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Reflection Groups on the Octave Hyperbolic Plane

✍ Scribed by Daniel Allcock

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
223 KB
Volume
213
Category
Article
ISSN
0021-8693

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✦ Synopsis

For two different integral forms K of the exceptional Jordan algebra we show that Aut K is generated by octave reflections. These provide ''geometric'' examples Ž . of discrete reflection groups acting with finite covolume on the octave or Cayley 2 Ž hyperbolic plane ‫ޏ‬H , the exceptional rank one symmetric space. The isometry . group of the plane is the exceptional Lie group F . Our groups are defined in 4Žy 20.

terms of Coxeter's discrete subring K K of the nonassociative division algebra ‫ޏ‬ and we interpret them as the symmetry groups of ''Lorentzian lattices'' over K K. We also show that the reflection group of the ''hyperbolic cell'' over K K is the rotation subgroup of a particular real reflection group acting on H 8 ( ‫ޏ‬H 1 . Part of our approach is the treatment of the Jordan algebra of matrices that are Hermitian with respect to any real symmetric matrix.

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