bio  website  math.umass.edu/~jeh 

location  U. Massachusetts, Amherst  
age  75  
visits  member for  5 years, 1 month 
seen  27 mins ago  
stats  profile views  15,692 
Moreorless retired professor at UMass Amherst.
Basically an algebraist with interests in Lie theory,
specifically representation theory and related
algebraic geometry.
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Root in positive Weyl chamber
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Root in positive Weyl chamber
@shu: Having gone back to the foundational material on root systems, I'm motivated to outline the full story more carefully. This is all fairly elementary, starting with the axioms, but it does need to be done systematically. Though some shortcuts are possible for the narrow question you've raised, the details in the outline are useful to know about. 
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Root in positive Weyl chamber
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Root in positive Weyl chamber
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Mar 28 
answered  Root in positive Weyl chamber 
Mar 27 
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Root in positive Weyl chamber
@shu: This example is correct, since the highest root happens to coincide with $\rho$. But it is apparently the only such case. (I echoed Sasha's comment too quickly, since the highest root can lie inside the chamber $K$. But it doesn't usually coincide with $\rho$. So I don't quite understand what you asking.) 
Mar 27 
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Subgroups generated by opposite root groups
Have you consulted the classic 1965 IHES paper by BorelTits on the structure of (isotropic) reductive groups over arbitrary fields, or the later papers by BruhatTits specializing to local fields? These are freely available online at numdam.org and could help to answer your questions. (Also, your boldface notation here is unhelpful. It's better to follow BorelTits.) 
Mar 27 
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Root in positive Weyl chamber
@shu: Your question isn't clear to me. The set $R \cap K$ is empty to begin with (as Sasha comments, only one or two roots can be dominant weights, and they don't lie in the interior of $K$). On the other hand, $\rho$ isn't a root but does lie in the (dominant) open Weyl chamber $K$. What are you trying to find a proof for without using classification? 
Mar 27 
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Simply connected Lie groups homeomorphic to R^n are solvable
@user61471: It would help to have more precise references for the "many proofs" you mention, for example in the short paper you mention in your comment: ams.org/mathscinetgetitem?mr=975639 
Mar 26 
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Is an $\mathfrak{sl}_2$triple determined up to Lie algebra automorphism by the adjoint representation?
@Dave: Yes, I'm only considering simple Lie algebras. I've also tried to clarify my answer yet again but will give up edits at this point. 
Mar 26 
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Is an $\mathfrak{sl}_2$triple determined up to Lie algebra automorphism by the adjoint representation?
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Mar 26 
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Is an $\mathfrak{sl}_2$triple determined up to Lie algebra automorphism by the adjoint representation?
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Mar 26 
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Is an $\mathfrak{sl}_2$triple determined up to Lie algebra automorphism by the adjoint representation?
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Mar 25 
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Is an $\mathfrak{sl}_2$triple determined up to Lie algebra automorphism by the adjoint representation?
@Dave: I was just having some second thoughts, so I edited my answer. I may still be oversimplifying. 
Mar 25 
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Is an $\mathfrak{sl}_2$triple determined up to Lie algebra automorphism by the adjoint representation?
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Mar 25 
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Automorphisms of B_n
This question should probably be asked on stackexchange.com rather than here. Assuming you are dealing with simple Lie algebras over a field such as $\mathbb{C}$, the automorphisms are well known and are all inner if there is no graph automorphism (as in type $B_n$). 
Mar 25 
answered  Is an $\mathfrak{sl}_2$triple determined up to Lie algebra automorphism by the adjoint representation? 
Mar 25 
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Prescribed spherical representations, symplectic group $Sp(n)$
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Mar 25 
answered  Prescribed spherical representations, symplectic group $Sp(n)$ 
Mar 25 
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Generators of invariant polynomials of semisimple Lie algebra
This looks like a nice higherlevel way to realize the algebraic results. By the way, the fact that the algebra of invariant polynomial functions is generated in all cases by powers of trace functions is quite classical. But Bourbaki's exercise makes a subtle refinement in the choice of a minimal set of generators for type $D_\ell$. 