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: addcf9e5cd178c0666ecda536035e7c74e96ebfe
Parent: be399204733e707534089919c1a5b212f0ac0335
Author: Pablo <pablo-escobar@riseup.net>
Date: Mon, 5 Sep 2022 18:30:22 +0000
Concertados mais alguns errinhos gramaticais e frases estranhas
Diffstat

1 file changed, 42 insertions, 47 deletions
Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/semisimple-algebras.tex	89	42	47
diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -125,6 +125,7 @@ handy later on, is\dots
   all \(Y \in \mathfrak{g}\) then \(X = 0\).
 \end{proposition}
 
+% TODO: Write a proper introduction?
 \section{Complete Reducibility}
 
 We are primarily interested in establishing\dots
@@ -133,33 +134,32 @@ We are primarily interested in establishing\dots
   Every representation of a semisimple Lie algebra is completely reducible.
 \end{theorem}
 
-% TODO: Cite myself? I don't really know anywhere else that does this
 Historically, this was first proved by Herman Weyl for \(K = \mathbb{C}\),
 using his knowledge of unitary representations of compact groups. Namely, Weyl
 showed that any finite-dimensional semisimple complex Lie algebra is
-(isomorphic to) the complexification of the Lie algebra of a simply connected
-compact Lie group, so that the category of its finite-dimensional
-representations is equivalent to that of the finite-dimensional smooth
-representations of such compact group -- under which Mashcke's theorem for
-compact groups applies. We refer the reader to (TODO: cite someone) for further
-details.
+(isomorphic to) the complexification of the Lie algebra of a unique simply
+connected compact Lie group, known as its \emph{compact form}. Hence that the
+category of the finite-dimensional representations of a given complex
+semisimple algebra is equivalent to that of the finite-dimensional smooth
+representations of its compact form, whose representations are known to be
+completely reducible.
 
 This proof, however, is heavily reliant on the geometric structure of
 \(\mathbb{C}\). In other words, there is no hope for generalizing this for some
 arbitrary \(K\). Hopefully for us, there is a much simpler, completely
 algebraic proof which works for algebras over any algebraically closed field of
-characteristic zero. The algebraic proof included in here mainly based on that
-of \cite[ch. 6]{kirillov}, and uses some basic homological algebra. Admittedly,
-much of the homological algebra used in here could be concealed from the reader
--- see \cite{humphreys} for instance -- which would make the exposition more
-accessible.
+characteristic zero. The algebraic proof included in here is mainly based on
+that of \cite[ch. 6]{kirillov}, and uses some basic homological algebra.
+Admittedly, much of the homological algebra used in here could be concealed
+from the reader, which would make the exposition more accessible -- see
+\cite{humphreys} for an elementary account, for instance.
 
 However, this does not change the fact the arguments used in this proof are
 essentially homological in nature. Hence we consider it more productive to use
 the full force of the language of homological algebra, instead of burring the
-reader in a pile of unmotivated elementary arguments. Furthermore, the
-homological algebra used in here is actually \emph{very basic}. In fact, all we
-need to know is\dots
+reader in a pile of unmotivated arguments. Furthermore, the homological algebra
+used in here is actually \emph{very basic}. In fact, all we need to know
+is\dots
 
 \begin{theorem}\label{thm:ext-exacts-seqs}
   There is a sequence of bifunctors \(\operatorname{Ext}^i :
@@ -213,7 +213,6 @@ need to know is\dots
   \end{center}
 \end{theorem}
 
-% TODO: Make the correspondance more precise?
 \begin{theorem}\label{thm:ext-1-classify-short-seqs}
   Given \(\mathfrak{g}\)-modules \(W\) and \(U\), there is a one-to-one
   correspondence between elements of \(\operatorname{Ext}^1(W, U)\) and
@@ -230,10 +229,10 @@ need to know is\dots
 \end{theorem}
 
 \begin{note}
-  This, of course, \emph{far} from a comprehensive account of homological
+  This is, of course, \emph{far} from a comprehensive account of homological
   algebra. Nevertheless, this is all we need. We refer the reader to
-  \cite{harder} for further details, or to part II of \cite{ribeiro} for a more
-  modern account using derived categories.
+  \cite{harder} for a complete exposition, or to part II of \cite{ribeiro} for
+  a more modern account using derived categories.
 \end{note}
 
