Thursday 26 November

Michel GOZE. Mulhouse U. [email protected]. Jean-Luc JACQUOT. Strasbourg ... Natacha MAKARENKO ... Michel RAUSCH de TRAUBENBERG Strasbourg.
119KB taille 17 téléchargements 572 vues
6th Alsacian Meeting in Mathematics and Physics Superalgebras, n-ary algebras and related structure : applications to physics Strasbourg, IPHC 26-27-28 Novembre 2009 Amphi-Gr¨ unewald-Bˆatiment 25 http://indico.in2p3.fr/conferenceDisplay.py?confId=2207

Thursday 26 November Chair: 8h45 9h00 9h45

D. Huss I. Antoniadis X. Bekaert

10h30 11H00

M.Valenzuella

11h45

G. Moultaka

12h15

Welcome-Presentation of IPHC Non-linear supersymmetry and D−branes dynamics Singletons & Higher-Spin Algebras: definitions and applications Coffe break Hidden relativistic (super)symmetries of the free Schr¨odinger equation Indeterminism versus determinism in quantum mechanics: where mathematics and physics failed to meet Lunch Chair:

14h15 15h00

R. Kerner J. Dudek

15h45 16h15

N. Mohammedi

17h00 17h45

M. Capdequi-Peyranere V. Abramov

18h30

Invariance groups of ternary Z3 graded structures Physics and mathematics behind nuclear super-fluidity phenomena Coffee break On the geometry of classically integrable two dimensional non linear sigma models Kowalevski’s analysis of the Swinging Atwood’s Machine Generalization of connection based on a graded q−differential algebra End of the day

1

Friday 27 November

9h00 9h45 10h30 11h00 11h45 12h30

E. Khukhro N. Makarenko L. Garcia-Vergnolle S. Benayadi

Applications of Lie algebras with finite cyclic grading Algebras of Lie type with almost regular automorphisms Coffee break Complex Structures on nilpotent Lie algebras Odd-quadratic Lie superalgebras Lunch Chair:

14h15

J. Polonyi

15h00

E. Paal

15h45 16h15 17h00 17h45

T. Popov M. Slupinski

Radiation backreaction: Consistency of classical field theory Operadic dynamics and quantization of 3d real Lie algebras over harmonic oscillator Coffee break Parastatistics Fock spaces The exceptional Lie algebra g2 and binary cubics End of the day

Saturday 28 November Chair: 9h00 9h45 10h30 10H45 11h30 12h15

G. Volkov S. Silvestrov

M. Rausch de Traubenberg M. Goze

Some physical applications of Rn complexification Quasi-Lie, hom-Lie and color Lie algebras and related Hom-algebraic structures Break Parafermions, ternary algebras and their associated superspace n-Lie algebras End

2

V. Abramov: Generalization of connection based on a graded qdifferential algebra We propose a generalization of a concept of connection form by means of a graded q-differential algebra, where q is a primitive Nth root of unity, and develop a concept of a curvature N-form for this generalization of connection form. The Bianchi identity for a curvature N-form is proved. We study an q-connection on module and prove that every projective module admits an q-connection. If module is equipped with a Hermitian structure we introduce a notion of a q-connection consistent with Hermitian structure.

I. Antoniadis: Non-linear supersymmetry and D-branes dynamics X. Bekaert: Singletons & Higher-Spin Algebras: definitions and applications Abstractly, singletons are defined as those unitary irreducible representations (UIRs) of the Poincar´e group that can be lifted to UIRs of the conformal group; the higher-spins algebras correspond to the corresponding realizations of the universal enveloping algebra of the conformal algebra. Mathematically, their explicit construction involve various ingredients interesting on their own: Metaplectic representation, Howe duality, ... Physically, they appear in a wide variety of areas: infinite-component Majorana equations, higher-spin multiplets, generalized symmetries, ... which will be reviewed from this perspective.

S. Benayadi: Odd-quadratic Lie superalgebras An odd-quadratic Lie superalgebra is a Lie superalgebra with a non-degenerate, supersymmetric, odd, and invariant bilinear form. We give examples and present some properties of odd-quadratic Lie superalgebras. We obtain description of odd-quadratic Lie superalgebras such that the even part is a reductive Lie algebra without using the notions of double extensions. We obtain also an other inductive descriptions of this superalgebras by using the concept of elementary generalized odd double extensions.

