lie-algebras-and-their-representations

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -362,47 +362,13 @@ 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 \(\mathfrak{g}\).
+  In other words, all weights of \(V\) lie in the same \(Q\)-coset in
+  \(\mfrac{\mathfrak{h}^*}{Q}\).
 \end{theorem}
 
-% TODOO: Turn this into a proper discussion of basis and give the idea of the
-% proof of existance of basis?
-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
-\(\mathfrak{h}^* \to \QQ\) such that \(Q\) lies outside of its kernel. This
-choice induces a partition \(\Delta = \Delta^+ \cup \Delta^-\) of the set of
-roots of \(\mathfrak{g}\) and once more we find\dots
-
-\begin{definition}
-  The elements of \(\Delta^+\) and \(\Delta^-\) are called \emph{positive} and
-  \emph{negative roots}, respectively. The subalgebra \(\mathfrak{b} =
-  \mathfrak{h} \oplus \bigoplus_{\alpha \in \Delta^+} \mathfrak{g}_\alpha\) is
-  called \emph{the Borel subalgebra associated with \(\mathfrak{h}\)}.
-\end{definition}
-
-\begin{theorem}
-  There is a weight vector \(v \in V\) that is killed by all positive root
-  spaces of \(\mathfrak{g}\).
-\end{theorem}
-
-% TODO: Here we may take a weight of maximal height, but why is it unique?
-% TODO: We don't really need to talk about height tho, we may simply take a
-% weight that maximizes B(gamma, lambda) in QQ
-% TODOO: Either way, we need to move this to after the discussion on the
-% integrality of weights
-\begin{proof}
-  It suffices to note that if \(\lambda\) is the weight of \(V\) lying the
-  furthest along the direction we chose and \(V_{\lambda + \alpha} \ne 0\) for
-  some \(\alpha \in \Delta^+\) then \(\lambda + \alpha\) is a weight that is
-  furthest along the direction we chose than \(\lambda\), which contradicts the
-  definition of \(\lambda\).
-\end{proof}
-
-Accordingly, we call \(\lambda\) \emph{the highest weight of \(V\)}, and we
-call any \(v \in V_\lambda\) \emph{a highest weight vector}. The strategy then
-is to describe all weight spaces of \(V\) in terms of \(\lambda\) and \(v\), as
-in theorem~\ref{thm:sl3-irr-weights-class}, and unsurprisingly we do so by
-reproducing the proof of the case of \(\mathfrak{sl}_3(K)\). Namely, we
-show\dots
+Again, we may levarage our knowlage of \(\mathfrak{sl}_2(K)\) to obtain further
+restrictions on the geometry of the space of weights of \(V\). Namelly, such as
+in the case of \(\mathfrak{sl}_3(K)\) we show\dots
 
 \begin{proposition}\label{thm:distinguished-subalgebra}
   Given a root \(\alpha\) of \(\mathfrak{g}\) the subspace
@@ -414,7 +380,7 @@ show\dots
 \begin{corollary}\label{thm:distinguished-subalg-rep}
   For all weights \(\mu\), the subspace
   \[
-    V_\mu[\alpha] = \bigoplus_k V_{\mu + k \alpha}
+    \bigoplus_k V_{\mu + k \alpha}
   \]
   is invariant under the action of the subalgebra \(\mathfrak{s}_\alpha\)
   and the weight spaces in this string match the eigenspaces of \(h\).
@@ -436,49 +402,161 @@ satisfies
 \end{align*}
 