 We are particular interested in the case where \(S = K\) is the trivial
@@ -363,10 +362,10 @@ we introduce a distinguished element of \(\mathcal{U}(\mathfrak{g})\), known as
   Whatever basis \(\{X_i\}_i\) we choose, the image of \(C\) under the
   canonical isomorphism \(\mathfrak{g} \otimes \mathfrak{g} \isoto \mathfrak{g}
   \otimes \mathfrak{g}^* \isoto \operatorname{End}(\mathfrak{g})\) is the
-  identity operator -- here the isomorphism \(\mathfrak{g} \otimes \mathfrak{g}
-  \isoto \mathfrak{g} \otimes \mathfrak{g}^*\) is given by tensoring the
-  identity \(\mathfrak{g} \to \mathfrak{g}\) with the isomorphism
-  \(\mathfrak{g} \isoto \mathfrak{g}^*\) induced by the Killing form \(B\).
+  identity operator\footnote{Here the isomorphism \(\mathfrak{g} \otimes
+  \mathfrak{g} \isoto \mathfrak{g} \otimes \mathfrak{g}^*\) is given by
+  tensoring the identity \(\mathfrak{g} \to \mathfrak{g}\) with the isomorphism
+  \(\mathfrak{g} \isoto \mathfrak{g}^*\) induced by the Killing form \(B\).}.
 \end{proof}
 
 \begin{proposition}
@@ -542,9 +541,7 @@ We are now finally ready to prove complete reducibility once and for all.
     \end{tikzcd}
   \end{center}
   of vector spaces. Since all maps involved are intertwiners, this is an exact
-  sequence of \(\mathfrak{g}\)-modules. Now by applying the functor
-  \(\operatorname{Hom}_{\mathfrak{g}}(K, -)\) to our sequence, we get long
-  exact sequence
+  sequence of \(\mathfrak{g}\)-modules. This then induces a long exact sequence
   \begin{center}
     \begin{tikzcd}
       0 \arrow{r} &
@@ -575,7 +572,8 @@ We are now finally ready to prove complete reducibility once and for all.
 
   % TODO: Define the action of g in Hom(V, W) beforehand
   Now notice \(\operatorname{Hom}(U, -)^{\mathfrak{g}} =
-  \operatorname{Hom}_{\mathfrak{g}}(U, -)\). Indeed,
+  \operatorname{Hom}_{\mathfrak{g}}(U, -)\). Indeed, given a
+  \(\mathfrak{g}\)-module \(S\) and a \(K\)-linear mapa \(T : U \to S\)
   \[
     \begin{split}
       T \in \operatorname{Hom}(U, S)^{\mathfrak{g}}
@@ -584,7 +582,8 @@ We are now finally ready to prove complete reducibility once and for all.
       & \iff T \in \operatorname{Hom}_{\mathfrak{g}}(U, S)
     \end{split}
   \]
-  for all \(\mathfrak{g}\)-module \(S\). We thus have a short exact sequence
+
+  We thus have a short exact sequence
   \begin{center}
     \begin{tikzcd}
       0 \arrow{r} &
@@ -609,19 +608,18 @@ We are now finally ready to prove complete reducibility once and for all.
   is a splitting of (\ref{eq:generict-exact-sequence}).
 \end{proof}
 
