I am interested in an explicit description of the principal homomorphism from $SL(2,\mathbb{C})$ to $G$, for each complex semisimple Lie group $G$. Does any one have specific references please? Kostant's original paper is of course great, and contains a lot, but I do not think it contains explicit descriptions for each specific example (which is what I really want at the moment).
Welcome to MathOverflow
MathOverflow is a question and answer site for professional mathematicians. It's built and run by you as part of the Stack Exchange network of Q&A sites. With your help, we're working together to build a library of detailed answers to every question about research level mathematics.
We're a little bit different from other sites. Here's how:
Ask questions, get answers, no distractions
This site is all about getting answers. It's not a discussion forum. There's no chit-chat.
Just questions...
...and answers.
up vote
Good answers are voted up and rise to the top.
The best answers show up first so that they are always easy to find.
accept
The person who asked can mark one answer as "accepted".
Accepting doesn't mean it's the best answer, it just means that it worked for the person who asked.
Where can I find explicit descriptions of principal $SL(2,\mathbb{C})$s?
up vote
14
down vote
favorite
2 Answers
up vote
4
down vote
accept
You may try exploring the reference
"Lie Algebras, Geometry, and Toda-Type Systems"
by Alexander V. Razumov, Mikhail V. Saveliev, Cambridge University Press.
answered Oct 23 '16 at 19:49
T. Amdeberhan
5,9111144
5,9111144
up vote
3
down vote
I would like to add to the suggested references, one more:
A.L. Onishchik, E.B. Vinberg, Lie Groups and Algebraic Groups, Springer, 1990. Exercise 4.2.28.
answered Oct 24 '16 at 7:42
Malkoun
35917
35917
Get answers to practical, detailed questions
Focus on questions about an actual problem you have faced. Include details about what you have tried and exactly what you are trying to do.
Ask about...
- Specific issues with research level mathematics
- Real problems or questions that you’ve encountered
Not all questions work well in our format. Avoid questions that are primarily opinion-based, or that are likely to generate discussion rather than answers.
Questions that need improvement may be closed until someone fixes them.
Don't ask about...
- Anything not directly related to research level mathematics
- Questions that are primarily opinion-based
- Questions with too many possible answers or that would require an extremely long answer
Tags make it easy to find interesting questions
All questions are tagged with their subject areas. Each can have up to 5 tags, since a question might be related to several subjects.
Click any tag to see a list of questions with that tag, or go to the tag list to browse for topics that interest you.
Where can I find explicit descriptions of principal $SL(2,\mathbb{C})$s?
up vote
14
down vote
I am interested in an explicit description of the principal homomorphism from $SL(2,\mathbb{C})$ to $G$, for each complex semisimple Lie group $G$. Does any one have specific references please? Kostant's original paper is of course great, and contains a lot, but I do not think it contains explicit descriptions for each specific example (which is what I really want at the moment).
asked Oct 23 '16 at 19:32
Malkoun
35917
35917
You earn reputation when people vote on your posts
Your reputation score goes up when others vote up your questions, answers and edits.
+5
question voted up
+10
answer voted up
+15
answer is accepted
+2
edit approved
As you earn reputation, you'll unlock new privileges like the ability to vote, comment, and even edit other people's posts.
| Reputation | Privilege |
|---|---|
| 15 | Vote up |
| 50 | Leave comments |
| 125 | Vote down (costs 1 rep on answers) |
At the highest levels, you'll have access to special moderation tools. You'll be able to work alongside our community moderators to keep the site focused and helpful.
| 2000 | Edit other people's posts |
|---|---|
| 3000 | Vote to close, reopen, or migrate questions |
| 10000 | Access to moderation tools |
Improve posts by editing or commenting
Our goal is to have the best answers to every question, so if you see questions or answers that can be improved, you can edit them.
Use edits to fix mistakes, improve formatting, or clarify the meaning of a post.
up vote
9
down vote
You may try exploring the reference
"Lie Algebras, Geometry, and Toda-Type Systems"
by Alexander V. Razumov, Mikhail V. Saveliev, Cambridge University Press.
answered Oct 23 '16 at 19:49
T. Amdeberhan
5,9111144
5,9111144
Thank you T. Amdeberhan! Unfortunately, I don't have access to an extensive library right now. And Google only shows some selected pages of that book (which seems indeed to contain what I want). Would you happen to have a reference that is available online (arxiv, or journal say)? - Malkoun Oct 23 '16 at 19:57
Unlock badges for special achievements
Badges are special achievements you earn for participating on the site. They come in three levels: bronze, silver, and gold.
| Student | First question with score of 1 or more |
| Editor | First edit |
| Good Answer | Answer score of 25 or more |
| Civic Duty | Vote 300 or more times |
| Famous Question | Question with 10,000 views |
Sign up to get started
Signing up allows you to:
- Earn reputation when you help others with questions, answers and edits.
- Select favorite tags to customize your home page.
- Claim your first badge: Informed
Looking for more in-depth information on the site? Visit the Help Center
MathOverflow is part of the Stack Exchange network
Like this site? Stack Exchange is a network of 162 Q&A sites just like it. Check out the full list of sites.
Use comments to ask for more information or clarify a question or answer.
You can always comment on your own questions and answers. Once you earn 50 reputation, you can comment on anybody's post.
Remember: we're all here to learn, so be friendly and helpful!