kirillov-KostantPoisson bracket

Theorem 12 [6] The classical enveloping algebras U(g) of simple Lie algebras g do not admitan associative differential calculus Ω(U(g)) in deformation theory with ad-invariance under gand classical dimensions

This was again proven using similar methods notably Kostantrsquos invariant theory We notethat an important first attempt at such a rigidity result was in [13] which in particular pointedto the role of preconnections Moreover attempts at such calculi at the quantum group levelhad been found to require extra dimensions in the cotangent bundle and this was understoodnow as a way to lsquoneutralisersquo the anomaly again much as for other anomalies in physics On theother hand [5 6] also provided an alternative using cochain twisting methods it was shownthat we can always keep classical dimensions and deform the classical picture provided we allownonassociative geometry In short if one wants a strict deformation-theoretic correspondenceon quantization we have

NCGrArr NAG

(noncommutative geometry implies nonassociative geometry) In the present paper we explorethis radical alternative further And if we want to be speculative then once we allownonassociative exterior algebras we should also allow nonassociative coordinate algebras iespaces themselves not only their differential geometry could be nonassociative

In entering this nonassociative world we also need tight controls The key idea is to work ina lsquoquasiassociativersquo setting in which coordinate algebras are nonassociative but in a controlledway by means of a multiplicative associator In mathematical terms this means that the algebrais associative but in some monoidal category different from that of vector spaces A theorem ofMaclane says that all constructions in such a category can be done as if associative ie withoutworrying about brackets One must then insert the associator ΦXYZ Xotimes(YotimesZ)rarr (XotimesY )otimesZbetween any three objects as needed in order to make sense of expressions and Maclanersquos theoremsays that in a monoidal category any different ways to insert Φ as needed will give the same resultWe focus in this article on a specific lsquotwist quantisation functorrsquo that results in noncommutativealgebras in such a monoidal category and also quantises all other covariant structures withreference to a classical symmetry


2

To first explain the background VG Drinfeld [12] showed that all quantum group envelopingalgebras Uq(g) could be obtained as follows start with the classical symmetry algebra U(g) but

viewed as a quasi-Hopf algebra with respect to a certain element φ isin U(g)otimes3 (a topologicaltensor product) obtained by solving the KZ equations Its category of representations is abraided monoidal category with Φ given by the action of φ (and braiding given by the action

of R = eht where q = eh2 and t is the split Casimir of g) Drinfeld showed that there exists a

certain cochain F isin U(g)otimes2 such that conjugating the classical coproduct of U(g) by F givesUq(g) Here φ also twists to some φF but F is chosen so that φF = 1 ndash an ordinary Hopfalgebra In [3 5] we developed a rather different application of the same data namely startwith completely classical data (U(g) φ = 1R = 1) and twist this by Drinfeldrsquos F (conjugatethe classical coproduct) The result U(g)F looks like Uq(g) but regarded as a quasi-Hopf algebrawith φF closely related to Drinfeldrsquos and RF = F21F

minus1 making its category of representationsa (symmetric) monoidal category There is also an important spin-off [15] if the classical objectU(g) acts on a classical coordinate algebra C(M) then because of the functorial nature of thetwist construction C(M) also gets lsquoquantizedrsquo to an algebra C(M)F typically noncommutativeand nonassociative because it lives in the category of representation of U(g)F Viewed in thatcategory it is in fact commutative and associative but the category is no longer the usual one dueto a nontrivial Φ Similarly the classical Ω(M) gets quantised to Ω(M)F as a calculus on C(M)F In this way any classical geometry can be systematically quantised with respect to a classicalsymmetry and a choice of cochain F ndash provided we can live with potential nonassociativity

Note that in geometry we are interested not in enveloping algebras but in coordinate algebrasTherefore while not essential we will convert the above to work with classical C(G) and withCq(G) viewed as a coquasiHopf algebra C(G)F and we will replace the action of U(g) on C(M)by a coaction of C(G) In particular G acts (C(G) coacts) on C(G) by left translation andinduces a cochain quantisation C(G)F covariant under C(G)F [3 6] We recall this less familiarsetting in the Preliminaries below Depending on the choice of cochain some of the twistedalgebras may remian accidentally associative ndash this would be the case for Drinfeldrsquos cochain andC(G)F ndash and in that case the nonassociativity is hidden But it is still present and appearstypically in the differential calculus even in this case This programme has already been carriedquite far and covers the general principles and the calculus while in the present article we nowstudy how the next main layers of geometry connections and curvature etc behave under suchcochain twists The short answer is that everything works as expected if everything is covarantunder the twisting classical symmetry This also points to the limitations of the twist approachit works too well ndash different but covariant ways to express classical constructions will all twistand only by having a general theory of noncommutative (and now nonassociative) geometry thatmakes sense beyond examples given by twisting will we be able to know which of these is mostnatural We will illustrate this with the Ricci tensor in Section 7 One could consider metricsfor example that are not invariant under the classical symmetry although we do not do so here

Finally for nonassociative geometry we need concrete examples to build up our experienceAbout the only example that most readers may have some experience with is the octonionsand these provided the first concrete example of the above lsquocochain quantisationrsquo in action [2]The octonions are a dimension 8 algebra but with a nonassociative product which is howeveralternative in the sense that if two elements are repeated then they associate so x(yx) = (xy)xetc The full picture coming out of quantum group theory amounts to the following One canchoose a basis ea labelled by a isin Z3

2 (a 3-vector with values in Z2) such that [2]

(eaeb)ec = φ(a b c)ea(ebec) φ(a b c) = (minus1)amiddot(btimesc) = (minus1)|abc|

This means that three elements associate if and only if their labels are linearly dependentas vectors over Z2 otherwise there is a minus1 This was obtained in [2] precisely as a cochain


3

quantization C(Z32)F of C(Z3

2) under the action of Z32 by left translation and a certain choice

of F One can then proceed to differential geometry on the octonions as coordinate algebra[16] Our reinterpretation of Drinfeldrsquos ideas to produce C(G)F is inspired by this and exactlyparallels the construction of the octonions but now for a Lie group The octonion examplealso illustrates that even though twisting is lsquoeasyrsquo in the sense that everything is somewhatautomatic by an equivalence of categories reflected in the fact that φ is cohomologically trivialthis does not mean that the resulting objects are uninteresting The octonions for exampleare a division algebra while the group algebra of Z3

2 is certainly not At the same time oncewe adopt our framework of geometry in monoidal categories we are not limited to such twistor coboundary examples but can consider more general ones It would also be interesting toapply nonassociative spectral triples to the standard model for example based now on octonionsrather than the two copies of the quaternions for the lsquointernal geometryrsquo in [11]

Note added Some of Tony Sudberyrsquos work was towards the noncommuative geometry comingout of quantum groups and it is therefore a pleasure to contribute the present article to thisFestschrift volume There is also a degree of irony In the late 1990s Tony left quantum groupsto work on the seemingly unrelated area of octonions and other topics Meanwhile those whostayed behind to work purely on quantum groups and their noncommutative geometry are nowlearning lessons on how to do it from the octonions

2 Preliminaries cochain twistsWe work over a field k but for physics we will often specialise this to C or R We will need to usequantum group or Hopf algebra methods and we refer to [15] for an introduction We recall thata quantum group H is a bialgebra equipped with a notion of lsquolinearised inversersquo or lsquoantipodersquoS H rarr H Here a bialgebra means that the algebra H is also a coalgebra in a compatibleway The concept of a coalgebra is just the same as the concept of an algebra but with arrowsreversed so there is a lsquocoproductrsquo ∆ H rarr HotimesH and lsquocounitrsquo H rarr k and we require ∆ (andhence ε) to be algebra homomorphisms There is a standard (Sweedler) notation ∆h = h(1)otimesh(2)

which we shall use for the result of ∆ as an element of HotimesH (summation understood)Suppose that H is a coalgebra We consider its category of left comodules HM and we write

the coaction on an object V as λ V rarr HotimesV or as v 7rarr v[minus1]otimesv[0] with summation understoodThe concept of a comodule here is just the same as that of a module but with arrows reversedIf H is a Hopf algebra we can use its algebra product to define a tensor product on HM by

λ(votimesw) = v[minus1]w[minus1]otimes(v[0]otimesw[0])

The identity in the category is just the underlying field k with λ k rarr Hotimesk being 1HotimesidNote that at this stage a bialgebra structure on H would suffice but it shall be convenient tohave a Hopf algebra later An algebra in the category HM is an object A with an associativeproduct micro AotimesArarr A and a unit 1A k rarr A and which are both H-comodule maps

If G is a finite or algebraic group we are interested in the point of view where H could beits function or coordinate algebra k(G) with its usual pointwise product If similarly A = k(X)is the algebra of functions on some set or algebraic variety X and G acts on X this would beexpressed in our algebraic terms as a coaction λ Ararr HotimesA For example in the finite groupcase

λ(f)(x) =sumgisinG

δg f(gx)

where δg is the Kronecker δ-function It may seem strange to do everything lsquobackwardsrsquo in termsof coactions but this will allow us to work with algebraic tensor products and algebraic groups


4

Our framework also allows A and H to be noncommutative in which case there would be noactual spaces XG respectively

Now we shall consider a different tensor product otimesF for HM We shall do this by changingthe product on H while leaving the coproduct unaltered As the coproduct is the same we canconsider the category HM to be the same we have only changed the tensor product We beginwith a linear map F HotimesH rarr k which is invertible in the sense that there is another mapFminus HotimesH rarr k so that

F (a(1)otimesb(1))Fminus(a(2)otimesb(2)) = Fminus(a(1)otimesb(1))F (a(2)otimesb(2)) = ε(a) ε(b)

and which obeys F (aotimes1) = F (1otimesa) = ε(a) Such a map is called a 2-cochain on H [15] Nowwe specify the tensor product otimesF on HM by the same vector space VotimesW as usual but withcoaction

λ(votimesFw) = v[minus1] w[minus1]otimes(v[0]otimesFw[0])

where the modified product HotimesH rarr H is given in terms of the original product by

a b = F (a(1)otimesb(1)) a(2)b(2) Fminus(a(3)otimesb(3))

This product may not be associative ndash in general HF = (H ) is a coquasi-Hopf algebra [15]but this is not a problem in defining otimesF Where it does appear as a complication is that thetrivial lsquoidentityrsquo map from (UotimesFV )otimesFW rarr UotimesF (VotimesFW ) is not a morphism Instead we canuse the nontrivial associator ΦUVW (UotimesFV )otimesFW rarr UotimesF (VotimesFW ) defined by

Φ((uotimesF v)otimesFw) = (partF )minus(u[minus1]otimesv[minus1]otimesw[minus1]) u[0]otimesF (v[0]otimesFw[0])

Φminus1(uotimesF (votimesFw)) = (partF )(u[minus1]otimesv[minus1]otimesw[minus1]) (u[0]otimesF v[0])otimesFw[0]

where partF (partF )minus HotimesHotimesH rarr k is defined by

(partF )minus(aotimesbotimesc) = F (a(1)otimesb(1))F (a(2)b(2)otimesc(1))Fminus(a(3)otimesb(3)c(2))F

minus(b(4)otimesc(3))

(partF )(aotimesbotimesc) = F (b(1)otimesc(1))F (a(1)otimesb(2)c(2))Fminus(a(2)b(3)otimesc(3))F

minus(a(3)otimesb(4))

It would be customary here to restrict to the case where F is a cocycle ie (partF )(aotimesbotimesc) =ε(a)ε(b)ε(c) This means that the new tensor product would also be trivially associated Howeverit is not really necessary to suppose this if the reader does not mind dealing with nontriviallyassociated tensor categories and we shall not do so

It will be useful to note that there is a natural transformation c between the two tensorproduct functors otimes and otimesF from HMtimes HMrarr HM given by

c(votimesw) = Fminus(v[minus1]otimesw[minus1]) v[0]otimesFw[0] (1)

which together with the identity map on objects and morphisms provids an equivalence ofmonoidal categories (HMotimes) and (HMotimesF ) The latter is the category of comodules ofHF = (H )

3 Twisting algebras and modulesAn algebra (Amicro) in the tensor category (HMotimes) is twisted to an algebra AF = (AmicroF ) in(HMotimesF ) by

microF (aotimesb) = a bull b = F (a[minus1]otimesb[minus1]) a[0]b[0] (2)


5

A quick check will show that this is associative in (HMotimesF ) remembering to change the orderof the bracketing using Φ Thus

(a bull b) bull c = (partF )minus(a[minus1]otimesb[minus1]otimesc[minus1]) a[0] bull (b[0] bull c[0]) (3)

We can also twist left or right A-modules V in (HMotimes) (meaning that the left or rightactions have to preserve the H-coaction) to get left or right modules of (AmicroF ) by

aF v = F (a[minus1]otimesv[minus1]) a[0]v[0] vFa = F (v[minus1]otimesa[minus1]) v[0]a[0]

for v isin V and a isin A and where we use and for the left and right actions respectively Theseobey the requirements of an action of (AmicroF ) in the category with otimesF ie inserting Φ in theform partFminus much as in (3) When we have both left and right actions on V we can require themto commute so (av)b = a(vb) We say that V isin AMA the category of bimodules If inaddition our bimodule is left covariant under H then twisting both actions gives us a bimoduleof (AmicroF ) with respect to otimesF ie

(aF v)F b = (partF )minus(a[minus1]otimesv[minus1]otimesb[minus1]) a[0]F (v[0]

F b[0])

This all works in the same way as for the product microF

For A-bimodules V and W we can quotient the tensor product VotimesW to get the tensorproduct VotimesAW over the algebra The vector space which we quotient by is spanned byeverything of the form vaotimesw minus votimesaw When our bimodules are in (HMotimesF ) we cananalogously quotient the vector space VotimesFW to give VotimesAFW by identifying the elementsgiven by going the two ways round the following diagram

VotimesF (AotimesFW )idotimesF F

Φminus1 ((QQQQQQQQQQQQQ VotimesFW

(VotimesFA)otimesFWFotimesF id

77ooooooooooo

(4)

In fact this gives the same vector space VotimesAW as in the untwisted case The easiest way to seethis is that the natural transformation c between the two tensor product functors otimes and otimesF(see equation (1)) sends elements of the form vaotimesw minus votimesaw exactly to the required relationsfor VotimesAF

W given by (4) Thus performing the quotient of vector spaces gives an isomorphismin the category

Now VotimesAW also has a standard A-bimodule structure given by

a(votimesw) = (av)otimesw (votimesw)a = votimes(wa)

Corresponding to this we have left and right actions of (AmicroF ) on VotimesAFW which are given at

the level of VotimesFW by the following compositions

AotimesF (VotimesFW )Φminus1

minusrarr (AotimesFV )otimesFW Fotimesidminusrarr VotimesFW

(VotimesFW )otimesA Φminusrarr VotimesF (WotimesFA)idotimesFminusrarr VotimesFW

The reader can check that these quotient to well defined maps on VotimesAFW These constructions

make sense for algebras and their bimodules in any k-linear monoidal category not just(HMotimesF ) and work the same way as for usual algebras due to Mac Lanersquos coherence theorem


6

ndash we need only express usual linear algebra constructions as compositions of maps and use thesame in the monoidal category with the relevant Φ inserted to make sense

It will be useful to note that the definition of the left action on the tensor product (a similarpicture holds for right) is made so that the following diagram commutes

VotimesW

c

AotimesVotimesW cAVotimesW

otimesidoo AotimesF (VotimesW )

idotimesF c

VotimesFW AotimesF (VotimesFW )Foo

(5)

This just says that F is the image of under the functorial equivalence from the category withotimes to the category with otimesF When all maps are lsquoquantizedrsquo by this equivalence then any equationexpressed as a commutative diagram between maps will map over to an analogous commutativediagram on the otimesF side This is similarly the abstract origin of the results in this paper

Such cochain twist ideas were applied to the octonions with H = A = k(Z32)sim=kZ3

2 and λ = ∆the lsquoleft regular coactionrsquo of any group on itself We use the second lsquogroup algebrarsquo form wherethe algebra is spanned by basis elements ea where a isin Z3

2 The cochain F and the twistedproduct of AF are

F (eaotimeseb) = (minus1)sum

ilej aibj+a1a2b3+a1b2a3+b1a2a3 ea bull eb = F (eaotimeseb)ea+b (6)

The product of H is not at it happens changed (it remains associative) but this is largelyan accident and its tensor category of comodules otimesF is nevertheless modified Technically itbecomes a cotriangular coquasiHopf algebra which happens to be a Hopf algebra The categorywith otimesF is a symmetric monodal one and in this category the octonions are both associativeand commutative Similarly if C(G) is the algebra of functions for a finite or algebraic groupand F any choice of cochain on it we have C(G)F with product which can happen to remainassociative (eg in the case of Drinfeldrsquos F ) We have a canonical coaction of C(G) on itselfby λ = ∆ and hence a typically nonassociative algebra C(G)F with the induced bull productNote the exact parallel with the octonion construction They are both examples of systematicquantisation by cochain twist

