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).
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$\begingroup$ Malkoun, what do you mean by explicit? At the level of Lie algebras, do you want more than what is outlined, e.g., in the homework set www2.bc.edu/mark-reeder/820hw4.pdf , especially parts f and g there? $\endgroup$– guestCommented Oct 23, 2016 at 20:53
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$\begingroup$ thank you guest. Also interested in the exceptional cases. Anyway, I guess I will work out what they look like using such root formulas. You are right. It is explicit enough. $\endgroup$– MalkounCommented Oct 23, 2016 at 21:13
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$\begingroup$ guest there is a small inconsistency in the indices in that file in part g). A typo of sorts. $\endgroup$– MalkounCommented Oct 23, 2016 at 21:14
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3 Answers
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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, 2016 at 19:49
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$\begingroup$ 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)? $\endgroup$– MalkounCommented Oct 23, 2016 at 19:57
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$\begingroup$ I'll let you know if I get a full online access. $\endgroup$ Commented Oct 23, 2016 at 20:02
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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.
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Bourbaki on Lie groups chapters 7-9 has a lot of info. on PRINCIPAL Sl2-triplets in complex semisimple Lie algebras (and more info in Exercises). Is that related to what you need ?
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$\begingroup$ Thank you Nazih, but I also do not seem to have online access to this book either. $\endgroup$– MalkounCommented Oct 23, 2016 at 20:33
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$\begingroup$ I will be very glad to lend you the book which belongs to AUB Jafet Library. What I like about Bourbaki books that most of his exercises are very significant. Example: Once, I found an exercise in Bourbaki which implies immediately Brown's Theorem on commuatators of complex semisimple Lie algebras. His idea that every Borel subalgebra B \subset [a, L] for some principal nil[potnet elment a of L.; so Brown's Theorem follows by conjugacy pf Borel subalgebras. (but Bourbaki did not refer to Brown paper around 1965). $\endgroup$ Commented Oct 24, 2016 at 20:03
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$\begingroup$ Thank you Nazih. I am sure it is an excellent reference, and I might borrow it from you at some point. I do recommend that you have a look at Onishchik and Vinberg, "Lie Groups and Algebraic Groups" too: it is full of exercises. You might like it too. $\endgroup$– MalkounCommented Oct 25, 2016 at 8:22
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$\begingroup$ Ok, I got access to the Bourbaki book(s) on Lie groups, via SpringerLink. They were not showing up initially because I was searching for "Bourbaki Lie groups", instead of spelling the word "groups" the French way, "groupes"! $\endgroup$– MalkounCommented Oct 25, 2016 at 11:39