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: c0974a025412d5bddbd016a1678eda368a4d4afa
Parent: 491b2cba922a3d39f7be1bc1bb8c1126be31cae1
Author: Pablo <pablo-escobar@riseup.net>
Date: Tue, 20 Dec 2022 13:02:39 +0000

Clarified the ortogonality relations in the weight diagrams of sl3

Also had to trim an example to accomodate for the extra room required by the clarification

Diffstat

2 files changed, 26 insertions, 19 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/semisimple-algebras.tex	3	2	1
Modified	sections/sl2-sl3.tex	42	24	18

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -280,7 +280,8 @@ hardly ever a non-degenerate form.
 \begin{note}
   Since \(B\) induces an isomorphism \(\mathfrak{h} \isoto \mathfrak{h}^*\), it
   induces a bilinear form \((B(X, \cdot), B(Y, \cdot)) \mapsto B(X, Y)\) in
-  \(\mathfrak{h}^*\). We denote this form by \(B\) as well.
+  \(\mathfrak{h}^*\). As in section~\ref{sec:sl3-reps}, we denote this form by
+  \(B\) as well.
 \end{note}
 
 We now have most of the necessary tools to reproduce the results of the

diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -650,14 +650,25 @@ As a first consequence of this, we show\dots
   \lambda([E_{2 3}, E_{3 2}]) \alpha_2 \in P\).
 \end{proof}
 
-There is a clear parallel between the case of \(\mathfrak{sl}_3(K)\) and that of
-\(\mathfrak{sl}_2(K)\), where we observed that the eigenvalues of the action of
-\(h\) all lied in the lattice \(P = \mathbb{Z}\) and were congruent modulo the
-sublattice \(Q = 2 \mathbb{Z}\). Among other things, this last result goes to
-show that the diagrams we have been drawing are in fact consistent with the
-theory we have developed. Namely, since all weights lie in the rational span of
-\(\{\alpha_1, \alpha_2, \alpha_3\}\), we may as well draw them in the Cartesian
-plane.
+There is a clear parallel between the case of \(\mathfrak{sl}_3(K)\) and that
+of \(\mathfrak{sl}_2(K)\), where we observed that the eigenvalues of the action
+of \(h\) all lied in the lattice \(P = \mathbb{Z}\) and were congruent modulo
+the sublattice \(Q = 2 \mathbb{Z}\).
+
+Among other things, this last result goes to show that the diagrams we have
+been drawing are in fact consistent with the theory we have developed. Namely,
+since all weights lie in the rational span of \(\{\alpha_1, \alpha_2,
+\alpha_3\}\), we may as well draw them in the Cartesian plane. In fact, the
+attentive reader may notice that \(B(E_{1 2}, E_{2 3}) = - \sfrac{1}{2}\), so
+that the angle -- with respect to the Killing form \(B\) -- between the root
+vectors \(E_{1 2}\) and \(E_{2 3}\) is precisely the same as the angle between
+the points representing their roots \(\alpha_1 - \alpha_2\) and \(\alpha_2 -
+\alpha_3\) in the Cartesion plane. Since \(\alpha_1 - \alpha_2\) and \(\alpha_2
+- \alpha_3\) span \(\mathfrak{h}^*\), this implies the diagrams we've been
+drawing are given by an isometry \(\mathbb{Q} P \isoto \mathbb{Q}^2\), where
+\(\mathbb{Q} P\) is endowed with the bilinear form defined by \((\alpha_i -
+\alpha_j, \alpha_k - \alpha_\ell) \mapsto B(E_{i j}, E_{k \ell})\) -- which we
+denote by \(B\) as well.
 
 To proceed we once more refer to the previously established framework: next we
 saw that the eigenvalues of \(h\) form an unbroken string of integers symmetric
@@ -687,16 +698,11 @@ and let \(\lambda\) be the weight lying the furthest in this direction.
 
 Its easy to see what we mean intuitively by looking at the previous picture,
 but its precise meaning is still allusive. Formally this means we will choose a
-linear functional \(f : \mathbb{Q} P \to \mathbb{Q}\) and pick the weight that maximizes
-\(f\). To avoid any ambiguity we should choose the direction of a line
-irrational with respect to the root lattice \(Q\). For instance if we choose
-the direction of \(\alpha_1 - \alpha_3\) and let \(f\) be the projection \(\mathbb{Q}
-P \to \mathbb{Q} \langle \alpha_1 - \alpha_3 \rangle \cong \mathbb{Q}\) then \(\alpha_1 - 2
-\alpha_2 + \alpha_3 \in Q\) lies in \(\ker f\), so that if a weight \(\lambda\)
-maximizes \(f\) then the translation of \(\lambda\) by any multiple of
-\(\alpha_1 - 2 \alpha_2 + \alpha_3\) must also do so. In others words, if the
-direction we choose is parallel to a vector lying in \(Q\) then there may be
-multiple choices the ``weight lying the furthest'' along this direction.
+linear functional \(f : \mathbb{Q} P \to \mathbb{Q}\) and pick the weight that
+maximizes \(f\). To avoid any ambiguity we should choose the direction of a
+line irrational with respect to the root lattice \(Q\) -- for if \(f\) is not
+irrational there may be multiple choices the ``weight lying the furthest''
+along this direction.
 
 \begin{definition}
   We say that a root \(\alpha\) is positive if \(f(\alpha) > 0\) -- i.e. if it