arXiv subjects
Publications and source records attributed to M. Rausch de Traubenberg.
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 Inönü-Wigner contractions.
Let $V$ be a two-dimensional vector space over a field $\mathbb F$ of characteristic not $2$ or $3$. We show there is a canonical surjection $ν$ from the set of suitably generic commutative algebra structures on $V$ modulo the action of $GL(V)$ onto the plane $\mathbb F^2$. In these coordinates, which are quotients of invariant quartic polynomials, properties such as associativity and the existence of zero divisors are described by simple algebraic conditions. The map $ν$ is a bijection over the complement of a degenerate elliptic curve $Γ$ and over $Γ$ we give an explicit parametrisation of the fibre in terms of Galois extensions of $\mathbb F$. Algebras in $ν^{-1}(Γ)$ are exactly those which admit non-trivial automorphisms. We show how $ν$ can be lifted to a map from the $SL(V)-$moduli space to an algebraic hypersurface $Γ'$ in a four-dimensional vector space whose equation is essentially the classical Eisenstein equation for the covariants of a binary cubic. This map is the restriction of a surjective map from the set of stable commutative algebras on $V$ modulo the action of $SL(V)$ onto $Γ'$.
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.
Parafermions of order two are shown to be the fundamental tool to construct ternary superspaces related to cubic extensions of the Poincaré algebra
Parafermions of order two and three are shown to be the fundamental tool to construct superspaces related to cubic and quartic extensions of the Poincaré algebra. The corresponding superfields are constructed, and some of their main properties analyzed in detail. In this context, the existence problem of operators acting like covariant derivatives is analyzed, and the associated operators are explicitly constructed.
We construct an universal enveloping algebra associated to the ternary extension of Lie (super)algebras called Lie algebra of order three. A Poincaré-Birkhoff-Witt theorem is proven is this context. It this then shown that this universal enveloping algebra can be endowed with a structure of Hopf algebra. The study of the dual of the universal enveloping algebra enables to define the parameters of the transformation of a Lie algebra of order three. It turns out that these variables are the variables which generate the three-exterior algebra.
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.
We study supersymmetric extensions of classical kinematical algebras from the point of view of contraction theory. It is shown that contracting the supersymmetric extension of the anti-de Sitter algebra leads to a hierarchy similar in structure to the classical Bacry-Lévy-Leblond classification
In these lectures we study some possible higher order (of degree greater than two) extensions of the Poincaré algebra. We first give some general properties of Lie superalgebras with some emphasis on the supersymmetric extension of the Poincaré 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 Poincaré 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 Poincaré 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 Poincaré algebra in $D-$space-time dimensions. Invariant Lagrangians are constructed.
We study and classify kinematical algebras which appear in the framework of Lie superalgebras or Lie algebras of order three. All these algebras are related through generalised Inonü-Wigner contractions from either the orthosymplectic superalgebra or the de Sitter Lie algebra of order three.
Previouly a possible extension of the complex number, together with its connected trigonometry was introduced. In this paper we focuss on the simplest case of ternary complex numbers. Then, some types of holomorphicity adapted to the ternary complex numbers and the corresponding results upon integration of differential forms are given. Several physical applications are given, and in particuler one type of holomorphic function gives rise to a new form of stationary magnetic field. The movement of a monopole type object in this field is then studied and shown to be integrable. The monopole scattering in the ternary field is finally studied.
In this paper we initiate a general classification for Lie algebras of order 3 and we give all Lie algebras of order 3 based on $\mathfrak{sl}(2,\mathbb C)$ and $\mathfrak{iso}(1,3)$ the Poincaré algebra in four-dimensions. We then set the basis of the theory of the deformations (in the Gerstenhaber sense) and contractions for Lie algebras of order 3.
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.
A systematic study of non-trivial cubic extensions of the four-dimensional Poincaré algebra is undertaken. Explicit examples are given with various techniques (Young tableau, characters etc).
We classify up to isomorphism all finite-dimensional Lie algebras that can be realised as Lie subalgebras of the complex Weyl algebra $A_1$. The list we obtain turns out to be discrete and for example, the only non-solvable Lie algebras with this property are: $sl(2)$, $sl(2)\times\mathbb C$ and $sl(2)\ltimes{\cal H}_3$. We then give several different characterisations, normal forms and isotropy groups for the action of $Aut (A_1)\times Aut (sl(2))$ on a particular class of realisations of $sl(2)$ in $A_1$.
We study briefly some properties of real Clifford algebras and identify them as matrix algebras. We then show that the representation space on which Clifford algebras act are spinors and we study in details matrix representations. The precise structure of these matrices gives rise to the type of spinors one is able to construct in a given space-time dimension: Majorana or Weyl. Properties of spinors are also studied. We finally show how Clifford algebras enable us to construct supersymmetric extensions of the Poincaré algebra. A special attention to the four, ten and eleven-dimensional space-times is given. We then study the representations of the considered supersymmetric algebras and show that representation spaces contain an equal number of bosons and fermions. Supersymmetry turns out to be a symmetry which mixes non-trivially the bosons and the fermions since one multiplet contains bosons and fermions together. We also show how supersymmetry in four and ten dimensions are related to eleven dimensional supersymmetry by compactification or dimensional reduction.
On the basis of recent results extending non-trivially the Poincaré symmetry, we investigate the properties of bosonic multiplets including $2-$form gauge fields. Invariant free Lagrangians are explicitly built which involve possibly $3-$ and $4-$form fields. We also study in detail the interplay between this symmetry and a U(1) gauge symmetry, and in particular the implications of the automatic gauge-fixing of the latter associated to a residual gauge invariance, as well as the absence of self-interaction terms.
We investigate a non-trivial extension of the $D-$dimensional Poincaré algebra. Matrix representations are obtained. The bosonic multiplets contain antisymmetric tensor fields. It turns out that this symmetry acts in a natural geometric way on these $p-$forms. Some field theoretical aspects of this symmetry are studied and invariant Lagrangians are explicitly given.