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Let V be a two-dimensional vector space over a field F of characteristic not 2 or 3. We show there is a surjection ν from the set of ‘generic’ commutative algebra structures on V modulo the action ...

International audience; The representation theory of three dimensional real and complex Lie groups is reviewed from the perspective of harmonic functions defined over certain appropriate manifolds. An explicit construction of all unitary representations is given. The realisations obtained are shown to be related with each other by either natural operations as real forms or Inönü–Wigner contractions.

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—Levy-Leblond classification.

Supersymmetry can be consistently generalized in one- and two-dimensional spaces, fractional supersymmetry being one of the possible extension. 2D fractional supersymmetry of arbitrary order F is explicitly constructed using an adapted superspace formalism. This symmetry connects the fractional spin states ( 0, 1 F , … , F−1 F 1 ). Besides the stress-momentum tensor, we obtain a conserved current of spin ( I + 1 F ).The coherence of the theory imposes strong constraints upon the commutation relations of the modes of the fields. The creation and annihilation operators turn out to generate alternative statistics, currently referred to as quons in the literature. We consider, with special attention, the consistence of the algebra, on the level of the Hilbert space and the Green functions. The central charges are generally irrational numbers except for the particular cases F = 2, 3, 4. A natural classification emerges according to the decomposition of F into its product of prime numbers leading to sub-systems with smaller symmetries.

The supersymmetric extension of the standard model with an additional gauge singlet is analysed in detail in the light of the recent experimental bounds on supersymmetric particles. The useful part of the parameter space and the particle spectrum are displayed. We find that once the recent bounds on the chargino mass are imposed, all other new particles practically satisfy the present experimental limits. Special attention is given to particles to be searched for in the future experiments. The singlet fields tend to decouple and give rise to an effective MSSM, enlarging the validity of many phenomenological analyses based in the minimal field content. However, in some ranges of the parameters the singlino is the lightest neutralino, which modifies the signature for susy particles. Simple analytical approximations are developed that qualitatively explain the numerical results.

Using para-Grassmann, or generalized Grassmann algebras, we define rational power of annihilation and creation operators, in order to extend supersymmetric quantum mechanics. This extension can be replaced, under some assumptions, in para-Grassmann quantum mechanics. We then apply this method to the construction of an equation which generalizes the (2+1)D Pauli equation to a particle of arbitrary spin s. This is done by means of a Hamiltonian defined as a sum of 2s monomials of degree 2s+1.

We consider a special extension of Clifford algebras and show that these generalized Clifford algebras are naturally equipped with a metric defined by a fundamental form of degreen which isSL(n, φ) ⊗SL(n,φ) invariant. Using the embedding of the quaternions in the generalized Clifford algebras, in the Hermitian limit, we obtain an algebraic description of the inclusion of the Minkowski space into the hyperspin manifold.

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., 14 pages, more details have been given, to appear in J. Phys. A

It is well known that the linearization of quadratic forms is accomplished using Dirac matrices. The general problem of linearization of any polynomial of degree n having p variables is considered. First the homogeneous polynomials are considered and it is shown that we only need to study two basic homogeneous forms, namely, the sum and the product one. The sum is linearized using matrices which turn out to be a matrix representation of a generalized Clifford algebra. The homogeneous form is linearized using matrices, the size of which is large for practical use. Some clues are given to reduce their size. Since any polynomial of degree n can be made homogeneous by introducing a supplementary variable, the method proposed is quite general. It constitutes an algorithm for the linearization of any polynomial.

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

We construct an universal enveloping algebra associated to the ternary extension of Lie (super)algebras called Lie algebra of order three. A Poincar\'e-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., Comment: 21 pages

As a first step towards a classification of the heterotic superstring solutions in the fermionic approach, we study a class of solutions characterized by complex fermions with ZN boundary conditions. The same solutions can be obtained by an equivalent modular invariant system of real fermions with Z2 boundary conditions. The proof is supplied by Riemann identities for Jacobi Θ-functions that we derive. The patterns of the gauge groups of the D-dimensional solutions are determined for 4≤D≤10, and their uniqueness in ten dimensions is checked for the different ZN boundary conditions. A construction of the E8×E8 supersymmetric solution based on nine complex fermions with Z3 periodicity is exhibited.

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 Inon\"u-Wigner contractions from either the orthosymplectic superalgebra or the de Sitter Lie algebra of order three., Comment: LaTeX, 16 pages

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., 11 pages contribution to the 5th International Symposium on Quantum Theory and Symmetries (QTS5)

The unitary representations of the three dimensional simple Lie groups are reconsidered from the perspective of harmonic functions acting on certain manifolds related to differential realisations of the groups themselves. By means of contractions of Lie groups, the procedure is also applied to the group E2 of rotations-translations in two dimensions.

