lie-algebras-and-their-representations

Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules

git clone: git://git.pablopie.xyz/lie-algebras-and-their-representations
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Commit: 0e094c9490e750a952ee5dd01bc4ac41ba2ad962
Parent: c45df8a9192cfe231dec86c4260918c99edd69de
Author: Pablo <pablo-escobar@riseup.net>
Date: Sat, 1 Jul 2023 19:36:03 +0000

Added some comments on the graph structure of 𝓑

Added some comments on the motivation behind the definition of the structure of oriented graph of 𝓑 in the case of sl(n)

Diffstat

1 file changed, 14 insertions, 5 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/coherent-families.tex	19	14	5

diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex
@@ -345,17 +345,26 @@ Proposition~\ref{thm:better-sl(n)-parameters}.
 % TODO: Explain that for each m ∈ 𝓑  there is a unique i such that so that
 % m_i - m_i+1 is not a positive integer. For m ∈ 𝓑 + this is i = 1, while for
 % m ∈ 𝓑 - this is i = n-1
-% TODOO: Explain the intuition behind defining the arrows like so: the point is
-% that if there is an arrow m(λ) → m(μ) then μ = σ_i ∙ λ for some i, which
-% implies L(μ) is contained in 𝓔𝔁𝓽(L(λ)) - so that L(μ) is also bounded and
-% 𝓔𝔁𝓽(L(λ)) ≅ 𝓔𝔁𝓽(L(μ))
+The issue here is that the relationship between \(\lambda, \mu \in P^+\) with
+\(m(\lambda), m(\mu) \in \mathscr{B}\) and \(\mExt(L(\lambda)) \cong
+\mExt(L(\mu))\) is more complicated than in the case of \(\mathfrak{sp}_{2
+n}(K)\). Nevertheless, Lemma~\ref{thm:lemma6.1} affords us a criteria for
+verifying that \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\). For \(\sigma =
+\sigma_i\) and the weight \(\lambda + \rho\), the hypothesis of
+Lemma~\ref{thm:lemma6.1} translates to \(m(\lambda)_i - m(\lambda)_{i+1} =
+(\lambda + \rho)(H_{\beta_i}) \notin \mathbb{N}\). If \(m(\lambda) \in
+\mathscr{B}\), this is equivalent to requiring that \(m(\lambda)\) is not
+ordered, but becomes ordered after removing its \(i\)-th term. This discussions
+losely inspires the following definition, which endows the set \(\mathscr{B}\)
+with the structure of a directed graph.
+
 \begin{definition}
   Given \(m, m' \in \mathscr{B}\), say there is an arrow \(m \to m'\) if there
   some \(i\) such that \(m_i - m_{i + 1}\) is \emph{not} a positive integer and
   \(m' = \sigma_i \cdot m\).
 \end{definition}
 
-It should then be obvious that\dots
+It should then be obvious from Lemma~\ref{thm:lemma6.1} that\dots
 
 \begin{proposition}\label{thm:arrow-implies-ext-eq}
   Let \(\lambda \notin P^+\) be such that \(L(\lambda)\) is bounded -- so that