 The elements \(E_\alpha, F_\alpha \in \mathfrak{g}\) are not uniquely
-determined by this condition, but \(H_\alpha\) is. The second statement of
-corollary~\ref{thm:distinguished-subalg-rep} imposes a restriction on the
-weights of \(V\). Namely, if \(\mu\) is a weight, \(\mu(H_\alpha)\) is an
-eigenvalue of \(h\) in some representation of \(\mathfrak{sl}_2(K)\), so
-that\dots
+determined by this condition, but \(H_\alpha\) is. As promised, the second
+statement of corollary~\ref{thm:distinguished-subalg-rep} imposes strong
+restrictions on the weights of \(V\). Namely, if \(\lambda\) is a weight,
+\(\lambda(H_\alpha)\) is an eigenvalue of \(h\) in some representation of
+\(\mathfrak{sl}_2(K)\), so it must be an integer. In other words\dots
+
+\begin{definition}\label{def:weight-lattice}
+  The lattice \(P = \{ \lambda \in \mathfrak{h}^* : \lambda(H_\alpha) \in
+  \mathbb{Z} \, \forall \alpha \in \Delta \} \subset \mathfrak{h}^*\) is called
+  \emph{the weight lattice of \(\mathfrak{g}\)}.
+\end{definition}
+
+\begin{proposition}\label{thm:weights-fit-in-weight-lattice}
+  The weights of an irreducible representation \(V\) of \(\mathfrak{g}\) is Lie
+  in the weight lattice \(P\).
+\end{proposition}
+
+We call the elements of \(P\) \emph{integral}.
+Proposition~\ref{thm:weights-fit-in-weight-lattice} is clearly analogous to
+corollary~\ref{thm:sl3-weights-fit-in-weight-lattice}. In fact, the weight
+lattice of \(\mathfrak{sl}_3(K)\) -- as in definition~\ref{def:weight-lattice}
+-- is precisely \(\mathbb{Z} \alpha_1 \oplus \mathbb{Z} \alpha_2 \oplus
+\mathbb{Z} \alpha_3\).
+
+To proceed further, we would like to take \emph{the highest weight of \(V\)} as
+in section~\ref{sec:sl3-reps}, but the meaning of \emph{highest} is again
+unclear in this situation. We could simply fix a linear function \(\mathbb{Q} P
+\to \mathbb{Q}\) -- as we did in section~\ref{sec:sl3-reps} -- and choose a
+weight \(\lambda\) of \(V\) that maximizes this functional, but at this point
+it is conveniant to introduce some additional tools to our arsenal. This tools
+are called \emph{basis}.
+
+\begin{definition}\label{def:basis-of-root}
+  A subset \(\Sigma = \{\beta_1, \ldots, \beta_k\} \subset \Delta\) of linearly
+  independant roots is called \emph{a basis for \(\Delta\)} if, given \(\alpha
+  \in \Delta\), there are \(n_1, \ldots, n_k \in \mathbb{N}\) such that
+  \(\alpha = \pm(n_1 \beta_1 + \cdots + n_k \beta_k)\).
+\end{definition}
+
+The interesting thing about basis for \(\Delta\) is that they allow us to
+compare weights of a given representation. At this point the reader should be
+asking himself: how? Definition~\ref{def:basis-of-root} doesn't exactly screem
+``comparison''. Well, the thing is that any choice of basis induces a partial
+order in \(Q\), where elements are ordered by their \emph{hights}.
+
+\begin{definition}
+  Let \(\Sigma = \{\beta_1, \ldots, \beta_k\}\) be a basis for \(\Delta\).
+  Given \(\alpha = n_1 \beta_1 + \cdots + n_2 \beta_2 \in Q\) with \(n_1,
+  \ldots, n_k \in \mathbb{Z}\), we call the number \(h(\alpha) = n_1 + \cdots +
+  n_k \in \mathbb{Z}\) \emph{the height of \(\alpha\)}. We say that \(\alpha
+  \preceq \beta\) if \(h(\alpha) \le h(\beta)\).
+\end{definition}
+
+\begin{definition}
+  Given a basis \(\Sigma\) for \(\Delta\), there is a canonical partition
+  \(\Delta^+ \cup \Delta^- = \Delta\), where \(\Delta^+ = \{ \alpha \in \Delta
+  : \alpha \succeq 0 \}\) and \(\Delta^- = \{ \alpha \in \Delta : \alpha
+  \preceq 0 \}\). The elements of \(\Delta^+\) and \(\Delta^-\) are called
+  \emph{positive} and \emph{negative roots}, respectively.
+\end{definition}
+
+\begin{definition}
+  Let \(\Sigma\) be a basis for \(\Delta\). The subalgebra \(\mathfrak{b} =
+  \mathfrak{h} \oplus \bigoplus_{\alpha \in \Delta^+} \mathfrak{g}_\alpha\) is
+  called \emph{the Borel subalgebra associated with \(\mathfrak{h}\) and
+  \(\Sigma\)}.
+\end{definition}
+
+It should be obvious that the binary relation \(\preceq\) in \(Q\) is a partial
+order. In addition, we may compare the elements of a given \(Q\)-coset
+\(\lambda + Q\) by comparing their difference with \(0 \in Q\). In other words,
+we say \(\lambda \preceq \mu\) if \(\lambda - \mu \preceq 0\) for \(\lambda \in
+\mu + Q\). In particular, since the weights of \(V\) all lie in a single
+\(Q\)-coset, we may compare them in this fashion. Given a basis \(\Sigma\) for
+\(\Delta\) we may take ``the highest weight of \(V\)'' as a maximal weight
+\(\lambda\) of \(V\). The obvious question then is: can we always find a basis
+for \(\Delta\)?
 
