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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 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 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.

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

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.

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

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.

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