# Ternary Numbers, Algebras of Reflexive Numbers and Berger Graphs

@article{Dubrovski2006TernaryNA, title={Ternary Numbers, Algebras of Reflexive Numbers and Berger Graphs}, author={Alexey Dubrovski and Guennadi Volkov}, journal={Advances in Applied Clifford Algebras}, year={2006}, volume={17}, pages={159-181} }

Abstract.The Calabi-Yau spaces with SU(n) holonomy can be studied by the algebraic way through the integer lattice where one can construct the Newton reflexive polyhedra or the Berger graphs. Our conjecture is that the Berger graphs can be directly related with the n-ary algebras. To find such algebras we study the n-ary generalization of the well-known binary norm division algebras,
$${\mathbb{R}}, {\mathbb{C}}, {\mathbb{H}}, {\mathbb{O}}$$
, which helped to discover the most important…

## 6 Citations

### Ternary SU(3)-group symmetry and its possible applications in hadron-quark substructure. Towards a new spinor-fermion structure

- MathematicsEPJ Web of Conferences
- 2019

The questions on the existence of the three color quark symmetry and three quark-lepton generations could have the origin associated with the new exotic symmetries outside the Cartan-Killing-Lie…

### A new Tzitzeica hypersurface and cubic Finslerian metrics of Berwald type

- Mathematics
- 2010

A new hypersurface of Tzitzeica type is obtained in all three forms: parametric, implicit and explicit. To a two-parameters family of cubic Tzitzeica surfaces we associate a cubic Finsler function…

### On the complexifications of the Euclidean $R^n$ spaces and the n-dimensional generalization of Pithagore theorem

- Mathematics
- 2010

We will discuss the following results C_n complexification of R(n) spaces, C_n structure and the invariant surfaces C_n holomorphicity and harmonicity. We also consider the link between C_n…

### Ternary 'Quaternions' and Ternary TU(3) algebra

- Mathematics
- 2010

To construct ternary "quaternions" following Hamilton we must introduce two "imaginary "units, $q_1$ and $q_2$ with propeties $q_1^n=1$ and $q_2^m=1$. The general is enough difficult, and we consider…

### The GUT of the light: On the Abelian Complexifications of the Euclidean R^n spaces

- Mathematics
- 2009

The new great development in Physics could be related to the excited progress of a new mathematics: ternary theory of numbers, ternary Pithagor theorem and ternary complex analysis, ternary algebras…

### Geometry of the Majorana Neutrino and New Symmetries

- Physics
- 2006

Experimental observation of Majorana fermion matter gives a new impetus to the understanding of the Lorentz symmetry and its extension, the geometrical properties of the ambient space-time structure,…

## References

SHOWING 1-10 OF 22 REFERENCES

### HUNTING FOR THE NEW SYMMETRIES IN CALABI–YAU JUNGLES

- Mathematics
- 2004

It was proposed that the Calabi–Yau geometry can be intrinsically connected with some new symmetries, some new algebras. In order to do so the Berger graphs corresponding to K3-fibre CYd (d≥3)…

### The A-D-E classification of minimal andA1(1) conformal invariant theories

- Mathematics
- 1987

We present a detailed and complete proof of our earlier conjecture on the classification of minimal conformal invariant theories. This is based on an exhaustive construction of all modular invariant…

### Calabi's conjecture and some new results in algebraic geometry.

- MathematicsProceedings of the National Academy of Sciences of the United States of America
- 1977

A proof of Calabi's conjectures on the Ricci curvature of a compact Kähler manifold is announced and some new results in algebraic geometry and differential geometry are proved, including that the only Köhler structure on a complex projective space is the standard one.

### Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties

- Mathematics
- 1993

We consider families ${\cal F}(\Delta)$ consisting of complex $(n-1)$-dimensional projective algebraic compactifications of $\Delta$-regular affine hypersurfaces $Z_f$ defined by Laurent polynomials…

### Geometry of the Majorana Neutrino and New Symmetries

- Physics
- 2006

Experimental observation of Majorana fermion matter gives a new impetus to the understanding of the Lorentz symmetry and its extension, the geometrical properties of the ambient space-time structure,…

### XXVIII. On Jacobi's Elliptic functions, in reply to the Rev. Brice Bronwin; and on Quaternions

- Physics, Mathematics
- 1845

### On lie k-algebras

- Mathematics
- 1995

Abstract We define the notion of a "Lie k -algebra" to be a ( k + 1)-ary skew-symmetric operation on a bigraded vector space which satisfies a certain relation of degree 2 k + 1. The notion of Lie…

### On Nambu–Poisson Manifolds

- Mathematics
- 1998

+ {g1, {f1, . . . , fn−1, g2}, g3, . . . , gn} + · · ·+ {g1, . . . , gn−1, {f1, . . . , fn−1, gn}} for all f1, . . . , fn−1, g1, . . . , gn ∈ C∞(M). We should note that (f1, . . . , fn−1) acts on…