-% TODO: Define what the semisimple form of a complex Lie algebra is in the
-% introduction
 We should point out that this last results are just the beginning of a well
-developed cohomology theory. For example, a similar argument using the Casimir
-element can be used to show that \(H^i(\mathfrak{g}, V) = 0\) for all
+developed cohomology theory. For example, a similar argument involving the
+Casimir element can be used to show that \(H^i(\mathfrak{g}, V) = 0\) for all
 non-trivial finite-dimensional irreducible \(V\), \(i > 0\). For \(K =
-\mathbb{C}\), the Lie algebra cohomology groups of an algebra \(\mathfrak{g}\)
-are intimately related with the topological cohomologies -- i.e. singular
-cohomology, de Rham cohomology, etc. -- of its simply connected form. We refer
-the reader to \cite{cohomologies-lie} for further details.
+\mathbb{C}\), the Lie algebra cohomology groups of an algebra \(\mathfrak{g} =
+\mathbb{C} \otimes \operatorname{Lie}(G)\) are intimately related with the
+topological cohomologies -- i.e. singular cohomology, de Rham cohomology, etc.
+-- of \(G\). We refer the reader to \cite{cohomologies-lie} for further
+details.
 
-Complete reducibility can be generalized to a certain extent for arbitrary --
-not necessarily semisimple -- \(\mathfrak{g}\) by considering the exact
+Complete reducibility can be generalized for arbitrary -- not necessarily
+semisimple -- \(\mathfrak{g}\), to a certain extent, by considering the exact
 sequence
 \begin{center}
   \begin{tikzcd}
@@ -633,10 +631,9 @@ sequence
   \end{tikzcd}
 \end{center}
 where \(\operatorname{Rad}(\mathfrak{g})\) is the sum of all solvable ideals of
-\(\mathfrak{g}\) -- i.e. a maximal solvable ideal -- for arbitrary
-\(\mathfrak{g}\). Of course, this sequence does not generally split, but it
-implies we can deduce information about the representations of \(\mathfrak{g}\)
-by studying those of its ``semisimple part''
+\(\mathfrak{g}\). Of course, this sequence does not split in general, but its
+exactness implies we can deduce information about the representations of
+\(\mathfrak{g}\) by studying those of its ``semisimple part''
 \(\mfrac{\mathfrak{g}}{\operatorname{Rad}(\mathfrak{g})}\). In practice this
 translates to\dots
 
@@ -1725,9 +1722,6 @@ and then analyze how the remaining elements of \(\mathfrak{g}\) act on this
 subspaces. The bigger \(\mathfrak{h}\) the simpler our problem gets, because
 there are fewer elements outside of \(\mathfrak{h}\) left to analyze.
 
-% TODO: Remove or adjust the comment on maximal tori
-% TODO: Turn this into a proper definition
-% TODO: Also define the associated Borel subalgebra
 Hence we are generally interested in maximal Abelian subalgebras \(\mathfrak{h}
 \subset \mathfrak{g}\), which leads us to the following definition.
 
@@ -1950,7 +1944,8 @@ then\dots
   all congruent module the root lattice \(Q = \ZZ \Delta\) of \(\mathfrak{g}\).
 \end{theorem}
 
-% TODO: Rewrite this: the concept of direct has no sence in the general setting
+% TODO: Rewrite this: the concept of direction has no sence in the general
+% setting
 To proceed further, as in the case of \(\mathfrak{sl}_3(K)\) we have to fix a
 direction in \(\mathfrak{h}^*\) -- i.e. we fix a linear function
 \(\mathfrak{h}^* \to \RR\) such that \(Q\) lies outside of its kernel. This
@@ -2092,7 +2087,7 @@ theorem~\ref{thm:weak-dominant-weight} we used for the case when \(\mathfrak{g}
 knowledge of the roots of \(\mathfrak{sl}_3(K)\). Instead, we need a new
 strategy for the general setting.
 
-% TODO: Add further details. turn this into a proper proof?
+% TODO: Add further details. Turn this into a proper proof?
 Alternatively, one could construct  a potentially infinite-dimensional
 representation of \(\mathfrak{g}\) whose highest weight is some fixed dominant
 integral weight \(\lambda\) by taking the induced representation