4 Twisting of -structuresWe now proceed to study the different layers of geometry but allowing our lsquocoordinate algebrasrsquoA to be potentially nonassociative by working in a monoidal category The first item pertainsto the right concept of real form or lowast-structure In noncommutative geometry and in physicsthis specifies which elements are to be represented as Hermitian or unitary in the sense that lowastis to be respected A categorical approch to this was introduced in [7] with the notion of a lsquostarobjectrsquo in a bar category We are only going to discuss it in the case relevant to cochain twistsand we refer the reader to [7] for the formal category theory

Thus we take H to be a lowast-Hopf algebra ie equipped with a conjugate-linear involutionh 7rarr hlowast on the algebra and required also to be a coalgebra map and order-reversing for theproduct Then for every object V in the comodule category HM we have another V As a setV is the same as V but we distinguish between elements by writing v isin V for v isin V Then theleft H-coaction is given by

λ(v) = (v[minus1])lowastotimesv[0]


7

We have natural isomorphisms Υ VotimesW rarrWotimesV and bb V rarr V defined by

Υ(votimesw) = wotimesv bb(v) = v

This provides the bar category associated to a Hopf lowast-algebra as a somewhat formal way ofspeaking about conjugate corepresentations

We now twist this construction by altering the map Υ to ΥF VotimesFW rarrWotimesFV as follows

ΥF (votimesFw) = F (v[minus2]otimesw[minus2])lowast Fminus(w[minus1]

lowastotimesv[minus1]lowast) w[0]otimesF v[0]

ΥFminus1(votimesFw) = F (v[minus2]lowastotimesw[minus2]

lowast) Fminus(w[minus1]otimesv[minus1])lowast w[0]otimesF v[0]

These formulae are made to make the following diagram commute

VotimesW Υ

c

WotimesVc

VotimesFW

ΥF WotimesFV

To check that this works we calculate

ΥF (ΥF (votimesFw)) = F (v[minus2]otimesw[minus2]) Fminus(w[minus1]

lowastotimesv[minus1]lowast)lowast ΥF (w[0]otimesF v[0])

= F (v[minus4]otimesw[minus4]) Fminus(w[minus3]

lowastotimesv[minus3]lowast)lowast F (w[minus2]

lowastotimesv[minus2]lowast)lowast

Fminus(v[minus1]otimesw[minus1]) v[0]otimesFw[0]

= votimesFw

which is one of the axioms of a bar category In this way we arrive at (HMotimesF ) as another barcategory [7] It can be viewed as the category of comodules of HF as a certain lowast-coquasi-Hopfalgebra

Next as in [7] we think of an H-comodule equipped with an antilinear involution categoricallyas a star object meaning V in HM equipped with a morphism V V rarr V so that V V = bbV We shall write v = vlowast to relate it to the usual lowast as an antilinear involution on a vector spaceV As V V rarr V is a morphism (ie a right H-comodule map) we deduce that

vlowast[minus1]otimesvlowast[0] = v[minus1]otimesv[0]lowast

We can then define a star algebra A in (HMotimes) as a star object A which has an associativeproduct micro(aotimesb) = ab so that microΥminus1(otimes) = micro which is a rather formal way to say (ab)lowast = blowastalowastPutting our notions this way of course now makes sense in any bar category

Proposition 41 If A is a star algebra (Amicro) in (HMotimes) then its cochain twist (AmicroF ) obeys

microF ΥFminus1(otimesF ) = microF and so is a star algebra in (HMotimesF ) Here the operation itself is notdeformed

Proof By the definitions

ΥFminus1(otimesF )(aotimesF b) = ΥFminus1(alowastotimesF blowast)= F (a[minus2]otimesb[minus2]) F

minus(b[minus1]lowastotimesa[minus1]

lowast)lowast b[0]lowastotimesFa[0]

lowast

microF ΥFminus1(otimesF )(aotimesF b) = F (a[minus1]otimesb[minus1]) b[0]lowast a[0]

lowast

= F (a[minus1]otimesb[minus1]) (a[0]b[0])lowast


8

= microF (aotimesF b)

Of interest in this construction is the relation between and the tensor product That isgiven that V and W are star objects is VotimesW a star object To simplify this we assume thatH is commutative and then we define (votimesw)lowast = vlowastotimeswlowast We define Γ VotimesW rarr WotimesV to be(minus1W otimes

minus1V ) Υ VotimesW which merely recovers usual transposition in the case of (HMotimes) In the

twisted case we define

(votimesFw) = F (v[minus2]otimesw[minus2]) Fminus(v[minus1]

lowastotimesw[minus1]lowast)lowast v[0]

lowastotimesFw[0]lowast (7)

and this corresponds to

ΓF (votimesFw) = F (v[minus2]otimesw[minus2]) Fminus(w[minus1]otimesv[minus1]) w[0]otimesF v[0]

This is a generalised braiding associated to star objects and in some cases it obeys the braidrelations [7]

The reader might want a notion of real elements that is preserved under tensor product sothat the tensor product of two real elements is real So if we have v = vlowast and w = wlowast thenwe would like (votimesFw)lowast = votimesFw From (7) we can see how to ensure that this happens we caninsist on the following extra condition on the cochain

F (aotimesb)lowast = F (alowastotimesblowast) (8)

5 Twisting differential calculi and connectionsSuppose that (ΩAdand) is a differential graded algebra in HM Here the exterior algebra is adirect sum of different degrees of differential forms with Ω0A = A and there is a wedge productand and an exterior derivative d ΩnA rarr Ωn+1A obeying d2 = 0 and a graded Leibniz rule Wesuppose that H coacts on A and that this coaction extends to differential forms with d and andrespecting the coaction or put another way that all structure maps live in HM

The differential calculus is twisted to (ΩA dandF ) where d is unchanged and

ξ andF η = F (ξ[minus1]otimesη[minus1]) ξ[0] and η[0]

We do not have to do any new work here we just apply the cochain twist machinery of Section 2to the algebra ΩA with its wedge product [3] cf [20]

Once we have a differential calculus we can define the notion of a connection or covariantderivative For any usual associative algebra A this is given by a map nabla V rarr Ω1AotimesAV whereV is a left A-module (more properly a projective module for a vector bundle in noncommutativegeometry) and required to obey

nabla(av) = daotimesAv + anabla(v) (9)

Its curvature is defined as a map Rnabla V rarr Ω2AotimesAV defined by

Rnabla = (dotimesidminus id andnabla)nabla = nabla[1]nabla (10)

where the extension to higher forms is provided by

nabla[n] = dotimesid + (minus1)nid andnabla ΩnAotimesAV rarr Ωn+1otimesAV (11)


9

Proposition 51 Given a covariant derivative nabla on a left A-module V which is also a leftH-comodule map

nablaF (v) = Fminus(ξ[minus1]otimesw[minus1]) ξ[0]otimesFw[0]

where nabla(v) = ξotimesw (or a sum of such terms) provides a well-defined covariant derivativenablaF V rarr Ω1AFotimesAF

V with V twisted to an AF -module as in Section 2

Proof We can say loosely that nablaF = c nabla or that nablaF is defined by commutativity of

VnablaF

nabla

Ω1AotimesFV

Ω1AotimesVc

88qqqqqqqqqq

(12)

but what we mean more precisely is a choice of representatives for the result of nabla allowing usto work in Ω1AotimesV followed by a check that nablaF then descends to otimesAF

if nabla descends to otimesA Abrief check will further show that this is a morphism in HM To check the left Liebniz rule webegin by writing for v isin V and b isin A

nablav = ξotimesw v[minus1]otimesnablav[0] = ξ[minus1]w[minus1]otimesξ[0]otimesw[0]

v[minus1]otimesnabla(bv[0]) = ξ[minus1]w[minus1]otimesbξ[0]otimesw[0] + v[minus1]otimesdbotimesv[0]

v[minus1]otimesnablaF (bv[0]) = Fminus(b[minus1]ξ[minus1]otimesw[minus1]) ξ[minus2]w[minus2]otimes(b[0]ξ[0]otimesFw[0])

+Fminus(b[minus1]otimesv[minus1]) v[minus2]otimes(db[0]otimesF v[0])

Now we calculate using the last equation and writing bull = F for clarity

nablaF (a bull v) = F (a[minus1]otimesv[minus1])nablaF (a[0]v[0])

= F (a[minus2]otimesξ[minus2]w[minus2])Fminus(a[minus1]ξ[minus1]otimesw[minus1]) (a[0]ξ[0]otimesFw[0])

+F (a[minus2]otimesv[minus2])Fminus(a[minus1]otimesv[minus1]) (da[0]otimesF v[0])

= F (a[minus2]otimesξ[minus2]w[minus2])Fminus(a[minus1]ξ[minus1]otimesw[minus1]) (a[0]ξ[0]otimesFw[0]) + daotimesF v

We now recognise the first term as

a bull nablaF v = Fminus(ξ[minus1]otimesw[minus1]) a bull (ξ[0]otimesFw[0])

= Fminus(ξ[minus2]otimesw[minus2]) (partF )(a[minus1]otimesξ[minus1]otimesw[minus1]) (a[0] bull ξ[0])otimesFw[0]

= Fminus(ξ[minus3]otimesw[minus2]) (partF )(a[minus2]otimesξ[minus2]otimesw[minus1]) F (a[minus1]otimesξ[minus1]) (a[0]ξ[0])otimesFw[0]


We next check what happens to maps which preserve the covariant derivatives between leftmodules That is given left modules and covariant derivatives (VnablaV ) and (UnablaU ) we havea left module map θ V rarr U for which (idotimesθ)nablaV = nablaU θ V rarr Ω1AotimesAU If we alsosuppose that θ V rarr U is an H-comodule map then it is immediate from Proposition 51 that(idotimesF θ)nablaFV = nablaFU θ V rarr Ω1AotimesAF

U We likewise extend the covariant derivative to maps

nablaF [n] = dotimesF id + (minus1)nid andnablaF ΩnAFotimesAFV rarr Ωn+1AFotimesAF

V

where we remember that we have to use the associator to calculate idandnablaF Then the curvatureis defined analogously to before as

RnablaF = nablaF [1]nablaF V rarr Ω2AFotimesAFV


10

Proposition 52 If the curvature of the covariant derivative (Vnabla) is given by Rnabla(v) = ωotimesw isinΩ2AotimesAV (or a sum of such terms) then the curvature of the corresponding connection (VnablaF )is given by

RnablaF (v) = Fminus(ω[minus1]otimesw[minus1]) ω[0]otimesFw[0]

Proof We compute the curvature taking nablav = ξotimesw as

nablaF [1]nablaF v = Fminus(ξ[minus1]otimesw[minus1]) nablaF [1](ξ[0]otimesFw[0])

= Fminus(ξ[minus1]otimesw[minus1]) dξ[0]otimesFw[0] minus Fminus(ξ[minus1]otimesw[minus1]) ξ[0] andF nablaFw[0]

Now we set nablaw = ηotimesu and

w[minus1]otimesnablaF (w[0]) = Fminus(η[minus1]otimesu[minus1]) η[minus2]u[minus2]otimes(η[0]otimesFu[0])

Now we calculate

nablaF [1]nablaF v = Fminus(ξ[minus1]otimesw[minus1]) dξ[0]otimesFw[0]

minus Fminus(ξ[minus1]otimesη[minus2]u[minus2]) Fminus(η[minus1]otimesu[minus1]) ξ[0] andF (η[0]otimesFu[0])

= Fminus(ξ[minus1]otimesw[minus1]) dξ[0]otimesFw[0]

minus Fminus(ξ[minus2]otimesη[minus3]u[minus3]) Fminus(η[minus2]otimesu[minus2]) (partF )(ξ[minus1]otimesη[minus1]otimesu[minus1])(ξ[0] andF η[0])otimesFu[0]

= Fminus(ξ[minus1]otimesw[minus1]) dξ[0]otimesFw[0] minus Fminus(ξ[minus1]η[minus1]otimesu[minus1]) (ξ[0] and η[0])otimesFu[0]

We compare this with the straight computation of Rnabla(v) and change notations to the form inthe statement We can say loosely that RnablaF = c Rnabla

6 Bimodule covariant derivativesIn applying these ideas to Riemannan geometry we will be in the case of V = Ω1A whichis a bimodule When V is a bimodule we may have a well-defined generalised braidingσ VotimesAΩ1Ararr Ω1AotimesAV such that

nabla(va) = (nablav)a+ σ(votimesAda)

A covariant derivative on a bimodule equipped with such an object if it exists is called abimodule covariant derivative We already saw in Section 2 how to twist anH-covariant bimoduleto a bimodule of (AmicroF ) We suppose that nabla and σ are also H-covariant ie expressed in thecategory HM

Proposition 61 If (Vnabla σ) is an H-covariant bimodule covariant derivative then (VnablaF σF )is a bimodule covariant derivative for (AmicroF ) where (writing σ(v[0]otimesξ[0]) = ηotimesw or a sum ofsuch representatives)

σF (votimesF ξ) = F (v[minus1]otimesξ[minus1]) Fminus(η[minus1]otimesw[minus1])η[0]otimesFw[0]

descends to the quotients as required

Proof One should either work with otimesA and otimesAFor representatives with otimes and otimesF and an

eventual quotient understood By definition

σF (votimesFda) = nablaF (vFa)minusnablaF (v)Fa= F (v[minus1]otimesa[minus1]) nablaF (v[0] a[0])minusnablaF (v)Fa


11

If we write nablav = ξotimesw then v[minus1]otimesnablav[0] = ξ[minus1]w[minus1]otimesξ[0]otimesw[0] so

σF (votimesFda) = F (v[minus1]otimesa[minus1]) nablaF (v[0] a[0])minus Fminus(ξ[minus1]otimesw[minus1]) (ξ[0]otimesFw[0])Fa

Now we use

v[minus1]otimesa[minus1]otimesnabla(v[0]a[0]) = v[minus1]otimesa[minus1]otimesnabla(v[0])a[0] + v[minus1]otimesa[minus1]otimesσ(v[0]otimesda[0])= ξ[minus1]w[minus1]otimesa[minus1]otimesξ[0]otimesw[0]a[0] + v[minus1]otimesa[minus1]otimesσ(v[0]otimesda[0])

v[minus1]otimesa[minus1]otimesnablaF (v[0]a[0]) = ξ[minus2]w[minus2]otimesa[minus2]otimesFminus(ξ[minus1]otimesw[minus1]a[minus1]) (ξ[0]otimesFw[0]a[0])

+ v[minus1]otimesa[minus1]otimesFminus(σ(1)(v[0]otimesda[0])[minus1]otimesσ(2)(v[0]otimesda[0])[minus1])

σ(1)(v[0]otimesda[0])[0]otimesσ(2)(v[0]otimesda[0])[0]

The first term in this equation cancels with nablaF (v)Fa giving the answer

The main purpose of the bimodule covariant derivative construction is to allow us totensor product covariant derivatives Given A-bimodule covariant derivatives (VnablaV σV ) and(WnablaW σW ) we have the following covariant derivative on VotimesAW defined by

nablaVotimesAW (votimesw) = nablaV (v)otimesw + (σVotimesAid)(votimesnablaWw) (13)

quotiented to VotimesAW By Proposition 61 we have a bimodule connection nablaF on VotimesAW as anAF -bimodule VotimesAF

W

Proposition 62 Given A-bimodule covariant derivatives (VnablaV σV ) and (WnablaW σW ) thetwisted nablaF on VotimesAF

W obeys

nablaFVotimesAFW = Φ(nablaFVotimesF id) + (σFVotimesF id)Φminus1(idotimesFnablaFW ))

where the composition on the right descends to the required quotient over AF

Proof Unwinding the definitions here nablaF is characterised by

VotimesW c

nabla

VotimesFW

nablaF

Ω1AotimesVotimesW c Ω1AotimesF (VotimesW )

idotimesF c Ω1AotimesF (VotimesFW )

(14)

where we again abuse notations as in previous proofs To check that the map is indeed welldefined over VotimesAF

W we check that the paths from the top left to bottom right corners of thefollowing diagrams are identical (up to the relations on VotimesAF

W in the bottom right corner)which can be done easily just by considering the left and bottom paths of the diagrams

VotimesAotimesW

otimesid

c (VotimesA)otimesFW

cotimesF id (VotimesFA)otimesFW

FotimesF id

VotimesW c

nabla

VotimesFW

nablaF




12

VotimesAotimesW

idotimes

c VotimesF (AotimesW )

idotimesF c VotimesF (AotimesFW )

idotimesF F

VotimesW c

nabla

VotimesFW

nablaF



We do not actually need to check this nor that we have a bimodule connection in view ofProposition 61 However it is instructive to do so for good measure For the left Liebniz rulewe use

AotimesVotimesW

otimesid

c AotimesF (VotimesW )

idotimesF c AotimesF (VotimesFW )

F

VotimesW c

nabla

VotimesFW

nablaF



(15)

