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: 4f800549d96246c5cdde1b5fcc18b8437dfc6374
Parent: 873e879ea2e71e57579cbf51693d0a892e87d211
Author: Pablo <pablo-escobar@riseup.net>
Date: Sun, 27 Nov 2022 15:17:31 +0000

Removed the macros for double-struck letters

Diffstat

5 files changed, 18 insertions, 52 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	preamble.tex	34	0	34
Modified	sections/complete-reducibility.tex	2	1	1
Modified	sections/mathieu.tex	6	3	3
Modified	sections/semisimple-algebras.tex	2	1	1
Modified	sections/sl2-sl3.tex	26	13	13

diff --git a/preamble.tex b/preamble.tex
@@ -95,40 +95,6 @@
 % Command to define the subtitle in the cover of the book
 \newcommand{\subtitle}[1]{\def\thesubtitle{#1}}
 
-% Double struck letters
-% The following is kept besides \AA being already defined because I don't care
-% abaut Å
-\renewcommand{\AA}{\mathbb{A}}   % Double struck A
-\newcommand{\BB}{\mathbb{B}}     % Double struck B
-\newcommand{\CC}{\mathbb{C}}     % Double struck C
-\newcommand{\DD}{\mathbb{D}}     % Double struck D
-\newcommand{\EE}{\mathbb{E}}     % Double struck E
-\newcommand{\FF}{\mathbb{F}}     % Double struck F
-\newcommand{\GG}{\mathbb{G}}     % Double struck G
-\newcommand{\HH}{\mathbb{H}}     % Double struck H
-\newcommand{\II}{\mathbb{I}}     % Double struck I
-\newcommand{\JJ}{\mathbb{J}}     % Double struck J
-\newcommand{\KK}{\mathbb{K}}     % Double struck K
-\newcommand{\LL}{\mathbb{L}}     % Double struck L
-\newcommand{\MM}{\mathbb{M}}     % Double struck M
-\newcommand{\NN}{\mathbb{N}}     % Double struck N
-\newcommand{\OO}{\mathbb{O}}     % Double struck O
-\newcommand{\PP}{\mathbb{P}}     % Double struck P
-\newcommand{\QQ}{\mathbb{Q}}     % Double struck Q
-\newcommand{\RR}{\mathbb{R}}     % Double struck R
-% The following is ommited because LaTeX keeps complaining about \SS not being
-% allowed in mathmode
-% \renewcommand{\SS}{\mathbb{S}} % Double struck S
-\newcommand{\TT}{\mathbb{T}}     % Double struck T
-\newcommand{\UU}{\mathbb{U}}     % Double struck U
-\newcommand{\VV}{\mathbb{V}}     % Double struck V
-\newcommand{\WW}{\mathbb{W}}     % Double struck W
-\newcommand{\XX}{\mathbb{X}}     % Double struck X
-\newcommand{\YY}{\mathbb{Y}}     % Double struck Y
-\newcommand{\ZZ}{\mathbb{Z}}     % Double struck Z
-\newcommand{\hh}{\mathds{h}}     % Double struck h
-\newcommand{\kk}{\mathds{k}}     % Double struck k
-
 % Use \blacksquare for \qed
 \renewcommand{\qedsymbol}{\ensuremath{\blacksquare}}

diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -86,7 +86,7 @@ this is not always the case. For instance\dots
   is a representation of the Lie algebra \(K[x]\). Notice \(V\) has a single
   nonzero proper subrepresentation, which is spanned by the vector \((1, 0)\).
   This is because if \((a + b, b) = \rho(x) \ (a, b) = \lambda (a, b)\) for
-  some \(\lambda \in \CC\) then \(\lambda = 1\) and \(b = 0\). Hence \(V\) is
+  some \(\lambda \in \mathbb{C}\) then \(\lambda = 1\) and \(b = 0\). Hence \(V\) is
   indecomposable -- it cannot be broken into a direct sum of 1-dimensional
   subrepresentations -- but it is evidently not irreducible.
 \end{example}

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -1234,7 +1234,7 @@ Explicitly\dots
     r
     F_{\beta_1}^{- i_1} \cdots F_{\beta_n}^{- i_n}
   \]
-  for all \(k_1, \ldots, k_n \in \NN\).
+  for all \(k_1, \ldots, k_n \in \mathbb{N}\).
 
   Since the binomial coefficients \(\binom{x}{k} = \frac{x (x -1) \cdots (x - k +
   1)}{k!}\) can be uniquely extended to polynomial functions in \(x \in K\), we
@@ -1261,8 +1261,8 @@ Explicitly\dots
     \mathfrak{h}^* & \to \Sigma^{-1} \mathcal{U}(\mathfrak{g})        \\
            \lambda & \mapsto \theta_\lambda(\theta_{-\lambda}(r)) - r
   \end{align*}
-  is a polynomial extension of the zero map \(\ZZ \beta_1 \oplus \cdots \oplus
-  \ZZ \beta_n \to \Sigma^{-1} \mathcal{U}(\mathfrak{g})\) and is therefore
+  is a polynomial extension of the zero map \(\mathbb{Z} \beta_1 \oplus \cdots \oplus
+  \mathbb{Z} \beta_n \to \Sigma^{-1} \mathcal{U}(\mathfrak{g})\) and is therefore
   identically zero.
 
