Noncommutative Structures in Mathematics and Physics pot

The concepts of noncommutative space-time and quantum groups have foundgrowing attention in quantum field theory and string theory.. Fad-deev and his collaborators had also interpreted t

Steven Duplij and Julius Wess

Noncommutative Structures in

Mathematics and Physics

Proceedings of theNATO Advanced Research Workshop

“NONCOMMUTATIVE STRUCTURES IN

MATHEMATICS AND PHYSICS”

Kiev, UkraineSeptember 24-28, 2000

Steven Duplij

Theory Group

Nuclear Physics Laboratory

Kharkov National University

PREFACE viii

D Leites, V Serganova Symmetries Wider Than Supersymmetry 13

P Grozman, D Leites An Unconventional Supergravity 41

E Bergshoeff, R Kallosh, A Van Proeyen Supersymmetry Of RS

D Galtsov, V Dyadichev D-branes And Vacuum Periodicity 61

P Kosi ´nski, J Lukierski, P Ma´slanka Quantum Deformations Of

Space-Time SUSY And Noncommutative Superfield Theory 79

D Leites, I Shchepochkina The Howe Duality And Lie Superalgebras 93

A Sergeev Enveloping Algebra Of GL(3) And Orthogonal Polynomials 113

S Duplij, W Marcinek Noninvertibility, Semisupermanifolds And

F Brandt An Overview Of New Supersymmetric Gauge Theories With

K Peeters, P Vanhove, A Westerberg SupersymmetricR4 Actions

And Quantum Corrections To Superspace Torsion Constraints 153

S Fedoruk, V G Zima Massive Superparticle With Spinorial Central

A Burinskii Rotating Super Black Hole as Spinning Particle 181

F Toppan ClassifyingN-extended 1-dimensional Super Systems 195

C Quesne Para, Pseudo, And Orthosupersymmetric Quantum

A Frydryszak Supersymmetric Odd Mechanical Systems And Hilbert

S Vacaru, I Chiosa, N Vicol Locally Anisotropic Supergravity And

Gauge Gravity On Noncommutative Spaces 229

T Kobayashi, J Kubo, M Mondrag´on, G Zoupanos Finiteness In

J Simon World Volume Realization Of Automorphisms 259

G Fiore, M Maceda, J Madore Some Metrics On The Manin Plane 271

V Lyubashenko Coherence Isomorphisms For A Hopf Category 283

V Mazorchuk On Categories Of Gelfand-Zetlin Modules 299

D Shklyarov, S Sinel’shchikov, L Vaksman Hidden Symmetry Of

Some Algebras Of q-differential Operators 309

P Jorgensen, D Proskurin, Y Samoilenko A Family Of∗-Algebras

Allowing Wick Ordering: Fock Representations And Universal

A U Klimyk Nonstandard Quantization Of The Enevloping Algebra

A Gavrilik Can the Cabibbo mixing originate from noncommutative

N Iorgov Nonclassical Type Representations Of Nonstandard

Quantization Of Enveloping Algebras U(so(n)), U(so(n,1)) and

K Landsteiner Quasiparticles In Non-commutative Field Theory 369

A Sergyeyev Time Dependence And (Non)Commutativity Of

B Dragovich, I V Volovich p-Adic Strings And Noncommutativity 391

G Djordjevi´c, B Dragovich, L Neˇsi´c Adelic Quantum Mechanics:

Nonarchimedean And Noncommutative Aspects 401

Y Kozitsky Gibbs States Of A Lattice System Of Quantum Anharmonic

D Vassiliev A Metric-Affine Field Model For The Neutrino 427

M Visinescu Generalized Taub-NUT Metrics And Killing-Yano

V Dzhunushaliev An Effective Model Of The Spacetime Foam 453

A Higuchi Possible Constraints On String Theory In Closed Space

A Alscher, H Grabert Semiclassical Dynamics OfSU(2) Models 475

The concepts of noncommutative space-time and quantum groups have foundgrowing attention in quantum field theory and string theory The mathematicalconcepts of quantum groups have been far developed by mathematicians andphysicists of the Eastern European countries Especially, V G Drinfeld fromUkraine, S Woronowicz from Poland and L D Faddeev from Russia have beenpioneering the field It seems to be natural to bring together these scientists withresearchers in string theory and quantum field theory of the Western Europeancountries From another side, supersymmetry, as one of examples of noncom-mutative structure, was discovered in early 70’s in the West by J Wess (one

of the co-Directors) and B Zumino and in the East by physicists from Ukraine

V P Akulov and D V Volkov Therefore, Ukraine seems to be a natural place tomeet

Supersymmetry is a very important and intriguing mathematical conceptwhich has become a basic ingredient in many branches of modern theoreticalphysics In spite of its still lacking physical evidence, its far-reaching theoret-ical implications uphold the belief that supersymmetry plays a prominent role

in the fundamental laws of nature At present the most promising hope for atruly supersymmetric unified and finite description of quantum field theory andgeneral relativity is superstring theory and its latest formulation, Witten’s M-theory Superstrings possess by far the largest set of gauge symmetries ever found

in physics, perhaps even large enough to eliminate all divergences in quantumgravity Not only does superstring’s symmetry include that of Einstein’s theory ofgeneral relativity and the Yang-Mills theory, it also includes supergravity and theGrand Unified Theories

One of the exciting new approaches to nonperturbative string theory involvesM-theory and duality, which, in fact, force theoretical physicists to reconsider thecentral role played by strings in supersymmetry In this revised new picture allfive superstring theories, which on first glance have entirely different propertiesand spectra, are now seen as different vacua of a same theory, M-theory Thisunification cannot, however, occur at the perturbative level, because it is preciselythe perturbative analysis which singles out the five different string theories Thehope is that when one goes beyond this perturbative limit, and takes into accountall non-perturbative effects, the five string theories turn out to be five differentdescriptions of the same physics In this context a duality is a particular relationapplying to string theories, which can map for instance the strong coupling re-gion of a theory to the weak coupling region of the same theory or of another

one, and vice versa, thus being an intrinsically non-perturbative relation In therecent years, the structure of M-theory has begun to be uncovered, with the es-sential tool provided by supersymmetry Its most striking characteristic is that itindicates that space-time should be eleven dimensional Because of the intrinsicnon-perturbative nature of any approach to M-theory, the study of the p-brane

solitons, or more simply ‘branes’, is a natural step to take The branes are extendedobjects present in M-theory or in string theories, generally associated to classicalsolutions of the respective supergravities

Quantum groups arise as the abstract structure underlying the symmetries ofintegrable systems Then the theory of quantum inverse scattering gives rise tosome deformed algebraic structures which were first explained by Drinfeld asdeformations of the envelopping algebras of the classical Lie algebras An analo-gous structure was obtained by Woronowicz in the context of noncommutative

C∗-algebras There is a third approach, due to Yu I Manin, where quantumgroups are interpreted as the endomorphisms of certain noncommutative algebraicvarieties defined by quadratic algebras, called quantum linear spaces L D Fad-deev and his collaborators had also interpreted the quantum groups from the point

of view of corepresentations and quantum spaces, furnishing a connection withthe quantum deformations of the universal enveloping algebras and the quantumdouble of Hopf algebras From the algebraic point of view, quantum groups areHopf algebras and the relation with the endomorphism algebra of quantum linearspaces comes from their corepresentations on tensor product spaces The usualconstruction of the coaction on the tensor product space involves the flip operatorinterchanging factors of the tensor product of the quantum linear spaces with thebialgebra This fact implies the commutativity between the matrix elements of

a representation of the endomorphism and the coordinates of the quantum ear spaces Moreover, the flip operator for the tensor product is also involved

lin-in many steps of the construction of quantum groups In the braided approach

toq-deformations the flip operator is replaced with a braiding giving rise to the

quasi-tensor category ofk-modules, where a natural braided coaction appears

The study of differential geometry and differential calculus on quantumgroups that Woronowicz initiated is also very important and worthwile to investi-gate Next step in this direction is consideration of noncommutative space-time as

a possible realistic picture of how space-time behaves at short distances Startingfrom such a noncommutative space as configuration space, one can generalize

it to a phase space where noncommutativity is already intrinsic for a quantummechanical system The definition of this noncommutative phase space is derivedfrom the noncommutative differential structure on the configuration space Thenoncommutative phase space is aq-deformation of the quantum mechanical phase

space and one can apply all the machinery learned from quantum mechanics

If one demands that space-time variables are modules or co-modules of the

q-deformed Lorentz group, then they satisfy commutation relations that make them

elements of a non-commutative space The action of momenta on this space isnon-commutative as well The full structure is determined by the (co-)moduleproperty It can serve as an explicit example of a non-commutative structure forspace-time This has the advantages that the q-deformed Lorentz group plays

the role of a kinematical group and thus determines many of the properties ofthis space and allows explicit calculations One can explicitly construct Hilbertspace representations of the algebra and find that the vectors in the Hilbert spacecan be determined by measuring the time, the three-dimensional distance, the q-

deformed angular momentum and its third component The eigenvalues of theseobservables form aq-lattice with accumulation points on the light-cone In a way

physics on the light-cone is best approximated by this q-deformation One can

consider the simplest version of aq-deformed Heisenberg algebra as an example

of a noncommutative structure, first derive a calculus entirely based on the algebraand then formulate laws of physics based on this calculus