Summarising the right side of diagram (15) as lsquorightrsquo we get two parts to (15) the first being

AotimesVotimesW

dotimesidotimesid



first right



which gives the right hand side dotimesF id by functoriality The second term is rather more difficult

Aotimes(VotimesW )

idotimesnabla



second right

AotimesΩ1AotimesVotimesW

otimesidotimesid



Functoriality of c applied to the top and bottom sides gives

Aotimes(VotimesW )

idotimesnabla

idotimesc Aotimes(VotimesFW ) c

AotimesF (VotimesFW )

second right


otimesidotimesid

Ω1AotimesVotimesW

idotimesc Ω1Aotimes(VotimesFW )c Ω1AotimesF (VotimesFW )


13

and since the bottom left two operators commute we get

Aotimes(VotimesW )

idotimesnabla



second right


idotimesidotimesc

AotimesΩ1Aotimes(VotimesFW )otimesid Ω1Aotimes(VotimesFW )

c Ω1AotimesF (VotimesFW )

Now use diagram (14) to get

Aotimes(Ω1AotimesF (VotimesFW ))

idotimescminus1

Aotimes(VotimesFW ) cidotimesnablaF

oo AotimesF (VotimesFW )

second right

AotimesΩ1Aotimes(VotimesFW )

otimesid Ω1Aotimes(VotimesFW )c Ω1AotimesF (VotimesFW )

More functoriality of c gives

AotimesF (Ω1Aotimes(VotimesFW ))

cminus1

AotimesF (Ω1AotimesF (VotimesFW ))idotimescminus1oo AotimesF (VotimesFW )

idotimesFnablaFoo

second right



Now a direct calculation will show that this is the required F (idotimesFnablaF AotimesF (VotimesFW ) rarrΩ1AotimesF (VotimesFW )

For the bimodule part of the bimodule covariant derivative consider

VotimesWotimesA

idotimes

c (VotimesW )otimesFA

cotimesF id (VotimesFW )otimesFA

F

VotimesW c

nabla

VotimesFW

nablaF



This splits into two terms the first of which is

VotimesWotimesA

idotimesidotimesd



first term

VotimesWotimesΩ1A

σVotimesW




14

By functoriality of d and the formula for σVotimesW this is

VotimesWotimesΩ1A

idotimesσW

(VotimesW )otimesFΩ1Acminus1

oo (VotimesFW )otimesFA

first term

cminus1otimesF doo

VotimesΩ1AotimesWσV otimesid



The second term is

VotimesWotimesA

nablaVotimesWotimesid



second term

Ω1AotimesVotimesWotimesA

idotimesidotimes

Ω1AotimesVotimesW c Ω1AotimesF (VotimesW )idotimesF c Ω1AotimesF (VotimesFW )

By using the functoriality of c this is

(Ω1AotimesVotimesW )otimesFA

cminus1

(VotimesW )otimesFAcotimesF id

nablaVotimesWotimesF idoo (VotimesFW )otimesFA

second term


idotimesidotimes


The first term of the equation in the statement is given by differentialting the v and can beread off the diagram as the following where we set nablav = ξotimesu

F (ξ[minus2]u[minus3]otimesw[minus3])Fminus(ξ[minus1]u[minus2]otimesw[minus2])F

minus(u[minus1]otimesw[minus1]) ξ[0]otimesF (u[0]otimesFw[0])

which gives Φ(nablaFVotimesF id) To find the second term we differentiate the w and substitute intodiagram (14) to get

VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

σotimesid

Ω1AotimesF (VotimesFW )

Ω1AotimesVotimesW c (Ω1AotimesV )otimesFW cotimesF id (Ω1AotimesFV )otimesFW

Φ

OO


15

By the functoriality of c we get

VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c


(VotimesΩ1A)otimesFW σotimesF id (Ω1AotimesV )otimesFW cotimesF id (Ω1AotimesFV )otimesFW

Φ

OO

and by the definition of σF

VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c


(VotimesΩ1A)otimesFW cotimesF id (VotimesFΩ1A)otimesFWσFotimesF id (Ω1AotimesFV )otimesFW

Φ

OO

Finally we use the definition of Φminus1 and nablaFW

Finally we need to see how the notion of -compatibilty behaves This is an important extracondition in noncommutative Riemannian geometry [8] and corresponds in our application toensuring reality constraints To keep things simple we again assume that the Hopf algebraH is now commutative (it then has S2 = id) and that (AΩlowastAd) is a classical differentialgeometry So we assume that A is commutative for bimodule A-covariant derivatives the mapσ is transposition and that and is given by antisymmetrisation

Proposition 63 Suppose for simplicity that we start in the classical situation with Acommutative If nabla is compatible with then nablaF is compatible with

Proof In classical differential geometry the condition that nabla is compatible with is

σΥminus1(otimes)nabla(v) = nabla(vlowast)

and this reduces to nabla(vlowast) = ξlowastotimeswlowast where nabla(v) = ξotimesw Now

σF ΥFminus1(otimesF )nablaF v = Fminus(ξ[minus1]otimesw[minus1])σF ΥFminus1(otimesF )(ξ[0]otimesFw[0])

= Fminus(ξ[minus1]otimesw[minus1])σF ΥFminus1(ξ[0]lowastotimesFw[0]

lowast)

= Fminus(wlowast[minus1]otimesξlowast[minus1])

lowast σF (w[0]lowastotimesF ξ[0]

lowast)


lowast F (w[minus2]lowastotimesξ[minus2]

lowast)lowast Fminus(ξ[minus1]lowastotimesw[minus1]

lowast)lowast(ξ[0]lowastotimesFw[0]

lowast)

= Fminus(ξ[minus1]lowastotimesw[minus1]


lowast)

= nablaF (vlowast)

7 Twisting of metrics and Riemannian structuresWe are now ready to put much of this together into an algebraic framework for Riemanniangeometry In this section we will start with differential calculus over an (associative) algebra Abut the example to bear in mind is of the smooth functions on a manifold As previously we take


16

a differential calculus ΩA in the category HM where (because of our previous simplifications)we assume that H is commutative We again assume that the connection nabla to be deformed isan H-comodule map To fit the twisting procedure we assume that the Riemannian metric isH-invariant Within the standard framework of real differential geometry star preservation isautomatic (as star is the identity) so we assume that the connection preserves star

We look at connections nabla Ω1Ararr Ω1AotimesAΩ1A where ΩA = oplusn ΩnA is the exterior algebraon A We use the formalism invented for (associative) noncommutative geometry but we aremainly interested in using it in the classical commutative case and then twisting it to generatenonassociative examples Thus the curvature Rnabla now has the meaning of Riemann curvatureand nabla will be some kind of Levi-Civita connection for a metric Torsion for any connection onΩ1A is defined as

Tornabla = dminus andnabla Ω1Ararr Ω2A (16)

so torsion-free makes senseTo formulate the metric and metric compatibility there are currently two approaches The

original one [18] is

g isin Ω1AotimesAΩ1A and(g) = 0 (17)

where the second expresses symmetry with respect to the notion of skew-symmetrization usedin the exterior algebra We also require some form of nondegeneracy best expressed in termsof projective modules or in a frame bundle approach Finally this setting has a remarkablesymmetry between tangent and cotangent bundles and dual to torsion (ie torsion on the dualbundle regarded as cotangent bundle) is a notion of cotorsion If we assume that the torsionvanishes then cotorsion reduces to

coTor = (andnablaotimesidminus (andotimesid)(idotimesnabla))g isin Ω2AotimesAΩ1A (18)

and we see that its vanishing is a weaker (skew-symmetrized) version of usual metriccompatibility A generalised Levi-Civita connection is then a torsion free cotorsion free (orskew-compatible) connection This weakening of the usual notion of metric compatibility seemsto be necessary in several examples where the generalised Levi-Civita connection then existsand is unique for the chosen metric If A is commutative we can of course impose nablag = 0 in theusual way extending it as a derivation to the two tensor factors but keeping the lsquoleft outputrsquo ofnabla to the far left where it can be evaluated against a vector field

A second more recent approach [8] makes sense for bimodule connections First of all thereis a useful weaker notion to torsion free namely that the torsion Tornabla be a bimodule map(we call this torsion-compatible) We also make use of bar categories and work with Hermitian

metrics g isin Ω1AotimesAΩ1A and in this case full metric compatibility nablag = 0 makes perfect senseThere is now a slightly different notion of cotorsion in place of (18)

coTor = (andotimesid)nablag isin Ω2AotimesAΩ1A (19)

In [8] we showed that these notions lead for example to a unique torsion free metric and lowast-compatible connection with classical limit on a class of metrics on Cq(SU2) with its left covariant3D calculus

When all of these structures are H-covariant we can twist them We have already seen howandnabla twist and checking our formulae for andF and nablaF we see that

TornablaF = dminus andFnablaF = Tornabla


17

is unchanged as a linear map Ω1Ararr Ω2A For g isin Ω1AotimesAΩ1A we take

gF = c(g) isin Ω1AFotimesAFΩ1AF

and the cotorsion or skew-metric compatibility computed in the category with otimesF takes theform

coTorF = (andFnablaFotimesF idminus (andFotimesF id)Φminus1(idotimesFnablaF )) gF isin Ω2AFotimesAFΩ1AF

where an associator is inserted The reader can easily check the following commutative diagram

Ω1AotimesΩ1Ac

X

Ω1AotimesFΩ1A

XF

Ω2AotimesΩ1A

c Ω2AotimesFΩ1A

where X = (andotimesid)(idotimesnabla) and XF = (andFotimesF id)Φminus1(idotimesFnablaF ) It follows that the twistedcotorsion vanishes if and only if the original cotorsion vanishes Similarly in the hermitianmetric framework The following result follows directly from this discussion

Proposition 71 Let g isin Ω1AotimesAΩ1A be a Riemannian metric for A invariant under H andassume for simplicity that we are in the classical situation Then a torsion free cotorsion freeconnection nabla on A which is an H-comodule map is twisted to a torsion free cotorsion freeconnection nablaF on AF

In particular the classical Levi-Civita connection will be covariant and hence twist asrequired We can also consider a Riemannian metric as an inner product 〈〉 Ω1AotimesAΩ1Ararr A(using nondegeneracy) which we assume is an H-comodule map We can use a little moregenerality to get the following result for bimodule connections

Lemma 72 Begin with A-bimodule connections (VnablaV ) and (WnablaW ) in HM Suppose thatwe have a map κ VotimesAW rarr A in HM which preserves the connections (using (Ad) as aconnection on A) ie

dκ = (idotimesAκ)nablaVotimesAW VotimesW rarr Ω1A

Then for the twisted connections we have

dκF = (idotimesAFκF )nablaFVotimesAF

W VotimesAFW rarr Ω1AF

where

κF (votimesFw) = F (v[minus1]otimesw[minus1])κ(v[0]otimesw[0])

descends to κF VotimesAFW rarr AF

Proof From (14) we have (where micro is multiplication)

VotimesW c

nabla

κ

xxrrrrrrrrrrrrVotimesFW

nablaF

A

d

Ω1AotimesVotimesW c

idotimesκ

Ω1AotimesF (VotimesW )idotimesF c

idotimesF κ


idotimesF κFuukkkkkkkkkkkkkk

Ω1A Ω1AotimesA c micro

oo Ω1AotimesFA

The result can be read off from the diagram


18

Corollary 73 Let g isin Ω1AotimesAΩ1A be a hermitian Riemannian metric on A invariant underH and assume for simplicity that we are in the classical situation and that F obeys (8) Then atorsion free bimodule connection nabla on A which is metric and star-preserving and an H-comodulemap is twisted to a star preserving torsion free bimodule connection nablaF on AF which preservesthe twisted Riemannian metric

Again we can take a classical connection which is trivially a bimodule connection with theflip map for σ and we will obtain a twisted one Clearly one can go on in this line One can usea metric to define an interior product of 1-forms on n-forms and use this to define all relevantquantities in physics including candidates for the Ricci tensor and the stress energy tensor ofscalar and other fields The problem at this point is not that we canrsquot do this but that theremay be several different covariant formalisations all with the same classical limit and they willall twist For this one really needs a theory that works beyond the twist examples Until thentwisting nevertheless provides a source of examples

This is illustrated by the Ricci tensor where the classical lsquoformularsquo of contracting Riemanncould be expressed in different ways depending on where we contract While not unique probablythe simplest formulation is as follows as a version similar to [18 19] First we need a splittingmap such that

i Ω2Ararr Ω1AotimesAΩ1A and i = id (20)

In the classical case this is i(ξ and η) = (ξotimesη minus ηotimesξ)2 Then we can define Ricci in our firstformulation of metric as

Ricci = (〈〉otimesid)(idotimesiotimesid)(idotimesRnabla)g isin Ω1AotimesAΩ1A (21)

where g isin Ω1AotimesAΩ1A and where we write its inverse as 〈〉 Ω1AotimesAΩ1A rarr A The use ofthe metric here is for convenience and is not visible classically as the metric and inverse metricare contracted It would be better to use that Ω1A is a projective module and assume theexistence of a suitable trace but this would involve more formalism We will check shortly thatin the classical case (21) is minus12 of the usual Ricci tensor ie the natural normalisation from astructural point of view is not quite the usual one The Ricci scalar S = 〈〉(Ricci) is likewiseminus12 the usual one

We now twist this In view of the way that andF is defined clearly we can set

iF (ξ and η) = c i(ξ and η) = Fminus(v[minus1]otimesw[minus1])v[0]otimesFw[0]

if i(ξ and η) = votimesw (or rather a sum of such terms) and descend to quotients For example if westart with classical geometry then clearly

iF (ξ and η) =1

2(Fminus(ξ[minus1]otimesη[minus1]) ξ[0]otimesF η[0] minus Fminus(η[minus1]otimesξ[minus1]) η[0]otimesF ξ[0])

We then compute the Ricci tensor defined analogously in the otimesF category

RicciF =(

(〈〉FotimesF id) Φminus1 (idotimesF iF )otimesF id)

Φminus1 (idotimesFRnablaF ) gF isin Ω1AFotimesAFΩ1AF

Proposition 74 RicciF = c(Ricci) The Ricci scalar SF = S is unchanged under twist

Proof To compare the result with classical geometry we use the following commutative


19

diagram applying the top line to g rarr gF

Ω1AotimesAΩ1A c

idotimesRnabla

Ω1AotimesFAΩ1A

idotimesFRnablaF

Ω1AotimesA(Ω2AotimesAΩ1A)

(idotimesc)c

=

Ω1AotimesFA(Ω2AotimesFAΩ1A)

Φminus1

(Ω1AotimesAΩ2A)otimesAΩ1A

(cotimesid)c

(idotimesi)otimesid

(Ω1AotimesFAΩ2A)otimesFAΩ1A

(idotimesF iF )otimesF id

(Ω1AotimesA(Ω1AotimesAΩ1A))otimesAΩ1A((idotimesc)cotimesid)c

=

(Ω1AotimesFA(Ω1AotimesFAΩ1A))otimesFAΩ1A

Φminus1otimesF id

((Ω1AotimesAΩ1A)otimesAΩ1A)otimesAΩ1A((cotimesid)cotimesid)c

(〈〉otimesid)otimesid

((Ω1AotimesFAΩ1A)otimesFAΩ1A)otimesFAΩ1A

(〈〉FotimesF id)otimesF id

Ω1AotimesAΩ1A c Ω1AotimesFAΩ1A

(22)

and filling in the cells as commutative in view of the definitions of the various twisted objects inrelation to the originals The method here is the same as for other results above and thereforethis point given with greater brevity

This kind of argument is an explicit elaboration of the statement that the twist functor isan equivalence of categories Nevertheless there may be readers who would like an even moreexplicit treatment with tensors as familiar in Riemannian geometry There are good reasonsnot to do this in noncommutative (and nonassociative) geometry namely we would need first toelaborate a notion of local coordinates We shall however check the definition of Ricci in (21)in the classical case We use standard formulae for vector fields U V and W and Christoffelsymbols

R(U V )(W )l = RlijkWi U j V k

R(U V )(W ) = nablaUnablaV W minusnablaVnablaU W minusnabla[UV ]W

(nablaV W )l = V k (W lk + ΓlkiW

i)

Rlijk =partΓlkipartxj

minuspartΓljipartxk

+ Γmki Γljm minus Γmji Γlkm

We use ξa for the differential form dxa in terms of coordinates on the manifold M Thennablaξa = minusΓabc ξ

botimesξc and

Rnabla(ξa) = (dotimesidminus id andnabla)(minusΓabc ξbotimesξc)

= minusΓabcd ξd and ξbotimesξc minus Γabc Γcpq ξ

b and ξpotimesξq

= minus (Γabcd + Γade Γebc) ξd and ξbotimesξc

= minus1

2Racdb ξ

d and ξbotimesξc

Using this to continue our calculation

g = gfa ξfotimesξa

(idotimesRnabla) g = minus1

2Racdb gfa ξ

fotimes(ξd and ξbotimesξc)