   Finally, let \(M\) be a \(\Sigma^{-1} \mathcal{U}(\mathfrak{g})\)-module

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -360,7 +360,7 @@ if we denote by \(\Delta\) the set of all roots of \(\mathfrak{g}\) then\dots
 
 \begin{theorem}\label{thm:weights-congruent-mod-root}
   The weights of an irreducible representation \(V\) of \(\mathfrak{g}\) are
-  all congruent module the root lattice \(Q = \ZZ \Delta\) of \(\mathfrak{g}\).
+  all congruent module the root lattice \(Q = \mathbb{Z} \Delta\) of \(\mathfrak{g}\).
   In other words, all weights of \(V\) lie in the same \(Q\)-coset
   \(t \in \mfrac{\mathfrak{h}^*}{Q}\).
 \end{theorem}

diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -56,17 +56,17 @@ Visually, we may draw
   \end{tikzcd}
 \end{center}
 
-This implies \(\bigoplus_{k \in \ZZ} V_{\lambda - 2 k}\) is an
+This implies \(\bigoplus_{k \in \mathbb{Z}} V_{\lambda - 2 k}\) is an
 \(\mathfrak{sl}_2(K)\)-invariant subspace, which goes to show
 \[
-  V = \bigoplus_{k \in \ZZ} V_{\lambda - 2 k},
+  V = \bigoplus_{k \in \mathbb{Z}} V_{\lambda - 2 k},
 \]
 and the eigenvalues of \(h\) all have the form \(\lambda - 2 k\) for some
 \(k\).
-Even more so, if \(a = \max \{ k \in \ZZ : V_{\lambda - 2 k} \ne 0 \}\) and
-\(b = \min \{ k \in \ZZ : V_{\lambda - 2 k} \ne 0 \}\) we can see that
+Even more so, if \(a = \max \{ k \in \mathbb{Z} : V_{\lambda - 2 k} \ne 0 \}\) and
+\(b = \min \{ k \in \mathbb{Z} : V_{\lambda - 2 k} \ne 0 \}\) we can see that
 \[
-  \bigoplus_{\substack{k \in \ZZ \\ a \le n \le b}} V_{\lambda - 2 k}
+  \bigoplus_{\substack{k \in \mathbb{Z} \\ a \le n \le b}} V_{\lambda - 2 k}
 \]
 is also an \(\mathfrak{sl}_2(K)\)-invariant subspace, so that the eigenvalues
 of \(h\) form an unbroken string
@@ -189,7 +189,7 @@ Other important consequences of proposition~\ref{thm:basis-of-irr-rep} are\dots
     0 = e 0 = e f^m v = m (\lambda + 1 - m) f^{m - 1} v
   \]
 
-  This implies \(\lambda + 1 - m = 0\) -- i.e. \(\lambda = m - 1 \in \ZZ\). Now
+  This implies \(\lambda + 1 - m = 0\) -- i.e. \(\lambda = m - 1 \in \mathbb{Z}\). Now
   since \(\{v, f v, f^2 v, \ldots, f^{m - 1} v\}\) is a basis for \(V\), \(m =
   \dim V\). Hence if \(n = \lambda = \dim V - 1\) then the eigenvalues of \(h\)
   are
@@ -270,7 +270,7 @@ clues on why \(h\) was a sure bet and the race was fixed all along.
 \section{Representations of \(\mathfrak{sl}_3(K)\)}\label{sec:sl3-reps}
 
 The study of representations of \(\mathfrak{sl}_2(K)\) reminds me of the
-difference between the derivative of a function \(\RR \to \RR\) and that of a
+difference between the derivative of a function \(\mathbb{R} \to \mathbb{R}\) and that of a
 smooth map between manifolds: it's a simpler case of something greater, but in
 some sense it's too simple of a case, and the intuition we acquire from it can
 be a bit misleading in regards to the general setting. For instance, I
@@ -485,7 +485,7 @@ representation of \(\mathfrak{sl}_3(K)\) deserve some special attention.
 Theorem~\ref{thm:sl3-weights-congruent-mod-root} can thus be restated as\dots
 
 \begin{definition}
-  The lattice \(Q = \ZZ \langle \alpha_i - \alpha_j : i, j = 1, 2, 3 \rangle\)
+  The lattice \(Q = \mathbb{Z} \langle \alpha_i - \alpha_j : i, j = 1, 2, 3 \rangle\)
   is called \emph{the root lattice of \(\mathfrak{sl}_3(K)\)}.
 \end{definition}
 
@@ -625,7 +625,7 @@ In general, we find\dots
 As a first consequence of this, we show\dots
 
 \begin{definition}
-  The lattice \(P = \ZZ \alpha_1 \oplus \ZZ \alpha_2 \oplus \ZZ \alpha_3\) is
+  The lattice \(P = \mathbb{Z} \alpha_1 \oplus \mathbb{Z} \alpha_2 \oplus \mathbb{Z} \alpha_3\) is
   called \emph{the weight lattice of \(\mathfrak{sl}_3(K)\)}.
 \end{definition}
 
@@ -667,7 +667,7 @@ As a first consequence of this, we show\dots
 
 There's a clear parallel between the case of \(\mathfrak{sl}_3(K)\) and that of
 \(\mathfrak{sl}_2(K)\), where we observed that the weights all lied in the
-lattice \(P = \ZZ\) and were congruent modulo the sublattice \(Q = 2 \ZZ\).
+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've been
 drawing are in fact consistent with the theory we've developed. Namely, since
 all weights lie in the rational span of \(\{\alpha_1, \alpha_2, \alpha_3\}\),
@@ -699,11 +699,11 @@ 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'll choose a
-linear functional \(f : \QQ P \to \QQ\) and pick the weight that maximizes
+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 \(\QQ
-P \to \QQ \langle \alpha_1 - \alpha_3 \rangle \cong \QQ\) then \(\alpha_1 - 2
+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