Laboratoire de Mathématiques Informatique et Applications

 

Colloque d’Algèbre

Algebra Workshop

Aspects algébriques et géométriques des algèbres de Lie

(Algebraic and geometric aspects of Lie algebras)

 

Mulhouse, 12-14 Juin 2008

Université de Haute Alsace

À la Faculté des Sciences et Techniques, Amphi A

18, rue des Frères Lumière, Mulhouse

 

Organisateurs : A. Makhlouf et M. Bordemann

 

Programme / Program

 

Jeudi 12 juin 2008

 

14h00 Martin  Markl (U. Prague)

 Lie elements in pre-Lie algebras, trees and cohomology operations

 

15h00  Said Benayadi (U. Metz)

 Superalgèbres de Lie avec métriques impaires

15h45 Pause

16h15 Friedrich Wagemann (U. Nantes)

 Déformations d'algèbres de Lie de dimension 10 en caractéristique 3

17h00 Paola PIU (U. Cagliari)

 Surfaces dans le groupe de Heisenberg H₃

 

 

Vendredi 13 juin 2008

 

8h30 Sergei  Silvestrov (U. Lund)

 Quasi Lie algebras, Hom Lie algebras, Hom-associative algebras and Hom-coalgebras

 

9h30 Martin Schlichenmaier (U. Luxembourg) 

 Almost-graded central extensions of Lax operator algebras

10h30 Pause

11h00 Yael Frégier (U. Luxembourg)

 Une décomposition de type Hodge-Lepage dans le cas contact

 

11h30 Serena Cicalo (U. Trento) 

 Constructing n-Engel Lie rings

12h00 Déjeuner

14h00 Ctirad  Klimcik (Marseille)

On integrability of the Yang-Baxter sigma model

 

15h00 Jose Ramon Gómez (U. Seville)

 Naturally graded p-filiform Leibniz and Lie algebras

15h45 Pause

16h15 Michel Rausch de Traubenberg (U. Strasbourg)

 Ternary algebras and groups

 

17h00 Faouzi Ammar (U. Sfax)

 Deforming the Lie superalgebra of contact vector fields-modules of symbols

 

17h45 Luisa M. Camacho Santana et Isabe Rodríguez García (U. Seville),

 On filiform and quasifiliform Leibniz algebras of maximal length

 

20h00 Repas à la Tour de l’Europe

 

 

Samedi 14 juin 2008

 

9h00 Dmitry Roytenberg (U. Utrecht)

 Cohomology and deformations of Courant-Dorfman algebras

 

10h00 Emily Burgunder (U. Montpellier)

 Une version symétrique du complexe de graphes de Kontsevich

 

10h45 Pause

 

11h00 Gianluca Bande (U. Cagliari) (en collaboration avec A. Hadjar)

 On normal contact pairs

 

11h45 Joakim Arnlind (U. Stockholm/ Potsdam)

 Representation theory for C-algebras of spheres and tori via graphs and dynamical systems

 

 

Résumés / Abstracts

 

 

Faouzi Ammar (U. Sfax)

Deforming the Lie superalgebra of contact vector fields-modules of symbols

 

 

Joakim Arnlind (U. Stockholm /Potsdam)

Representation theory for C-algebras of spheres and tori via graphs and dynamical systems

Abstract:  To construct noncommutative analogues of a manifold, a common procedure is to replace the (commutative) function algebra (of smooth functions on the manifold) by a noncommutative algebra A. This can be done in many ways, but for manifolds endowed with a symplectic form, there is a natural correspondence, relating Poisson-brackets of functions to commutators in A. We introduce C-algebras as noncommutative analogues of compact surfaces of arbitrary genus, and for a class of spheres and tori we completely characterize the representation theory of the corresponding C-algebras.  To achieve this, a graph method has been developed that leads to a classification of representations in terms of ``loops'' and  ``strings''. Moreover, to every surface there is an associated dynamical map s: R²->R² whose periodic orbits and N-strings one has to understand in order to construct explicit matrix representations. The constructed C-algebras can also be viewed as non-linear deformations  (in the universal enveloping algebra) of a three dimensional Lie algebra. It is interesting to note that some of these deformations lead to algebras which have irreducible representations of any  dimension.