20

When we apply i we use the fact that we already have antisymmetry due to the symmetries ofRacdb and so do not have to explicitly antisymmetrise Referring to (21) we have

Ricci = minusRacdb gfa (〈〉otimesid) (idotimesi)(ξfotimesξd and ξb)otimesξc2= minusRacdb gfa (〈〉otimesid) (ξfotimes(ξdotimesξb))otimesξc2= minusRacdb gfa gfd ξbotimesξc2= minusRacab ξbotimesξc2

= minus1

2Rcbξ

botimesξc

in our conventions (with otimesA understood throughout) The minus sign has its roots in the fact that wework with Riemann as an operator on forms not on vector fields The factor of 12 has its rootsin the fact that Rnabla has values in Ω2A rather than in antisymmetric tensors Doing the samecomputations using the twisted objects similarly gives RicciF = minus1

2RcbFminus(ξb[minus1]otimesξ

c[minus1])ξ

b[0]otimes

F ξc[0]

as it shouldIn view of the last proposition we see that the Einstein tensor Ricciminus 2

ngS (the trace-reversal

of Ricci) also twists as EinsteinF = c(Einstein) We can similarly extend the metric and i tointerior products a Hodge -operation etc in particular to define the stress-energy tensor T ofmatter fields of various types as a composition of maps all of which twist Then (in suitablenormalisation)

Einstein = T hArr EinsteinF = TF

Details will be elaborated elsewhere but the same remarks apply that this approach of itselfworks too well we need experience and insight from examples that go beyond twisting in orderto know which of various ways to do these constructions is the most natural

8 Some examplesExamples of interest fall into two groups The first consists of examples where the algebra ofcoordinates is associative and of independent interest In this case knowing that such examplesare twists opens the door to associated constructions for these algebras Quite often in theliterature however it is not realised that even when the algebra happens to be associative thisdoes not mean that the nonassociativity has gone away Rather as we explained in [6] it ismerely hidden and typically resurfaces elsewhere for example in the differential geometry Thesecond class of examples are lsquomanufacturedrsquo in the sense that the coordinate algebra typicallynonassociative is defined for the first time by twisting and then studied Since twisting worksautomatically the content here is to have interesting classical data consisting of a geometry witha symmetry which could be a discrete one and an interesting cochain F We shall look in detailat examples of both types Let us note that although it is not our focus we can also applyour theory to the case where F is a cocycle and we stay within associative but noncommutativegeometry for example our results imply noncommutative conformal geometry on θ-deformedtwistor spaces in the approach of [9]

81 Drinfeld twist examplesWe have already covered the first class of examples in [5 6] namely the coordinate Hopf algebrasCq(G) associated to semisimple Lie algebras g using a different point of view on Drinfeldrsquos twistelement F that obtains this as C(G)F We have explained the situation in the introductionUnfortunately the form of F is not really known other than to lowest order It indeed takes theform

F (aotimesb) = ε(a)ε(b) + λf(aotimesb) +O(λ2)


21

where f isin gotimesg and λ is the deformation parameter For matrix groups a Lie algebra element ξis evaluated against matrix element functions tij isin C(G) by

ξ(tij) = ρij(ξ)

where ρ is a representation and then extended to products via the additive coproduct on ξIn Drinfeldrsquos theory F is given as a formal power-series of enveloping algebra elements but itcan be similarly evaluated on elements to give formal power-series in λ This was sufficient forthe deformation-theory analysis in [5] but it is expected that Drinfeldrsquos F defined in this way isactually finite and hence defined for numerical values of λ The reason is that F is quite similarin character to the lsquouniversal R-matrixrsquo which makes Uq(g) formally quasitriangular and thisdoes dualise to a well-defined linear functional making Cq(G) coquasitriangular over the fieldThe underling reason is that the relevant elements in the power-series are nilpotent and hencesome power of them vanish against any finite product of representative functions Moreoverfrom Drinfeldrsquos theory one knows that f21 minus f = rminus the antisymmetric part of the Drinfeld-Sklyanin solution r of the modified classical Yang-Baxter equations (which is known explicitly)In this context it was shown in [5] that while C(G)F happens to be associative this necessarilydoes not apply to Ω(G)F

We note that although the HF construction means rather more its algebra can be viewed asan example of the ( )F algebra cochain twist To do this take A = C(G) andH = C(G)otimesC(G)cop

where left and right translation are viewed as a left coaction of H The standard Ω(G) is left-and-right translation invariant (bicovariant) and hence indeed twists by

F (aotimesbotimescotimesd) = F (aotimesc)Fminus(botimesd)

where F on the right is Drinfeldrsquos F viewed as a linear functional We therefore have a twistedcalculus Ω(G)F for Cq(G) as another point of view on [5]

In order to apply our further twisting theory we take the classical metric defined by theKilling form K

g = K(ωotimesAω)

where ω isin Ω1(G)otimesg is the left Maurer-Cartan form and K gotimesg rarr C is supposed to benondegenerate and ad-invariant This is both left and right invariant and hence invariant underH The Levi-Civita connection determined by this is

nablaω = minus1

2[ωotimesAω]

where we take the Lie bracket of the values of ω The latter obeys the Maurer-Cartan equations

dω +1

2[ω and ω] = 0

and hence Tornabla = 0 That nabla is metric compatible follows easily from ad-invariance of K and ashort computation yields Rnabla and Ricci prop g Now because K is ad-invariant the metric definedas above in terms of the left-invariant Maurer-Cartan form is biinvariant ie both left andright translations on the group are isometries Hence H = C(G)otimesC(G)cop coacts and leaves themetric invariant and the results in Section 7 apply Therefore we now also have a metric gF andLevi-Civita connection nablaF etc

The above is an example of hidden nonassociativity in that the algebra AFsim=Cq(G) isassociative We can also do the one-sided twist using only left-covariance of all the samestructures and working directly with the dual version of Drinfeldrsquos F on H = C(G) (not doubledup) Then as in [3] we have AF = C(G)F a nonassociative algebra now with Ω(AF ) = Ω(G)F gF and nablaF all defined by twisting It remains however the case that working with Drinfeldrsquoscochain is difficult due to lack of formulae


22

82 Podles sphere and nonassociative black holesIn this section we look at g = sl2 and cochain F of the form

F = 1 +

infinsumn=1

cn(λ)(xotimesy)n = f(xotimesy)

for some function f Here [x y] = h [h x] = 2x [h y] = minus2y are the sl2 relations Althoughfor practical purposes one can think of F isin U(su2)otimesU(su2) we actually think of F in theconvolution algebra of linear functionals C(SU2)otimesC(SU2) rarr C or even better as an elementof the smaller space (C(SU2)lowast)otimes2 ie as a tensor product of linear functionals Here x (say) isregarded as such a functional and note that

xn(ti1j1 middot middot middot timjm) = 0

for all n gt m ge 1 due to x2(tij) = 0 in the defining representation With these remarks we aregoing to work formally with F viewed in U(su2)otimesU(su2) but everything can be made precise inthe above way Clearly we have (εotimesid)F = (idotimesε)F = 1 as needed for a cochain of this formand we require F to be invertible For this we require that the power-series f has a non-zeroradius of convergence about the origin

According to our theory any such cochain induces a noncommutative and nonassociativegeometry on any classical manifold which has an SU2 symmetry For Riemannian geometry itmeans any manifold M with an su2-triple of Killing vectors fields ξi obeying

[ξi ξj ] = minusεijkξk

and we set h = 2ıξ3 x = minusξ+ and y = minusξminus where ξplusmn = ξ1 plusmn ıξ2 to match our aboveconventions Note that since we based our twists on left coactions this corresponds to a rightaction of U(su2) now However this Hopf algebra is isomorphic to its opposite so we can workwith familiar left actions as stated For our global theory we should suppose that these vectorfields integrate to a right action of SU2 and that everything is algebraic (so that we have acoaction) However none of these is really necessary in deformation theory where we work onlywith power-series

Proposition 81 We let M = S2 defined as usual with coordinate relationsumx2i = 1 and

ξi = εijkxjpartpartxk

A nontrivial F of the form above can never induce an associative product

However if c2 = c21 the product of C(S2)F coincides on the generators with that of the Podles

sphere defined by generators K z zlowast with relations

zK = q2Kz zlowastz = 1minusK2 zzlowast = 1minus q2K2

where q2 = 1minus 2c1 We assume that c1 is real and 6= 1 12

Proof We work in complex coordinates xplusmn = x1 plusmn ıx2 and begin by computing the action as

x

x+

xminusx3

= ı

0minus2x3

x+

y

x+

xminusx3

= ı

2x3

0minusxminus

We compute AF from a bull b = middot(F(aotimesb)) = middotf(xotimesy)(aotimesb) to obtain

x3 bull x3 = x23 minus c1x+(minusxminus) x+ bull x+ = x2

+ xminus bull xminus = x2minus


23

x+ bull xminus = x+xminus xminus bull x+ = xminusx+ minus c1(minus2x3)2x3 minus 4c2x+(minusxminus)

x+ bull x3 = x+x3 x3 bull x+ = x3x+ minus c1x+2x3

xminus bull x3 = xminusx3 minus c1(minus2x3)(minusxminus) x3 bull xminus = x3xminus

Now from the last four relations we see that xplusmn indeed q2-commute with x3 as required forK prop x3 and z prop xminus From the first relation and x+xminus = r2 minus x2

3 for a sphere of radius r we seethat

x3 bull x3 = x23 + c1(r2 minus x2

3) = (1minus c1)x23 + r2c1

and hence x23(1minus c1) = x3 bull x3 minus r2c1 We also see that

x+ bull xminus = x+xminus = r2 minus x23 =

(r2 minus x3 bull x3)

(1minus c1)

Accordingly we set z = xminusradic

1minus c1 and K = x3r and have the 2nd relation of the Podlessphere Note now that if we assume associativity among the generators

xminus bull (x+ bull xminus) = xminus bull(r2 minus x3 bull x3)

1minus c1=

xminusr2

1minus c1minus (1minus 2c1)2

1minus c1(x3 bull x3) bull xminus

= (xminus bull x+) bull xminus =(AminusBx3 bull x3)

(1minus c1)bull xminus

forces A = r2 and B = (1 minus 2c1)2 (in other words the 3rd of the Podles sphere relations isdictated by the first two and associativity) Comparing with what we computed for xminus bull x+ weneed c2

1 = c2 and in this case we obtain the 3rd relation as well A look at bull of some higherdegree monomials however confirms that in this case bull is nevertheless not associative

We have included this example because it is striking that while nonassociative thenonassociativity is almost hidden and on the generators gives us a Podles sphere ie a lsquomildlynonassociativersquo Podles sphere Notice that the true Podles sphere is Cq(SU2)-covariant whichsuggests that our form of F has features similar to the Drinfeld cochain for sl2 Also the truePodles sphere has recently been interpreted as a q-fuzzy sphere or more geometrically as a slicein the q-hyperboloid of unit Lorentzian distance in Minkowski space [17] and indeed arises in aneffective description of 3D quantum gravity with cosmological constant [21]

We can apply this construction to spheres of any radius and to manifolds nested by spheressuch as R3 the entire hyperboloid in Minkowski space and the Schwarzschild black-hole Clearlyit will induce a lsquomildly nonassociativersquo Podles or q-fuzzy sphere in place of each classical sphereIn particular as the rotational symmetry is an isometry of the Schwarzschild solution the metricis invariant and we have gF nablaF on these spaces by our construction

It is worth noting that these results are in the spirit of but rather different from recentattempts to apply projective Drinfeld twists to quantize black holes [22] based on a certaindifferent F = f(xotimesy) in [1] where cn are generated by a beta-function As we have seen usingsuch a cochain in a standard way cannot give an associative algebra Rather the point of viewis quite different F acts on S2 = SU2U(1) via right-translation to induce a left-translationSU2-invariant bidifferential operator Here U(su2) itself does not act on C(SU2U(1)) in sucha way at all so this does not fit our framework However the failure of F to be a cocycle iscontained (φ = partF has an h to the right in its middle factor and this acts trivially) and asa result the quantizaiton is accidentally associative giving a fuzzy-sphere and a residual SU2

symmetry Although the framework is different from our more conventional use of Drinfeldtwists the lesson is the same that even though the algebra quantizing each sphere is associativethere remains a hidden nonassociativity in the model as F is not exactly a cocycle


24

83 Octonionic-S7 as second quantisationIn this section we give a first concrete example of nontrivial nonassociative Riemannian geometrywith curvature of interest because it lies in the same category as the one in which the octonionsO live in the approach of [2] This is the category of Z3

2-graded spaces and we use the samecochain F that lsquoquantizesrsquo k(Z3

2) to obtain O = k(Z32)F as explained in Section 3 We again

work algebraically over a field k and in the sequel we will assume this to be of characteristiczero (the reader can keep in mind k = R) In physics when the result of first quantisation is(in some form) used as the basis of another (infinite dimensional) quantization one speaks ofsecond quantization In the same spirit we regard the result of the first (finite) quantization asa space in its own right and look at the submanifold of unit length octonions This submanifoldis S7 and we lsquorequantizersquo this via a twist albeit quite mildly by an action of Z3

2 These methodscould be used to similarly twist any geometry admitting a finite group symmetry

As usual the effort is all in the preparation We recall that the description of octonions weuse has basis ea labelled by a isin Z3

2 The norm on the octonions in this basis is the usualeuclidean one Hence our natural description of S7 is with coordinates xa again labelled bya isin Z3

2 and the relationsum

a x2a = 1 Following [14] we also have a coproduct counit and antipode

∆xc =suma+b=c

xaotimesxbF (a b) εxa = δa0 Sxa = F (a a)xa

making k(S7) into a Hopf coquasigroup It fails to be an ordinary Hopf algebra just becauseS7 is not a group Notice that all these maps and the algebra relations themselves respect a Z3

2

grading defined by |xa| = a ie H = kZ32sim=k(Z3

2) coacts on everything hereNext in [14] Hopf-algebra-like methods are used to define the differential geometry of S7

and we continue this programme now to compute its Riemannian geometry The startingpoint is that a natural differential structure defined by the ideal (k(S7)+)2 as for algebraicgroups is parallelizable with the basis of lsquoleft-invariantrsquo 1-forms ωi where ωi = Sxi(1)dx

i(2) =sum

a F (a i)xadxa+i is defined as for Hopf algebras and computes as stated Our convention isthat middling indices i j k lm n run in the range 1 middot middot middot 7 This induces lsquoleft-invariant vectorfieldsrsquo parti and lsquoLie-like structure constantsrsquo cijk isin k(S7) by

df =sumi

(partif)ωi dωi + cijkωjωk = 0

which can be computed as

parti = minussuma

F (a i)xa+i part

partxa

cijk = minusF (i j)F (i+ j k)suma

φ(a i j)φ(a i+ j k)F (a i+ j + k)xaxa+i+j+k

Notice that the former do not close under commutator and the latter are not constants butfunctions both effects are due to the nonassociativity of the product on S7 Moreover

Proposition 82 The lsquoKilling formrsquo is cmincnjm = minus6δij (summation of repeated indices

understood) Using δij from now on to lower indices cijk is totally antisymmetric

We refer to [14] for the methods used to obtain these results However F and φ are givenby particular formulae hence all our assertions can be checked by direct computation usingMathematica Being antisymmetric in the outer two indices then means that the metric

g = δijωiotimesωj


25

is preserved bynablaωi = minuscijkωjotimesωk

asnablag = minuscimnδijωmotimesωnotimesωj minus cjmnδijωiotimesωmotimesωn = 0

In calculations we leave otimesA understood Here nabla acts as a derivation and its left output (whichmight be evaluated against a vector field) is kept to the far left and all otimes products are overk(S7) so we can bring coefficients to the left as well This connection is also manifestly torsionfree and is therefore the Levi-Civita connection for the natural metric constructed in this wayThese steps are exactly as for the Lie group case in Section 81 but now cijk are functions anddo not define a Lie algebra (they define a Malrsquotsev algebra at the identity of S7) and we havegiven the computations in a basis rather than more abstractly

It remains to compute the Riemann and Ricci curvatures to make sure we have the usualconstant curvature metric From (10) we compute

Rnablaωi = (dotimesidminus id andnabla)(minuscijkωjotimesωk)

= minusdcijkωjotimesωk + cijkc

jmnω

mωnotimesωk minus cijkωjckmnωmotimesωn

= minus(cimjc

jnk minus cijkcjmn + partmc

ink

)ωmωnotimesωk

To obtain the Ricci curvature we apply a map i that lifts the 2-form-valued part of Rnabla to anantisymmetrised element of Ω1otimesΩ1 (this is one way to be able to define an interior product)We then contract the first leg of the metric and the first leg of the lifting with the inverse metric〈ωi ωj〉 = δij as in (21)

