Explain to me in the simplest terms triality?

[Lapidus]

Anybody here how could explain to me in the simplest terms triality?

I know some very littel Lie and representation theory. As I understand there are vector and spinor representations for so(n). If n is even, there are two spin reps and one vec rep, if n is odd there is one spin rep and one vec rep.

For example, for n=4 there are the two 2-dim Weyl spinors and the one 4-dim vec representation.

so(8) is special, since the dim of the two spin reps and the one vec rep is ithe same, 8.

Now, how do these two spinors look for so(8)?

They say triality gives the symmetry of Spin(8), not of so(8), it is an automorphism of Spin(8). What does that mean?

I asked the question here, not in the math section, because most of the time I do not understand mathematicans when they talk about Lie algebras and representations...

thank you

 

[petergreat]

Anybody here how could explain to me in the simplest terms triality?

For Spin(8) the vector representation and the two spinor representations are equivalent, i.e. related by similarity transformations A -> BAB^(-1).

They say triality gives the symmetry of Spin(8), not of so(8), it is an automorphism of Spin(8). What does that mean?

Spin(8) is the double cover (i.e. 2-to-1 equivalent) of SO(8), in the same way that SL(2,C) is the double cover of SO(3,1) in the case of Dirac spinors. Automorphism just means similarity transformation.

Now, how do these two spinors look for so(8)?

Similar to Dirac spinors, You need 8 gamma matrices satisfying the 8-dimensional (Euclidean) Clifford Algebra. Then the generators are given by (-1/4)[gamma_i, gamma_j]. With an appropriate basis, the upper and lower components separate into two spinor representations, similar to how a Dirac spinor separates into two irreducible (left and right) Weyl spinors.

 

[Lapidus]

Thanks, Petergreat!

For Spin(8) the vector representation and the two spinor representations are equivalent, i.e. related by similarity transformations A -> BAB^(-1).

So A could be one of the two spin reps, B could be the other spin rep or the vec rep, right?
Are A and B 8-dim matrices? How do I know which of the eigth elements (matrices) to take from A and B, while checking A -> BAB^(-1)? What does B^(-1) mean, the inverse?

Similar to Dirac spinors, You need 8 gamma matrices satisfying the 8-dimensional (Euclidean) Clifford Algebra.

These 8 gamma matrices are 8-dimensional, are they?

again many thanks

 

[petergreat]

The gamma matrices are 16-dimensional, but the 16-dimensional representation space decompose into two 8-dimensional spinor representations. Yes I mean inverse by ^(-1). I must admit I don't know the mathematical details of triality (yet), other than the fact that it's a symmetry of the roots system of Spin(8). I'm sure other people can help, but I suppose you won't understand it without learning about the mathematics of roots systems in Lie theory.

 

[garrett]

http://arxiv.org/abs/0910.1828"

 

[Lapidus]

Why can't someone do the world a favor and write a text about Clifford algebras, spinors, octonions, triality, Dynkin diagrams, exceptional lie groups and all that jazz, which would be readable and understandable by a non-expert? Is that asked too much?

(I read once John Baez saying that one day he will write such a book. But I am tired of waiting... )

 

[weburbia]

Even if he has not written the book he has a lot on his website.

http://math.ucr.edu/home/baez/octonions/octonions.html