Clifford Algebras and Possible Kinematics

We review Bacry and Levy-Leblond's work on possible kinematics as applied to 2-dimensional spacetimes, as well as the nine types of 2-dimensional Cayley-Klein geometries, illustrating how the Cayley-Klein geometries give homogeneous spacetimes for all but one of the kinematical groups. We then construct a two-parameter family of Clifford algebras that give a unified framework for representing both the Lie algebras as well as the kinematical groups, showing that these groups are true rotation groups. In addition we give conformal models for these spacetimes.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Associative Submanifolds of the 7-Sphere

Associative submanifolds of the 7-sphere S^7 are 3-dimensional minimal submanifolds which are the links of calibrated 4-dimensional cones in R^8 called Cayley cones. Examples of associative 3-folds are thus given by the links of complex and special Lagrangian cones in C^4, as well as Lagrangian submanifolds of the nearly K\"ahler 6-sphere. By classifying the associative group orbits, we exhibit the first known explicit example of an associative 3-fold in S^7 which does not arise from other geometries. We then study associative 3-folds satisfying the curvature constraint known as Chen's equality, which is equivalent to a natural pointwise condition on the second fundamental form, and describe them using a new family of pseudoholomorphic curves in the Grassmannian of 2-planes in R^8 and isotropic minimal surfaces in S^6. We also prove that associative 3-folds which are ruled by geodesic circles, like minimal surfaces in space forms, admit families of local isometric deformations. Finally, we construct associative 3-folds satisfying Chen's equality which have an S^1-family of global isometric deformations using harmonic 2-spheres in S^6.Comment: 42 pages, v2: minor corrections, streamlined and improved exposition, published version; Proceedings of the London Mathematical Society, Advance Access published 17 June 201

Cayley-Dickson Algebras and Finite Geometry

Given a $2^N$-dimensional Cayley-Dickson algebra, where $3 \leq N \leq 6$, we first observe that the multiplication table of its imaginary units $e_a$, $1 \leq a \leq 2^N -1$, is encoded in the properties of the projective space PG$(N-1,2)$ if one regards these imaginary units as points and distinguished triads of them $\{e_a, e_b, e_c\}$, $1 \leq a < b <c \leq 2^N -1$ and $e_ae_b = \pm e_c$, as lines. This projective space is seen to feature two distinct kinds of lines according as $a+b = c$ or $a+b \neq c$. 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$(N-1,2)$, 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 $\left({N+1 \choose 2}_{N-1}, {N+1 \choose 3}_{3}\right)$-configuration ${\cal C}_N$; in particular, ${\cal C}_3$ (octonions) is isomorphic to the Pasch $(6_2,4_3)$-configuration, ${\cal C}_4$ (sedenions) is the famous Desargues $(10_3)$-configuration, ${\cal C}_5$ (32-nions) coincides with the Cayley-Salmon $(15_4,20_3)$-configuration found in the well-known Pascal mystic hexagram and ${\cal C}_6$ (64-nions) is identical with a particular $(21_5,35_3)$-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 ${\cal C}_{N-1}$ occurs as a geometric hyperplane of ${\cal C}_N$. Finally, a brief examination of the structure of generic ${\cal C}_N$ leads to a conjecture that ${\cal C}_N$ is isomorphic to a combinatorial Grassmannian of type $G_2(N+1)$.Comment: 26 pages, 20 figures; V2 - the basis made explicit, a footnote and a couple of references adde

On the cohomology of some exceptional symmetric spaces

This is a survey on the construction of a canonical or "octonionic K\"ahler" 8-form, representing one of the generators of the cohomology of the four Cayley-Rosenfeld projective planes. The construction, in terms of the associated even Clifford structures, draws a parallel with that of the quaternion K\"ahler 4-form. We point out how these notions allow to describe the primitive Betti numbers with respect to different even Clifford structures, on most of the exceptional symmetric spaces of compact type.Comment: 12 pages. Proc. INdAM Workshop "New Perspectives in Differential Geometry" held in Rome, Nov. 2015, to appear in Springer-INdAM Serie

Associative Geometries. II: Involutions, the classical torsors, and their homotopes

For all classical groups (and for their analogs in infinite dimension or over general base fields or rings) we construct certain contractions, called "homotopes". The construction is geometric, using as ingredient involutions of associative geometries. We prove that, under suitable assumptions, the groups and their homotopes have a canonical semigroup completion.Comment: V2: terminology changed ("torsor" instead of "groud"); some improvements in Chapter 3; to appear in Journal of Lie Theor