My Math Forum  

Go Back   My Math Forum > College Math Forum > Abstract Algebra

Abstract Algebra Abstract Algebra Math Forum


Reply
 
LinkBack Thread Tools Display Modes
October 16th, 2010, 05:04 PM   #1
Senior Member
 
Joined: Jan 2009
From: Russia

Posts: 113
Thanks: 0

Homomorphism of Lie groups as groups but not as manifolds

I was wondering if there exists an example of a homomorphism (or even better isomorphism) between Lie groups as groups, but not as manifolds. If anyone could give such an example, that would be of great value!
lime is offline  
 
October 18th, 2010, 02:18 AM   #2
Senior Member
 
Joined: Sep 2008

Posts: 150
Thanks: 5

Re: Homomorphism of Lie groups as groups but not as manifold

Hi lime,

Well, what does the abstract group look like? It is just some infinite dimensional vector space over . Now the same is true for the vectorgoup . It is easy to see, that both dimesions are the same as cardinal numbers, thus there is an isomophism as -vector spaces thus in particular one as groups. But of course they don't have the same dimension.

It is a bit harder with compact Lie groups, but at least there should be many more automophisms of as an abstract group then as a Lie group. You get lots of abstract morpisms for free realising that , where V is again an infinite dimensional vector space.

best
Peter
Peter is offline  
October 18th, 2010, 02:05 PM   #3
Senior Member
 
Joined: Jan 2009
From: Russia

Posts: 113
Thanks: 0

Re: Homomorphism of Lie groups as groups but not as manifold

Peter thank you for your answer. That's exactly what I was looking for.

Non-compact space example was absolutely clear to me, though I'm still a bit confused with your counterexample for compact Lie groups. Do you say that there must be a group isomorphism

?
If yes then how to construct it? Why the last one is compact? Why these are different manifolds? Would be very helpful if you elucidate here a bit.
lime is offline  
October 25th, 2010, 07:30 AM   #4
Senior Member
 
Joined: Sep 2008

Posts: 150
Thanks: 5

Re: Homomorphism of Lie groups as groups but not as manifold

Sorry for the late answer.

The second example was not entirely written out, and I was meaning to say: is has automorphisms as an abstract group, which are not continuous or even differentiable.

I don't know any example of two compact Lie groups, which are abstractly isomorphic, but not as Lie groups. What you can do, is to show, that is isomorphic to as an abstract group but not as a Lie group. (Basicly you have to show as abstract groups for a infinite dimensional vector space V.

I will think a bit, if I can come up with an example of two different compact Lie groups.

rgds
Peter
Peter is offline  
Reply

  My Math Forum > College Math Forum > Abstract Algebra

Tags
groups, homomorphism, lie, manifolds



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Abelian X-Groups and Noetherian (Abelian) X-Groups Math Amateur Abstract Algebra 0 October 29th, 2013 03:46 PM
Groups gaussrelatz Abstract Algebra 2 February 1st, 2013 02:09 PM
About minimal normal groups and subnormal groups Sheila496 Abstract Algebra 0 October 20th, 2011 09:45 AM
First Homomorphism Thm on Quotient Groups matheart Abstract Algebra 1 May 4th, 2011 11:53 PM
groups young_gun Algebra 6 May 28th, 2008 05:33 PM





Copyright © 2019 My Math Forum. All rights reserved.