Alternate Clifford basis for Octonions...?

Higher-dimensional geometry (previously "Polyshapes").

Alternate Clifford basis for Octonions...?

Postby Paul » Fri Jan 07, 2005 10:56 pm

Hello all,

I've been studying the octionions... off and on...

Lately, I've been experimenting with some of the various pictorial representations of the octonion multiplication table and how each relates to the possible bases for the octonions...

This pictorial representation of the octonion multiplication table is my favorite so far...

Image

where the arrows define the positive direction around the Hamilton triangle... so, if you multiply two distinct basis elements in one of the seven Hamilton triangles in a counter-clockwise direction their product is positive... clockwise, negative.

The general rule is usually expressed:

Image

It seems that when the octonions are represented in Clifford Algebra they're either represented as paravectors in Image, or 1-vectors in Image. The variation in the adjustment necessary to the Clifford product to represent the octonion product is relatively minor.

I also notice that many authors build up the octonions through the Cayley-Dickson doubling process:

Image

which yields to this equivalent pictorial representation of the octonion multiplication table:

Image

The complex numbers have two Clifford representations... Cl<sub>0,1</sub> where i is a 1-vector, and Cl<sub>2</sub><sup>+</sup> where i is a 2-vector. The quaternions also have two Clfford representations... {1,i,j,ij} in Cl<sub>0,2</sub> where i, j are 1-vectors and ij = k is a 2-vector, and {1,i,j,k} in Cl<sub>3</sub><sup>+</sup> where i,j,k are all 2-vectors.

I've not seen a Clifford Algebra representation that appears to more directly correspond to the basis of the octonions as represented above, i.e. O = {1,i,j,k,l,il,jl,kl}... Similar to the representation we have of the quaternions for {1,i,j,ij} in Cl<sub>0,2</sub> where i, j are 1-vectors and ij = k is a 2-vector. Some representation where il, jl, and kl are composed of some Clifford elements that are i, j, k, l...?

Has anyone seen any such Clifford representation of the octionions? Can such a Clifford representation be constructed?
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Postby Paul » Sun Jan 16, 2005 10:21 pm

Hello all,

Instead of starting a new thread, I wanted to ask if anyone here... maybe Pat... can help me with a couple threads I started at the Math Message Board that either I've stopped getting any responses to... or, having gotten any responses to at all.

SO(n), SU(n), Spin(n), etc.

Spinors and Trialities...?

Allot of my confusions seem to revolve understanding these various groups, and their structures...

Does anyone know of somewhere where relatively easy-to-understand reference for definitions, examples, and such of these groups...?

The problem is... that it seems I can't find places, neither on the Internet, or in books, that really tells me exactly what such and such is... in this or that context...

I know there's more abstract definitions and such... and those are helpful. But, in another sense, it'd also be nice for someone to say... okay, this matrices of this form serve as a basis for O(n). And... you multiply these matrices by... I don't know... 1-vectors represented as column vectors...? What do you multiply the basis matrices by... that is, what are the various options?

Also, for instance... what elements of the Clifford Algebra do the basis matrices for O(n) correspond to?

Also... in particular, I kinda get the impression... maybe, it seems difficult for me to determine with my limited mathematical background... that say, Spin(n) is the even Clifford Algebra... but, when people say 'spinors' they usually just me the bivectors of a Clifford Algebra... which form a basis for Spin(n), but are not all of Spin(n)...? Is this correct...?

Of course, again... this is just represention of Spin(n) in Clifford Algebra. If one wanted to say... Spin(n) is such and such... it'd probably be more accurate to say something more abstract... like Spin(n) is the group of certain rotations of such and such...

Anyway... I'm not sure where my confusions start, and end here... so, what I've just said about Spin(n) may not be entirely accurate.

However, I've come to the conclusion that if I want to understand the stuff I'm interested in right now, I really need to understand about these groups...

Any help would be appreciated.
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Postby pat » Mon Jan 17, 2005 2:15 pm

I'll have to look, but I suspect Clifford Algebras and the Classical Groups may have stuff to say about what you're asking. But, I haven't really looked at that book yet. I'll check later today.

In the meantime, I don't have a really good reference for you.

GO(n) [also: O(n) those this looks like Big-O notation from Computer Science] is the group (under matrix multiplication) of real-valued, orthogonal matrices with determinant +1 or -1. An orthogonal matrix A is one which satisifies: A<sup>T</sup>A = I. SO(n) is the subgroup of GO(n) of those elements with determinant +1.

GU(n) [also: U(n), but this looks like the group (under multiplication modulo 'n') of integers relatively prime to 'n'] is the group (under matrix multiplication) of complex-valued, unitary matrices with determinant +1 or -1. A unitary matrix A is one which satisfies A<sup>H</sup>A = I.... where A<sup>H</sup> means transpose the matrix and take the complex conjugate of each element. SU(n) is the subgroup of GU(n) of those elements with determinant +1.

Spin(n), I'm not totally clear on off of the top of my head. They are an extension of the Pauli Spin Matrices (which are the n=3 case). I believe that Spin(2n) is isomorphic to SU(n) or Spin(n) is isomorphic to SO(2n) or some such thing. And, as you mentioned, when people talk about Spin(n) they are very often thinking about the standard basis elements... or more accurately, representing an element as a vector assuming the standard basis elements.
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Postby wendy » Tue Jan 18, 2005 11:07 pm

From what i was told, we have:

O(n) = orthogonal unit-matrices = rotations and reflections of a sphere.

SO(n) = semi-orthognonal = rotations of a sphere.

i was discussing about the great circles on a glomochorix (4-sphere-surface) making a one-to-one mapping with the bi-glomohedric prism (a kind of surface in 6D, made by the product of the surface of two three-spheres).
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Postby pat » Tue Jan 18, 2005 11:11 pm

I've never seen 'semi-orthogonal'. I've just seen 'special orthogonal' meaning only those orthogonal transformations with preserve orientation. i.e. no flipping, only rotating.
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Postby wendy » Tue Jan 18, 2005 11:31 pm

special and semi- here mean the same thing: that is a halving of symmetry by the removal of a reflection group. Much of what i do with geometry comes from polytopes, where the style is to use semi- rather than special.

BTW, i developed a lot of this in isolation, and the terminology is somewhat different.

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