Ricci = δpi〈ωp (iotimesid)Rnablaωi〉

= minusδpi〈ωp1

2

(cimjc

jnk minus cinjcjmk minus 2cijkc

jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink

)ωnotimesωk = minus3g

using the symmetries of c and the Killiing form identity Here the term particink is antisymmetric

in n k hence it must vanish as the Ricci tensor is necessarily symmetric This turns out to bethe case as one can compute using Mathematica from the formulae already given Hence this S7

is an Einstein space as it should be (the usual answer here for a unit 7-sphere would be 6g theminus sign and factor of 12 is due to our conventions as explained at the end of Section 7) Inshort the main difference that the cijk are functions does show up in the full Riemann tensorbut not in Ricci

We have given an algebraic derivation of the usual geometry of S7 using Hopf coquasigroupmethods This is actually a uniform approach that applies to all the parallellizable spheres Nownotice that |ωi| = i (here d does not change the Z3

2-degree) and |cijk| = i+ j + k Hence g haszero degree and nabla does change the degree ie everything is covariant under the grading Hopfalgebra kZ3

2 and we can apply our cochain twist We obtain k(S7)F and a calculus Ω(S7)F onit with everything nonassociative The product xa bull xb = F (a b)xaxb obeys

(xa bull xb) bull xc = φ(a b c)xa bull (xb bull xc) xb bull xa = R(a b)xa bull xb

where φ(a b c) = minus1 if and only if a b c are linearly independent as before and R(a b) = minus1 ifand only if a b are linearly independent and otherwise 1 again as for the octonions The second


26

feature means that the algebra is altercommutative in the sense that elements anticommute ifand only if they have distinct non-zero degrees Otherwise they commute This is not howeverthe octonion algebra For products of generators we have degree (and the same for their bullproducts)

|(x0)m(x1)n1 middot middot middot (x7)n7 | = d(N) = (n4 + n5 + n6 + n7 n2 + n3 + n6 + n7 n1 + n3 + n5 + n7)

where N = (n1 middot middot middot n7) and the degree is computed modulo 2 to give an element of Z32 We

have enumerated the xa in binary order x000 x001 middot middot middot x111 for brevity Then

(prod

(xi)ni) bull (prod

(xj)mj ) = F (d(N) d(M))prod

(xi)ni+mi

and associativity remains via φ on the relevant degrees (so three monomials fail to associate ifand only if their Z3

2-degrees are linearly independent) and commutativity remains via R (so twomonomials fail commute if and only if they have different non-zero degrees) The generator x0

has zero degree has unchanged product with everything and any power of it associates witheverything Finally we have the sphere relation which now takes the form

1 = x0 bull x0 minussumi

xi bull xi

The even powers xi bull xi = minus(xi)2 and 1 x0 generate a commutative associative subalgebra ofdegree 0 ie which commutes and associates with everything else Similarly we can computethe differential calculus obtained by twisting the classical one The products

xa bull dxb = F (a b)xadxb dxa bull dxb = F (a b)dxadxb

obey parallel formulae controlled by φR Explicitly

0 = x0 bull dx0 minussumi

xi bull dxi

with relations involving x0 or dx0 unchanged as is dxabulldxa = 0 for all a The changed relationsfor distinct i j k are

xi bull dxj = minusdxj bull xi dxi bull dxj = dxj bull dxi (xi bull dxj) bull xk = φ(i j k)xi bull (dxj bull dxk)

and similar In degree three we have

(dxi bull dxj) bull dxk = minusdxi bull (dxj bull dxk)

if and only if i+ j + k 6= 0By our theory the metric and connection also twist

gF = ω0otimesω0 minussumi

ωiotimesωi nablaFωi =sumjk

F (i k)cijkωiotimesωk

where we made use of F (i + k k) = F (i k)F (k k) = minusF (i k) to simplify We write thesummations explicitly for clarity The inverse metric 〈〉F also acquires minus1 values on thei = 1 middot middot middot 7 diagonal entries and

iF (ωi bull ωj) =1

2(ωiotimesωj + ωjotimesωi) i 6= j


27

because F (i j) = minusF (j i) when we compute iF (ωiωj) which we can then cancel as the samefactor occurs in ωi bullωj Following through the calculation of RicciF in terms of these quantitiesand remembering the associator one can verify

RicciF = minus3gF

as implied by Propostion 74 (we again have an Einstein space with constant curvature) Thisprovides a useful check of our nonassociative theory

Further insight into this lsquooctonionicrsquo S7 can be gained by looking at the same constructionfor second-quantized quaternions and complex numbers The quaternions are constructed in thesame way but with Z2

2-grading F has the same form but without the lsquocubicrsquo term As a resultF is a cocycle and φ = 1 The above geometry gives the classical Riemannian geometry of S3

viewed as the unit quaternions and k(S3)F is associative and altercommutative It thereforehas bosonic central generators 1 x0 and noncommutative generators x1 x2 x3 with relations

x1 x2 = x2 x3 = x3 x1 = 0 1 = (x0)2 minus (x1)2 minus (x2)2 minus (x3)2

where the bullet product is understood The noncommutative generators are lsquoGrassman-likersquobut do not anticommute with themselves indeed the algebra is commutative in the categoryof Z2

2-graded spaces with a specific symmetric braiding different from that of the categoryof supervector spaces If we go down one level more to Z2-gradings we have one degree 1generator x1 and 1 x0 bosonic As a result everything is commutative and associative andk(S1)F = k[x0 x1] modulo the relation 1 = (x0)2minus(x1)2 ie a hyperbola Again one has metricgF and connection nablaF with constant curvature So all of these lsquotwisted spheresrsquo are actuallysome kind of hyperbolae One could view them as baby versions of the q-deformation twist inSection 82 which also turned spheres into mildly nonassociative q-hyperboloids

References[1] Alekseev A and Lachowska A 2005 Invariant lowast-products on coadjoint orbits and the Shapovalov pairing

Comment Math Helv 80 795-810[2] Albuquerque H and Majid S 1999 Quasialgebra structure of the octonions J Algebra 220 188-224[3] Akrami S E and Majid S 2004 Braided cyclic cocycles and nonassociative geometry J Math Phys 45

3883-3911[4] Amelino-Camelia G and Majid S 2000 Waves on noncommutative spacetime and gamma-ray bursts Int

J Mod Phys A 15 4301-4324[5] Beggs E J and Majid S 2006 Semiclassical differential structures Pac J Math 224 1-44[6] Beggs E J and Majid S (in press) Quantization by cochain twists and nonassociative differentials J

Math Phys[7] Beggs E J and Majid S 2009 Bar categories and star operations Alg and Representation Theory 12

103-152[8] Beggs E J and Majid S 2009 -Compatible connections in noncommutative Riemannian geometry

Preprint arXiv09040539[9] Brain S J and Majid S 2008 Quantisation of twistor theory by cocycle twist Commun Math Phys 284

713-774[10] Connes A 1994 Noncommutative Geometry (London and San Diego Academic Press)[11] Connes A Chamseddine A and Marcolli M 2007 Gravity and the standard model with neutrino mixing

Adv Theor Math Phys 11 991[12] Drinfeld V G 1990 Quasi-Hopf algebras Leningrad Math J 1 1419-1457[13] Hawkins E 2004 Noncommutative rigidity Commun Math Phys 246 211-235[14] Klim J and Majid S (in press) Hopf quasigroups and the algebraic 7-sphere J Algebra[15] Majid S 2000 Foundations of Quantum Group Theory (Cambridge Cambridge University Press 1995)[16] Majid S 2005 Gauge theory on nonassociative spaces J Math Phys 46 103519[17] Majid S 2008 q-Fuzzy spheres and quantum differentials on Bq[SU2] and Uq(su2) Preprint

arXiv08124942


28

[18] Majid S 2002 Riemannian geometry of quantum groups and finite groups with nonuniversal differentialsCommun Math Phys 225 131-170

[19] Majid S 2005 Noncommutative Riemannian and spin geometry of the standard q-sphere Commun MathPhys 256 255-285

[20] Majid S and Oeckl R 1999 Twisting of quantum differentials and the Planck scale Hopf algebra CommunMath Phys 205 617-655

[21] Majid S and Schroers B 2009 q-Deformation and semidualisation in 3D quantum gravity J Phys A 42425402

[22] Schupp P and Solodukhin S N 2009 Exact black hole solutions in noncommutative gravity PreprintarXiv09062724


29

Nonassociative Riemannian geometry by twisting

Edwin Beggs1 and Shahn Majid2

1 Department of Mathematics Swansea UniversitySingleton Parc Swansea SA2 8PP UK2 School of Mathematical Sciences Queen Mary University of London327 Mile End Rd London E1 4NS

E-mail ejbeggsswanseaacuk smajidqmulacuk

Abstract Many quantum groups and quantum spaces of interest can be obtained by cochain(but not cocycle) twist from their corresponding classical object This failure of the cocycle con-dition implies a hidden nonassociativity in the noncommutative geometry already known to bevisible at the level of differential forms We extend the cochain twist framework to connectionsand Riemannian structures and provide examples including twist of the S7 coordinate algebrato a nonassociative hyperbolic geometry in the same category as that of the octonions

Received 17 November 2009

1 IntroductionThe idea of noncommutative geometry of course is to replace points in an actual space by acoordinate algebra These are generally generated by variables x y z say enjoying algebraicproperties for addition multiplication and other relations paralleling the numbers for which theycould be viewed as placeholders Thus solving x2 + y2 + z2 = 1 over R would bring one to theactual points of a sphere but one could also consider this equation more abstractly with x y zas generators of an algebra And if in place of the usual xy = yx etc we have some other non-commutation relations then clearly these generators could never be realised as numbers Overthe last two decades it has become obvious that the assumption of commutativity is a historicalaccident to do with the fact that classical mechanics was discovered before quantum mechanicsand there is no particular reason certainly from a mathematical perspective not to allownoncommutavity as a generalisation of usual geometry more applicable to the quantum worldThere are also physical reasons to think that plausibly the first quantum gravity corrections togeometry should be expressed as such noncommutativity [21] Accordingly a great deal of efforthas been put into developing such a geometry both in Connesrsquo lsquospectral triplersquo approach and ina lsquoquantum groupsrsquo approach modelled around quantum groups such as the deformations Cq(G)and their homogeneous spaces in the first instance Both approaches are by now somewhatmature and the quantum groups approach includes specific testable predictions for Planck scalephysics [4] Meanwhile the spectral triples approach includes specific predictions and massrelations for the standard model [11]

In this article we go one step further and consider geometry that is nonassociative in thesame algebraic sense While noncommutative geometry is motivated from quantum theory andhence somewhat familiar this is not so for nonassociativity If we simply drop associativity then


ccopy 2010 IOP Publishing Ltd 1






NCGrArr NAG




2












3












λ(f)(x) =sumgisinG

δg f(gx)



4























5







F b[0])






77ooooooooooo

(4)













6



VotimesW

c



idotimesF c


(5)












7










VotimesW Υ

c

WotimesVc

VotimesFW

ΥF WotimesFV








= votimesFw











lowast


lowast



8
























9






VnablaF

nabla

Ω1AotimesFV

Ω1AotimesVc

88qqqqqqqqqq

(12)




















V




10








Now we calculate

















11



Now we use









W


W obeys




VotimesW c

nabla

VotimesFW

nablaF



(14)




VotimesAotimesW

otimesid



FotimesF id

VotimesW c

nabla

VotimesFW

nablaF




12

VotimesAotimesW

idotimes



idotimesF F

VotimesW c

nabla

VotimesFW

nablaF




AotimesVotimesW

otimesid



F

VotimesW c

nabla

VotimesFW

nablaF



(15)


AotimesVotimesW

dotimesidotimesid



first right




Aotimes(VotimesW )

idotimesnabla



second right


otimesidotimesid




Aotimes(VotimesW )

idotimesnabla



second right


otimesidotimesid

Ω1AotimesVotimesW



13


Aotimes(VotimesW )

idotimesnabla



second right


idotimesidotimesc





idotimescminus1



second right





cminus1


idotimesFnablaFoo

second right





VotimesWotimesA

idotimes



F

VotimesW c

nabla

VotimesFW

nablaF




VotimesWotimesA

idotimesidotimesd



first term

VotimesWotimesΩ1A

σVotimesW




14


VotimesWotimesΩ1A

idotimesσW



first term

cminus1otimesF doo




The second term is

VotimesWotimesA




second term


idotimesidotimes




cminus1



second term


idotimesidotimes






VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

σotimesid



Φ

OO


15


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO









lowast)



lowast)





lowast)



lowast)

= nablaF (vlowast)



16

















17






Ω1AotimesΩ1Ac

X

Ω1AotimesFΩ1A

XF

Ω2AotimesΩ1A

c Ω2AotimesFΩ1A









where




VotimesW c

nabla

κ


nablaF

A

d


idotimesκ


idotimesF κ




oo Ω1AotimesFA



18











iF (ξ and η) =1



RicciF =(






19


Ω1AotimesAΩ1A c

idotimesRnabla

Ω1AotimesFAΩ1A

idotimesFRnablaF


(idotimesc)c

=


Φminus1


(cotimesid)c

(idotimesi)otimesid




=


Φminus1otimesF id






(22)






i)





botimesξc and



b and ξpotimesξq


= minus1

2Racdb ξ

d and ξbotimesξc




2Racdb gfa ξ



20



= minus1

2Rcbξ

botimesξc



c[minus1])ξ

b[0]otimes

F ξc[0]










21


ξ(tij) = ρij(ξ)







g = K(ωotimesAω)


nablaω = minus1

2[ωotimesAω]


dω +1

2[ω and ω] = 0




22


F = 1 +

infinsumn=1
















x

x+

xminusx3

= ı

0minus2x3

x+

y

x+

xminusx3

= ı

2x3

0minusxminus





23







3) = (1minus c1)x23 + r2c1




(1minus c1)




1minus c1=

xminusr2












24










∆xc =suma+b=c



2




i(2) =sum


df =sumi



parti = minussuma

F (a i)xa+i part

partxa









25







jmnω


= minus(cimjc


ink

)ωmωnotimesωk



= minusδpi〈ωp1

2

(cimjc


jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink











26





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29






NCGrArr NAG




2












3












λ(f)(x) =sumgisinG

δg f(gx)



4























5







F b[0])






77ooooooooooo

(4)













6



VotimesW

c



idotimesF c


(5)












7










VotimesW Υ

c

WotimesVc

VotimesFW

ΥF WotimesFV








= votimesFw











lowast


lowast



8
























9






VnablaF

nabla

Ω1AotimesFV

Ω1AotimesVc

88qqqqqqqqqq

(12)




















V




10








Now we calculate

















11



Now we use









W


W obeys




VotimesW c

nabla

VotimesFW

nablaF



(14)




VotimesAotimesW

otimesid



FotimesF id

VotimesW c

nabla

VotimesFW

nablaF




12

VotimesAotimesW

idotimes



idotimesF F

VotimesW c

nabla

VotimesFW

nablaF




AotimesVotimesW

otimesid



F

VotimesW c

nabla

VotimesFW

nablaF



(15)


AotimesVotimesW

dotimesidotimesid



first right




Aotimes(VotimesW )

idotimesnabla



second right


otimesidotimesid




Aotimes(VotimesW )

idotimesnabla



second right


otimesidotimesid

Ω1AotimesVotimesW



13


Aotimes(VotimesW )

idotimesnabla



second right


idotimesidotimesc





idotimescminus1



second right





cminus1


idotimesFnablaFoo

second right





VotimesWotimesA

idotimes



F

VotimesW c

nabla

VotimesFW

nablaF




VotimesWotimesA

idotimesidotimesd



first term

VotimesWotimesΩ1A

σVotimesW




14


VotimesWotimesΩ1A

idotimesσW



first term

cminus1otimesF doo




The second term is

VotimesWotimesA




second term


idotimesidotimes




cminus1



second term


idotimesidotimes






VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

σotimesid



Φ

OO


15


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO









lowast)



lowast)





lowast)



lowast)

= nablaF (vlowast)



16

















17






Ω1AotimesΩ1Ac

X

Ω1AotimesFΩ1A

XF

Ω2AotimesΩ1A

c Ω2AotimesFΩ1A









where




VotimesW c

nabla

κ


nablaF

A

d


idotimesκ


idotimesF κ




oo Ω1AotimesFA



18











iF (ξ and η) =1



RicciF =(






19


Ω1AotimesAΩ1A c

idotimesRnabla

Ω1AotimesFAΩ1A

idotimesFRnablaF


(idotimesc)c

=


Φminus1


(cotimesid)c

(idotimesi)otimesid




=


Φminus1otimesF id






(22)






i)





botimesξc and



b and ξpotimesξq


= minus1

2Racdb ξ

d and ξbotimesξc




2Racdb gfa ξ



20



= minus1

2Rcbξ

botimesξc



c[minus1])ξ

b[0]otimes

F ξc[0]










21


ξ(tij) = ρij(ξ)







g = K(ωotimesAω)


nablaω = minus1

2[ωotimesAω]


dω +1

2[ω and ω] = 0




22


F = 1 +

infinsumn=1
















x

x+

xminusx3

= ı

0minus2x3

x+

y

x+

xminusx3

= ı

2x3

0minusxminus





23







3) = (1minus c1)x23 + r2c1




(1minus c1)