Bringing together scientists from quantum field theory, string theory and tum gravity with researchers in noncommutative geometry, Hopf algebras andquantum groups as well as experts on representation theory of these algebrashad a stimulating effect on each side and will lead to new developments Ineach field there is a highly developed knowledge by experts which can only betransformed to another field only by having close personal contact through dis-cussions, talks and reports We hope that common projects can be found such thatworking in these projects the detailed techniques can be learned from each other.The Workshop has promoted the development of new directions in the field ofmodern theoretical and mathematical physics combining the efforts of scientistsfrom NATO, East European countries and NIS

quan-We are greatly indebted to the NATO Division of Scientific Affairs for funding

of our meeting and to the National Academy of Sciences of Ukraine for help in itslocal organizing It is also a great pleasure to thank all the people who contributed

to the successful organization of the Workshop, especially members of the LocalOrganizing Committee Profs N Chashchyn and P Smalko Finally, we wouldlike to thank all the participants for creating an excellent working atmosphere andfor outstanding contributions to this volume

Editors

are considered as elements of an algebra overC subject to the relations:

This characterizesRn as a commutative space The relations generate a 2-sided

ideal IR From the algebraic point of view, we deal with the algebra freelygenerated by the elementsxiand divided by the idealIR:

f(x1, , xn) = X∞

r i =0

fr1 rn(x1)r1 · · · (xn)rn

Multiplication is the pointwise multiplication of these functions.

The monomials of fixed degree form a finite-dimensional subspace of the bra This algebraic concept can be easily generalized to non-commutative spaces

alge-We consider algebras freely generated by elements ˆx1, ˆxn, again calling themcoordinates But now we change the relations to arrive at non-commutative spaces:

Rˆx,ˆx: [ˆxi, ˆxj] = iθij(ˆx) (5)Following L.Landau, non-commutativity carries a hat Now we deal with thealgebra:

In the following we impose one more condition on the algebra: the dimension

of the subspace of homogeneous polynomials should be the same as for muting coordinates This is the so called Poincare-Birkhof-Witt property (PBW).Only algebras with this property will be considered, among them are the algebraswhereθij is a constant:

com-Canonical structure, ref [2]:

In order to establish the isomorphism we choose a particular basis in the tor space of homogeneous polynomials in the non-commuting variables ˆx and

vec-characterize the elements of Aˆx by the coefficient functions in this basis Thecorresponding element in the algebra Ax of commuting variables is supposed

to have the same coefficient function The particular form of this isomorphismdepends on the basis chosen The vector space isomorphism can be extended to

an algebra isomorphism To establish it we compute the coefficient function of theproduct of two elements inAˆxand map it toAx This defines a product inA§that

we denote as diamond product (♦ product) The algebra with this ♦ product we

call♦Ax There is a natural isomorphism:

the canonical structure we obtain the Moyal-Weyl * product, ref [6], if we startfrom the basis of completely symmetrized monomials:

For the Lie structure we can use the Baker-Campbell-Hausdorf formula:

eik·ˆxeip·ˆx = ei(k+p+12 g(k,p))·ˆx (12)This definesg(k, p)

The product of fields will always be the * product To formulate field equations

we introduce derivatives On the algebraAˆx this can be done on purely algebraicgrounds We have to extend the algebraAˆx by algebraic elements ˆ∂i, ref [7] A

generalized Leibniz rule will play the role of algebraic relations.

Leibniz rule:

(ˆ∂if ˆg) = ( ˆ∂ˆ if)ˆg + Oˆ l

i( ˆf)ˆ∂lˆg : Rˆx, ˆ∂ (16)From the law of associativity in Aˆx follows that the operation O has to be an

Finally R∂, ˆˆ∂ relations have to be defined As conditions we consider the ˆ∂

subalgebra, demand PBW and derive consistency relations from R∂, ˆˆ∂ and theLeibniz rule as before Derivatives defined that way induce a map fromAˆxtoAˆx:

ˆ

(ˆ∂if) = ˆ∂ˆ if − Oˆ l

i( ˆf)ˆ∂l

This algebraic concept of derivatives has been explained in ref[] and applied

to quantum planes Following the same strategy derivatives can be defined for thecanonical structure as well

For the rest of this talk we will restrict ourselves to the canonical case only.The Leibniz rule for the canonical case is the usual one:

It satisfies all the consistency relations As explained above, the derivatives induce

a map on the algebraAˆx:

ˆ

f ∈ Aˆx : ˆf → [ˆ∂i, ˆf] ∈ Aˆx (20)This is the relation that we shall use to define derivatives on fields For this purpose

we map ˆ∂ to♦Ax From (20) follows that it becomes the usual derivative in♦Ax:

commutes with all coordinates For invertible θij this can be used to define theaction of the derivative entirely inAˆx

ThisR∂, ˆˆ∂ relation satisfies all the requirements of (ref7)

To formulate a Lagrangian field theory we have to learn how to integrate.Whereas it was easier to formulate derivatives on objects of Aˆx it is easier toformulate integration on objects of♦Ax For the canonical structure we define:

has this property.

In general we can construct Hilbert space representations of the algebra anddefine the integral as the trace This will lead to infinite sums that can be inter-preted as Riemannian sums for an integral and lead to the respective measure forthe integration

To formulate a gauge theory on a non-commutative space we start with fieldsψ(x)

that are elements of♦Aˆxand again span a representation of the Lie algebra (33)

We demand the transformation law:

in analogy to (34) But now we cannot demand α to be Lie algebra valued, we

shall assume it to be enveloping algebra valued:

α(x) = α0a(x)Ta+ α1ab(x) : TaTb : + · · · + αn−1a1 an(x) : Ta1 · · · Tan: + · · ·

(41)This is in analogy to (35) We have adopted the:: notation for the basis elements

of the enveloping algebra We shall use the symmetrized polynomials as a basis:

: TaTb : = 12(TaTb+ TbTa) etc

In analogy to (36) we find

(δαδβ− δβδα)ψ = [α∗, β] ∗ ψ (43)Naturally,[α∗, β] will be an enveloping algebra valued element of♦Ax

The unpleasant fact of the definition (41) of an enveloping algebra valuedtransformation parameter is that it depends on an infinite set of parameter fields

αn(x) In physics we would have to deal with an infinite set of fields when

defining a covariant derivative, something we try to avoid However, a gaugetransformation can be realized by transformation parameters that depend on x

via the parameter fieldα0(x), the gauge field ai,a(x) and their derivatives only In

the notation of eqn (41) we have

αna1 an+1(x) = αna1 an+1(αa0(x), a0i,a(x), ∂iα0a(x), ) (44)Transformation parameters that are restricted that way we shall denote Λα0(x)

These parameters can be constructed in such a way that eqn (36) holds:

δα0ψ(x) = iΛα0 (x)(x) ∗ ψ(x),(δα0δβ0 − δβ0δα0)ψ = δα0 ×β 0ψ, (45)

We shall constructΛα0 in a power series expansion inθ To illustrate the method

we expandΛα0 to first order inθ

Λα0 = α0aTa+ θijΛ1α0 ,ij+ , (47)

To be consistent we expand the * product in (46) also to first order in θ and

compare powers of θ the θ-independent term defines α0× β0 as we have used

it in (45) This had to be expected, this order is exactly the commutative case Tofirst order we obtain the equation:

We see that Λ1 is of second order in the generators T of the Lie algebra The

structure of eqn (46) allows a solution whereΛn, the term in (47) of ordern − 1

inθ, is a polynomial of order n in T

Λα0 = α0aTa+12θij(∂iα0a)aj,b : TaTb : + (50)

In a next step in the formulation of a gauge theory we introduce covariantderivatives Eqn (24) shows that we can relate this problem to the construction ofcovariant coordinates We try to define such coordinates with the help of a gaugefield, in the same way as we did it for derivatives in eqn (37):

This leads to a transformation law for the gauge fieldAi(x):

δAi= −i[xi ∗, Λα0] + i[Λα0 ∗, Ai] (53)

We have to assume that Ai is enveloping algebra valued but we try to make anansatz where all the coefficient functions only depend onai,aand its derivatives:

Now we expand (53) in θ, demand Ai,nto be a polynomial of order n in θ and

δα0F˜ij = i[Λα0 ∗, ˜Fij] (58)This can be verified from (53) and the definition of ˜F

To first order inθ we find:

˜

Fij = Fij,aTa+ θln(Fil,aFjn,l− (59)

1

2al,a(2∂nFij,b+ an,cFij,dfecd)) : TaTb : + (60)

We see that new “contact” terms appear in the field strength ˜F

A good candidate for a Lagrangian is

The trace is taken in the representation space of the generatorsT The Lagrangian

(61) is not invariant because the * product is not commutative:

This action depends on the gauge field ai,a and its derivatives only It can beconsidered as a gauge-invariant object ifai,a transforms according to (39) thisimplies thatW satisfies the Ward identities.

