Joy Christian wrote:gill1109 wrote:Very exciting talk by Anthony Lasenby just now at AGACSE! Working in Cl(1, 3)(R), which he calls the STA (Space-Time Algebra), he has shown that you can define a product and a norm *in fairly simple terms using the given GA structure* which convert this 8-dimensional space into ... wait for it ... a normed division algebra! ie ... the octonions!

It was pointed out in the discussion that you could also do this with Cl(3, 1)(R) which also has some further attractive advantages. For instance: it contains *two* copies of (a representation of) SU(3), not just one, and this enables some of the things that Anthony has been doing to be re-done in a nicer way. More symmetries. The Cl(1, 3)(R) approach suffers from some kind of built in chirality. Handedness. The Cl(3, 1)(R) approach gives full symmetry.

Lounesto has shown all that and more twenty years ago in his extensive studies linking the four of the five normed division algebras with various Clifford algebras. The fifth, the even subalgebra of the algebra Cl(4,0), is already a Clifford algebra.

See Lounesto's paper here:

Advances in Applied Clifford Algebras, Vol. 11 No. 2, 191--213 (2001). What Lasenby presented is nothing new. It is all well known for at least twenty years.

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Thanks, I will check that out. Lasenby knows the literature very well and did mention errors in earlier published work. He will be publishing a paper soon on his present understanding of these things.

You say "five division algebras". But there are only four division algebras. (According to modern definitions thereof)

PS I checked Lounseto's paper. I did not notice the connection with Cl(3, 1) or with Cl(1, 3).

Of course, you can start with any real algebra based on an 8-dimensional vector space (ie, you have a multiplication on top of R^8 as a real vector space with the usual compatibility conditions) and then define a multiplication and a norm making it a division algebra, in terms of some chosen basis. It's a different matter to start with Cl(1, 3) and a particular basis of Cl(1, 3) and define a new multiplication and a new norm

in simple terms of the existing Clifford algebra structure such that you now also have the octonions superimposed [url]in a very harmonious way[/url] on the GA.

I am sure that Anthony knows Lounesto's work, and if he says that he is presenting something very new in his talk, then as a serious mathematician he certainly means what he says.