1minus c1=

xminusr2












24










∆xc =suma+b=c



2




i(2) =sum


df =sumi



parti = minussuma

F (a i)xa+i part

partxa









25







jmnω


= minus(cimjc


ink

)ωmωnotimesωk



= minusδpi〈ωp1

2

(cimjc


jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink











26





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29












3












λ(f)(x) =sumgisinG

δg f(gx)



4























5







F b[0])






77ooooooooooo

(4)













6



VotimesW

c



idotimesF c


(5)












7










VotimesW Υ

c

WotimesVc

VotimesFW

ΥF WotimesFV








= votimesFw











lowast


lowast



8
























9






VnablaF

nabla

Ω1AotimesFV

Ω1AotimesVc

88qqqqqqqqqq

(12)




















V




10








Now we calculate

















11



Now we use









W


W obeys




VotimesW c

nabla

VotimesFW

nablaF



(14)




VotimesAotimesW

otimesid



FotimesF id

VotimesW c

nabla

VotimesFW

nablaF




12

VotimesAotimesW

idotimes



idotimesF F

VotimesW c

nabla

VotimesFW

nablaF




AotimesVotimesW

otimesid



F

VotimesW c

nabla

VotimesFW

nablaF



(15)


AotimesVotimesW

dotimesidotimesid



first right




Aotimes(VotimesW )

idotimesnabla



second right


otimesidotimesid




Aotimes(VotimesW )

idotimesnabla



second right


otimesidotimesid

Ω1AotimesVotimesW



13


Aotimes(VotimesW )

idotimesnabla



second right


idotimesidotimesc





idotimescminus1



second right





cminus1


idotimesFnablaFoo

second right





VotimesWotimesA

idotimes



F

VotimesW c

nabla

VotimesFW

nablaF




VotimesWotimesA

idotimesidotimesd



first term

VotimesWotimesΩ1A

σVotimesW




14


VotimesWotimesΩ1A

idotimesσW



first term

cminus1otimesF doo




The second term is

VotimesWotimesA




second term


idotimesidotimes




cminus1



second term


idotimesidotimes






VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

σotimesid



Φ

OO


15


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO









lowast)



lowast)





lowast)



lowast)

= nablaF (vlowast)



16

















17






Ω1AotimesΩ1Ac

X

Ω1AotimesFΩ1A

XF

Ω2AotimesΩ1A

c Ω2AotimesFΩ1A









where




VotimesW c

nabla

κ


nablaF

A

d


idotimesκ


idotimesF κ




oo Ω1AotimesFA



18











iF (ξ and η) =1



RicciF =(






19


Ω1AotimesAΩ1A c

idotimesRnabla

Ω1AotimesFAΩ1A

idotimesFRnablaF


(idotimesc)c

=


Φminus1


(cotimesid)c

(idotimesi)otimesid




=


Φminus1otimesF id






(22)






i)





botimesξc and



b and ξpotimesξq


= minus1

2Racdb ξ

d and ξbotimesξc




2Racdb gfa ξ



20



= minus1

2Rcbξ

botimesξc



c[minus1])ξ

b[0]otimes

F ξc[0]










21


ξ(tij) = ρij(ξ)







g = K(ωotimesAω)


nablaω = minus1

2[ωotimesAω]


dω +1

2[ω and ω] = 0




22


F = 1 +

infinsumn=1
















x

x+

xminusx3

= ı

0minus2x3

x+

y

x+

xminusx3

= ı

2x3

0minusxminus





23







3) = (1minus c1)x23 + r2c1




(1minus c1)




1minus c1=

xminusr2












24










∆xc =suma+b=c



2




i(2) =sum


df =sumi



parti = minussuma

F (a i)xa+i part

partxa









25







jmnω


= minus(cimjc


ink

)ωmωnotimesωk



= minusδpi〈ωp1

2

(cimjc


jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink











26





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29












λ(f)(x) =sumgisinG

δg f(gx)



4























5







F b[0])






77ooooooooooo

(4)













6



VotimesW

c



idotimesF c


(5)












7










VotimesW Υ

c

WotimesVc

VotimesFW

ΥF WotimesFV








= votimesFw











lowast


lowast



8
























9






VnablaF

nabla

Ω1AotimesFV

Ω1AotimesVc

88qqqqqqqqqq

(12)




















V




10








Now we calculate

















11



Now we use









W


W obeys




VotimesW c

nabla

VotimesFW

nablaF



(14)




VotimesAotimesW

otimesid



FotimesF id

VotimesW c

nabla

VotimesFW

nablaF




12

VotimesAotimesW

idotimes



idotimesF F

VotimesW c

nabla

VotimesFW

nablaF




AotimesVotimesW

otimesid



F

VotimesW c

nabla

VotimesFW

nablaF



(15)


AotimesVotimesW

dotimesidotimesid



first right




Aotimes(VotimesW )

idotimesnabla



second right


otimesidotimesid




Aotimes(VotimesW )

idotimesnabla



second right


otimesidotimesid

Ω1AotimesVotimesW



13


Aotimes(VotimesW )

idotimesnabla



second right


idotimesidotimesc





idotimescminus1



second right





cminus1


idotimesFnablaFoo

second right





VotimesWotimesA

idotimes



F

VotimesW c

nabla

VotimesFW

nablaF




VotimesWotimesA

idotimesidotimesd



first term

VotimesWotimesΩ1A

σVotimesW




14


VotimesWotimesΩ1A

idotimesσW



first term

cminus1otimesF doo




The second term is

VotimesWotimesA




second term


idotimesidotimes




cminus1



second term


idotimesidotimes






VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

σotimesid



Φ

OO


15


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO









lowast)



lowast)





lowast)



lowast)

= nablaF (vlowast)



16

















17






Ω1AotimesΩ1Ac

X

Ω1AotimesFΩ1A

XF

Ω2AotimesΩ1A

c Ω2AotimesFΩ1A









where




VotimesW c

nabla

κ


nablaF

A

d


idotimesκ


idotimesF κ




oo Ω1AotimesFA



18











iF (ξ and η) =1



RicciF =(






19


Ω1AotimesAΩ1A c

idotimesRnabla

Ω1AotimesFAΩ1A

idotimesFRnablaF


(idotimesc)c

=


Φminus1


(cotimesid)c

(idotimesi)otimesid




=


Φminus1otimesF id






(22)






i)





botimesξc and



b and ξpotimesξq


= minus1

2Racdb ξ

d and ξbotimesξc




2Racdb gfa ξ



20



= minus1

2Rcbξ

botimesξc



c[minus1])ξ

b[0]otimes

F ξc[0]










21


ξ(tij) = ρij(ξ)







g = K(ωotimesAω)


nablaω = minus1

2[ωotimesAω]


dω +1

2[ω and ω] = 0




22


F = 1 +

infinsumn=1
















x

x+

xminusx3

= ı

0minus2x3

x+

y

x+

xminusx3

= ı

2x3

0minusxminus





23







3) = (1minus c1)x23 + r2c1




(1minus c1)




1minus c1=

xminusr2












24










∆xc =suma+b=c



2




i(2) =sum


df =sumi



parti = minussuma

F (a i)xa+i part

partxa









25







jmnω


= minus(cimjc


ink

)ωmωnotimesωk



= minusδpi〈ωp1

2

(cimjc


jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink











26





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29























5







F b[0])






77ooooooooooo

(4)













6



VotimesW

c



idotimesF c


(5)












7










VotimesW Υ

c

WotimesVc

VotimesFW

ΥF WotimesFV








= votimesFw











lowast


lowast



8
























9






VnablaF

nabla

Ω1AotimesFV

Ω1AotimesVc

88qqqqqqqqqq

(12)




















V




10








Now we calculate

















11



Now we use









W


W obeys




VotimesW c

nabla

VotimesFW

nablaF



(14)




VotimesAotimesW

otimesid



FotimesF id

VotimesW c

nabla

VotimesFW

nablaF




12

VotimesAotimesW

idotimes



idotimesF F

VotimesW c

nabla

VotimesFW

nablaF




AotimesVotimesW

otimesid



F

VotimesW c

nabla

VotimesFW

nablaF



(15)


AotimesVotimesW

dotimesidotimesid



first right




Aotimes(VotimesW )

idotimesnabla



second right


otimesidotimesid




Aotimes(VotimesW )

idotimesnabla



second right


otimesidotimesid

Ω1AotimesVotimesW



13


Aotimes(VotimesW )

idotimesnabla



second right


idotimesidotimesc





idotimescminus1



second right





cminus1


idotimesFnablaFoo

second right





VotimesWotimesA

idotimes



F

VotimesW c

nabla

VotimesFW

nablaF




VotimesWotimesA

idotimesidotimesd



first term

VotimesWotimesΩ1A

σVotimesW




14


VotimesWotimesΩ1A

idotimesσW



first term

cminus1otimesF doo




The second term is

VotimesWotimesA




second term


idotimesidotimes




cminus1



second term


idotimesidotimes






VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

σotimesid



Φ

OO


15


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO









lowast)



lowast)





lowast)



lowast)

= nablaF (vlowast)



16

















17






Ω1AotimesΩ1Ac

X

Ω1AotimesFΩ1A

XF

Ω2AotimesΩ1A

c Ω2AotimesFΩ1A









where




VotimesW c

nabla

κ


nablaF

A

d


idotimesκ


idotimesF κ




oo Ω1AotimesFA



18











iF (ξ and η) =1



RicciF =(






19


Ω1AotimesAΩ1A c

idotimesRnabla

Ω1AotimesFAΩ1A

idotimesFRnablaF


(idotimesc)c

=


Φminus1


(cotimesid)c

(idotimesi)otimesid




=


Φminus1otimesF id






(22)






i)





botimesξc and



b and ξpotimesξq


= minus1

2Racdb ξ

d and ξbotimesξc




2Racdb gfa ξ



20



= minus1

2Rcbξ

botimesξc



c[minus1])ξ

b[0]otimes

F ξc[0]










21


ξ(tij) = ρij(ξ)







g = K(ωotimesAω)


nablaω = minus1

2[ωotimesAω]


dω +1

2[ω and ω] = 0




22


F = 1 +

infinsumn=1
















x

x+

xminusx3

= ı

0minus2x3

x+

y

x+

xminusx3

= ı

2x3

0minusxminus





23







3) = (1minus c1)x23 + r2c1




(1minus c1)




1minus c1=

xminusr2












24










∆xc =suma+b=c



2




i(2) =sum


df =sumi



parti = minussuma

F (a i)xa+i part

partxa









25







jmnω


= minus(cimjc


ink

)ωmωnotimesωk



= minusδpi〈ωp1

2

(cimjc


jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink











26





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29







F b[0])






77ooooooooooo

(4)













6



VotimesW

c



idotimesF c


(5)












7










VotimesW Υ

c

WotimesVc

VotimesFW

ΥF WotimesFV








= votimesFw











lowast


lowast



8
























9






VnablaF

nabla

Ω1AotimesFV

Ω1AotimesVc

88qqqqqqqqqq

(12)




















V




10








Now we calculate

















11



Now we use









W


W obeys




VotimesW c

nabla

VotimesFW

nablaF



(14)




VotimesAotimesW

otimesid



FotimesF id

VotimesW c

nabla

VotimesFW

nablaF




12

VotimesAotimesW

idotimes



idotimesF F

VotimesW c

nabla

VotimesFW

nablaF




AotimesVotimesW

otimesid



F

VotimesW c

nabla

VotimesFW

nablaF



(15)


AotimesVotimesW

dotimesidotimesid



first right




Aotimes(VotimesW )

idotimesnabla



second right


otimesidotimesid




Aotimes(VotimesW )

idotimesnabla



second right


otimesidotimesid

Ω1AotimesVotimesW



13


Aotimes(VotimesW )

idotimesnabla



second right


idotimesidotimesc





idotimescminus1



second right





cminus1


idotimesFnablaFoo

second right





VotimesWotimesA

idotimes



F

VotimesW c

nabla

VotimesFW

nablaF




VotimesWotimesA

idotimesidotimesd



first term

VotimesWotimesΩ1A

σVotimesW




14


VotimesWotimesΩ1A

idotimesσW



first term

cminus1otimesF doo




The second term is

VotimesWotimesA




second term


idotimesidotimes




cminus1



second term


idotimesidotimes






VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

σotimesid



Φ

OO


15


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO









lowast)



lowast)





lowast)



lowast)

= nablaF (vlowast)



16

















17






Ω1AotimesΩ1Ac

X

Ω1AotimesFΩ1A

XF

Ω2AotimesΩ1A

c Ω2AotimesFΩ1A









where




VotimesW c

nabla

κ


nablaF

A

d


idotimesκ


idotimesF κ




oo Ω1AotimesFA



18











iF (ξ and η) =1



RicciF =(






19


Ω1AotimesAΩ1A c

idotimesRnabla

Ω1AotimesFAΩ1A

idotimesFRnablaF


(idotimesc)c

=


Φminus1


(cotimesid)c

(idotimesi)otimesid




=


Φminus1otimesF id






(22)






i)





botimesξc and



b and ξpotimesξq


= minus1

2Racdb ξ

d and ξbotimesξc




2Racdb gfa ξ



20



= minus1

2Rcbξ

botimesξc



c[minus1])ξ

b[0]otimes

F ξc[0]










21


ξ(tij) = ρij(ξ)







g = K(ωotimesAω)


nablaω = minus1

2[ωotimesAω]


dω +1

2[ω and ω] = 0




22


F = 1 +

infinsumn=1
















x

x+

xminusx3

= ı

0minus2x3

x+

y

x+

xminusx3

= ı

2x3

0minusxminus





23







3) = (1minus c1)x23 + r2c1




(1minus c1)




1minus c1=

xminusr2












24










∆xc =suma+b=c



2




i(2) =sum


df =sumi



parti = minussuma

F (a i)xa+i part

partxa









25







jmnω


= minus(cimjc


ink

)ωmωnotimesωk



= minusδpi〈ωp1

2

(cimjc


jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink











26





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29



VotimesW

c



idotimesF c


(5)












7










VotimesW Υ

c

WotimesVc

VotimesFW

ΥF WotimesFV








= votimesFw











lowast


lowast



8
























9






VnablaF

nabla

Ω1AotimesFV

Ω1AotimesVc

88qqqqqqqqqq

(12)




















V




10








Now we calculate

















11



Now we use









W


W obeys




VotimesW c

nabla

VotimesFW

nablaF



(14)




VotimesAotimesW

otimesid



FotimesF id

VotimesW c

nabla

VotimesFW

nablaF




12

VotimesAotimesW

idotimes



idotimesF F

VotimesW c

nabla

VotimesFW

nablaF




AotimesVotimesW

otimesid



F

VotimesW c

nabla

VotimesFW

nablaF



(15)


AotimesVotimesW

dotimesidotimesid



first right




Aotimes(VotimesW )

idotimesnabla



second right


otimesidotimesid




Aotimes(VotimesW )

idotimesnabla



second right


otimesidotimesid

Ω1AotimesVotimesW



13


Aotimes(VotimesW )

idotimesnabla



second right


idotimesidotimesc





idotimescminus1



second right





cminus1


idotimesFnablaFoo

second right





VotimesWotimesA

idotimes



F

VotimesW c

nabla

VotimesFW

nablaF




VotimesWotimesA

idotimesidotimesd



first term

VotimesWotimesΩ1A

σVotimesW




14


VotimesWotimesΩ1A

idotimesσW



first term

cminus1otimesF doo




The second term is

VotimesWotimesA




second term


idotimesidotimes




cminus1



second term


idotimesidotimes






VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

σotimesid



Φ

OO


15


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO









lowast)



lowast)





lowast)



lowast)

= nablaF (vlowast)



16

















17






Ω1AotimesΩ1Ac

X

Ω1AotimesFΩ1A

XF

Ω2AotimesΩ1A

c Ω2AotimesFΩ1A









where




VotimesW c

nabla

κ


nablaF

A

d


idotimesκ


idotimesF κ




oo Ω1AotimesFA



18











iF (ξ and η) =1



RicciF =(






19


Ω1AotimesAΩ1A c

idotimesRnabla

Ω1AotimesFAΩ1A

idotimesFRnablaF


(idotimesc)c

=


Φminus1


(cotimesid)c

(idotimesi)otimesid




=


Φminus1otimesF id






(22)






i)





botimesξc and



b and ξpotimesξq


= minus1

2Racdb ξ

d and ξbotimesξc




2Racdb gfa ξ



20



= minus1

2Rcbξ

botimesξc



c[minus1])ξ

b[0]otimes

F ξc[0]










21


ξ(tij) = ρij(ξ)







g = K(ωotimesAω)


nablaω = minus1

2[ωotimesAω]


dω +1

2[ω and ω] = 0




22


F = 1 +

infinsumn=1
















x

x+

xminusx3

= ı

0minus2x3

x+

y

x+

xminusx3

= ı

2x3

0minusxminus





23







3) = (1minus c1)x23 + r2c1




(1minus c1)