 

 

Gianluca Bande (U. Cagliari)  (en collaboration avec A. Hadjar)

On normal contact pairs

Abstract: We consider manifolds endowed with a contact pair and an additional field of endomorphisms, in the same spirit of almost contact geometry. To such structures is naturally associated an almost complex structure which allows us to define normal contact pairs. Then we use flat bundles to generalize Morimoto's Theorem on product of almost contact manifolds and we construct some examples on Boothby--Wang fibrations over contact-symplectic manifolds. In particular, we obtain new constructions of complex manifolds.

 

 

Said Benayadi (U. Metz)

Superalgèbre de Lie avec métriques impaires

 

Abstract: Si ( f g ) est une super-algèbre de Lie munie d'une forme bilinéaire B qui est supersymétrique, invariante, non dégénérée et impaire, alors on dira que ({ f g},B) est une super-algèbre de Lie avec métriques impaires (ou une super-algèbre de Lie quadratique-impaire). Nous donnerons des exemples de telles super-algèbres de Lie et nous présenterons quelques résultats sur la structure de ces super-algèbres de Lie. Enfin,  nous donnerons une  description inductive des super-algèbres de Lie  avec métriques impaires.

 

 

Emily Burgunder (U. Montpellier)

Une version symétrique du complexe de graphes de Kontsevich

 
Abstract: Kontsevich a démontré que l'homologie de Lie des champs de vecteurs d'une variété formelle peut se calculer à partir d'un complexe de graphes. On démontre que l'homologiede Leibiz de cette algèbre peut-être reconstruite à partir d'un nouveau complexe de graphes muni d'une action des groupes symétriques. De plus, on montre que l'isomorphisme que l'on obtient est un isomorphisme de bigèbres Zinbiel-associatives.

 

 

Luisa Mª Camacho Santana et Isabe Rodríguez García (U. Sevilla)

On filiform and quasifiliform Leibniz algebras of maximal length

 

Abstract: The study of families of nilpotent Lie algebras with a gradation with a large number of subspaces facilitates the study of some cohomological properties for such algebras (see [1, 3, 4, 5]). For such a “length" of the gradation, the main interest is in algebras whose length is as large as possible. In the natural gradation on nilpotent Leibniz algebras, the subspaces in the gradation and the existence of an appropriate homogeneous basis (needed to obtain the classification) are a natural consequence of the central descending sequence of the considered algebras. However, it introduces a restriction by fixing the number of subspaces in the gradation by means of the nilindex. The gradations with n subspaces are, in a certain way, the finite connected gradations with the greatest possible number of non-zero subspaces, which will be called maximum length gradations. The algebras with maximum length gradations will be called maximum length algebras. The nilpotent Lie algebras of maximum length have studied in [3], [4]. This work is devoted of study of the filiform and quasifiliform Leibniz algebras which admit gradation of maximal number of non null homogeneous spaces. These results are published in [2].

[1] J.M. Cabezas, J.R. Gómez, A. Jiménez-Merchán, Family of p-filiform Lie algebras, in: Y. Khakimdjanov, M. Goze, S. Ayupov (Eds.), Algebra and Operator Theory, Kluwer Academic Publishers, Dordrecht, 1998, pp. 93-102.

[2] J.M. Cabezas, L.M. Camacho, I.M. Rodríguez, On filiform and Quasifiliform Leibniz algebras of maximun length, Journal of Lie Theory, 18(2), 2008, 335-350.

[3] J.R. Gómez, A. Jiménez-Merchán, J. Reyes, Filiform Lie algebras of maximum length, Extracta Math., 16(3) (2001) 405-421.

[4] J.R. Gómez, A. Jiménez-Merchán, J. Reyes, Quasi-filiform Lie algebras of great length., Accepted for publication Journal of Algebra, (2008).