The Lagrangian expanded to all orders inθ, is a non-local object It remains to be

seen if it is acceptable for a quantum field theory or if it has to be viewed as aneffective Lagrangian, ref [8]

References

1. B Jurˇco, S Schraml, P Schupp and J Wess, Enveloping algebra-valued gauge

transforma-tions for non-abelian gauge groups on non-commutative spaces, Eur Phys J C 17, (2000)

B deWit, J Hoppe, H Nicolai, Nucl.Phys B305 [FS 23] (1988) 545.

D Kabat, W Taylor IV, Spherical membranes in Matrix theory, Adv.Theor.Phys 2 (1998)

181-206, (hep-th 9711078).

4 J Wess,q-deformed Heisenberg Algebras, in H Gausterer, H Grosse and L Pittner, eds.,

Pro-ceedings of the 38 Internationale Universit¨atswochen f¨ur Kern- und Teilchenphysik, no 543

in Lect Notes in Phys., Springer-Verlag, 2000, Schladming, January 1999, math-ph/9910013.

5. F Bayen, M Flato, C Fronsdal, A Lichnerowicz, D Sternheimer, Deformation theory and

quantization I Deformations of symplectic structures, Ann Physics 111, 61 (1978).

M Kontsevitch, Deformation quantization of Poisson manifolds, I,

q-alg/9709040.

D Sternheimer, Deformation Quantization: Twenty Years After, math/9809056.

6. H Weyl, Quantenmechanik und Gruppentheorie, Z Physik 46, 1 (1927); The theory of

groups and quantum mechanics, Dover, New-York (1931), translated from Gruppentheorie und Quantenmechanik, Hirzel Verlag, Leipzig (1928).

J E Moyal, Quantum mechanics as a statistical theory, Proc Cambridge Phil Soc 45, 99

(1949).

7. J Wess and B Zumino, Covariant differential calculus on the quantum hyperplane, Nucl.

Phys Proc Suppl 18B (1991) 302.

8. L Bonora, M Schnabl, M M Sheikh-Jabbari and A Tomasiello, Noncommutative SO(n) and Sp(n) gauge theories, hep-th/0006091.

I Chepelev, R Roiban, Convergence Theorem for Non-commutative Feynman Graphs and Renormalization, hep-th/0008090.

A Bichl, J.M Grimstrup, V Putz, M Schweda, Perturbative Chern-Simons Theory on commutativeR 3, hep-th/0004071.

non-A Bichl, J.M Grimstrup, H Grosse, L Popp, M Schweda, R Wulkenhaar, The Superfield Formalism Applied to the Non-commutative Wess-Zumino Model, hep-th/0007050.

Abstract We observe that supersymmetries do not exhaust all the symmetries of the

super-manifolds On a generalization of supermanifolds (called metamanifolds), the “functions” form a

metaabelean algebra, i.e., the one for which [[x, y], z] = 0 with respect to the usual commutator.

The superspaces considered as metaspaces admit symmetries wider than supersymmetries

Conjec-turally, infinitesimal transformations of these metaspaces constitute Volichenko algebras which we

introduce as inhomogeneous subalgebras of Lie superalgebras The Volichenko algebras are ral generalizations of Lie superalgebras being 2-step filtered algebras They are non-conventional deformations of Lie algebras bridging them with Lie superalgebras.

natu-1 Introduction: Towards noncommutative geometry

This is an elucidation of our paper [31] In 1990 we were unaware of [42] towhich we now would like to add later papers [14], and [2], and papers citedtherein pertaining to this topic Observe also an obvious connection of Volichenkoalgebras with structures that become more and more fashionable lately, see [22];Volichenko algebras are one of the ingredients in the construction of simple Liealgebras over fields of characteristic 2, cf [23]

1.1 The gist of idea To describe physical models, the least one needs is a

triple (X, F (X), L), consisting of the “phase space” X, the sheaf of functions

on it, locally represented by the algebra F (X) of sections of this sheaf, and a

Lie subalgebra L of the Lie algebra of of differentiations of F (X) considered

∗ Instead of J Naudts contribution by the editor S Duplij’s request

† D.L is thankful to an NFR grant for partial financial support, to V Molotkov, A Premet and

S Majid for help.

‡ mleites@matematik.su.se

§ serganov@math.berkeley.edu

as vector fields on X Here X can be recovered from F (X) as the collection

Spec(F (X)), called the spectrum and consisting of maximal or prime ideals of

F (X) Usually, X is endowed with a suitable topology

After the discovery of quantum mechanics the attempts to replaceF (X) with

the noncommutative (“quantum”) algebraA became more and more popular The

first successful attempt was superization [25], [5] the road to which was prepared

in the works of A Weil, Leray, Grothendiek and Berezin, see [11] It turns out thathaving suitably generalized the notion of the tensor product and differentiation(by inserting certain signs in the conventional formulas) we can reproduce onsupermanifolds all the characters of differential geometry and actually obtain amuch reacher and interesting plot than on manifolds This picture proved to be

a great success in theoretical physics since the language of supermanifolds andsupergroups is a “natural” for a uniform description of bose and fermi particles.Today there is no doubt that this is the language of the Grand Unified Theories ofall known fundamental forces

Observe that physicists who, being unaware of [25], rediscovered groups and superspaces (Golfand–Likhtman, Volkov–Akulov, Neveu–Schwartz,Stavraki) were studying possibilities to enlarge the group of symmetries (or ratherthe Lie algebra of infinitesimal symmetries) of the known objects (in particular,objects described by Maxwell and Dirac equations) Their efforts did not drawmuch attention (like our [25] and [31]) until Wess and Zumino [43] understoodand showed to others some of the whole series of wonders one can obtain bymeans of supersymmetries

super-Here we show that the supergroups are not the largest possible symmetries ofsuperspaces; there are transformations that preserve more noncommutativity thanjust a “mere” supercommutativity To be able to observe that there are symmetriesthat unify bose and fermi particles we had to admit a broader point of view onour Universe and postulate that we live on a supermanifold Here (and in [31])

we suggest to consider our supermanifolds as paticular case of metamanifolds,

introduced in what follows

How noncommutative should F (X) be? To define the space

correspond-ing to an arbitrary algebra is very hard, see Manin’s gloomy remarks in [33],where he studies quadratic algebras as functions on “perhaps, nonexisting”noncommutative projective spaces

Manin’s idea that there hardly exists one uniform definition suitable for anynoncommutative algebra (because there are several quite distinct types of them)was supported by A Rosenberg’s studies; he managed to define several types ofspectra in order to interpret ANY algebra as the algebra of functions on a suitablespectrum, see preprints of his two books [27], no 25, and nos 26, 31 (the latter be-ing expanded as [35]) In particular, there IS a space corresponding to a quadratic(or “quadraticizable”) algebra such as the so-called “quantum” deformationUq(g)

ofU(g), see [12]

Observe that in [33] Manin also introduced and studied symmetries of commutative superalgebras wider than supersymmetries, but he only consideredthem in the context of quadratic algebras (not all relations of a supercommu-tative suepralgebra are quadratic or quadraticizable) Regrettably, nobody, asfar as we know, investigated consequences of Manin’s approach to enlargingsupersymmetries.

super-Unlike numerous previous attempts, Rosenberg’s theory is more natural; still,

it is algebraic, without any real geometry (no differential equations, integration,etc.) For some noncommutative algebras certain notions of differential geome-try can be generalized: such is, now well-known, A Connes geometry, see [10],and [34] Arbitrary algebras seem to be too noncommutative to allow to do anyphysics

In contrast, the experience with the simplest non-commutative spaces, the perspaces, tells us that all constructions expressible in the language of differentialgeometry (these are particularly often used in physics) can be carried over tothe super case Still, supersymmetry has, as we will show, certain shortcomings,which disappear in the theory we propose

su-Specifically, we continue the study started under Berezin’s influence in [25](later suppressed under the same influence in [5], [26]), of algebras just slightlymore general than supercommutative superalgebras, namely their arbitrary, notnecessarily homogeneous, subalgebras and quotients Thanks to Volichenko’s the-orem F (F is for “functions”, see [27], no 17 and Appendix below) such algebras

are precisely metaabelean ones, i.e., those that satisfy the identity

[x, [y, z]] = 0 (here [·, ·] is the usual commutator) (1.1)

As in noncommutative geometries, we think of metaabelean algebras as

“func-tions” on a what we will call metaspace.