On the basis of recent results extending non-trivially the Poincar\'e 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., Comment: LaTeX, 27 pages

We consider a four dimensional space-time symmetry which is a non trivial extension of the Poincar\'e algebra, different from supersymmetry and not contradicting {\sl a priori} the well-known no-go theorems. We investigate some field theoretical aspects of this new symmetry and construct invariant actions for non-interacting fermion and non-interacting boson multiplets. In the case of the bosonic multiplet, where two-form fields appear naturally, we find that this symmetry is compatible with a local U(1) gauge symmetry, only when the latter is gauge fixed by a `t Hooft-Feynman term., Comment: LaTex, 25 pages use macro amsmath and amsfonts.Section 5 slightly extended, some references corrected

We consider the problem of representing the Kac-Moody algebra $\mathfrak{g}(N)$ specified by an $r\times r$ indecomposable generalised Cartan matrix $N$ as vector fields on the torus ${{\bb C}^*}^r$. It is shown that, if the representations are of a certain form, this is possible if and only if $\mathfrak{g}(N)\cong sl(r+1,{\bb C})$ or $\tilde{sl}(r,{\bb C})$. For $sl(r+1,{\bb C})$ and $\tilde{sl}(r,{\bb C})$, discrete families of representations are constructed. These generalise the well-known discrete families of representations of $sl(2,{\bb C})$ as regular vector fields on ${\bb C}^*$, Comment: 20 pages, typo corrected in the title

$F-$Lie algebras are natural generalisations of Lie algebras (F=1) and Lie superalgebras (F=2). When $F>2$ not many finite-dimensional examples are known. In this paper we construct finite-dimensional $F-$Lie algebras $F>2$ by an inductive process starting from Lie algebras and Lie superalgebras. Matrix realisations of $F-$Lie algebras constructed in this way from $\mathfrak{su}(n), \mathfrak{sp}(2n)$ $\mathfrak{so}(n)$ and $\mathfrak{sl}(n|m)$, $\mathfrak{osp}(2|m)$ are given. We obtain non-trivial extensions of the Poincar�� algebra by In��n��-Wigner contraction of certain $F-$Lie algebras with $F>2$., 20 pages, LateX

Multicomplex numbers of order n have an associated trigonometry (multisine functions with (n-1) parameters) leading to a natural extension of the sine-Gordon model. The parameters are constrained from the requirement of local current conservation. In two dimensions for n < 6 known integrable models (deformed Toda and non-linear sigma, pure affine Toda...) with dual counterparts are obtained in this way from the multicomplex space MC itself and from the natural embedding $\MC_n \subset \MMC_m, n < m$. For $ n \ge 6$ a generic constraint on the space of parametersis obtained from current conservation at first order in the interaction Lagragien., 11 pages, no figure, LaTex with amsmath accepted by Phys. Lett. B

A generalization of super-Lie algebras is presented. It is then shown that all known examples of fractional supersymmetry can be understood in this formulation. However, the incorporation of three dimensional fractional supersymmetry in this framework needs some care. The proposed solutions lead naturally to a formulation of a fractional supersymmetry starting from any representation D of any Lie algebra g. This involves taking the Fth-roots of D in an appropriate sense. A fractional supersymmetry in any space-time dimension is then possible. This formalism finally leads to an infinite dimensional extension of g, reducing to the centerless Virasoro algebra when g=sl(2,R)., Comment: 23 pages, 1 figure, LaTex file with epsf.sty

Linearization of homogeneous polynomials of degree n and k variables leads to generalized Clifford algebras. Multicomplex numbers are then introduced in analogy to complex numbers with respect to usual Clifford algebra. In turn multicomplex extensions of trigonometric functions are constructed in terms of `compact' and `non-compact' variables. It gives rise to the natural extension of the d-dimensional sine-Gordon field theory in the n-dimensional multicomplex space. In dimension 2, the cases n=1,2,3,4 are identified as the quantum integrable Liouville, sine-Gordon and known deformed Toda models. The general case is discussed., LaTex,amstex, 10 pages; title changed, references added, some parts suppressed

Non-trivial extensions of the three dimensional Poincar\'e algebra, beyond the supersymmetric one, are explicitly constructed. These algebraic structures are the natural three dimensional generalizations of fractional supersymmetry of order $F$ already considered in one and two dimensions. Representations of these algebras are exhibited, and unitarity is explicitly checked. It is then shown that these extensions generate symmetries which connect fractional spin states or anyons. Finally, a natural classification arises according to the decomposition of $F$ into its product of prime numbers leading to sub-systems with smaller symmetries., Comment: 17 pages, LaTex, the algebraic structure is given with more details, unitarity of the respresentation is explicitly checked, references has been added, one author has been added, to appear in Mod. Phys. Lett. A

We discuss the Higgs sector of the supersymmetric standard model extended by a gauge singlet for the range of parameters, which is compatible with universal soft supersymmetry breaking terms at the GUT scale. We present results for the masses, couplings and decay properties of the lightest Higgs bosons, in particular with regard to Higgs boson searches at LEP. The prospects differ significantly from the ones within the MSSM., Comment: 12 pages (Plain Tex), 7 figs

A group theory justification of one dimensional fractional supersymmetry is proposed using an analogue of a coset space, just like the one introduced in $1D$ supersymmetry. This theory is then gauged to obtain a local fractional supersymmetry {\it i.e.} a fractional supergravity which is then quantized {\it \`a la Dirac} to obtain an equation of motion for a particle which is in a representation of the braid group and should describe alternative statistics. A formulation invariant under general reparametrization is given, by means of a curved fractional superline., Comment: 15 pages, latex, no figure

Cubic extensions of the Poincaré algebra can be constructed in consistency with physical assumptions, and provide new insights for the description of phenomena. In this paper we review some recent results associated to these structures: algebraic construction, construction of invariant Lagrangians and construction of their associated group.

Superstring solutions on twisted group manifolds are investigated, chiefly in the case of fermionizable Wess-Zumino-Witten models that yield space-time N=1 supersymmetry. Both outer and inner automorphisms are admitted as boundary conditions. The representations of the twisted current algebras are studied—their mass spectra match perfectly their fermionic realizations. The supersymmetric solutions are characterized and classified. A few especial ones are identified with Z4 and Z3 orbifolds by using the equality of the fermion equivalents.

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., LaTeX 28 pages

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 Poincar\'e 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., Comment: Title and presentation changed