[5] M. Goze, Y. Khakimdjanov, Nilpotent Lie Algebras, Kluwers Academic Publishers, Dordrecht, 1996.

 

Serena Cicalo (U. Trento) 

Constructing n-Engel Lie rings

 

Abstract: By a result of Zelmanov every finitely generated Lie ring that satisfies the n-Engel identity is nilpotent, and hence finite dimensional. In this talk we present methods for constructing a basis of the "freest" n-Engel Lie ring (over the integers) with t generators. We use Groebner basis type methods in free algebras, along with a linearizations of the n-Engel identity, valid over the integers.

 

Yael Frégier (U. Luxembourg)

Une décomposition de type Hodge-Lepage dans le cas contact

Abstract : Nous montrons comment décomposer  l'algèbre des tenseurs symétriques en présence d'une variété de contact projective. Nous introduisons pour cela deux opérateurs qui forment une représentation de l'algèbre sl(2,R), présentant de troublantes analogies avec la construction de la décomposition de Hodge-Lepage de l'algèbre extérieure d'un espace vectoriel symplectique.

 

 

Jose Ramon Gómez (U. Seville)

Naturally graded p-filiform Leibniz and Lie algebras

 

Abstract: The knowledge of the naturally graded algebras of a family of non associative algebras is important because it contributes to obtain relevant information about the structure of the family, about its irreducible components and about some cohomologic problems.

In the cohomological study of the variety of nilpotent Lie algebra laws obtained by Vergne [7], the classification of “naturally" graded filiform Lie algebras plays a fundamental role. The obtained classification allows to express a filiform algebra easily as an algebra of maximal nilindex among those having the same dimension. Goze and Khakimdjanov, using the naturally graded filiform algebras, gave in [6] the geometric description of the characteristically nilpotent filiform Lie algebras.

Cabezas, Gómez and Jiménez-Merchán [3] generalize the notion of filiform algebras to p-filiform algebras, which correspond to nilpotent algebras of Goze invariant (n-p,1,…,1) where n=dim( ); in this context, the filiform algebras are the 1-filiform algebras and the quasi-filiform algebras are the 2-filiform algebras.

Analogously to the notion of p-filiform Lie algebras (p>0), the p-filiform Leibniz algebras (p≥0) are the nilpotent n-dimensional Leibniz algebras with the characteristic sequence (n-p,1,1,...,1). Naturally graded p-filiform Lie algebras are known for p>0 (the admisible values of p) [4], [5], [7].

Leibniz algebras appear as a generalization of Lie algebras. To obtain the classification of nilpotent Leibniz algebras in general seems unapproachable. But it is possible to select more restrictive families with interesting properties. In particular, there are specially interesting to obtain the classification of naturally graded algebras.

The present conference offers the classification of naturally graded p-filiform Leibniz and Lie algebras in arbitrary finite dimension $n$, for all value admissible of p. In order to classify naturally graded p-filiform Lie algebras, the dimension n must be ≥ 3p-1. For all the expressions to be true it is also necessary that n ≥ p+8. We also study the cases where n <max{3p-1,p+8}. There are some difficulties and other algebras appear: the exceptional algebras. We obtain that for n≥max{3p-1,p+8}, there are non split algebras.

The study of p-filiform Leibniz non Lie algebras is solved for p=0 and it only exists one single algebra [1], for p=1 [2]. In this conference we get the classification of naturally graded non Lie p-filiform Leibniz algebras in arbitrary finite dimension.

[1] Sh.A. Ayupov, B.A. Omirov, On a Description of Irreducible Component in the set of nilpotent Leibniz Algebras containing the Algebra of maximal nilindex, and classification of graded filiform Leibniz algebras, Computer Algebra in Scientific Computing CASC, Springer, 21-34, 2000.

[2] Sh.A. Ayupov, B.A. Omirov, On Leibniz algebras, Algebra and operators theory, Proceeding of the coloquium in Tashkent 1997. Kluwer Academic Publishers, 1-13, 1998.

