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NEW?From: Steven
Subject: General
Date/Time 2008-07-08 16:20:18
Remote IP: 66.21.151.155
MessageThere is nothing new about generalizing quaternions.
The most natural extension is to Cayley numbers,
where you lose associativity as well as commutativity, and
any further extension you are reduced to an algebra that is
only quasi-associative . . . This isn't so nice.
In other words--using the "obvious" generalization--as "n" increases,
the algebraic structure becomes necessarily cumbersome.
It is therefore not obvious that this becomes simpler than using
local charts on the smooth structure of S^n and just recognizing
that the geodesics on S^n are great circles of S^{n-1} to obtain
the new coordinates.
Did you have a different idea in mind of a "generalized quaternion"?
S
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