35 research outputs found
A Classification of the Projective Lines over Small Rings II. Non-Commutative Case
A list of different types of a projective line over non-commutative rings
with unity of order up to thirty-one inclusive is given. Eight different types
of such a line are found. With a single exception, the basic characteristics of
the lines are identical to those of their commutative counterparts. The
exceptional projective line is that defined over the non-commutative ring of
order sixteen that features ten zero-divisors and it most pronouncedly differs
from its commutative sibling in the number of shared points by the
neighbourhoods of three pairwise distant points (three versus zero), that of
"Jacobson" points (zero versus five) and in the maximum number of mutually
distant points (five versus three).Comment: 2 pages, 1 tabl
Cayley-Dickson Algebras and Finite Geometry
Given a -dimensional Cayley-Dickson algebra, where , we
first observe that the multiplication table of its imaginary units , , is encoded in the properties of the projective space
PG if one regards these imaginary units as points and distinguished
triads of them , and , as lines. This projective space is seen to feature two distinct kinds
of lines according as or . Consequently, it also exhibits
(at least two) different types of points in dependence on how many lines of
either kind pass through each of them. In order to account for such partition
of the PG, the concept of Veldkamp space of a finite point-line
incidence structure is employed. The corresponding point-line incidence
structure is found to be a binomial -configuration ; in particular,
(octonions) is isomorphic to the Pasch -configuration,
(sedenions) is the famous Desargues -configuration,
(32-nions) coincides with the Cayley-Salmon -configuration found
in the well-known Pascal mystic hexagram and (64-nions) is
identical with a particular -configuration that can be viewed as
four triangles in perspective from a line where the points of perspectivity of
six pairs of them form a Pasch configuration. We also draw attention to a
remarkable nesting pattern formed by these configurations, where occurs as a geometric hyperplane of . Finally, a brief
examination of the structure of generic leads to a conjecture that
is isomorphic to a combinatorial Grassmannian of type .Comment: 26 pages, 20 figures; V2 - the basis made explicit, a footnote and a
couple of references adde
Twin "Fano-Snowflakes" Over the Smallest Ring of Ternions
Given a finite associative ring with unity, , any free (left) cyclic
submodule (FCS) generated by a modular ()-tuple of elements of
represents a point of the -dimensional projective space over . Suppose
that also features FCSs generated by ()-tuples that are
unimodular: what kind of geometry can be ascribed to such FCSs? Here, we
(partially) answer this question for when is the (unique)
non-commutative ring of order eight. The corresponding geometry is dubbed a
"Fano-Snowflake" due to its diagrammatic appearance and the fact that it
contains the Fano plane in its center. There exist, in fact, two such
configurations -- each being tied to either of the two maximal ideals of the
ring -- which have the Fano plane in common and can, therefore, be viewed as
twins. Potential relevance of these noteworthy configurations to quantum
information theory and stringy black holes is also outlined.Comment: 6 pages, 1 table, 1 figure; v2 -- standard representation of the ring
of ternions given, 1 figure and 3 references added; v3 -- published in SIGMA
(Symmetry, Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
A Classification of the Projective Lines over Small Rings
A compact classification of the projective lines defined over (commutative)
rings (with unity) of all orders up to thirty-one is given. There are
altogether sixty-five different types of them. For each type we introduce the
total number of points on the line, the number of points represented by
coordinates with at least one entry being a unit, the cardinality of the
neighbourhood of a generic point of the line as well as those of the
intersections between the neighbourhoods of two and three mutually distant
points, the number of `Jacobson' points per a neighbourhood, the maximum number
of pairwise distant points and, finally, a list of representative/base rings.
The classification is presented in form of a table in order to see readily not
only the fine traits of the hierarchy, but also the changes in the structure of
the lines as one goes from one type to the other. We hope this study will serve
as an impetus to a search for possible applications of these remarkable
geometries in physics, chemistry, biology and other natural sciences as well.Comment: 7 pages, 1 figure; Version 2: classification extended up to order 20,
references updated; Version 3: classification extended up to order 31, two
more references added; Version 4: references updated, minor correctio
A Jacobson Radical Decomposition of the Fano-Snowflake Configuration
The Fano-Snowflake, a specific -unimodular projective lattice
configuration associated with the smallest ring of ternions
(arXiv:0803.4436 and 0806.3153), admits an interesting partitioning with
respect to the Jacobson radical of . The totality of 21 free
cyclic submodules generated by non-unimodular vectors of the free left
-module are shown to split into three
disjoint sets of cardinalities 9, 9 and 3 according as the number of Jacobson
radical entries in the generating vector is 2, 1 or 0, respectively. The
corresponding "ternion-induced" factorization of the lines of the Fano plane
sitting in the middle of the Fano-Snowflake (6 -- 7 -- 3) is found to from the natural one, i. e., from that with respect to the
Jacobson radical of the Galois field of two elements (3 -- 3 -- 1).Comment: 7 pages, 3 figure
'Magic' Configurations of Three-Qubit Observables and Geometric Hyperplanes of the Smallest Split Cayley Hexagon
Recently Waegell and Aravind [J. Phys. A: Math. Theor. 45 (2012), 405301, 13
pages] have given a number of distinct sets of three-qubit observables, each
furnishing a proof of the Kochen-Specker theorem. Here it is demonstrated that
two of these sets/configurations, namely the and ones, can uniquely be extended into geometric hyperplanes of the
split Cayley hexagon of order two, namely into those of types and in the
classification of Frohardt and Johnson [Comm. Algebra 22 (1994), 773-797].
Moreover, employing an automorphism of order seven of the hexagon, six more
replicas of either of the two configurations are obtained
Veldkamp-Space Aspects of a Sequence of Nested Binary Segre Varieties
Let be a Segre variety that is -fold direct product of projective
lines of size three. Given two geometric hyperplanes and of
, let us call the triple the
Veldkamp line of . We shall demonstrate, for the sequence , that the properties of geometric hyperplanes of are fully
encoded in the properties of Veldkamp {\it lines} of . Using this
property, a complete classification of all types of geometric hyperplanes of
is provided. Employing the fact that, for , the
(ordinary part of) Veldkamp space of is , we shall
further describe which types of geometric hyperplanes of lie on a
certain hyperbolic quadric that
contains the and is invariant under its stabilizer group; in the
case we shall also single out those of them that correspond, via the
Lagrangian Grassmannian of type , to the set of 2295 maximal subspaces
of the symplectic polar space .Comment: 16 pages, 8 figures and 7 table