M. Capdequi-Peyranere: Kowalevski’s analysis of the Swinging Atwood’s Machine We study the Kowalevski expansions near singularities of the √ swinging Atwood’s machine. We show that in the integrable case these expansions are Laurent series in t, so–called weak Painlev expansions, and that similar expansions arise in many cases where the system is not integrable, contrary to some conjectures.

J. Dudek Physics: and mathematics behind nuclear super-fluidity phenomena In low-energy sub-atomic physics the so-called ‘pairing’ interactions play a distinct role; theory articles addressing the properties of the N-nucleon (many-body) systems that ignore this component in the manybody interactions are often down-graded as ‘not sufficiently realistic in the XXIst century’. The corresponding Hamiltonian, while manifesting analogies with the one responsible for the super-conductivity in condensedmatter physics has certain specificity when applied to the space-localised many-nucleon systems. We intend to address some aspects of the sub-atomic world specificity of the pairing hamiltonians: we will introduce the symmetry properties of the Hamiltonians in question in terms of the U(n) and SO(n) symmetries applying specifically to the closed and open systems, respectively. After recalling briefly the Gelfand-Tsetlin scheme of constructing the basis states for the related irreps we will introduce the system of 3 mutually commuting operators generating what we call ”P-symmetries” of the most general pairing Hamiltonian. 3

The P-symmetries allow to formally block-diagonalize the matrices of the Hamiltonians whose dimensions (in the realistic spaces required in this domain of physics) may easily rich 1020 up to 1080 many-body basis states. Will will effectively construct the solutions allowing to solve the problems of this type e x a c t l y in the spaces of dimensions up to 1010 and very good approximations for the spaces of ‘astronomical sizes’ as well. We will demonstrate how to attack the problem from the totally opposite view point: obtaining the same solutions using the stochastic simulation technique. Beginning with the image of the pair-scattering in the nucleonic medium we will demonstrate how to obtain an equivalent information by Monte-Carlo type methods which trade the huge-computer memory requirements for the very small memory space but repetitive use of fast processors (memory volume request replaced by fact processor). The two concepts determine the succes in the exact simulations of certain basic properties of bound many-body nucleonic systems in the large scale, massive simulation programs. Finally we will show how the algebraic considerations on the one hand and the stochastic ones on the other can be confronted with the traditional Nobel-Prize BCS (Bardeen-Cooper-Schrieffer) solutions and what is the evidence for the pairing and superfluid phenomena in the sub-atomic universe. We will show how all that compares with the results of sophisticated measurements obtained with the help of the billion-worth world-leading devices...

L. Garcia-Vergnolle: Complex Structures on nilpotent Lie algebras The study of complex manifolds has interested many authors in different fields in mathematics and physics. A complex structure J over a real even-dimensional Lie algebra g is an endomorphism of g which satisfies the Nijenhuis condition and J 2 = −Id. The classification of complex structures has only been completely obtained in dimensions 2 and 4. In dimension 6, Salamon found all the nilpotent Lie algebras provided with a complex structure. The first general result is the nonexistence of complex structures over nilpotent Lie algebras maximal nilindex, also called filiform. Generalized complex geometry introduced by Hitchin and developed by Gualtieri and Cavalcanti contains complex and symplectic geometry as extremal special cases. We will show that generalized complex structures represent an important tool in the study of complex structures over nilmanifolds.

M. Goze: n-Lie algebras The more general notion of n-Lie algebras is related to strong homotopy algebras. It corresponds to an n-skew symmetric product satisfying the n- Jacobi condition : X (−1)(σ)[[vσ(1) , · · · , vσ(n) ], vσ(n+1) , · · · , vσ(2n+1) ] = 0. σ∈Σ2n−1

We can present, for this class of algebras, the Maurer Cartan calculus, a general cohomological operadic approach, a theory of deformations. But the notions of nilpotency, solvability, simplicity is more difficult to introduce. The problem comes back to the inner derivations. The adjoint operators are not necesarily inner derivations. If we consider the subclass of n-Lie algebras whose adjoint operators are inner derivations, we obtain the n-Lie algebras in the sense of Filippov. Note that this last notion is older than the strong homotopy n-Lie algebras. We study some classes of nilpotent Filippov n-Lie algebras. We introduce the notion of filiformity. In a second part, we present some relations between n-Lie algebras and the Nambu mechanic. We hape to give a link between these algebras and the notions of k-symplectic structures which has permited to present some regular solutions of the Nambu equations.

R. Kerner: Invariance groups of ternary Z3 -graded structures We present several cubic and ternary algebraic structures, and endow them with Z3 grading. The automorphisms of certain strutures of this type define quite naturally the SL(2, C) and SU (3) goups. We shall also discuss possible relationship of these algebraic structures with the physics of quarks.

4

E. Khukhro: Applications of Lie algebras with finite cyclic grading Let L = ⊕n−1 i=0 be a (Z/nZ)-graded Lie ring (algebra), where the Li are additive subgroups (subspaces) satisfying [Li , Lj ] ⊆ Li+j(modn) . Theorems of Higman, Kostrikin, and Kreknin assert that if L0 = 0, then L is soluble (for n prime, nilpotent) of n-bounded (i. e. bounded in terms of n) derived length (class). Hence the same follows for a Lie ring M with a regular (i. e. without nontrivial fixed points) automorphism ϕ of order  n: after adjoining a primitive nth root of unity ω we obtain M = M+ M1 + · · · + Mn−1 for Mi = x ∈ M |ϕ(x) = ω i x , where [Mi , Mj ] ⊆ Mi+j(modn) and M0 = 0 (the fact that the sum is not direct in general is inessential). A similar assertion easily follows for (locally) nilpotent groups with a regular automorphism of prime order. But there is an open problem whether an analogue of Kreknin’s theorem holds for such groups with a regular automorphism of arbitrary finite order. Nevertheless, Kreknin’s theorem was successfully applied to finite p-groups with an automorphism of order pk and to pro-p-groups of given coclass in the papers of Jaikin-Zapirain, Khukhro, Medvedev, Shalev, Shalev-Zel’manov. Makarenko and Khukhro proved that if dim L0 = r (or |L0 | = r), then L contains a soluble (for n prime, nilpotent) ideal of n-bounded derived length (nilpotency class) and of (n, r)-bounded codimension. Khukhro applied this result to Lie rings and periodic nilpotent groups with an “almost regular” automorphism of prime order n; Medvedev lifted the periodicity condition for groups. Suppose that there are only d nonzero components among the grading compo- nents Li . Shalev and Khukhro proved that is L0 = 0, then L is soluble (for n prime, nilpotent) of d-bounded derived length (class). These results were applied to groups of bounded rank with almost regular automorphisms. In the works of Makarenko, Khukhro, Shumyatsky the condition L0 = 0 was replaced by dim L0 = r (or |L0 | = r): then L contains a soluble (for n prime, nilpotent) ideal of d-bounded derived length (class) and of (d, r)-bounded codimension. There results were applied to generalize Jacobson’s theorem on Lie algebras with a nilpotent algebra of derivations to the case of “almost without nontrivial constants”. Suppose that for some m for k 6= 0 we have |{i|[Lk , Li ] = 0}| ≤ m, i. e. each component Lk for k 6= 0 commutes with all but at most m components. Khukhro proved that if L0 = 0, then L is soluble (for n prime, nilpotent) of m-bounded derived length (class), and if dim L0 = r (or |L0 | = r), then L contains a soluble (for n prime, nilpotent) ideal of m-bounded derived length (class) and of (n, r)-bounded codimension. These results were applied to nilpotent groups with Frobenius groups of automorphisms.

N. Makarenko: Algebras of Lie type with almost regular automorphisms Let G be a group. A G-graded algebra A over a field F is called a Lie type algebra if for any g, h, k ∈ G there exist constants α 6= 0, β ∈ F such that a(bc) = α(ab)c + β(ac)b for all a ∈ Ag , b ∈ Ah , c ∈ Ak . The examples of Lie type algebras are Lie algebras, associative algebras, Lie superal- gebras, Leibniz(Loday) algebras, color Lie superalgebras, quantum Lie algebras, etc. Let G be a finite cyclic group and A a G-graded Lie type algebra. We proved that if the unity homogeneous component Ae is finite-dimensional, then A is almost soluble. In particular case, where Ae = 0, the algebra A is soluble. These are generalizations of the theorem of Kreknin (1963) and the theorem of Khukhro Makarenko (2004) on graded Lie algebras. As corollaries we obtain the theorems on Lie type algebras with regular and almost regular automorphisms.

N. Mohammedi: On the geometry of classically integrable two dimensional non linear sigma models The target space geometry and the Lie group structure behind the classical integrability of two-dimensional non-linear sigma models are highlighted.

5

G. Moultaka: Indeterminism versus determinism in quantum mechanics : where mathematics and physics failed to meet I review an old and partly forgotten story, hidden variables in quantum mechanics, taking as a prototype example the de Broglie-Bohm approach. I recall in what sense this approach differs from the standard quantum mechanics, how it avoids the main conceptual problems of the latter yet leading to the same empirical results, and how the celebrated von Neumann’s no-go theorem fails to forbid it. I then touch upon the problematic extension to relativistic quantum field theory, where differences between the standard and the de Broglie-Bohm approach could translate into observably distinctive phenomena. In particular, I argue how the notion of departure from ’quantum equilibrium’ could be a natural setting to discuss alternatives to the Higgs sector as a restorer of unitarity in electroweak interactions.

E. Paal, J. Virkepu: Operadic dynamics and quantization of 3d real Lie algebras over harmonic oscillator Operadic Lax representations for the harmonic oscillator (HO) are used to construct the dynamical deformations and quantum counterparts of 3d real Lie algebras. It is shown that the energy conservation is related to the Jacobi identities of the dynamically deformed algebras. Based on this observation, it is proved that the dynamical deformations of 3d real Lie algebras over HO are Lie algebras. The quantum counterparts of the 3d real Lie algebras over the harmonic oscillator are constructed and their Jacobi operators are calculated. Many resulting quantum counterparts of the 3d real Lie algebras turn out to be anomalous. It is discussed how the anomaly is related to quantization of the 3-space.

J. Polonyi: Radiation backreaction : Consistency of classical field theory Accelerating charges are loosing energy and momentum therefore, a radiation backreaction force, the Abraham-Lorentz force is acting upon them. The origin of this force which is not encoded in the set of Maxwell equation is the last open chapter of classical electrodynamics. The problem becomes simpler by considering point charges but it has already been recognized more than a half century ago that the AbrahamLorentz force leads to unacceptable, self accelerating runaway trajectories. It has been found recently that gravitational backreaction force generates time dependent mass in General Relativity. The mathematical origin of this effect can already be identified in a toy model of a point particle coupled to a free, massive scalar field in flat space-time.

T. Popov: Parastatistics Fock spaces We consider parabosonic and parafermionic algebras focusing on the structure of the Fock-like spaces of given parastatistics order. These are unitary irreducible representations with lowest weight of osp(1|2n) and so(2n + 1), respectively. More generally, parastatistics algebras with n parabosons and m parafermions are isomorphic to osp(1+2m|2n). The states in parastatistics Fock-like spaces are labelled by Super Semistandard Young Tableaux. Character formulas on Fock-like spaces yield new identities which are row and column length restrictions of some classical Schur function identities.

M. Rausch de Traubenberg: Parafermions, ternary algebras and their associated superspace Lie algebras of order F (or F −Lie algebras) are possible generalisations of Lie algebras (F = 1) and Lie superalgebras (F = 2). An F −Lie algebra admits a ZF −gradation, the zero-graded part being a Lie algebra. 6

An F −fold symmetric product (playing the role of the anticommutator in the case F = 2) expresses the zero graded part in terms of the non-zero graded part. These structures have been used to implement new non-trivial extensions of the Poincar´e algebra and a group associated to these types of algebras was defined. In this talk we construct explicitly a differential realisation of a given cubic extension of the Poincar´e algebra by means of parafermions of order two. This means in particular that parafermionic variables enable us to define an associated superspace together with corresponding superfields.