Observe that the conventional superspaces considered as metaspaces and

La-grangians on them have additional symmetries as compared with supersymmetry.

1.2 The notion of Volichenko algebras Volichenko’s Theorem F gives

us a natural generalization of the supercommutativity It remains to define theanalogs of the tensor product and study differentiation (e.g., Volichenko’s ap-proach, see§3) We conjecture that the analogs of Lie algebras in the new setting

are Volichenko algebras defined here as nonhomogeneous subalgebras of Lie

superalgebras

Supersymmetry had been already justified for physicists when cians’ attention was drawn to it by the list of simple finite dimensional Liesuperalgebras: bar one exception it was discrete and looked miraculously like thelist of simple Lie algebras Our list of simple Volichenko algebras is similar Ourmain mathematical result is the classification (under a technical hypothesis) ofsimple finite dimensional (and vectorial) Volichenko algebras, see [40], [31]

mathemati-Remarkably, Volichenko algebras are just deformations of Lie algebras though

in an entirely new sense: in a category broader than that of Lie algebras or Liesuperalgebras This feature of Volichenko algebras could be significant for paras-tatistics because once we abandon bose-fermi statistics, there seem to be too many

ad hoc ways to generalize Our classification asserts that within the natural context

of simple Volichenko algebras the set of possibilities is discrete or has at most parameter (hence, anyway, describable!) It is important because it suggests thepossibility of associating distinct types of particles to representations of thesestructures

1-Our generalization of supersymmetry and its implications for parastatisticsappear to be complementary to works on braid statistics in two dimensions [15]

in the context of [13], see also [19] We expect them to tie up at some stage.Examples of what looks like nonsimple Volichenko algebras recently appeared

in another context in [2], [36], [42] and [14]

1.3 An intriguing example: the general Volichenko algebravglµ(p|q) Let

the spaceh of vglµ(p|q) be the space of (p + q) × (p + q)-matrices divided into

the two subspaces as follows:

Herehˆ1is a naturalhˆ0-module with respect to the bracket of matrices; fixa, b ∈ C

such thata : b = µ ∈ CP1and define the multiplicationhˆ1× hˆ1 −→ hˆ0 by theformula

[X, Y ] = a[X, Y ]−+ b[X, Y ]+for any X, Y ∈ hˆ1 (1.3.2)

(The subscript− or + indicates the commutator and the anticommutator,

respec-tively.) As we sill see, h is a simple Volichenko algebra for any a, b except for

ab = 0 when it becomes isomorphic to either the Lie algebra gl(p + q) or the Lie

superalgebragl(p|q) To show that vglµ(p|q) is indeed a Volichenko algebra, we

have to realize it as a subalgebra of a Lie superalgebra This is done in heading 2

of Theorem 2.7

2 Metaabelean algebra as the algebra of “functions” Volichenko algebra as

an analog of Lie algebra

2.1 Symmetries broader than supersymmetries It was the desire to broaden

the notion of a group that lead physicists to supersymmetry However, in viewingsupergroups as transformations of superspaces we consider only even, “statistics-preserving”, maps: nonhomogeneous “statistics-mixing” maps between super-algebras are explicitly excluded and this is why and how odd parameters ofsupergroups appear, cf [3], [11]

On the one hand, this is justified: since we consider graded objects whyshould we consider transformations that preserve these objects as abstract ones

but destroy the grading? It would be inconsistent on our part, unless we decide toconsider the grading or “parity” as one considers the electric charge of a nucleon:

in certain problems we ignore it

On the other hand, if such parity violating transformations exist, they deserve

to be studied, to disregard them is physically and mathematically an artificialrestriction

We would like to broaden the notion of supergroups and superalgebras to allowfor the possibility of statistics-changing maps Soon after Berezin published hisdescription of automorphisms of the Grassmann algebra [4] it became clear thatBerezin missed nonhomogeneous automorphisms, but the complete description ofautomorphisms was unknown for a while In 1977, L Makar-Limanov gave us acorrect description of such automorphisms (private communication) A Kirillovrediscovered it while editing [3], Ch.1; for automorphisms in presence of evenvariables see [28]

Recall the answer: the generic finite transformation of a supercommutativesuperalgebra F of functions in n even generators x1, , xn and m odd ones

θ1, , θmis of the form (herepmis the parity ofm, i.e., either 0 or 1)

where fi, Fi and fi1 i 2k

i , and also gi1 i 2k+1

j are even superfields, whereas

fi1 i 2k+1

i , gj and gi1 i 2k

j and also g, Fi are odd superfields (A mathematician,see [11], would say that the odd superfields (underlined once) represent the pa-rameters corresponding to Λ-points with nonzero odd part of the background

supercommutative superalgebra Λ.) Notice that one g serves all the θj Thetwice underlined factors account for the extra symmetry ofF as compared with

supersymmetry

Comment When the number of odd variables is even, as is usually the

case in modern models of Minkowski superspace, there is only one extra tional parameter,g Therefore, on such supermanifolds, the notion of a boson is

func-coordinate-free, whereas that of a fermion depends on coordinates

Summing up, (this is our main message to the reader)

supersymmetry is not the most broad symmetry of

supercommutative superalgebras

2.2 Two complexifications Another quite unexpected flaw of

supersymme-try is that the category of supercommutative superalgebras is not closed with

respect to complexification It certainly is ifC is understood naively, as a purely

even space Declaring √

−1 to be odd, we make C into a nonsupercommutative

superalgebra This associative superalgebra over R is denoted by Q(1; R), see

for superalgebras An infinite dimensional representation of Q(1) is crucial in

A Connes’ noncommutative differential geometry In short, the odd complexstructure on superspaces is an important one

How to modify definition of supermanifold to incorporate the above tures?

struc-Conjecturally, the answer is to consider arbitrary, not necessarily geneous subalgebras and quotients of supercommutative superalgebras Thesealgebras are, clearly, metaabelean algebras But how to describe arbitrary metaa-belean algebras? In 1975 D.L discussed this with V Kac and Kac conjectured (see[26]) that considering metaabelean algebras we do not digress far from supercom-mutative superalgebras, namely, every metaabelean algebra is a subalgebra of asupercommutative superalgebra Therefore, the most broad notion of morphisms

homo-of supercommutative superalgebras should only preserve their metaabeleannessbut not parity (Since C, however understood, is metaabelean, we get a category

of algebras closed with respect to all algebra morphisms and complexifications.)Volichenko proved more than Kac’ conjecture (Appendix) Namely, he provedthat any finitely generated metaabelean algebra admits an embedding into a uni-versal supercommutative superalgebra and developed an analogue of Taylor seriesexpansion

Until Volichenko’s results, it was unclear how to work with metaabelean bras: are there any analogues of differential equations, or integral, in other words,

alge-is there any “real life” on metaspaces [26]? Thanks to Volichenko, we can nowconsider pairs

(a metaabelean algebra, its ambient supercommutative superalgebra)

and corresponding projections “superspace −→ metaspace” when we consider

these algebras as algebras of functions

It is interesting to characterize metaabelean algebras which are quotients of

supercommutative superalgebras: in this case the corresponding metaspace can

be embedded into the superspace and we can consider the induced structures(Lagrangeans, various differential equations, etc.)

But even if we would have been totally unable to work with metaspaces whichare not superspaces, it is manifestly useful to consider superspaces as metaspaces

In so doing, we retain all the paraphernalia of the differential geometry for sure,and in addition get more transformations of the same entities

For example, it is desirable to make use of the formula (first applied byArnowitt, Coleman and Nath)

Ber X = exp str log X

which extends the domain of the berezinian (superdeterminant) to geneous matrices X Then we can consider the additional nonhomogeneous

nonhomo-transformations, like the ones described in (2.1) All supersymmetric Lagrangeansadmit metasymmetry wider than supersymmetry

Remark In mathematics and physics, spaces are needed almost exclusively to integrate over

them or consider limits in analytic questions In problems where integration is not involved we need sheaves of sections of various bundles over the spaces rather than the spaces themselves Gauge fields, Lagrangeans, etc are all sections of coherent sheaves, corresponding to sections of vector bundles Now, almost 30 years after the definition of the scheme of a metaabelean algebra (metavariety or metaspace) had been delivered at A Kirillov’s seminar ([25]), there is still no accepted definition of nice (“morally coherent” as Manin says) sheaves over such a scheme even for superspaces (for a discussion see [8]) As to candidates for such sheaves see Rosenberg’s books

on noncommutative geometry [27], nos 25, 26, 31 and [35]) and §9 in [8] This §9 is, besides all, a

possible step towards “compactification in odd directions”.

