4
$\begingroup$

What is the reason for the BBG category $\mathcal{O}$ failing to be closed under extensions i.e which of the 3 axioms of $\mathcal{O}$ does not hold under taking extensions?

Is there a prototype of a counterexample?

$\endgroup$

1 Answer 1

4
$\begingroup$

You can usually extend two modules from $\mathcal{O}$ by a module which is not semisimple for the Cartan subalgebra (i.e. fails to be a weight module). See Exercise 3.1. in [J. E. Humphreys, Representations of semisimple Lie algebras in the BGG category $\mathcal{O}$. Graduate Studies in Mathematics, 94].

$\endgroup$
1
  • $\begingroup$ Thank you very much! $\endgroup$
    – KKD
    May 27, 2020 at 11:26

Your Answer

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.