S. Silvestrov: Quasi-Lie, hom-Lie and color Lie algebras and related Hom-algebraic structures Recent flow of works on Hom-algebra structures have been originated in the works by the author and his students introducing Quasi-Lie algebras encompassing in a natural way Lie superalgebras, color Lie algebras and a new class, of Hom-Lie algebras. Motivating examples included also various algebras of discrete and twisted vector fields arising from q-deformed vertex operators structures and q-deferential calculus, and various classes of multiparameter deformations of associative and non-associative algebras appearing in other contexts in Mathematics, Mathematical Physics and Engineering. Among examples arising within quasi-Lie algebras framework are known and new one-parameter and multi-parameter deformations of infinite-dimensional Lie algebras of Witt and Virasoro type some of which appear in the context of conformal field theory, string theory and deformed vertex models, multi-parameter families of quadratic and almost quadratic algebras that include for natural special “limiting” choices of parameters algebras appearing in non-commutative algebraic geometry, as well as universal enveloping algebras of Lie algebras, Lie superalgebras and color Lie algebras. Common unifying feature for all these algebras is appearance of some twisted generalizations of Jacoby identities providing new structures of interest for investigation from the side of non-associative algebras, generalizations of associative algebras, generalizations of Hopf algebras, non-commutative differential calculi, generalized central extensions and generalizations of homological algebra. In this talk, I will provide a review on the Hom-algebra structures, hom-Lie algebras and quasi-Lie algebras. I will also describe some related n-ary Hom-algebra generalizations of Nambu algebras, Nambu-Lie algebras, associative algebras and Lie algebras.

M. Slupinski: The exceptional Lie algebra g2 and binary cubics Let P (x, y) be a binary cubic, i.e., a homogeneous polynomial of degree three in two variables. To P (x, y) one can naturally associate a scalar DP (the discriminant), a binary quadratic form qO (x, y) and another binary cubic GP (x, y). In 1844 G. Eisenstein proved the relation 4qP (x, y)3 = GP (x, y)2 − DP P (x, y)2 . In the talk I will give a purely symplectic formulation of this relation and show that it is a special case of a more general relation satisfied by the symplectic covariants of a symplectic module which one can associate to any simple Lie algebra. From this point of view, the classical Eisenstein relation comes from the exceptionial Lie algebra g2 . This is joint work with R. J. Stanton.

M.Valenzuella: Hidden relativistic (super)symmetries of the free Schr¨ odinger equation It is shown that representation spaces of the non-relativistic (mass-central-extended) Galileo and Schr¨odinger algebras in d-dimensions are also modules of the Lie algebra hd ⊕s sp(2d) and the Lie superalgebra osp(1|2d). Their generators form a set of observables of the Schr¨odinger equation. Hence, new Lie symmetries of the Schr¨ odinger equation are revealed, as well as, that it is supersymmetric.

7

G. Volkov: Some physical applications of Rn complexification Following to the new ideas of the Abelian complexification of Rn spaces founded by N. Fleury, M. Rausch de Traunbenberg and R. M. Yamaleev we suggest some further physical applications which could very important for the Standard Model of elementary particles and for the Standard Cosmology Model.

8

PARTICIPANTS Viktor ABRAMOV Ignatios ANTONIADIS Xavier BEKAERT Said BENAYADI Michel CAPDEQUI-PEYRANERE

Tartu (Estonia) CERN Tours U. Metz U. Montpellier U.

Jerzy DUDEK Jean-Yves FORTIN Benjamin FUKS Lucia GARCIA-VERGNOLLE Michel GOZE Jean-Luc JACQUOT Richard KERNER Evgeny KHUKHRO Natacha MAKARENKO Noureddine MOHAMMEDI Herv MOLIQUE Gilbert MOULTAKA

Strasbourg U. Nancy U. Strasbourg U. Madrid, Spain and Mulhouse U. Mulhouse U. Strasbourg U. Paris, Jussieu Novosibirsk, Russia Mulhouse U. Tours U. Strasbourg U Montpellier U.

Fr´ed´eric NOWACKI Eugene PAAL Vincent PANGON Mathieu PLANAT Janos POLONYI Todor POPOV Michel RAUSCH de TRAUBENBERG Elisabeth REMM Nicolas RIVIER Sergei SILVESTROV Alicja SIWEK Dominique SPEHLER Marcus SLUPINSKI Mauricio VALENZUELLA Guennadi VOLKOV Karima ZAZOUA

Strasbourg U. Tartu, Etonia Strasbourg U. Strasbourg U. Strasbourg U. Sofia, Bulgaria Strasbourg Mulhouse U. Strasbourg U. Lund, Sweden Strasbourg U. Strasbourg U Strasbourg U. Tours U. CERN and Peterbourg, Russia Strasbourg U.

9

[email protected] [email protected] [email protected] [email protected] Michel.CAPDEQUI-PEYRANERE @lpta.univ-montp2.fr [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] natalia− [email protected] [email protected] [email protected] Gilbert.Moultaka @LPTA.univ-montp2.fr [email protected] [email protected] [email protected] [email protected] [email protected]¿ [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]