Is the Leech Lattice a Lie Group?

[Aztral]

Leech lattice is a 'lie group?"

My understanding of Lie groups is non-existent.

But I'm trying to understand if the Leech lattice is a 'lie group?"

 

[henry_m]

A Lie group is essentially a continuous group, which means that its elements are described by a number of smoothly varying parameters.

As I understand it, a lattice is a discrete subgroup of Rⁿ under addition, so it is not continuous and thus has no Lie group structure.

 

[Aztral]

Hi,

Ya..I'm getting most of my info off wikipedia (garbage-in garbage-out).

I was trying to extrapolate from E8 (which I thought was a Lie group) to to the Leech lattice. On wikipedia the entry for E8 says..."E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248."

I've now found other references saying E8 is a Lie Algebra...grrr. (as well of a list of the Lie Groups).

Anywayz, I've downloaded a book on Abstract Algebra and I'm hoping that sheds some light on Lattices, Lie Groups, etc. :)

 

[henry_m]

Hi,

Ya..I'm getting most of my info off wikipedia (garbage-in garbage-out).

I was trying to extrapolate from E8 (which I thought was a Lie group) to to the Leech lattice. On wikipedia the entry for E8 says..."E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248."

I've now found other references saying E8 is a Lie Algebra...grrr. (as well of a list of the Lie Groups).

Anywayz, I've downloaded a book on Abstract Algebra and I'm hoping that sheds some light on Lattices, Lie Groups, etc. :)

OK, I now see what's going on here, but it takes quite a bit to develop it. In outline, it goes like this:

You start with a Lie group, which is a group described by n continuous parameters.
You can investigate the Lie group by looking at its local structure, near the identity element. This gives rise to an n-dimensional vector space with a 'Lie bracket', which quantifies how elements of the group fail to commute. This is a Lie algebra.
To classify Lie algebras (specifically 'simple' Lie algebras), you can use a method of 'roots', where roots are a finite number of vectors in r-dimensional euclidean space with certain properties.
Finally, we can take all possible integer linear combinations of these root vectors to get a lattice. This is a group under addition, but as far as I can tell it is not related to the original Lie Group we started with.

Hope that sheds a little light. I'd recommend either:
i) Getting a good book on Lie algebras and working through this fully.
ii) Take a definition of the lattice your interested which doesn't need all this stuff and don't worry about it.

 

[Aztral]

Thanks for the overview henry_m! I've always found a bit of a synopsis about what I'm about to study keeps me more focused :)

I'm kind of starting a bit before i). I'm trying to get up to speed on "groups" in general before proceeding on to Lie Groups. I've download a few pdf books.

Anyway, thanks again!