Re: R (the set of real numbers) in LaTeX?

Mariano Suárez-Alvarez wrote:
In 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
.

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