[3] J.M. Cabezas, J.R. Gómez, A. Jiménez-Merchán, Family of p-filiform Lie algebras, In Algebra and Operator Theory, 1997. Ed. Y. Khakimdjanov, M. Goze, Sh. Ayupov, 93-102, Kluwer Academic Puplishers, 1998.

[4] J.M. Cabezas, E. Pastor, Naturally graded p-filiform Lie algebras in arbitrary finite dimension, Journal of Lie Theory, 15, 379-391, 2005.

[5] J.R. Gómez, A. Jiménez-Merchán, Naturally Graded Quasi-Filiform Lie Algebras, Journal of Algebra, 256, 211-228, 2002.

[6] M. Goze, Y. Khakimdjanov, Sur les algèbres de Lie nilpotentes admettant un tore de derivations, Manuscripta Math. 84, 115-224, 1994.

[7] M. Vergne, Cohomologie des algèbres de Lie nilpotentes. Application´`a l'étude de la varieté des algèbres de Lie nilpotentes, Bull. Soc. Math. France, 98, 81-116, 1970.

 

 

Ctirad  Klimcik (U. Marseille)

On integrability of the Yang-Baxter sigma-model

We prove the integrability of the  Yang-Baxter $\sigma$-model which is the Poisson-Lie   deformation of the principal chiral model. We find also an explicit one-to-one map  transforming every solution of the principal chiral model into a solution of the deformed  model. With the help of this map, the standard procedure of the dressing of the principal chiral solutions can be directly transferred into  the deformed Yang-Baxter context.

 

Martin Markl  (U. Prague)

Lie elements in pre-Lie algebras, trees and cohomology operations

Abstract: We give a simple characterization of Lie elements in free pre-Lie algebras as elements of the kernel of a map between spaces of trees. We explain how this result is related to natural operations on the Chevalley-Eilenberg complex of a Lie algebra. We also indicate a possible relation to Loday's theory of triplettes.

 

 

Paola PIU (U. Cagliari)

Surface dans le groupe de Heisenberg H₃

Abstract: Le group de Heisenberg H₃ est la variété riemannienne (R³, ds²) où ds² = dx² + dy² + [dz + 1/2 (ydx − xdy)] ²    est invariante par rotation autour de l’axe  z. On veut faire une excursion dans les résultats et donner une description des surfaces SO(2)-invariant . Nous ferons une description complète des surfaces SO(2)-invariantes avec courbure gaussienne ou moyenne constante et de celles qui sont surfaces minimales. On montrera la grande similitude que l’on a avec l’espace euclidien et montrera les surfaces qui n’ont pas une surface analogue dans R³. Les courbes profils seront illustrées et les figures correspondantes seront caractérisées géometriquement.

[1] Francesco Mercuri, Stefano Montaldo, Paola Piu, Weierstrass representation formulae of minimal surfaces in H₃ and H² ×R. Acta Math. Sinica, 22 (2006), 1603-1612.

[2] Paola Piu, Aristide Sanini, One-parameter subgroups and minimal surfaces in the Heisenberg group. Note di Matematica Lecce Vol 18 - n. 1, pg.143-153 (1998).

[3] Renzo Caddeo, Paola Piu, Andrea Ratto, Rotational surfaces in H₃ with constant Gauss curvature, Bollettino U.M.I. (7) , 9-B, pg. 341-357, (1996).

[4] Paola Piu, Sottogruppi totalmente geodetici dei gruppi di Lie 2-nilpotenti, Atti XV Congresso U.M.I., Padova, 11-16 Settembre 1995.

[5] Renzo Caddeo, Paola Piu, Andrea Ratto, SO(2)- invariant minimal and constant mean curvature surfaces in 3-dimensional homogeneous spaces, Manuscripta Math. 87, pg. 1-12 (1995).

[6] Michel Goze, Paola Piu, Une caractérisation Riemannienne du groupe de Heisenberg, Geometriae Dedicata, 50), pg. 27-36, (1994.