2.3 A description of Volichenko algebras It seemed natural [26] to get for

Lie superalgebras a result similar to Volichenko’s theorem F, i.e., to describearbitrary subalgebras of Lie superalgebras Shortly before his untimely death

I Volichenko (1955-88) announced such a description (Theorem A, here A is for

(Lie) “algebra”) In his memory then, a Volichenko algebra is a nonhomogeneous

subalgebrah of a Lie superalgebra g The adjective “Lie” before a (super)algebra

indicates that the algebra is not associative, likewise the adjective “Volichenko”reminds that the algebra is neither associative nor should it satisfy Jacobi or super-Jacobi identities Thus, a Volichenko algebrah is a non-homogeneous subspace of

a Lie superalgebrag closed with respect to the superbracket of g How to describe

h by identities, i.e., in inner terms, without appealing to any ambient?

Theorem A (I Volichenko, 1987) LetA be an algebra with multiplication

denoted by juxtaposition Define the Jordan elementsa ◦ b := ab + ba and Jacobi

elementsJ(a, b, c) := a(bc) + c(ab) + b(ca) Suppose that

(a)A is Lie admissible, i.e., A is a Lie algebra with respect to the new product

defined by the bracket (not superbracket)[a, b] = ab − ba;

(b)the subalgebraA(JJ)generated by all Jordan and Jacobi elements belongs

to the anticenter ofA, in other words

ax + xa = 0 for any a ∈ A(JJ), x ∈ A;

(c)a(xy) = (ax)y + x(ay) for any a ∈ A(JJ), x, y ∈ A

Then

(1)Any (not necessarily homogeneous) subalgebra h of a Lie superalgebra g

satisfies the above conditions (a) — (c)

(2)IfA satisfies (a) — (c), then there exists a Lie superalgebra SLie (A) such

thatA is a subsuperalgebra (closed with respect to the superbracket) of SLie (A)

Heading (1) is subject to a direct verification.

Clearly, the parts of conditions (b) and (c) which involve Jordan (resp Jacobi)elements replace the superskew-commutativity (resp Jacobi identity) Condition(a) ensures thatA is closed in SLie(A) with respect to the bracket in the ambient

Discussion If true, Volichenko’s theorem A would have disproved a

pes-simistic conjecture of V Markov cited in [26]: the minimal set of polynomialidentities that single out nonhomogeneous subalgebras of Lie superalgebras isinfinite I Volichenko did not investigate under which conditions a finite di-mensional Volichenko algebraA can be embedded into a finite dimensional Lie

superalgebrag; which is, perhaps, the quotient of SLie(A) modulo an ideal

Volichenko’s scrap papers were destroyed after his death and no hint of hisideas remains Several researchers tried to refute it and A Baranov succeeded Heshowed [1] that Volichenko’s theorem V is wrong as stated: one should add at leastone more relation of degree 5 First, following Volichenko, Baranov introducedinstead ofJ(a, b, c) more convenient linear combinations of the Jacobi elements

j(a, b, c) = [a, b ◦ c] + [b, c ◦ a] + [c, a ◦ b] for a, b, c ∈ A

Then Baranov rewrote identities (a)–(c) in the following equivalent but moretransparent form (i)–(v):

(i)[a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0;

(ii)a ◦ b ◦ c = 0;

(iii)j(a, b, c) ◦ d = 0;

(iv)[a ◦ b, c ◦ d] = [a ◦ b, c] ◦ d + [a ◦ b, d] ◦ c;

(v)[j(a, b, c), c ◦ d] = [j(a, b, c), c] ◦ d + [j(a, b, c), d] ◦ c

Baranov’s new identity independent of (i) – (v) is of degree 5 and is somewhatimplicit; it involves 49 monomials and no lucid expression for it is found yet.True or false, Volichenko’s theorem A does not affect our results, since we donot appeal to an intrinsic definition of Volichenko algebras

2.4 On simplicity of Volichenko algebras As we will see, the notion of

Volichenko algebra is a totally new type of deformation of the usual Lie algebra

It also generalizes the notion of a Lie superalgebra in a sence that the Lie algebras areZ/2-graded algebras (i.e., they are of the form g = ⊕

super-i=¯0,¯1gisuch that

[gi, gj] ⊂ gi+j) whereas Volichenko algebras are only 2-step filtered ones (i.e.,they are of the form h = ⊕

i=ˆ0,ˆ1hi as spaces andhˆ0 is a subalgebra There are,however, several series of examples when Volichenko algebras are Z/2-graded

(e.g.,vglµ(p|q))

Hereafterg is a Lie superalgebra over C and h ⊂ g a subspace which is not a

subsuperspace closed with respect to the superbracket ing For notations of simple

complex finite dimensional Lie superalgebras, the list of known simpleZ-graded

infinite dimensional Lie superalgebras of polynomial growth over C and R, and

their gradings see [20], [38], [27], [37], [30] A Volichenko algebra is said to be

simple if it has no two-sided ideals and its dimension is6= 1

Remark P Deligne argued that for an algebra such as a Volichenko one, modules over which

have no natural two-sided structure, the above definition seems to be too restrictive: one should

define simplicity by requiring the absence of one-sided ideals As it turns out, none of the simple

Volichenko algebras we list in what follows has one-sided ideals, so we will stick to the above (at first glance, preliminary) definition: it is easier to work with.

Lemma.For any simple Volichenko algebrah, h ⊂ g0, there exists a simpleLie subsuperalgebrag ⊂ g0that containsh

So, we can (and will) assume that the ambientg of a simple Volichenko

al-gebra is simple In what follows we will see that under a certain condition for asimple Volichenko algebrah its simple ambient Lie superalgebra g is unique here

is this condition:

2.5 The “epimorphy” condition Denote bypi : g −→ gi, wherei = ¯0, ¯1,

the projections to homogeneous components A Volichenko algebrah ⊂ g will be

called epimorphic ifp0(h) = g¯0 Not every Volichenko subalgebra is epimorphic:for example, the two extremes, Volichenko algebras with the zero bracket and freeVolichenko algebras, are not epimorphic, generally All simple finite dimensionalVolichenko algebras known to us are, however, epimorphic

Hypothesis.Every simple Volichenko algebra is epimorphic

A case study of various simple Lie superalgebras of low dimensions revealsthat they do not contain non-epimorphic simple Volichenko algebra Still, we cannot prove this hypothesis but will adopt it for it looks very natural at the moment

Lemma.Leth ⊂ g be an epimorphic Volichenko algebra and f : g¯0−→ g¯1 alinear map that determinesh, i.e.,

h = hf := {a + f(a) | a runs over g¯0}

Then

1)f is a 1-cocycle from C1(g¯0; g¯1);

2)f can be uniquely extended to a derivation of g (also denoted by f) such

thatf(f(g¯0)) = 0

Example Recall, that the odd elementx of any Lie superalgebra is called a

homologic one if[x, x] = 0, cf [41] Let x ∈ g¯1be such that

whereC(g) is the center of g Clearly, the map f = ad (x) satisfies Lemma 2.4 if

x satisfies (2.5.1), i.e., is homologic modulo center

A homologic modulo center elementx will be said to ensure nontriviality (of

the algebra

hx= {a + [a, x] | a runs over g¯0}) (2.5.2)

if

[[g¯0, x], [g¯0, x]] 6= 0,

i.e., if there exist elementsa, b ∈ g¯0such that

The meaning of this notion is as follows Leta, b ∈ h, a = a0+ a1,b = b0+ b1,where a1 = [a0, x], b1 = [b0, x] for some x ∈ g¯1 Notice that forx satisfying(2.5.1) we have

If (2.5.3) holds, we have

[a, b] = [a0, b0] + [a1, b1] + [a0, b1] + [a1, b0] = ([a0, b0] + [a1, b1]) + [[a0, b0], x]

(2.5.5)

It follows from (2.5.4) and (2.5.5) that if x is homologic modulo center, then hx

is closed under the bracket ofg; if this x does not ensure nontriviality, then hxisjust isomorphic tog¯0

In other words, an epimorphic Volichenko algebra is a deformation of the Liealgebrag¯0in a totally new sence: not in the class of Lie algebras, nor in that of Liesuperalgebras but in the class of Volichenko algebras whose intrinsic description

is to be given (To see that an epimorphic Volichenko algebra hx is a result of adeformation of sorts, multiplyx by an even parameter, t If t were odd, we would

have obtained a deformation ofg¯0in the class of Lie superalgebras.)

