Need help understanding a Lie Algebra

[topsquark]

I've been skimming my String Theory text and I've been having a hard time understanding a couple of things.

For now I'm simply going to ask...What is the Lie Algebra E8? (Not to be confused with E(8), 8 dimensional Euclidian space.) I've read the Wiki article and another on a different site, but I have not found a clear explanation about what the group actually is.

Thanks!
-Dan

 

[Euge]

I've been skimming my String Theory text and I've been having a hard time understanding a couple of things.

For now I'm simply going to ask...What is the Lie Algebra E8? (Not to be confused with E(8), 8 dimensional Euclidian space.) I've read the Wiki article and another on a different site, but I have not found a clear explanation about what the group actually is.

Thanks!
-Dan

Hi topsquark,

There are different objects or models which are considered E8. Each has real dimension 248. Since you haven't specified which form, l'll consider the real split-form of E8. This form can be viewed as the symmetry group of Spin(16)/SU(8). As far as I know, the character table for E8 has not been fully computed.

 

[topsquark]

Hi topsquark,

There are different objects or models which are considered E8. Each has real dimension 248. Since you haven't specified which form, l'll consider the real split-form of E8. This form can be viewed as the symmetry group of Spin(16)/SU(8). As far as I know, the character table for E8 has not been fully computed.

Spin(16)/SU(8) sounds like the one I'm looking for. (I actually need E8 x E8 heterotic, but I'll be happy with baby-steps for the nonce.)

I'll likely be back to this later after I've been able to follow up on it.

Thanks again!
-Dan

 

[topsquark]

Okay, I'm going to have to put all this on the back burner for now. I couldn't find a decent explanation in my String Theory text that explicitly covers why we would use E8, except that it is a large group. I have to re-peruse my Abstract Algebra preparatory to hitting my graduate Algebra text.

In the meantime I have a question about the quotient spaces where we are talking about Lie Algebras.

Denoting P(d,R,K) as the Poincare group (translations, rotations, and boosts in Special Relativity). Let's consider P/SO(3). Am I correct in saying the the fiber over a rotation about \(\displaystyle \hat{ k}\) by an angle \(\displaystyle \theta\) is the set of P(d, R',K) where R' is the rotation \(\displaystyle \hat{k},~\theta\) and subsequent rotations \(\displaystyle \theta + 2n \pi\) about \(\displaystyle \hat{k}\)?

I'm a little foggy on this so please let me know if my notation and/or comments about it are less than clear.

Thanks!
-Dan

 

[Euge]

Okay, I'm going to have to put all this on the back burner for now. I couldn't find a decent explanation in my String Theory text that explicitly covers why we would use E8, except that it is a large group. I have to re-peruse my Abstract Algebra preparatory to hitting my graduate Algebra text.

In the meantime I have a question about the quotient spaces where we are talking about Lie Algebras.

Denoting P(d,R,K) as the Poincare group (translations, rotations, and boosts in Special Relativity). Let's consider P/SO(3). Am I correct in saying the the fiber over a rotation about \(\displaystyle \hat{ k}\) by an angle \(\displaystyle \theta\) is the set of P(d, R',K) where R' is the rotation \(\displaystyle \hat{k},~\theta\) and subsequent rotations \(\displaystyle \theta + 2n \pi\) about \(\displaystyle \hat{k}\)?

I'm a little foggy on this so please let me know if my notation and/or comments about it are less than clear.

Thanks!
-Dan

Hi Dan,

You mentioned having a question about quotients of Lie algebras, but you only considered Lie groups. Were you trying to deal with the Poincare group, or Poincare algebra (i.e., the Lie algebra of the Poincare group)? Also, by $SO(3)$ did you in fact mean the Lorentz subgroup $SO(3,1)$?

 

[topsquark]

Hi Dan,

You mentioned having a question about quotients of Lie algebras, but you only considered Lie groups. Were you trying to deal with the Poincare group, or Poincare algebra (i.e., the Lie algebra of the Poincare group)? Also, by $SO(3)$ did you in fact mean the Lorentz subgroup $SO(3,1)$?

The fact that I am somewhat perplexed by your questions is rather telling, don't you think?

Overall I am interested in E8 and why it is used in String Theory. The quotient group Spin(16)/SU(8) was brought up and I am trying to understand the structure of this group. Spinors are more complicated than scalars so I was trying to come up with a simpler example to see if I am understanding things correctly. The only quotient groups that I have used, if I recall correctly, are with discrete groups, such as the classic example \(\displaystyle \mathbb{Z} / n \mathbb{Z}\). Apparently I didn't do a good job choosing P/SO(3). What I meant by using SO(3) is the rotation group in 3 dimensions. Since the Poincare group contains a rotation subgroup I thought it would make sense to talk about the quotient group of P with SO(3).

(groans) I've got a long way to go on this one...

Thanks for the help!
-Dan

 

[Euge]

The fact that I am somewhat perplexed by your questions is rather telling, don't you think?

Overall I am interested in E8 and why it is used in String Theory. The quotient group Spin(16)/SU(8) was brought up and I am trying to understand the structure of this group. Spinors are more complicated than scalars so I was trying to come up with a simpler example to see if I am understanding things correctly. The only quotient groups that I have used, if I recall correctly, are with discrete groups, such as the classic example \(\displaystyle \mathbb{Z} / n \mathbb{Z}\). Apparently I didn't do a good job choosing P/SO(3). What I meant by using SO(3) is the rotation group in 3 dimensions. Since the Poincare group contains a rotation subgroup I thought it would make sense to talk about the quotient group of P with SO(3).

(groans) I've got a long way to go on this one...

Thanks for the help!
-Dan

The relationship between $E8$ and physics is that it contains four subgroups that relate to the gravitational, weak, strong, and electromagnetic forces. You have already mentioned a relationship between $E8$ and string theory: the $E8 \times E8$ heterotic. The other kind of heterotic has gauge group $SO(32)$.

Due to the mathematical complexity of $E8$ and the string theory related to it, I think you should review some algebraic topology (such as the fundamental group, covering space theory, and fiber bundles) and representations of groups to get a more comfortable grasp of these concepts.

 

[topsquark]

The relationship between $E8$ and physics is that it contains four subgroups that relate to the gravitational, weak, strong, and electromagnetic forces. You have already mentioned a relationship between $E8$ and string theory: the $E8 \times E8$ heterotic. The other kind of heterotic has gauge group $SO(32)$.

Due to the mathematical complexity of $E8$ and the string theory related to it, I think you should review some algebraic topology (such as the fundamental group, covering space theory, and fiber bundles) and representations of groups to get a more comfortable grasp of these concepts.

Thanks for the suggestions. As it happens I'm working on some of that now. 8)

-Dan