 \begin{proposition}
-  The weights \(\mu\) of an irreducible representation \(V\) of
-  \(\mathfrak{g}\) are so that \(\mu(H_\alpha) \in \ZZ\) for each \(\alpha \in
-  \Delta\).
+  There is a basis \(\Sigma\) for \(\Delta\).
 \end{proposition}
 
-Once more, the lattice \(P = \{ \lambda \in \mathfrak{h}^* : \lambda(H_\alpha)
-\in \ZZ, \forall \alpha \in \Delta \}\) is called \emph{the weight lattice of
-\(\mathfrak{g}\)}, and we call the elements of \(P\) \emph{integral}. Finally,
-another important consequence of theorem~\ref{thm:distinguished-subalgebra}
-is\dots
+The intuition behind the proof of this proposition is similar to our original
+idea of fixing a direction in \(\mathfrak{h}^*\) in the case of
+\(\mathfrak{sl}_3(K)\). Namely, one can show that \(B(\alpha, \beta) \in
+\mathbb{Z}\) for all \(\alpha, \beta \in \Delta\), so that the Killing form
+\(B\) restricts to a nondegenerate form \(\mathbb{Q} \Delta \times \mathbb{Q}
+\Delta \to \mathbb{Q}\). We can then fix a nonzero vector \(\gamma \in
+\mathbb{Q} \Delta\) and consider the orthogonal projection \(f : \mathbb{Q}
+\Delta \to \mathbb{Q} \gamma \cong \mathbb{Q}\). We say a root \(\alpha \in
+\Delta\) is \emph{positive} if \(f(\alpha) > 0\), and we call a positive root
+\(\alpha\) \emph{simple} if it cannot be written as the sum two other positive
+roots. The subset \(\Sigma \subset \Delta\) of all simple roots is a basis for
+\(\Delta\), and all other basis can be shown to arise in this way.
+
+Fix some basis \(\Sigma\) for \(\Delta\), with corresponding decomposition
+\(\Delta^+ \cup \Delta^- = \Delta\). Let \(\lambda\) be a maximal weight of
+\(V\). We call \(\lambda\) \emph{the highest weight of \(V\)}, and we call any
+nonzero \(v \in V_\lambda\) \emph{a highest weight vector}. The strategy then
+is to describe all weight spaces of \(V\) in terms of \(\lambda\) and \(v\), as
+in theorem~\ref{thm:sl3-irr-weights-class}. Unsurprisingly we do so by
+reproducing the proof of the case of \(\mathfrak{sl}_3(K)\).
+
+First, we note that any highest weight vector \(v \in V_\lambda\) is
+annihilated by all positive root spaces, for if \(\alpha \in \Delta^+\) then
+\(E_\alpha v \in V_{\lambda + \alpha}\) must be zero -- or otherwise we would
+have that \(\lambda + \alpha\) is a weight with \(\lambda \prec \lambda +
+\alpha\). In particular,
+\[
+  \bigoplus_{k \in \mathbb{Z}}   V_{\lambda - k \alpha}
+  = \bigoplus_{k \in \mathbb{N}} V_{\lambda - k \alpha}
+\]
+and \(\lambda(H_\alpha)\) is the right-most eigenvalue of the action of \(h\)
+in the \(\mathfrak{sl}_2(K)\)-module \(\bigoplus_k V_{\lambda - k \alpha}\).
+
+This has a number of important consequences. For instance\dots
 
