Levi Theorem

  • #1
Hello!

I've got big problems with understanding abstract algebra, the way we deal with it in the seminar on Lie algebras. In just four weeks we progressed up to Levi and Malcev theorems, which are actually the culmination, the say, of classical Lie algebras theory. I didn't think, that the material would become so dense and abstract in so small amount of time. So it's my fault in the end. But I need help, because I have to make a presentation of these two theorems.

I'd like to note that I've got absolutely no background in Lie group theory. They said that some knowledge of linear algebra would suffice...

I attached a pdf with the statement of Levi's theorem and the treatment of the first case when kernel is not a minimal ideal.

What I don't understand is:
1. why the formula for the dimensions holds.
2. why [itex]\beta_1(s)[/itex] isomorphic to s.
3. why [itex]dim(ker\alpha_2)=dim(n_1)[/itex]

and why is there beta with tilde present. Is it a typo?
 
Last edited:

Answers and Replies

  • #2
morphism
Science Advisor
Homework Helper
2,015
4
What I don't understand is:
1. why the formula for the dimensions holds.
By the first isomorphism theorem,
[tex]\dim \ker \alpha_1 = \dim \mathfrak{g}/\mathfrak{n}_1 - \dim \mathfrak{s} = \dim \mathfrak{g}/\mathfrak{n}_1 - \dim \mathfrak{g}/\mathfrak{n} = \dim \mathfrak{n} - \dim \mathfrak{n}_1.[/tex]
The last equality follows from the fact that dim(v/w)=dimv-dimw.
2. why [itex]\beta_1(s)[/itex] isomorphic to s.
[itex]\beta_1[/itex] has a left inverse, and hence is injective.
3. why [itex]dim(ker\alpha_2)=dim(n_1)[/itex]
By definition, [itex]\ker\alpha_2 = \{x : x + \mathfrak{n}_1 = \mathfrak{n}_1\} = \mathfrak{n}_1[/itex].
and why is there beta with tilde present. Is it a typo?
The tilde is there probably because the domain [itex]\beta_2[/itex] is being adjusted to all of [itex]\mathfrak{g}/\mathfrak{n}_1[/itex]. But don't worry about it - it's mostly irrelevant.
 
Last edited:

Related Threads on Levi Theorem

  • Last Post
Replies
4
Views
3K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
1
Views
10K
Replies
6
Views
1K
Replies
4
Views
6K
Replies
10
Views
1K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
2K
Top