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Starting from the four normed division algebras - the real numbers, complex numbers, quaternions and octonions - a systematic procedure gives a 3-cocycle on the Poincare Lie superalgebra in dimensions 3, 4, 6 and 10. A related procedure…
Dimensional reduction of theories involving (super-)gravity gives rise to sigma models on coset spaces of the form G/H, with G a non-compact group, and H its maximal compact subgroup. The reverse process, called oxidation, is the…
A classification of hadrons and their interactions at low energies according to SU(4) allows to identify combinations of the fifteen mesons $\pi$, $\omega$ and $\rho$ within the spin-isospin decomposition of the regular representation…
Graded contractions of certain non-toral $\mathbb{Z}_2^3$-gradings on the simple Lie algebras $\mathfrak{so}(7,\mathbb C)$ and $\f{so}(8,\mathbb C)$ are classified up to two notions of equivalence. In particular, there arise two large…
Singletons are those unitary irreducible modules of the Poincare or (anti) de Sitter group that can be lifted to unitary modules of the conformal group. Higher-spin algebras are the corresponding realizations of the universal enveloping…
The methods of Information geometry have been glowing up to develop various subjects of theoretical physics, including quantum information systems. The present article has two purposes. The first one is to develop general theory of…
We obtain an infinite family of orthogonal hypergeometric groups, which are higher rank arithmetic groups. We also list cases of arithmetic hypergeometric groups whose real Zariski closure is O(2,3).
We study the symmetry group of the geodesic equations of the spatial solutions of the space-time generated by a noninertial rotating system of reference. It is a seven dimensional Lie group, which is neither solvable nor nilpotent. The…
We describe the different classes of $\mathrm{Spin(7)}$ structures in terms of spinorial equations. We relate them to the spinorial description of $\mathrm{G}_2$ structures in some geometrical situations. Our approach enables us to analyze…
Integrable systems in low dimensions, constructed through the symmetry reduction method, are studied using phase portrait and variable separation techniques. In particular, invariant quantities and explicit periodic solutions are…
In this paper we revisit the concept of conformality in the sense of Gauss in the context of octonions and Clifford algebras. We extend a characterization of conformality in terms of a system of partial differential equations and…
The classification of the octonionic realizations of the one-dimensional extended supersymmetries is here furnished. These are non-associative realizations which, albeit inequivalent, are put in correspondence with a subclass of the already…
The Poisson structure is constructed for a model in which spatial coordinates of configuration space are noncommutative and satisfy the commutation relations of a Lie algebra. The case is specialized to that of the group SU(2), for which…
For small dimensional Lie algebra's there are many so-called accidental isomorphisms which give rise to double covers of special orthogonal groups - Spin groups - which happen to coincide with groups already belonging to another…
We construct an explicit example of a generalized Lie 3-algebra from the octonions. In combination with the result of arXiv:0807.0808, this gives rise to a three-dimensional N=2 Chern-Simons-matter theory with exceptional gauge group G_2…
Object of investigation are almost hypercomplex manifolds with Hermitian-Norden metrics of the lowest dimension. The considered manifolds are constructed on 4-dimensional Lie groups. It is established a relation between the classes of a…
This paper is designed to attract people who work on real hyperbolic manifolds to consider thinking about discrete subgroups of higher rank Lie groups. To that end, we breezily discuss some applications of the ideas from the theory of…
Several classes of irreducible orthogonal representations of compact Lie groups that are of importance in Differential Geometry have the property that the second osculating spaces of all of their nontrivial orbits coincide with the…
The existing data appears to provide hints of an underlying high scale theory. These arise from the gauge coupling unification, from the smallness of the neutrino masses, and via a non-vanishing muon anomaly. An overview of high scale…
We consider a Dirac equation set on an extended spin space that contains fermion and boson solutions. At given dimension, it determines the scalar symmetries. The standard field equations can be equivalently written in terms of such degrees…