+% TODO: Change the notation for T_alpha
 \begin{corollary}
   If \(\alpha \in \Delta^+\) and \(T_\alpha : \mathfrak{h}^* \to
   \mathfrak{h}^*\) is the reflection in the hyperplane perpendicular to
-  \(\alpha\) with respect to the Killing form,
-  corollary~\ref{thm:distinguished-subalg-rep} implies that all \(\nu \in P\)
-  lying inside the line connecting \(\mu\) and \(T_\alpha \mu\) are weights --
-  i.e. \(V_\nu \ne 0\).
+  \(\alpha\) with respect to the Killing form, the weights of \(V\) occuring in
+  the line joining \(\lambda\) and \(T_\alpha\) are precisely the \(\mu \in P\)
+  lying between \(\lambda\) and \(T_\alpha \lambda\).
 \end{corollary}
 
 \begin{proof}
-  It suffices to note that \(\nu \in V_\mu[\alpha]\) -- see appendix D of
-  \cite{fulton-harris} for further details.
+  Notice that any \(\mu \in P\) in the line joining \(\lambda\) and \(T_\alpha
+  \lambda\) has the form \(\mu = \lambda - k \alpha\) for some \(k\), so that
+  \(V_\mu\) corresponds the eigenspace associated with the eigenvalue
+  \(\lambda(H_\alpha) - 2k\) of the action of \(h\) in \(\bigoplus_k V_{\lambda
+  - k \alpha}\). If \(\mu\) lies between \(\lambda\) and \(T_\alpha \lambda\)
+  then \(k\) lies between \(0\) and \(\lambda(H_\alpha)\), in which case
+  \(V_\mu \neq 0\) and therefore \(\mu\) is a weight.
+
+  On the other hand, if \(\mu\) does not lie between \(\lambda\) and \(T_\alpha
+  \lambda\) then either \(k < 0\) or \(k > \lambda(H_\alpha)\). Suppose \(\mu\)
+  is a weight. In the first case \(\mu \succ \lambda\), a contradiction. On the
+  second case the fact that \(V_\mu \ne 0\) implies \(V_{\lambda  + (k -
+  \lambda(H_\alpha)) \alpha} \ne 0 = V_{T_\alpha \mu}\), which contradicts the
+  fact that \(V_{\lambda + \ell \alpha} = 0\) for all \(\ell \ge 0\).
 \end{proof}
 
+This is entirely analogous to the situation of \(\mathfrak{sl}_3(K)\), where we
+found that the weights of the irreducible representations formed a continuous
+string symmetric with respect to the lines \(K \alpha\) with \(B(\alpha_i -
+\alpha_j, \alpha) = 0\). As in the case of \(\mathfrak{sl}_3(K)\), the same
+class of arguments leads us to the conclusion\dots
+
 \begin{definition}
   We refer to the group \(\mathcal{W} = \langle T_\alpha : \alpha \in \Delta^+
   \rangle \subset \operatorname{O}(\mathfrak{h}^*)\) as \emph{the Weyl group of
   \(\mathfrak{g}\)}.
 \end{definition}
 
