Fibonacci, Golden Ratio, and Vector Bundles

There is a family of vector bundles over the moduli space of stable curves that, while first appearing in theoretical physics, has been an active topic of study for algebraic geometers since the 1990s. By computing the rank of the exceptional Lie algebra g2 case of these bundles in three different ways, a family of summation formulas for Fibonacci numbers in terms of the golden ratio is derived.

Octonions, triality, the exceptional Lie algebra F4 and polar actions on the Cayley hyperbolic plane

Using octonions and the triality property of Spin(8), we find explicit formulae for the Lie brackets of the exceptional simple real Lie algebras [Formula: see text] and [Formula: see text], i.e. the Lie algebras of the isometry groups of the Cayley projective plane and the Cayley hyperbolic plane. As an application, we determine all polar actions on the Cayley hyperbolic plane which leave a totally geodesic subspace invariant.

Icosahedral Supersymmetry

The Exceptional Lie Algebra E6 used by the Author as a basis forthe Standard Model of the Elementary Particles is a subalgebra of the Lie algebra E8 which in turn is the Lie algebra of the icosahedral group by the McKay correspondence. It is possible to introduce a mass proportional toan entropy given by the the number of permutations of the elements of E6, E8 labeled by the Weyl group W. In this way the masses of the top-quark pair uu and electron are derived without any appeal to QCD and a mass of approximately 19 TeV is predicted for supersymmetric particles.

ELECTRONIC STRUCTURE OF GRAPHENE AND GERMANENE BASED ON DOUBLE HEXAGONAL STRUCTURE

In this paper, we study the electronic structure of monolayer materials based on a double hexagonal geometry with (1×1) and [Formula: see text] superstructures. Inspired from the two-dimensional root system of an exceptional Lie algebra called G2, this hexagonal atomic configuration involves two hexagons of unequal side length at angle 30°. The principal unit hexagonal cell contains twelve atoms instead of the usual configuration involving only six ones relying only on the (1×1) superstructure. Using ab initio calculations based on FPLO9.00-34 code, we investigate numerically the graphene and the germanene with the double hexagonal geometry. In particular, we find that the usual electronic properties and the lattice parameters of such materials are modified. More precisely, the lattice parameters are increased. It has been shown that, in the single hexagonal geometry, the grapheme and the germanene behave as a gapless semiconductor and a semi-metallic, respectively. In double hexagonal geometry however, both materials becomes metallic.