498 research outputs found
The Octonions
The octonions are the largest of the four normed division algebras. While
somewhat neglected due to their nonassociativity, they stand at the crossroads
of many interesting fields of mathematics. Here we describe them and their
relation to Clifford algebras and spinors, Bott periodicity, projective and
Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also
touch upon their applications in quantum logic, special relativity and
supersymmetry.Comment: 56 pages LaTeX, 11 Postscript Figures, some small correction
Fine gradings on the simple Lie algebras of type
Some fine gradings on the exceptional Lie algebras ,
and are described. This list tries to be
exhaustive.Comment: Note di Matematica, 201
Composition algebras and the two faces of
We consider composition and division algebras over the real numbers: We note
two r\^oles for the group : as automorphism group of the octonions and
as the isotropy group of a generic 3-form in 7 dimensions. We show why they are
equivalent, by means of a regular metric. We express in some diagrams the
relation between some pertinent groups, most of them related to the octonions.
Some applications to physics are also discussed.Comment: 11 pages, 3 figure
Classification of simple linearly compact Kantor triple systems over the complex numbers
Simple finite dimensional Kantor triple systems over the complex numbers are
classified in terms of Satake diagrams. We prove that every simple and linearly
compact Kantor triple system has finite dimension and give an explicit
presentation of all the classical and exceptional systems.Comment: 46 pages, 3 tables; v2: Major revision of the introduction; v3: Final
version to appear in Journal of Algebr
Borcherds symmetries in M-theory
It is well known but rather mysterious that root spaces of the Lie
groups appear in the second integral cohomology of regular, complex, compact,
del Pezzo surfaces. The corresponding groups act on the scalar fields (0-forms)
of toroidal compactifications of M theory. Their Borel subgroups are actually
subgroups of supergroups of finite dimension over the Grassmann algebra of
differential forms on spacetime that have been shown to preserve the
self-duality equation obeyed by all bosonic form-fields of the theory. We show
here that the corresponding duality superalgebras are nothing but Borcherds
superalgebras truncated by the above choice of Grassmann coefficients. The full
Borcherds' root lattices are the second integral cohomology of the del Pezzo
surfaces. Our choice of simple roots uses the anti-canonical form and its known
orthogonal complement. Another result is the determination of del Pezzo
surfaces associated to other string and field theory models. Dimensional
reduction on corresponds to blow-up of points in general position
with respect to each other. All theories of the Magic triangle that reduce to
the sigma model in three dimensions correspond to singular del Pezzo
surfaces with (normal) singularity at a point. The case of type I and
heterotic theories if one drops their gauge sector corresponds to non-normal
(singular along a curve) del Pezzo's. We comment on previous encounters with
Borcherds algebras at the end of the paper.Comment: 30 pages. Besides expository improvements, we exclude by hand real
fermionic simple roots when they would naively aris
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