1minus c1=

xminusr2












24










∆xc =suma+b=c



2




i(2) =sum


df =sumi



parti = minussuma

F (a i)xa+i part

partxa









25







jmnω


= minus(cimjc


ink

)ωmωnotimesωk



= minusδpi〈ωp1

2

(cimjc


jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink











26





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29










VotimesW Υ

c

WotimesVc

VotimesFW

ΥF WotimesFV








= votimesFw











lowast


lowast



8
























9






VnablaF

nabla

Ω1AotimesFV

Ω1AotimesVc

88qqqqqqqqqq

(12)




















V




10








Now we calculate

















11



Now we use









W


W obeys




VotimesW c

nabla

VotimesFW

nablaF



(14)




VotimesAotimesW

otimesid



FotimesF id

VotimesW c

nabla

VotimesFW

nablaF




12

VotimesAotimesW

idotimes



idotimesF F

VotimesW c

nabla

VotimesFW

nablaF




AotimesVotimesW

otimesid



F

VotimesW c

nabla

VotimesFW

nablaF



(15)


AotimesVotimesW

dotimesidotimesid



first right




Aotimes(VotimesW )

idotimesnabla



second right


otimesidotimesid




Aotimes(VotimesW )

idotimesnabla



second right


otimesidotimesid

Ω1AotimesVotimesW



13


Aotimes(VotimesW )

idotimesnabla



second right


idotimesidotimesc





idotimescminus1



second right





cminus1


idotimesFnablaFoo

second right





VotimesWotimesA

idotimes



F

VotimesW c

nabla

VotimesFW

nablaF




VotimesWotimesA

idotimesidotimesd



first term

VotimesWotimesΩ1A

σVotimesW




14


VotimesWotimesΩ1A

idotimesσW



first term

cminus1otimesF doo




The second term is

VotimesWotimesA




second term


idotimesidotimes




cminus1



second term


idotimesidotimes






VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

σotimesid



Φ

OO


15


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO









lowast)



lowast)





lowast)



lowast)

= nablaF (vlowast)



16

















17






Ω1AotimesΩ1Ac

X

Ω1AotimesFΩ1A

XF

Ω2AotimesΩ1A

c Ω2AotimesFΩ1A









where




VotimesW c

nabla

κ


nablaF

A

d


idotimesκ


idotimesF κ




oo Ω1AotimesFA



18











iF (ξ and η) =1



RicciF =(






19


Ω1AotimesAΩ1A c

idotimesRnabla

Ω1AotimesFAΩ1A

idotimesFRnablaF


(idotimesc)c

=


Φminus1


(cotimesid)c

(idotimesi)otimesid




=


Φminus1otimesF id






(22)






i)





botimesξc and



b and ξpotimesξq


= minus1

2Racdb ξ

d and ξbotimesξc




2Racdb gfa ξ



20



= minus1

2Rcbξ

botimesξc



c[minus1])ξ

b[0]otimes

F ξc[0]










21


ξ(tij) = ρij(ξ)







g = K(ωotimesAω)


nablaω = minus1

2[ωotimesAω]


dω +1

2[ω and ω] = 0




22


F = 1 +

infinsumn=1
















x

x+

xminusx3

= ı

0minus2x3

x+

y

x+

xminusx3

= ı

2x3

0minusxminus





23







3) = (1minus c1)x23 + r2c1




(1minus c1)




1minus c1=

xminusr2












24










∆xc =suma+b=c



2




i(2) =sum


df =sumi



parti = minussuma

F (a i)xa+i part

partxa









25







jmnω


= minus(cimjc


ink

)ωmωnotimesωk



= minusδpi〈ωp1

2

(cimjc


jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink











26





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29
























9






VnablaF

nabla

Ω1AotimesFV

Ω1AotimesVc

88qqqqqqqqqq

(12)




















V




10








Now we calculate

















11



Now we use









W


W obeys




VotimesW c

nabla

VotimesFW

nablaF



(14)




VotimesAotimesW

otimesid



FotimesF id

VotimesW c

nabla

VotimesFW

nablaF




12

VotimesAotimesW

idotimes



idotimesF F

VotimesW c

nabla

VotimesFW

nablaF




AotimesVotimesW

otimesid



F

VotimesW c

nabla

VotimesFW

nablaF



(15)


AotimesVotimesW

dotimesidotimesid



first right




Aotimes(VotimesW )

idotimesnabla



second right


otimesidotimesid




Aotimes(VotimesW )

idotimesnabla



second right


otimesidotimesid

Ω1AotimesVotimesW



13


Aotimes(VotimesW )

idotimesnabla



second right


idotimesidotimesc





idotimescminus1



second right





cminus1


idotimesFnablaFoo

second right





VotimesWotimesA

idotimes



F

VotimesW c

nabla

VotimesFW

nablaF




VotimesWotimesA

idotimesidotimesd



first term

VotimesWotimesΩ1A

σVotimesW




14


VotimesWotimesΩ1A

idotimesσW



first term

cminus1otimesF doo




The second term is

VotimesWotimesA




second term


idotimesidotimes




cminus1



second term


idotimesidotimes






VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

σotimesid



Φ

OO


15


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO









lowast)



lowast)





lowast)



lowast)

= nablaF (vlowast)



16

















17






Ω1AotimesΩ1Ac

X

Ω1AotimesFΩ1A

XF

Ω2AotimesΩ1A

c Ω2AotimesFΩ1A









where




VotimesW c

nabla

κ


nablaF

A

d


idotimesκ


idotimesF κ




oo Ω1AotimesFA



18











iF (ξ and η) =1



RicciF =(






19


Ω1AotimesAΩ1A c

idotimesRnabla

Ω1AotimesFAΩ1A

idotimesFRnablaF


(idotimesc)c

=


Φminus1


(cotimesid)c

(idotimesi)otimesid




=


Φminus1otimesF id






(22)






i)





botimesξc and



b and ξpotimesξq


= minus1

2Racdb ξ

d and ξbotimesξc




2Racdb gfa ξ



20



= minus1

2Rcbξ

botimesξc



c[minus1])ξ

b[0]otimes

F ξc[0]










21


ξ(tij) = ρij(ξ)







g = K(ωotimesAω)


nablaω = minus1

2[ωotimesAω]


dω +1

2[ω and ω] = 0




22


F = 1 +

infinsumn=1
















x

x+

xminusx3

= ı

0minus2x3

x+

y

x+

xminusx3

= ı

2x3

0minusxminus





23







3) = (1minus c1)x23 + r2c1




(1minus c1)




1minus c1=

xminusr2












24










∆xc =suma+b=c



2




i(2) =sum


df =sumi



parti = minussuma

F (a i)xa+i part

partxa









25







jmnω


= minus(cimjc


ink

)ωmωnotimesωk



= minusδpi〈ωp1

2

(cimjc


jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink











26





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29






VnablaF

nabla

Ω1AotimesFV

Ω1AotimesVc

88qqqqqqqqqq

(12)




















V




10








Now we calculate

















11



Now we use









W


W obeys




VotimesW c

nabla

VotimesFW

nablaF



(14)




VotimesAotimesW

otimesid



FotimesF id

VotimesW c

nabla

VotimesFW

nablaF




12

VotimesAotimesW

idotimes



idotimesF F

VotimesW c

nabla

VotimesFW

nablaF




AotimesVotimesW

otimesid



F

VotimesW c

nabla

VotimesFW

nablaF



(15)


AotimesVotimesW

dotimesidotimesid



first right




Aotimes(VotimesW )

idotimesnabla



second right


otimesidotimesid




Aotimes(VotimesW )

idotimesnabla



second right


otimesidotimesid

Ω1AotimesVotimesW



13


Aotimes(VotimesW )

idotimesnabla



second right


idotimesidotimesc





idotimescminus1



second right





cminus1


idotimesFnablaFoo

second right





VotimesWotimesA

idotimes



F

VotimesW c

nabla

VotimesFW

nablaF




VotimesWotimesA

idotimesidotimesd



first term

VotimesWotimesΩ1A

σVotimesW




14


VotimesWotimesΩ1A

idotimesσW



first term

cminus1otimesF doo




The second term is

VotimesWotimesA




second term


idotimesidotimes




cminus1



second term


idotimesidotimes






VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

σotimesid



Φ

OO


15


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO









lowast)



lowast)





lowast)



lowast)

= nablaF (vlowast)



16

















17






Ω1AotimesΩ1Ac

X

Ω1AotimesFΩ1A

XF

Ω2AotimesΩ1A

c Ω2AotimesFΩ1A









where




VotimesW c

nabla

κ


nablaF

A

d


idotimesκ


idotimesF κ




oo Ω1AotimesFA



18











iF (ξ and η) =1



RicciF =(






19


Ω1AotimesAΩ1A c

idotimesRnabla

Ω1AotimesFAΩ1A

idotimesFRnablaF


(idotimesc)c

=


Φminus1


(cotimesid)c

(idotimesi)otimesid




=


Φminus1otimesF id






(22)






i)





botimesξc and



b and ξpotimesξq


= minus1

2Racdb ξ

d and ξbotimesξc




2Racdb gfa ξ



20



= minus1

2Rcbξ

botimesξc



c[minus1])ξ

b[0]otimes

F ξc[0]










21


ξ(tij) = ρij(ξ)







g = K(ωotimesAω)


nablaω = minus1

2[ωotimesAω]


dω +1

2[ω and ω] = 0




22


F = 1 +

infinsumn=1
















x

x+

xminusx3

= ı

0minus2x3

x+

y

x+

xminusx3

= ı

2x3

0minusxminus





23







3) = (1minus c1)x23 + r2c1




(1minus c1)




1minus c1=

xminusr2












24










∆xc =suma+b=c



2




i(2) =sum


df =sumi



parti = minussuma

F (a i)xa+i part

partxa









25







jmnω


= minus(cimjc


ink

)ωmωnotimesωk



= minusδpi〈ωp1

2

(cimjc


jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink











26





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29








Now we calculate

















11



Now we use









W


W obeys




VotimesW c

nabla

VotimesFW

nablaF



(14)




VotimesAotimesW

otimesid



FotimesF id

VotimesW c

nabla

VotimesFW

nablaF




12

VotimesAotimesW

idotimes



idotimesF F

VotimesW c

nabla

VotimesFW

nablaF




AotimesVotimesW

otimesid



F

VotimesW c

nabla

VotimesFW

nablaF



(15)


AotimesVotimesW

dotimesidotimesid



first right




Aotimes(VotimesW )

idotimesnabla



second right


otimesidotimesid




Aotimes(VotimesW )

idotimesnabla



second right


otimesidotimesid

Ω1AotimesVotimesW



13


Aotimes(VotimesW )

idotimesnabla



second right


idotimesidotimesc





idotimescminus1



second right





cminus1


idotimesFnablaFoo

second right





VotimesWotimesA

idotimes



F

VotimesW c

nabla

VotimesFW

nablaF




VotimesWotimesA

idotimesidotimesd



first term

VotimesWotimesΩ1A

σVotimesW




14


VotimesWotimesΩ1A

idotimesσW



first term

cminus1otimesF doo




The second term is

VotimesWotimesA




second term


idotimesidotimes




cminus1



second term


idotimesidotimes






VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

σotimesid



Φ

OO


15


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO









lowast)



lowast)





lowast)



lowast)

= nablaF (vlowast)



16

















17






Ω1AotimesΩ1Ac

X

Ω1AotimesFΩ1A

XF

Ω2AotimesΩ1A

c Ω2AotimesFΩ1A









where




VotimesW c

nabla

κ


nablaF

A

d


idotimesκ


idotimesF κ




oo Ω1AotimesFA



18











iF (ξ and η) =1



RicciF =(






19


Ω1AotimesAΩ1A c

idotimesRnabla

Ω1AotimesFAΩ1A

idotimesFRnablaF


(idotimesc)c

=


Φminus1


(cotimesid)c

(idotimesi)otimesid




=


Φminus1otimesF id






(22)






i)





botimesξc and



b and ξpotimesξq


= minus1

2Racdb ξ

d and ξbotimesξc




2Racdb gfa ξ



20



= minus1

2Rcbξ

botimesξc



c[minus1])ξ

b[0]otimes

F ξc[0]










21


ξ(tij) = ρij(ξ)







g = K(ωotimesAω)


nablaω = minus1

2[ωotimesAω]


dω +1

2[ω and ω] = 0




22


F = 1 +

infinsumn=1
















x

x+

xminusx3

= ı

0minus2x3

x+

y

x+

xminusx3

= ı

2x3

0minusxminus





23







3) = (1minus c1)x23 + r2c1




(1minus c1)




1minus c1=

xminusr2












24










∆xc =suma+b=c



2




i(2) =sum


df =sumi



parti = minussuma

F (a i)xa+i part

partxa









25







jmnω


= minus(cimjc


ink

)ωmωnotimesωk



= minusδpi〈ωp1

2

(cimjc


jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink











26





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29



Now we use









W


W obeys




VotimesW c

nabla

VotimesFW

nablaF



(14)




VotimesAotimesW

otimesid



FotimesF id

VotimesW c

nabla

VotimesFW

nablaF




12

VotimesAotimesW

idotimes



idotimesF F

VotimesW c

nabla

VotimesFW

nablaF




AotimesVotimesW

otimesid



F

VotimesW c

nabla

VotimesFW

nablaF



(15)


AotimesVotimesW

dotimesidotimesid



first right




Aotimes(VotimesW )

idotimesnabla



second right


otimesidotimesid




Aotimes(VotimesW )

idotimesnabla



second right


otimesidotimesid

Ω1AotimesVotimesW



13


Aotimes(VotimesW )

idotimesnabla



second right


idotimesidotimesc





idotimescminus1



second right





cminus1


idotimesFnablaFoo

second right





VotimesWotimesA

idotimes



F

VotimesW c

nabla

VotimesFW

nablaF




VotimesWotimesA

idotimesidotimesd



first term

VotimesWotimesΩ1A

σVotimesW




14


VotimesWotimesΩ1A

idotimesσW



first term

cminus1otimesF doo




The second term is

VotimesWotimesA




second term


idotimesidotimes




cminus1



second term


idotimesidotimes






VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

σotimesid



Φ

OO


15


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO









lowast)



lowast)





lowast)



lowast)

= nablaF (vlowast)



16

















17






Ω1AotimesΩ1Ac

X

Ω1AotimesFΩ1A

XF

Ω2AotimesΩ1A

c Ω2AotimesFΩ1A









where




VotimesW c

nabla

κ


nablaF

A

d


idotimesκ


idotimesF κ




oo Ω1AotimesFA



18











iF (ξ and η) =1



RicciF =(






19


Ω1AotimesAΩ1A c

idotimesRnabla

Ω1AotimesFAΩ1A

idotimesFRnablaF


(idotimesc)c

=


Φminus1


(cotimesid)c

(idotimesi)otimesid




=


Φminus1otimesF id






(22)






i)





botimesξc and



b and ξpotimesξq


= minus1

2Racdb ξ

d and ξbotimesξc




2Racdb gfa ξ



20



= minus1

2Rcbξ

botimesξc



c[minus1])ξ

b[0]otimes

F ξc[0]










21


ξ(tij) = ρij(ξ)







g = K(ωotimesAω)


nablaω = minus1

2[ωotimesAω]


dω +1

2[ω and ω] = 0




22


F = 1 +

infinsumn=1
















x

x+

xminusx3

= ı

0minus2x3

x+

y

x+

xminusx3

= ı

2x3

0minusxminus





23







3) = (1minus c1)x23 + r2c1




(1minus c1)




1minus c1=

xminusr2












24










∆xc =suma+b=c



2




i(2) =sum


df =sumi



parti = minussuma

F (a i)xa+i part

partxa









25







jmnω


= minus(cimjc


ink

)ωmωnotimesωk



= minusδpi〈ωp1

2

(cimjc


jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink











26





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29

VotimesAotimesW

idotimes



idotimesF F

VotimesW c

nabla

VotimesFW

nablaF




AotimesVotimesW

otimesid



F

VotimesW c

nabla

VotimesFW

nablaF



(15)


AotimesVotimesW

dotimesidotimesid



first right




Aotimes(VotimesW )

idotimesnabla



second right


otimesidotimesid




Aotimes(VotimesW )

idotimesnabla



second right


otimesidotimesid

Ω1AotimesVotimesW



13


Aotimes(VotimesW )

idotimesnabla



second right


idotimesidotimesc





idotimescminus1



second right





cminus1


idotimesFnablaFoo

second right





VotimesWotimesA

idotimes



F

VotimesW c

nabla

VotimesFW

nablaF




VotimesWotimesA

idotimesidotimesd



first term

VotimesWotimesΩ1A

σVotimesW




14


VotimesWotimesΩ1A

idotimesσW



first term

cminus1otimesF doo




The second term is

VotimesWotimesA




second term


idotimesidotimes




cminus1



second term


idotimesidotimes






VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

σotimesid



Φ

OO


15


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO









lowast)



lowast)





lowast)



lowast)