Remark It is easy to show making use of formula (2.5.5) why it is impossible

to consider any other (inconsistent with parity)Z/2-grading (call it deg) of g and

deform in a similar way the Lie subsuperalgebra of elements of degree 0 withrespect todeg

Any epimorphic Volichenko algebrahx⊂ g is naturally filtered: it contains as

as subalgebra the Lie algebraann (x) = {a ∈ g¯0 | [x, a] = 0}

Problems 1) We have a sandwich: between Hopf (super)algebras, U(hx)

and U(g), a non-Hopf algebra, U(h) (the subalgebra of U(g) generated by h),

is squeezed How to measure its “non-Hopfness”? This invariant seems to be ofinterest

2) It is primarily real algebras and their representations that arise in tions So what are these notions for Volichenko algebras?

applica-We do not know at the moment the definition of a representation of a

Volichenko algebra even for epimorphic ones To say “a representation of a Volichenko algebra is a through map: the composition of an embedding h ⊂ g

into a minimal ambient and a representationg −→ gl(V )” is too restrictive: the

adjoint representation and homomorphisms of Volichenko algebras are ruled out.3) If we abandon the technical hypothesis on epimorphy, do we obtain anysimple Volichenko algebras? (Conjecture: we do not.)

4) Describe Volichenko algebras intrinsically, via polynomial identities Thisseems to be a difficult problem

5) Classify simple Volichenko subalgebras of the other known simple Liesuperalgebras of interest, e.g., of polynomial growth, cf [16], [17].

2.6 Vectorial Volichenko superalgebras For a vector field D = Pfr∂r

fromvect(m|n) = derC[x, θ], define its inverse order with respect to the

nonstan-dard (ifm 6= 0) grading induced by the grading of C[x, θ] (for which deg xi= 0

and deg θj = 1 for all i and j) and inv.ord(fr) is the least of the degrees of

monomials in the power series expansion offr

There are two major types of Lie (super)algebras and their subalgebras: theones realized by matrices and the ones realized by vector fields The former oneswill be refered to as matrix ones, the latter ones as vectorial algebras

2.6.1 Lemma. Let h ⊂ g be a simple epimorphic vectorial Volichenko

algebra, i.e., a subalgebra of a simple vectorial Lie superalgebra Then in therepresentation h = hf we have f(·) = [·, x], where x is homologic andinv.ord(x) = −1

2.6.2 Lemma.LetG be the Lie group with the Lie algebra g¯0, letG0 be theLie group with the Lie algebrag0of linear vector fields with respect to the stan-dard (see [37]) grading; letAut G0 be the group of automorphisms ofG0 Table

2.7.2 contains all, up to (Aut G0)-action, homologic elements of the minimal

inverse order in the vectorial Lie superalgebras In particular, forsvect0(2n) there

are none

2.7 Theorem. A simple epimorphic finite dimensional Volichenko algebra

h ⊂ g can be only one of the following h = hx, where:

1)x is an element from Table 2.7.2 or an element from Table 2.7.1 satisfying

the condition ensuring non-triviality ifg 6= psq(n);

2)ifg = psq(n), then either x is an element from Table 2.7.1 satisfying the

condition ensuring non-triviality orx = antidiag (X, X), where

X = diag (a1p, b1n−p) with ap + b(n − p) = 0

Now, the final touch:

Proposition.Simple epimorphic Volichenko algebras from Tables1, 2 have

no one-sided ideals

Table 2.7.1 Homologic elements x and the condition when x ensures

nontriviality of h for matrix Lie superalgebras g

psl(n|n) same as for sl(n|n) and also antidiag(1 n , 1 n) (as above)

osp(2m|2n) the image of the above x p ∈ sl(m|n) ⊂ osp(2m|2n)

2p ≤ min (m, n) (p > 0)

osp(2m + 1|2n) the image of the above x

under the embedding osp(2m|2n) ⊂ osp(2m + 1|2n) spe(n) antidiag(B, C), where B = diag(1 p , 0 n−p ),

ag2, ab 3 , the root vector corresponding to

osp(4|2; α) an isotropic (odd) simple root (never)

In Table 2.7.2 we have listed not only homologic elements — that is to sayVolichenko subalgebras — of finite dimensional simple Lie superalgebras of vec-tor fields but also simple Volichenko subalgebras of all nonexceptional simple Liesuperalgebras of vector fields, for their list see [30]

Table 2.7.2 Homologic elements x of minimal inverse order in simple Lie

superalgebras g of vector fields

3 Appendix Volichenko’s theorem F and elements of Calculus on matamenifolds

3.1 In what follows all the algebras are associative with unit over a field K,char K 6= 2 We will deal with two important PI-varieties of algebras (the

varieties singled out by polynomial identities):

– the varietyC of supercommutative superalgebras;

– the varietyG generated (by tensoring and passing to quotients) by the

Grass-mann algebraΛ(∞) of countably many indeterminates (its natural Z/2-grading

ignored)

The varietyG plays a significant role in the theory of varieties of associative

algebras ([21]) It is known that ifchar K = 0 it is distinguished by the identity

(1.1) Ifchar K 6= 0, the identity Xp = 0 should be added

I Volochenko wrote: “As pointed out by D Leites [26], in the conventionalsupermanifold theoryit seems too restrictive that not all subalgebras or quotients

of superalgebras are considered as algebras of functions on supermanifolds butonly the graded (homogenous) ones It is tempting to construct a variant of Calcu-lus which enables one to operate with arbitrary subalgebras, ideals and quotients Definition of the category of topological spaces ringed by such general algebras

is obvious, cf [25], where the algebraic case is considered

It remained unclear, however, how to uniformly describe such algebras Forinstance, do they constitute a variety? Leites recalls a conjecture of Kac (1975)

that such algebras are metaabelean, i.e., satisfy the identity(1.1) The conjecture

is a well-known fact of the theory of varieties of associative algebras, cf [24].From the context of [25], however, it is clear that the actual problem is, first ofall, how to describe a variety of not necessarily homogeneous subalgebras which

a priori can be less thanG

Actually, I will not only prove that any algebraG ∈ G can be embedded into

a commutative superalgebra but will also prove the existence of a universal (in anatural sence) enveloping algebraUC(G) from the class C of all the supercommu-

tative superalgebras and give an explicit realization ofUC(G) Therefore, we can,

in principle, reduce the study of homomorphisms of algebras from G to that of

their enveloping superalgebras fromC

I hope that this is (at least partly) an answer to Leites’ questionhow to work

with algebras fromG and the corresponding ‘supermanifolds’”

3.2 LetKC[X, Y ] be the algebra determined by the system of indeterminates

X ∪ Y = (Xi)i∈I∪ (Yj)j∈J and relations

Xi 1Xi 2− Xi 2Xi 1 = 0, XiYj− YjXi = 0, Yj 1Yj 2 + Yj 2Yj 1 = 0

fori, i1, i2 ∈ I, and j, j1, j2∈ J This algebra possesses a natural parity: p(Xi) =

¯0, p(Yj) = ¯1 for i ∈ I, j ∈ J

LetI = J; let KG[Z] be a non-graded subalgebra in KC[X, Y ] generated by

all the elementsZi = Xi+ Yi(i ∈ I)

Statement. KG[Z] is a free algebra in the variety G and the elements Zi

(i ∈ I) are its free generators In other words, let KA[T ] be a free associative

algebra with free generatorsT1, T2, If f(Z1, , Zn) = 0 in KG[Z] for somef(T1, , Tn) ∈ KA[T ], then f(a1, , an) = 0 for any a1, , an∈ KC[X, Y ]

3.3 Setd = P

i∈IYi ∂

∂X i

Statement.The polynomial f(X, Y ) ∈ KC[X, Y ] belongs to KG[Z] if and

only ifdf = f¯1, or, equivalently,df¯0 = f¯1

3.4 A relation between ideals ofKG[Z] and KC[X, Y ]

Statement.LetA be an ideal of KG[Z] and ¯A the ideal of KC[X, Y ] generated

byA¯0∪ A¯1= {f¯0, f¯1 : f ∈ A} Then ¯A ∩ KG[Z] = A

Now, let ˜G = G¯0⊕ G¯1be a linear superspace, where eachGiis a copy of ouralgebraG from G Consider the subalgebra KG[G] ⊂ KC[ ˜G] generated by all the

elements g¯0 + g¯1, whereg ∈ G Clearly, KC[ ˜G] ' KC[X, Y ], where X and Y

are bases inG¯0andG¯1, respectively, andKG[G] ' KG[Z] Then G is isomorphic

to the quotient ofKG[Z] modulo the ideal A generated by all the elements of the

form

(g¯0+ g¯1)(h¯0+ h¯1) − ((gh)¯0+ (gh)¯1)

