0
$\begingroup$

1.Let $\pi$ be an integrable representation of a Kac Moody algebra $g(A)$ on a vector space $V$. For $i = 1,2 ,...,n$ set $r_i^{\pi} = (exp fi)(exp (-ei))(exp fi)$. Then how to prove that $r_i^{\pi}(V_{\lambda}) = V_{r_i(\lambda)}$. Where $r_i$ is a fundamental reflection.

2.How to prove that $w(\alpha_i) = \alpha_j \implies w( \alpha_i^{\vee}) = \alpha_j^{\vee}$ for a $w \in W$?

First doubt is Lemma 3.8 from Kac's book on infinite dimensional Lie algebras and second doubt is statement (3.10.3) in the same book.

Thanks for your valuable time.

Improve this question
$\endgroup$
4
  • $\begingroup$ When you say "doubt", do you mean "question"? $\endgroup$
    – S. Carnahan ♦
    Aug 9, 2015 at 17:11
  • $\begingroup$ Sorry. yes. I mean first question. $\endgroup$
    – GA316
    Aug 10, 2015 at 1:18
  • $\begingroup$ @Vladimir I am not getting your question. please be little eloborate $\endgroup$
    – GA316
    Jan 27, 2016 at 14:25
  • $\begingroup$ Dear GA316, I have deleted my remark. I apologize a lot. $\endgroup$
    – Vladimir
    Jan 28, 2016 at 16:38

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.