[7] Paola Piu , Sur certains types de distributions non-integrables totalement géodésiques, Thèse de Doctorat, Univ. de Haute Alsace, (1988) 1.

 

 

Michel  Rausch de Traubenberg (U. Strasbourg)

Ternary algebras and groups

 

Abstract: 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 Z_F-gradation, the zero-graded part being a Lie algebra. 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 Poincaré algebra, but no group associated to these types of algebras were defined. In this talk we firstly characterise the universal enveloping algebra of an elementary Lie algebra of order three, and we show that it can be endowed with a Hopf algebra structure. This will enable us to define an appropriate group associated to Lie algebras of order three. We then construct  explicitly (when F=3)  a group associated to Lie algebras of order three. It turns out that the natural variables which appear in this construction are variables θ^I satisfying the relations:

θ^i θ^j θ^k θ^j θ^k θ^i + θ^k θ^I θ^j + θ^i θ^k θ^j

+ θ^j θ^i θ^k + θ^k θ^j θ^i = 0,

i.e. the variables $\theta^i$ generate the three-exterior algebra. An explicit matrix representation of a group associated to a peculiar Lie algebra of order three is constructed.

 

 

Dmitry Roytenberg (U. Utrecht)

Cohomology and deformations of Courant-Dorfman algebras

 

Abstract: Courant-Dorfman algebras are a higher analogue of Poisson and Lie-Rinehart algebras; they appear naturally in Poisson geometry, generalized geometry (in the sense of Hitchin - Gualtieri), and string theory. We describe a graded Lie (in fact, Poisson) algebra whose Maurer- Cartan elements correspond to Courant-Dorfman structures. This yields an explicit formula for the differential in the standard deformation complex of a Courant-Dorfman algebra; it resembles the well-known Cartan formula for the cochain complex of a Lie algebra with coefficients in a module, but contains additional cubic terms. In case of Courant-Dorfman algebras of geometric origin, an equivalent simple construction can be given in terms of a differential graded symplectic manifold.

 

 

Martin Schlichenmaier (U. Luxembourg)

Almost-graded central extensions of Lax operator algebras

 

Abstract: Lax operator algebras constitute a new class of infinite dimensional Lie algebras of geometric origin. More precisely, they are algebras of matrices whose entries are meromorphic functions on a compact Riemann surface. They generalize classical current algebras and current algebras of Krichever-Novikov type. Lax operators for gl(n) were introduced  by Krichever. In joint works of Krichever and Sheinman their algebraic structures was revealed and extended to more general groups. These algebras are almost-graded. In this talk we recall their definition and present classification and uniqueness results for almost-graded central extensions for this new class of algebras. The presented results are  joint work with Oleg Sheinman.

 

 

Sergei Silvestrov  (U. Lund)

Quasi Lie algebras, Hom Lie algebras, Hom-associative algebras and Hom-coalgebras

Abstract: In this talk I will review main constructions, examples and some new results from the theory of Quasi Lie algebras, Hom Lie algebras, Hom-associative algebras and Hom-coalgebras, quasi Lie deformations of Lie algebras and superalgebras and their quasi Lie cental extensions. New results presented in this talk are joint with Abdenacer Makhlouf.

 

 

Friedrich Wagemann  (U. Nantes)

Déformations d'algèbres de Lie de dimension 10 en caractéristique 3

Abstract: Les algèbres de Lie classiques n'admettent de déformations non triviales en caractéristique 0 ou plus grand ou égal à 5, par contre, en caractéristique 2 et 3, il existe des algèbres de Lie classiques non rigides. Dans ce travail en commun avec Sofiane Bouarroudj (University of the United Arab Emirates, Al Ain), nous étudions via MATHEMATICA ordre par ordre une déformation infinitésimale à 5 paramètres d'une algèbre de Lie de dimension 10 en caractéristique 3 afin de comprendre pourquoi l'algèbre de Lie n'admet que des vraies déformations à 3 paramètres, tandis que le H² est de dimension 5. Ceci se base sur un travail de Kostrikin et Kusnetsov de 1995.