The universal C-enveloping of G is the quotient UC[G] of KC[ ˜G] modulo the

ideal ¯A generated by the elements of the form

g¯0h¯0+ g¯1h¯1− (gh)¯0, g¯0h¯1+ g¯1h¯0− (gh)¯1

Any elementg ∈ G is identified with the image of g¯0 + g¯1 under the canonicalepimorphismKC[ ˜G] → UC[G]

InKC[ ˜G], same as in KC[X, Y ], there is defined the derivation: d(g¯0) = g¯1,

d(g¯1 = 0 for any g ∈ G Since ¯A is d-invariant, it follows that d induces a

canonical derivation ofUC[G] which we will also denote by d

Proposition.The elementf of UC[G] belongs to G if and only if df¯0= f¯1

3.5 An explicit description of the supercommutative envelope: Theorem

F. The universal C-enveloping UC(G) of the algebra G of G is isomorphic to

the supercommutative superalgebraS = G(+)⊕ Ω1

G (+) /C whose even component

G(+)isG considered with the Jordan product x◦y = 1

2(xy+yx) and the odd

com-ponentΩ1

G (+) /C considered as aG(+)-module is the module of differentials, i.e.,the quotient of the freeG(+)-module with basis(dx)x∈Gmodulo the submodulegenerated by

d(x + y) − dx − dy, d(x ◦ y) − xdy − yd for x, y ∈ G(+), and dc for c ∈ C,

whereC be the subalgebra (with unit) in G and in G(+)generated by the elements

of the form[x, y] for x, y ∈ G.The product of odd elements is determined by the

dx · dy = 12[x, y] (x, y ∈ G)

3.6 The Taylor formula Hereafterchar K = 0, the set of indices I is either

N or {1, 2, , n} For arbitrary c1, , cp ∈ KG[Z] (p ∈ N) set

symm(c1, , cp) = 1

p!

X σ∈S p

cσ(1) cσ(p)

The expressions of this form will be called an s-monomial (in c1, , cp)

Determine also an a-monomial inc1, , c2qby setting

The elements of the formcm(m ∈ M) will be called sa-monomials in ci(i ∈ I)

PropositionThesa-monomials Zm(m ∈ M) constitute a basis of KG[Z]

Hereafter we assume thatI = {1, 2, , n} For m = (α, β) set δ(m) = q

and letm! = (−1)δ(m)d1! dn!, where diis the degree ofsymm(Zα1, , Zαp)

∂mf¯0(a)

∂Zm (Z − a)m

1. Baranov, A A., Volichenko algebras and nonhomogeneous subalgebras of Lie superalgebras.

(Russian) Sibirsk Mat Zh 36 (1995), no 5, 998–1009; translation in Siberian Math J 36

(1995), no 5, 859–868

2 Quesne C., Vansteenkiste N C λ-extended oscillator algebra and parasupersymmetric quantum mechanics Czechoslovak J Phys 48 (1998), no 11, 1477–1482; Beckers J., De- bergh N., Quesne C., Parasupersymmetric quantum mechanics with generalized deformed parafermions, hep-th/9604132; Helv Phys Acta 69, 1996, 60–68 e-Print Archive: hep-

th/9604132; Beckers J., Debergh N., Nikitin A.G., On parasupersymmetries and relativistic

description for spin one particles 1, 2 Fortsch Phys 43, 1995, 67–96; id., More on supersymmetry of the Schroedinger equation, Mod Phys Lett A8, 1993, 435–444; Beckers

para-J., Debergh N., From relativistic vector mesons in constant magnetic fields to nonrelativistic

(pseudo)supersymmetries, Int J Mod Phys A10, 1995, 2783–2797

3. Berezin F., Analysis with anticommuting variables, Kluwer, 1987

4. Berezin F., Automorphisms of the Grassmann algebra Mat Zametki 1, 1967, 269–276

5. Berezin F A., Leites D A., Supermanifolds, Sov Math Doklady 16 (1975), 1976, 1218–1222

6. Bernstein J., Leites D., The superalgebra Q(n), the odd trace and the odd determinant, C R.

Acad Bulgare Sci., 35 (1982), no 3, 285–286

7. Bokut’ L A., Makar-Limanov L G., A base of the free metaabelean associative algebra, Sib.

Math J., 32, 1991, 6, 910–915

8. Buchweitz R.-O., Cohen–Macauley modules and Tate-cohomology over Gorenstein rings.

Univ Toronto, Dept Math preprint, 1988

9. Corwin L., Ne’eman Y., Sternberg S., Graded Lie algebras in mathematics and physics, Rev.

Mod Phys 47, 1975, 573–609

10. Connes A., Noncommutative geometry Academic Press, Inc., San Diego, CA, 1994 xiv+661

pp.

11. Deligne P et al (eds.) Quantum fields and strings: a course for mathematicians Vol 1, 2.

Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997 AMS, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ, 1999 Vol 1: xxii+723 pp.; Vol 2: pp i–xxiv and 727–1501

12. Drinfeld V.G., Quantum groups, Proc Int Congress Math., Berkeley, 1, 1986, 798–820

13. Doplicher S., Roberts J.E., Why is there a field algebra with compact gauge group describing

the superselection structure in particle physics, Commun Math Phys 131, 1990, 51–107

14. Filippov A T., Isaev A P., Kurdikov A B., Para-Grassmann analysis and quantum groups,

Modern Phys Lett A7, no 23, 1992, 2129–2214; id., Para-Grassmann differential calculus,

Teoret Mat Fiz 94, no 2, 1993, 213–231; translation in Theoret and Math Phys 94, no.

2, 1993,150–165; id., Para-Grassmann extensions of the Virasoro algebra Internat J Modern Phys A 8,1993, no 28, 4973–5003

15 Fredenhagen K., Rehren K.H., Schroer B., Superselection sectors with braid statistics and exchange algebras, Commun Math Phys 125, 1989, 201–226

16. Feigin B., Leites D., Serganova V., Kac–Moody superalgebras In: Markov M et al (eds) Group–theoretical methods in physics (Zvenigorod, 1982), v 1, Nauka, Moscow, 1983, 274–

278 (Harwood Academic Publ., Chur, 1985, Vol 1–3 , 631–637)

17. Grozman P., Leites D., From supergravity to ballbearings In: Wess J., Ivanov E (eds.)

Procedings of the Internatnl seminar in the memory of V Ogievetsky, Dubna 1997, Springer

Lect Notes in Physics, 524,1999, 58–67

18. Grozman P., Leites D., Lie superalgebras of supermatrices of complex size Their izations and related integrable systems In: Vasilevsky N et al (eds.) Proc Internatnl Symp.

general-Complex Analysis and related topics, Mexico, 1996, Birkhauser Verlag, 1999, 73–105

19 Jing N., Ge Mo-Lin, Wu Yong-Shi, New quantum groups associated with a “nonstandard” braid group representation, Lett Math Phys., 21, 1991, 193–204

20. Kac V., Lie superalgebras, Adv Math., 26, 1977, 8–96

21. Kemer A R Varieties of Z-graded algebras Math USSR Izvestiya, 1984, v 48, No 5,

1042–1059 (Russian)

22. Kinyon M., Weinstein A., Leibniz Algebras, Courant Algebroids, and Multiplications on Reductive Homogeneous Spaces, 24 pages math.DG/0006022

23. Kochetkov Yu., Leites D., Simple Lie algebras in characteristic 2 recovered from

superalge-bras and on the notion of a simple finite group Proceedings of the International Conference

on Algebra, Part 2 (Novosibirsk, 1989), 59–67, Contemp Math., 131, Part 2, Amer Math Soc., Providence, RI, 1992

24. Krakowski D., Reyev A The polynomial identities of the Grassmann algebra Trans Amer.

Math Soc., 1973, v 181, 429–438.

25. Leites D., Spectra of graded-commutative rings Uspehi Matem Nauk, 29, no 3, 1974,

157–158 (in Russian)

26. Leites D., Introduction to supermanifold theory, Russian Math Surveys, 35, 1980, 1, 1-57.

An expanded version is: Supermanifold theory, Karelia Branch of the USSR Acad of Sci.,

Petrozavodsk, 1983, 200p (in Russian, an expanded version in English is [27])

27. Leites D (ed.), Seminar on supermanifolds, no 1–34, 2100 pp Reports of Dept of Math.

of Stockholm Univ., 1986–1990; Leites D., Lie superalgebras, Modern Problems of

Mathe-matics Recent developments, 25, VINITI, Moscow, 1984, 3–49 (Russian; English transl in: JOSMAR 30 (6), 1985, 2056).

