0
$\begingroup$

In their paper Lepowsky and Mcmollum sketch theory of weights in a more general setting. Here is their definition of a weight space:

If $A$ is a subset of $\mathfrak g$ and $\lambda$ is a function from $A$ into $k$, $V^{\lambda}_A$ is the set of $x\in\mathfrak g$ such that some power of $\pi(a)-\lambda(a)$ annihilates $x$ for all $a\in A$. The function $\lambda$ such that $A$ into $k$, $V^{\lambda}_A \neq 0$ is called aweight vector.

here $\mathfrak g$ is a lie algebra over a field $k$ and $\pi: \mathfrak g \rightarrow End(V) $ a representation of $\mathfrak g$ on a vctor space $V$.

My question : Shouldn't $V^{\lambda}_A $ be the set of $x\in V$ instead of $x\in\mathfrak g$ in the first line?

Improve this question
$\endgroup$
2
  • $\begingroup$ Yes. It looks like its just a typo. $\endgroup$
    – Noah White
    Commented Oct 21, 2014 at 15:38
  • 1
    $\begingroup$ Yes, this seems to be an obvious misprint, probably due to the authors' interest in the adjoint representation. If in doubt, check with Lepowsky, who is at Rutgers. [By the way, your own formulation could use some proofreading.] $\endgroup$
    – Jim Humphreys
    Commented Oct 21, 2014 at 15:42

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .