Parafermions, ternary algebras and their associated superspace

Parafermions of order two are shown to be the fundamental tool to construct ternary superspaces related to cubic extensions of the Poincar\'e algebraComment: Talk given at the VIII. International Workshop Lie Theory and its applications in physics, 15 - 21 June 2009, Varna, Bulgari

Color Lie algebras and Lie algebras of order F

The notion of color algebras is generalized to the class of F-ary algebras, and corresponding decoloration theorems are established. This is used to give a construction of colored structures by means of tensor products with Clifford-like algebras. It is moreover shown that color algebras admit realisations as q=0 quon algebras.Comment: LaTeX, 16 page

Unitary representations of three dimensional Lie groups revisited: An approach via harmonic functions

Harmonic functions of the three dimensional Lie groups defined on certain manifolds related to the Lie groups themselves and carrying all their unitary representations are explicitly constructed. The realisations of these Lie groups are shown to be related with each other by either natural operations as real forms or In\"on\"u-Wigner contractions.Comment: The title was changed; More details are given for the constuction of harmonic functions 19 page

Ternary algebras and groups

We construct explicitly groups associated to specific ternary algebras which extend the Lie (super)algebras (called Lie algebras of order three). It turns out that the natural variables which appear in this construction are variables which generate the three-exterior algebra. An explicit matrix representation of a group associated to a peculiar Lie algebra of order three is constructed considering matrices with entry which belong to the three exterior algebra.Comment: 11 pages contribution to the 5th International Symposium on Quantum Theory and Symmetries (QTS5

Unexpected Features of Supersymmetry with Central Charges

It is shown that N=2 supersymmetric theories with central charges present some hidden quartic symmetry. This enables us to construct representations of the quartic structure induced by superalgebra representations.Comment: 14 pages, more details have been given, to appear in J. Phys.

Towards the Unification of Gravity and other Interactions: What has been Missed?

Faced with the persisting problem of the unification of gravity with other fundamental interactions we investigate the possibility of a new paradigm, according to which the basic space of physics is a multidimensional space ${\cal C}$ associated with matter configurations. We consider general relativity in ${\cal C}$. In spacetime, which is a 4-dimensional subspace of ${\cal C}$, we have not only the 4-dimensional gravity, but also other interactions, just as in Kaluza-Klein theories. We then consider a finite dimensional description of extended objects in terms of the center of mass, area, and volume degrees of freedom, which altogether form a 16-dimensional manifold whose tangent space at any point is Clifford algebra Cl(1,3). The latter algebra is very promising for the unification, and it provides description of fermions.Comment: 11 pages; Talk presented at "First Mediterranean Conference on Classical and Quantum Gravity", Kolymbari, Crete, Greece, 14-18 September 200

Some Results on Cubic and Higher Order Extensions of the Poincar\'e Algebra

In these lectures we study some possible higher order (of degree greater than two) extensions of the Poincar\'e algebra. We first give some general properties of Lie superalgebras with some emphasis on the supersymmetric extension of the Poincar\'e algebra or Supersymmetry. Some general features on the so-called Wess-Zumino model (the simplest field theory invariant under Supersymmetry) are then given. We further introduce an additional algebraic structure called Lie algebras of order F, which naturally comprise the concepts of ordinary Lie algebras and superalgebras. This structure enables us to define various non-trivial extensions of the Poincar\'e algebra. These extensions are studied more precisely in two different contexts. The first algebra we are considering is shown to be an (infinite dimensional) higher order extension of the Poincar\'e algebra in $(1+2)-$dimensions and turns out to induce a symmetry which connects relativistic anyons. The second extension we are studying is related to a specific finite dimensional Lie algebra of order three, which is a cubic extension of the Poincar\'e algebra in $D-$space-time dimensions. Invariant Lagrangians are constructed.Comment: Mini course given at the Workshop higher symmetries in physics, Madrid, Spain, November 6-8, 200

On Deformations of n-Lie algebras

The aim of this paper is to review the deformation theory of $n$-Lie algebras. We summarize the 1-parameter formal deformation theory and provide a generalized approach using any unital commutative associative algebra as a deformation base. Moreover, we discuss degenerations and quantization of $n$-Lie algebras.Comment: Proceeding of the conference Dakar's Workshop in honor of Pr Amin Kaidi. arXiv admin note: text overlap with arXiv:hep-th/9602016 by other author

On supersymmetric quantum mechanics

This paper constitutes a review on N=2 fractional supersymmetric Quantum Mechanics of order k. The presentation is based on the introduction of a generalized Weyl-Heisenberg algebra W_k. It is shown how a general Hamiltonian can be associated with the algebra W_k. This general Hamiltonian covers various supersymmetrical versions of dynamical systems (Morse system, Poschl-Teller system, fractional supersymmetric oscillator of order k, etc.). The case of ordinary supersymmetric Quantum Mechanics corresponds to k=2. A connection between fractional supersymmetric Quantum Mechanics and ordinary supersymmetric Quantum Mechanics is briefly described. A realization of the algebra W_k, of the N=2 supercharges and of the corresponding Hamiltonian is given in terms of deformed-bosons and k-fermions as well as in terms of differential operators.Comment: Review paper (31 pages) to be published in: Fundamental World of Quantum Chemistry, A Tribute to the Memory of Per-Olov Lowdin, Volume 3, E. Brandas and E.S. Kryachko (Eds.), Springer-Verlag, Berlin, 200