= nablaF (vlowast)



16

















17






Ω1AotimesΩ1Ac

X

Ω1AotimesFΩ1A

XF

Ω2AotimesΩ1A

c Ω2AotimesFΩ1A









where




VotimesW c

nabla

κ


nablaF

A

d


idotimesκ


idotimesF κ




oo Ω1AotimesFA



18











iF (ξ and η) =1



RicciF =(






19


Ω1AotimesAΩ1A c

idotimesRnabla

Ω1AotimesFAΩ1A

idotimesFRnablaF


(idotimesc)c

=


Φminus1


(cotimesid)c

(idotimesi)otimesid




=


Φminus1otimesF id






(22)






i)





botimesξc and



b and ξpotimesξq


= minus1

2Racdb ξ

d and ξbotimesξc




2Racdb gfa ξ



20



= minus1

2Rcbξ

botimesξc



c[minus1])ξ

b[0]otimes

F ξc[0]










21


ξ(tij) = ρij(ξ)







g = K(ωotimesAω)


nablaω = minus1

2[ωotimesAω]


dω +1

2[ω and ω] = 0




22


F = 1 +

infinsumn=1
















x

x+

xminusx3

= ı

0minus2x3

x+

y

x+

xminusx3

= ı

2x3

0minusxminus





23







3) = (1minus c1)x23 + r2c1




(1minus c1)




1minus c1=

xminusr2












24










∆xc =suma+b=c



2




i(2) =sum


df =sumi



parti = minussuma

F (a i)xa+i part

partxa









25







jmnω


= minus(cimjc


ink

)ωmωnotimesωk



= minusδpi〈ωp1

2

(cimjc


jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink











26





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29


Aotimes(VotimesW )

idotimesnabla



second right


idotimesidotimesc





idotimescminus1



second right





cminus1


idotimesFnablaFoo

second right





VotimesWotimesA

idotimes



F

VotimesW c

nabla

VotimesFW

nablaF




VotimesWotimesA

idotimesidotimesd



first term

VotimesWotimesΩ1A

σVotimesW




14


VotimesWotimesΩ1A

idotimesσW



first term

cminus1otimesF doo




The second term is

VotimesWotimesA




second term


idotimesidotimes




cminus1



second term


idotimesidotimes






VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

σotimesid



Φ

OO


15


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO









lowast)



lowast)





lowast)



lowast)

= nablaF (vlowast)



16

















17






Ω1AotimesΩ1Ac

X

Ω1AotimesFΩ1A

XF

Ω2AotimesΩ1A

c Ω2AotimesFΩ1A









where




VotimesW c

nabla

κ


nablaF

A

d


idotimesκ


idotimesF κ




oo Ω1AotimesFA



18











iF (ξ and η) =1



RicciF =(






19


Ω1AotimesAΩ1A c

idotimesRnabla

Ω1AotimesFAΩ1A

idotimesFRnablaF


(idotimesc)c

=


Φminus1


(cotimesid)c

(idotimesi)otimesid




=


Φminus1otimesF id






(22)






i)





botimesξc and



b and ξpotimesξq


= minus1

2Racdb ξ

d and ξbotimesξc




2Racdb gfa ξ



20



= minus1

2Rcbξ

botimesξc



c[minus1])ξ

b[0]otimes

F ξc[0]










21


ξ(tij) = ρij(ξ)







g = K(ωotimesAω)


nablaω = minus1

2[ωotimesAω]


dω +1

2[ω and ω] = 0




22


F = 1 +

infinsumn=1
















x

x+

xminusx3

= ı

0minus2x3

x+

y

x+

xminusx3

= ı

2x3

0minusxminus





23







3) = (1minus c1)x23 + r2c1




(1minus c1)




1minus c1=

xminusr2












24










∆xc =suma+b=c



2




i(2) =sum


df =sumi



parti = minussuma

F (a i)xa+i part

partxa









25







jmnω


= minus(cimjc


ink

)ωmωnotimesωk



= minusδpi〈ωp1

2

(cimjc


jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink











26





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29


VotimesWotimesΩ1A

idotimesσW



first term

cminus1otimesF doo




The second term is

VotimesWotimesA




second term


idotimesidotimes




cminus1



second term


idotimesidotimes






VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

σotimesid



Φ

OO


15


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO









lowast)



lowast)





lowast)



lowast)

= nablaF (vlowast)



16

















17






Ω1AotimesΩ1Ac

X

Ω1AotimesFΩ1A

XF

Ω2AotimesΩ1A

c Ω2AotimesFΩ1A









where




VotimesW c

nabla

κ


nablaF

A

d


idotimesκ


idotimesF κ




oo Ω1AotimesFA



18











iF (ξ and η) =1



RicciF =(






19


Ω1AotimesAΩ1A c

idotimesRnabla

Ω1AotimesFAΩ1A

idotimesFRnablaF


(idotimesc)c

=


Φminus1


(cotimesid)c

(idotimesi)otimesid




=


Φminus1otimesF id






(22)






i)





botimesξc and



b and ξpotimesξq


= minus1

2Racdb ξ

d and ξbotimesξc




2Racdb gfa ξ



20



= minus1

2Rcbξ

botimesξc



c[minus1])ξ

b[0]otimes

F ξc[0]










21


ξ(tij) = ρij(ξ)







g = K(ωotimesAω)


nablaω = minus1

2[ωotimesAω]


dω +1

2[ω and ω] = 0




22


F = 1 +

infinsumn=1
















x

x+

xminusx3

= ı

0minus2x3

x+

y

x+

xminusx3

= ı

2x3

0minusxminus





23







3) = (1minus c1)x23 + r2c1




(1minus c1)




1minus c1=

xminusr2












24










∆xc =suma+b=c



2




i(2) =sum


df =sumi



parti = minussuma

F (a i)xa+i part

partxa









25







jmnω


= minus(cimjc


ink

)ωmωnotimesωk



= minusδpi〈ωp1

2

(cimjc


jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink











26





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO


VotimesW c

idotimesnablaW

VotimesFW

second

VotimesΩ1AotimesW

c



Φ

OO









lowast)



lowast)





lowast)



lowast)

= nablaF (vlowast)



16

















17






Ω1AotimesΩ1Ac

X

Ω1AotimesFΩ1A

XF

Ω2AotimesΩ1A

c Ω2AotimesFΩ1A









where




VotimesW c

nabla

κ


nablaF

A

d


idotimesκ


idotimesF κ




oo Ω1AotimesFA



18











iF (ξ and η) =1



RicciF =(






19


Ω1AotimesAΩ1A c

idotimesRnabla

Ω1AotimesFAΩ1A

idotimesFRnablaF


(idotimesc)c

=


Φminus1


(cotimesid)c

(idotimesi)otimesid




=


Φminus1otimesF id






(22)






i)





botimesξc and



b and ξpotimesξq


= minus1

2Racdb ξ

d and ξbotimesξc




2Racdb gfa ξ



20



= minus1

2Rcbξ

botimesξc



c[minus1])ξ

b[0]otimes

F ξc[0]










21


ξ(tij) = ρij(ξ)







g = K(ωotimesAω)


nablaω = minus1

2[ωotimesAω]


dω +1

2[ω and ω] = 0




22


F = 1 +

infinsumn=1
















x

x+

xminusx3

= ı

0minus2x3

x+

y

x+

xminusx3

= ı

2x3

0minusxminus





23







3) = (1minus c1)x23 + r2c1




(1minus c1)




1minus c1=

xminusr2












24










∆xc =suma+b=c



2




i(2) =sum


df =sumi



parti = minussuma

F (a i)xa+i part

partxa









25







jmnω


= minus(cimjc


ink

)ωmωnotimesωk



= minusδpi〈ωp1

2

(cimjc


jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink











26





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29

















17






Ω1AotimesΩ1Ac

X

Ω1AotimesFΩ1A

XF

Ω2AotimesΩ1A

c Ω2AotimesFΩ1A









where




VotimesW c

nabla

κ


nablaF

A

d


idotimesκ


idotimesF κ




oo Ω1AotimesFA



18











iF (ξ and η) =1



RicciF =(






19


Ω1AotimesAΩ1A c

idotimesRnabla

Ω1AotimesFAΩ1A

idotimesFRnablaF


(idotimesc)c

=


Φminus1


(cotimesid)c

(idotimesi)otimesid




=


Φminus1otimesF id






(22)






i)





botimesξc and



b and ξpotimesξq


= minus1

2Racdb ξ

d and ξbotimesξc




2Racdb gfa ξ



20



= minus1

2Rcbξ

botimesξc



c[minus1])ξ

b[0]otimes

F ξc[0]










21


ξ(tij) = ρij(ξ)







g = K(ωotimesAω)


nablaω = minus1

2[ωotimesAω]


dω +1

2[ω and ω] = 0




22


F = 1 +

infinsumn=1
















x

x+

xminusx3

= ı

0minus2x3

x+

y

x+

xminusx3

= ı

2x3

0minusxminus





23







3) = (1minus c1)x23 + r2c1




(1minus c1)




1minus c1=

xminusr2












24










∆xc =suma+b=c



2




i(2) =sum


df =sumi



parti = minussuma

F (a i)xa+i part

partxa









25







jmnω


= minus(cimjc


ink

)ωmωnotimesωk



= minusδpi〈ωp1

2

(cimjc


jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink











26





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29






Ω1AotimesΩ1Ac

X

Ω1AotimesFΩ1A

XF

Ω2AotimesΩ1A

c Ω2AotimesFΩ1A









where




VotimesW c

nabla

κ


nablaF

A

d


idotimesκ


idotimesF κ




oo Ω1AotimesFA



18











iF (ξ and η) =1



RicciF =(






19


Ω1AotimesAΩ1A c

idotimesRnabla

Ω1AotimesFAΩ1A

idotimesFRnablaF


(idotimesc)c

=


Φminus1


(cotimesid)c

(idotimesi)otimesid




=


Φminus1otimesF id






(22)






i)





botimesξc and



b and ξpotimesξq


= minus1

2Racdb ξ

d and ξbotimesξc




2Racdb gfa ξ



20



= minus1

2Rcbξ

botimesξc



c[minus1])ξ

b[0]otimes

F ξc[0]










21


ξ(tij) = ρij(ξ)







g = K(ωotimesAω)


nablaω = minus1

2[ωotimesAω]


dω +1

2[ω and ω] = 0




22


F = 1 +

infinsumn=1
















x

x+

xminusx3

= ı

0minus2x3

x+

y

x+

xminusx3

= ı

2x3

0minusxminus





23







3) = (1minus c1)x23 + r2c1




(1minus c1)




1minus c1=

xminusr2












24










∆xc =suma+b=c



2




i(2) =sum


df =sumi



parti = minussuma

F (a i)xa+i part

partxa









25







jmnω


= minus(cimjc


ink

)ωmωnotimesωk



= minusδpi〈ωp1

2

(cimjc


jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink











26





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29











iF (ξ and η) =1



RicciF =(






19


Ω1AotimesAΩ1A c

idotimesRnabla

Ω1AotimesFAΩ1A

idotimesFRnablaF


(idotimesc)c

=


Φminus1


(cotimesid)c

(idotimesi)otimesid




=


Φminus1otimesF id






(22)






i)





botimesξc and



b and ξpotimesξq


= minus1

2Racdb ξ

d and ξbotimesξc




2Racdb gfa ξ



20



= minus1

2Rcbξ

botimesξc



c[minus1])ξ

b[0]otimes

F ξc[0]










21


ξ(tij) = ρij(ξ)







g = K(ωotimesAω)


nablaω = minus1

2[ωotimesAω]


dω +1

2[ω and ω] = 0




22


F = 1 +

infinsumn=1
















x

x+

xminusx3

= ı

0minus2x3

x+

y

x+

xminusx3

= ı

2x3

0minusxminus





23







3) = (1minus c1)x23 + r2c1




(1minus c1)




1minus c1=

xminusr2












24










∆xc =suma+b=c



2




i(2) =sum


df =sumi



parti = minussuma

F (a i)xa+i part

partxa









25







jmnω


= minus(cimjc


ink

)ωmωnotimesωk



= minusδpi〈ωp1

2

(cimjc


jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink











26





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29


Ω1AotimesAΩ1A c

idotimesRnabla

Ω1AotimesFAΩ1A

idotimesFRnablaF


(idotimesc)c

=


Φminus1


(cotimesid)c

(idotimesi)otimesid




=


Φminus1otimesF id






(22)






i)





botimesξc and



b and ξpotimesξq


= minus1

2Racdb ξ

d and ξbotimesξc




2Racdb gfa ξ



20



= minus1

2Rcbξ

botimesξc



c[minus1])ξ

b[0]otimes

F ξc[0]










21


ξ(tij) = ρij(ξ)







g = K(ωotimesAω)


nablaω = minus1

2[ωotimesAω]


dω +1

2[ω and ω] = 0




22


F = 1 +

infinsumn=1
















x

x+

xminusx3

= ı

0minus2x3

x+

y

x+

xminusx3

= ı

2x3

0minusxminus





23







3) = (1minus c1)x23 + r2c1




(1minus c1)




1minus c1=

xminusr2












24










∆xc =suma+b=c



2




i(2) =sum


df =sumi



parti = minussuma

F (a i)xa+i part

partxa









25







jmnω


= minus(cimjc


ink

)ωmωnotimesωk



= minusδpi〈ωp1

2

(cimjc


jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink











26





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29



= minus1

2Rcbξ

botimesξc



c[minus1])ξ

b[0]otimes

F ξc[0]










21


ξ(tij) = ρij(ξ)







g = K(ωotimesAω)


nablaω = minus1

2[ωotimesAω]


dω +1

2[ω and ω] = 0




22


F = 1 +

infinsumn=1
















x

x+

xminusx3

= ı

0minus2x3

x+

y

x+

xminusx3

= ı

2x3

0minusxminus





23







3) = (1minus c1)x23 + r2c1




(1minus c1)




1minus c1=

xminusr2












24










∆xc =suma+b=c



2




i(2) =sum


df =sumi



parti = minussuma

F (a i)xa+i part

partxa









25







jmnω


= minus(cimjc


ink

)ωmωnotimesωk



= minusδpi〈ωp1

2

(cimjc


jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink











26





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29


ξ(tij) = ρij(ξ)







g = K(ωotimesAω)


nablaω = minus1

2[ωotimesAω]


dω +1

2[ω and ω] = 0




22


F = 1 +

infinsumn=1
















x

x+

xminusx3

= ı

0minus2x3

x+

y

x+

xminusx3

= ı

2x3

0minusxminus





23







3) = (1minus c1)x23 + r2c1




(1minus c1)




1minus c1=

xminusr2












24










∆xc =suma+b=c



2




i(2) =sum


df =sumi



parti = minussuma

F (a i)xa+i part

partxa









25







jmnω


= minus(cimjc


ink

)ωmωnotimesωk



= minusδpi〈ωp1

2

(cimjc


jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink











26





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29


F = 1 +

infinsumn=1
















x

x+

xminusx3

= ı

0minus2x3

x+

y

x+

xminusx3

= ı

2x3

0minusxminus





23







3) = (1minus c1)x23 + r2c1




(1minus c1)




1minus c1=

xminusr2












24










∆xc =suma+b=c



2




i(2) =sum


df =sumi



parti = minussuma

F (a i)xa+i part

partxa









25







jmnω


= minus(cimjc


ink

)ωmωnotimesωk



= minusδpi〈ωp1

2

(cimjc


jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink











26





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29







3) = (1minus c1)x23 + r2c1




(1minus c1)




1minus c1=

xminusr2












24










∆xc =suma+b=c



2




i(2) =sum


df =sumi



parti = minussuma

F (a i)xa+i part

partxa









25







jmnω


= minus(cimjc


ink

)ωmωnotimesωk



= minusδpi〈ωp1

2

(cimjc


jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink











26





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29










∆xc =suma+b=c



2




i(2) =sum


df =sumi



parti = minussuma

F (a i)xa+i part

partxa









25







jmnω


= minus(cimjc


ink

)ωmωnotimesωk



= minusδpi〈ωp1

2

(cimjc


jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink











26





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29







jmnω


= minus(cimjc


ink

)ωmωnotimesωk



= minusδpi〈ωp1

2

(cimjc


jmn + partmc

ink minus partncimk

)ωm〉ωnotimesωk

= minus1

2

(ciijc

jnk minus c

injc

jik minus 2cijkc

jin + partic

ink minus partnciik

)ωnotimesωk

=1

2

(cinjc

jki minus particink











26





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29





(prod

(xi)ni) bull (prod


(xi)ni+mi





xi bull xi





xi bull dxi













27


RicciF = minus3gF

















arXiv08124942


28







29


RicciF = minus3gF

















arXiv08124942


28







29







29

Documents

Nonassociative Riemannian geometry by twisting