Real versus Complex quaternions

Subject: Re: Quaternions
From: Pertti Lounesto <lounesto@dopey.hut.fi>
Date: 1997/05/01
Message-Id: <w0i3es7kbf4.fsf@dopey.hut.fi>
Newsgroups: sci.physics.electromag,sci.physics.relativity

"Srdjan Budisin" <srdjan@EUnet.yu> wrote:
-One of the fundamental quastions which need to be answered in order to
- explain our Universe is: "Why is our Universe 3- dimensional?".

One explanation is that only in dimension n=3 the potential 1/r^(n-2)
of the electron gives rise to a closed orbit, Kepler's ellipse.

"Srdjan Budisin" <srdjan@EUnet.yu> wrote:
-In mathematics it is well known that only real numbers, complex numbers and
-quaternions form mathematical structures known as Algebras. It is proven
-that there can be no algebra with more than 4 dimensions.

Reals R, complexes C and quaternions H are the only real associative
division algebras. To fix terminology: an algebra A is a linear space
with a bilinear product AxA->A. In a division algebra D the equations
ax=b and ya=b have unique solutions x,y for all non-zero a,b \in D.

"Srdjan Budisin" <srdjan@EUnet.yu> wrote:
-Quaternions were invented by Hamilton (England) in last century.

Hamilton was Irish.

"Srdjan Budisin" <srdjan@EUnet.yu> wrote:
-Maybe, Maxwell equations can be formulated in therms of quatrnions?
-Maybe, Special Theory of Relativity can be formulated using Quaternions.
-How about Quantum Mechanics?

No. No. Yes, if you mean real quaternions.
Yes, Yes, Yes, for complex quaternions (called also biquaternions).

sweetser@alum.mit.edu (Doug B Sweetser) wrote:
: There are only three finite dimensional division algebras,
: the reals, the complex numbers and the quaternions (if you drop
: associativity - a BIG deal - you can add octonions to the list).
: A division algebra means that an inverse will always exist.

A counterexample (due to Benjamin Tilly): Consider a 3- dimensional
real algebra with basis {1,i,j} such that 1 is the unity and
i^2=j^2=-1 but ij=ji=0. The algebra is commutative and non- associtive
but admits inverses, although the inverses of xi+yj are not unique.

--
Pertti Lounesto http://www.math.hut.fi/~lounesto