28. Leites D., Selected problems of supermanifold theory Duke Math J., 54, no 2, 1987, 649–656

29. Leites D., Poletaeva E., Supergravities and contact type structures on supermanifolds Second

International Conference on Algebra (Barnaul, 1991), 267–274, Contemp Math., 184, Amer.

Math Soc., Providence, RI, 1995

30. Leites D., Shchepochkina I., Classification of simple Lie superalgebras of vector fields, to appear; id., How to quantize antibracket, Theor and Math Physics, to appear

31. Leites D., Serganova I., Metasymmetry and Volichenko algebras, Phys Lett B, 252, 1990,

no 1, 91–96

32. Majid S., Quasitriangular Hopf algebras and Yang-Baxter equations, Int J Mod Phys A5,

1990, 1–91

33. Manin Yu Quantum groups and non-commutative geometry, CRM, Montreal, 1988

34. Manin Yu Topics in non-commutative geometry, Rice Univ., 1989

35 Rosenberg A., Noncommutative algebraic geometry and representations of quantized algebras Mathematics and its Applications, 330 Kluwer, Dordrecht, 1995 xii+315 pp.

36. Rubakov V., Spiridonov V., Parasupersymmetric quantum mechanics, Mod Phys Lett A, 3,

no 14, 1988, 1337–1347

37. Shchepochkina I., The five exceptional simple Lie superalgebras of vector fields (Russian)

Funktsional Anal i Prilozhen 33 (1999), no 3, 59–72, 96 translation in Funct Anal Appl 33

(1999), no 3, 208–219 (hep-th 9702121); id., The five exceptional simple Lie superalgebras of

vector fields and their fourteen regradings Represent Theory, (electronic) 3 (1999), 373–415

38. Serganova V., Classification of real forms of simple finite-dimensional Lie superalgebras and

symmetric superspaces Funct Anal Appl 1983, 17, no 3, 46–54

39 Serganova V., Automorphisns of Lie superalgebras of string theories Funct Anal Appl 1985,

v 19, no.3, 75–76

40 Serganova V., Simple Volichenko algebras Proceedings of the International Conference on Algebra, Part 2 (Novosibirsk, 1989), 155–160, Contemp Math., 131, Part 2, Amer Math Soc., Providence, RI, 1992

41. Shander V., Vector fields and differential equations on supermanifolds (Russian) Funktsional.

Anal i Prilozhen 14 (1980), no 2, 91–92

42. Spiridonov V., Dynamical parasypersymmetries in Quantum systems In: Tavkhelidze et al.

Proc Int Seminar “Quarcs 1990” , Telavi, May 1990, World Sci., 1991

43. Wess, J., Zumino, B., Supergauge transformations in four dimensions Nuclear Phys B70

(1974), 39–50; Wess J., Supersymmetry-supergravity Topics in quantum field theory and gauge theories (Proc VIII Internat GIFT Sem Theoret Phys., Salamanca, 1977), pp 81–

125, Lecture Notes in Phys., 77, Springer, Berlin-New York, 1978; Wess J., Zumino B.,

Superspace formulation of supergravity Phys Lett B 66 (1977), no 4, 361–364; Wess

J., Supersymmetry/supergravity Concepts and trends in particle physics (Schladming, 1986), 29–58, Springer, Berlin, 1987; Wess J., Bagger J., Supersymmetry and supergravity Princeton

Series in Physics Princeton University Press, Princeton, N.J., 1983 i+180 pp

KELLOGG STELLE∗

Theoretical Physics Group, Imperial College London SW7 2BW, UK

Abstract We show how the Randall-Sundrum geometry, which has been proposed as a scenario

for the universe realized as a 3-brane embedded in a 5-dimensional spacetime, arises naturally as an

S 5dimensional reduction of a supersymmetric 3-brane of type IIB supergravity However, a closerinspection of the D = 10 delta-function sources for this solution reveals a more complex situation:

in addition to the anticipated positive and negative shells of 3-brane source, there is also a non-brane stress-tensor delta-function The latter singularity may be interpreted as arising from a patching of two discs of D = 10 spacetime coincident with the inner and outer brane locations.

The idea that our universe might be realized as a 3-brane embedded in ahigher-dimensional spacetime has been considered at various times in recentyears [1–5] In the context of string duality, it was specifically the construction

of Hoˇrava and Witten [6, 7] realizing heterotic to M-theory duality via an orbifoldcompactification that set a pattern for this scenario In particular, one may obtain

a 3-brane solution to M-theory reduced on a Calabi-Yau manifold down to fivespacetime dimensions [8–10] This solution has parallel 3-brane universes facingeach other across a transverse fifth dimension, located at the fixed planes of theHoˇrava-WittenS1/Z2 orbifold The 3-branes are magnetically charged and satu-rate a BPS bound, so are supersymmetric The solution is supported by aD = 5

scalar field which has a higher-dimensional interpretation as the volume modulus,

or “breathing mode” of the compactifying space This scalar field acquires a tential as a result of 4-form fluxes being turned on in the compactified dimensions.The dimensional reduction is thus an example of a generalized (aka Scherk-Schwarz) reduction with non-trivial field strengths turned on in the compactifyingspace

po-Interest in such pictures became very much heightened when Randall and drum showed [11, 12] that in such a brane-world universe, gravity could behave

Sun-as if it were effectively 4-dimensional even though the distance between the two3-branes might be taken to infinity, provided the bulk geometry near the brane welive on is aD = 5 anti de Sitter space Specifically, a model was considered that

∗ k.stelle@ic.ac.uk

involved two segments of a pure AdS5spacetime patched together,

ds2

with a “kink” atz = 0 corresponding to a positive-tension δ-function stress-tensor

source In Ref [12] it was shown that this gives rise to a “binding” of gravity to the

D = 5 spacetime region near the (3+1) dimensional braneworld, with an effective

Newtonian gravitational potential plus eventual measurable corrections,

pergravity has the properties needed to flow correctly to a fixed point at locationsfar from the RS brane, and this was encoded in a “no-go theorem” [13, 14] As

is frequently the case with no-go theorems, however, the main result may be todirect attention towards the underlying assumptions that need to be relaxed Thekey one in this case concerns the nature of the supporting scalar

Even before the Randall-Sundrum work on our universe as a braneworld bedded in aD = 5 spacetime, a general study had been made [15] of the spherical

em-dimensional reductions of various supergravity theories and of the branes anddomain walls that exist in these reduced theories For the specific case of the

S5 reduction of type IIB supergravity down to D = 5, it was shown that the

familiar D3-brane geometry of D = 10 type IIB theory dimensionally reduces

to a 3-brane inD = 5, supported by the “breathing mode” scalar modulus that

determines the volume of the compactifying S5 This works in a very similarway to the breathing-mode supported 3-brane in the Calabi-Yau reduction of M-theory [8–10] It should be noted, however, that the breathing mode for an S5reduction does not itself belong to the massless supergravity multiplet Instead, thebreathing mode belongs to a massive spin-two multiplet, as is appropriate, sincethe dimensional reduction turns on a flux in the internalS5 directions, and thisgives the breathing modeϕ a scalar potential that allows this mode to support a

brane solution Without this potential, the breathing mode could not support a brane solution But the massive character of this mode places it outside the class ofmodes normally considered inD = 5 compactifications of supergravity theories

3-The importance of this mode for realizing the Randall-Sundrum braneworld as asupergravity construction was recognized in Refs [16, 17], although a main focus

was still on the difficulty of realizing RS geometries as a fully “smoothed-out”solitonic solution.

To see how a construction analogous to the M-theory 3-brane solution can bemade inS5reduced type IIB theory, consider a simplified theory just retaining the

D = 10 metric and the self-dual 5-form field strength H[5],

Rµν = 961 (H[5])2µν

where the equations of motion for the five-form are implied by the Bianchi identity

dH ≡ 0 taken together with the H[5] = ∗H[5] duality relation Dimensionallyreducing onS5, one makes the Kaluza-Klein ansatz

in the reduction ansatz (5), while the one with the coefficientR5comes from theEinstein-Hilbert action in the five compactified directions, since S5is not Ricci-flat The coefficientR5is equal to the constant Ricci scalar of the internalS5.The presence of two potential terms in (6) with opposite signs enables a partic-ularly simple and maximally symmetric solution to theD = 5 reduced theory In

this case, one can find a solution with a constant breathing-mode scalarϕ = ϕ∗,with

Solving this D = 5 Einstein equation with a cosmological term, one finds the

AdS5×S5“vacuum” of theS5compactified theory The existence of this vacuummakes this a simpler situation than the one obtained in M-theory reduced on aCalabi-Yau manifold, where only a single potential term is obtained, and where

no maximally-symmetric solution inD = 5 is found

In addition to the AdS5 × S5 solution (7), one can also search for brane lutions with less symmetry, but which tend asymptotically in appropriate regions