Post by Mariano Suárez-AlvarezPost by Michaël GrünewaldIn some areas, it is customary to use boldface R, C, H, O for the four
real normed algebras (reals, complexes, quaternions and octonions), and
the blackboard face R, C, H, O for their complexified counterparts.
Interesting. What areas are those?
The four real normed algebras are important objects with respect to the
classification of Jordan Algebras. This classifications looks the same
as the theory of representations of a reductive group, the basic pieces
are called simple Jordan algebras and each simple Jordan algebra can be
realized as sub algebra of the matrix algebra with scalars in R, C, H or
O (and the A*B = (AB + BA)/2 product, commutative but non associative).
Jordan algebras were introduced to formalize some properties of
operators in quantum physics. IIRC (from Kevin Mc Crimmon's book) this
is not a brillant success from the physicist point of view, but Jordan
algebras are very interesting objects for the mathematician. As an
example of their marvellous properties, one gets all the exceptional Lie
groups (E6 -- E8, F4 and G2) as automorphism groups with respect to some
generic construction applied to Jordan algebras. Thus the exceptional
groups do not look so exceptional from this point of view. Another
example, is that the Severi varieties can be constructed out of Jordan
algebras (again, with a generic construction).
If you feel curious about all of this, and want some more general
information, I warmly recommand John Baez's introduction to octonions,
and more especially the ``Octonionic projective geometry part'':
http://math.ucr.edu/home/baez/octonions/node8.html
It's a very pleasing presentation, with historical notes, pictures and
not much technicites.
--
Cheers,
Michaël