-This is entirely analogous to the situation of \(\mathfrak{sl}_3(K)\), where we
-found that the weights of the irreducible representations were symmetric with
-respect to the lines \(K \alpha\) with \(B(\alpha_i - \alpha_j, \alpha) = 0\).
-Indeed, the same argument leads us to the conclusion\dots
-
 \begin{theorem}\label{thm:irr-weight-class}
   The weights of an irreducible representation \(V\) of \(\mathfrak{g}\) with
   highest weight \(\lambda\) are precisely the elements of the weight lattice
@@ -489,7 +567,13 @@ Indeed, the same argument leads us to the conclusion\dots
 
 Now the only thing we are missing for a complete classification is an existence
 and uniqueness theorem analogous to theorem~\ref{thm:sl2-exist-unique} and
-theorem~\ref{thm:sl3-existence-uniqueness}. Lo and behold\dots
+theorem~\ref{thm:sl3-existence-uniqueness}. It is already clear from the
+previous discussion that if \(\lambda\) is the highest weight of \(V\) then
+\(\lambda(H_\alpha) \ge 0\) for all positive roots \(\alpha\). Another way of
+putting it is to say that having \(\lambda(H_\alpha) \ge 0\) for all \(\alpha
+\in \Delta^+\) is a necessary condition for the existance of irreducible
+representations with highest weight given by \(\lambda\). Surprisingly, this
+condition is also sufficient. In other words\dots
 
 \begin{definition}
   An element \(\lambda\) of \(P\) such that \(\lambda(H_\alpha) \ge 0\) for all
@@ -749,7 +833,6 @@ are really interested in is\dots
   of the maximal subrepresentation of \(M(\lambda)\), the projection \(v^+ +
   N(\lambda) \in V\) is nonzero.
 
-  % TODO: Why is V_mu = M(lambda)_mu + N(lambda)? Turn this into a proposition?
   We now claim that \(v^+ + N(\lambda) \in V_\lambda\). Indeed,
   \[
     H (v^+ + N(\lambda))

diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -221,7 +221,7 @@ To conclude our analysis all it's left is to show that for each \(n\) such
 \(V\) does indeed exist and is irreducible. Surprinsingly, we have already
 encountered such a \(V\).
 
-\begin{theorem}\label{thm:irr-rep-of-sl2-exists}
+\begin{theorem}\label{thm:sl2-exist-unique}
   For each \(n \ge 0\) there exists a unique irreducible representation of
   \(\mathfrak{sl}_2(K)\) whose left-most eigenvalue of \(h\) is \(n\).
 \end{theorem}
@@ -299,7 +299,7 @@ chapter, but for now we note that perhaps the most fundamental property of
 annihilated by \(e\)} -- that being the generator of the right-most eigenspace
 of \(h\). This was instrumental to our explicit description of the irreducible
 representations of \(\mathfrak{sl}_2(K)\) culminating in
-theorem~\ref{thm:irr-rep-of-sl2-exists}.
+theorem~\ref{thm:sl2-exist-unique}.
 
 Our fist task is to find some analogue of \(h\) in \(\mathfrak{sl}_3(K)\), but
 it's still unclear what exactly we are looking for. We could say we're looking
@@ -602,8 +602,8 @@ As a first consequence of this, we show\dots
   called \emph{the weight lattice of \(\mathfrak{sl}_3(K)\)}.
 \end{definition}
 
-\begin{corollary}
-  Every weight \(\lambda\) of \(V\) lies in the weight lattice \(\lambda\).
+\begin{corollary}\label{thm:sl3-weights-fit-in-weight-lattice}
+  Every weight \(\lambda\) of \(V\) lies in the weight lattice \(P\).
 \end{corollary}
 
 \begin{proof}
@@ -643,8 +643,8 @@ There's a clear parallel between the case of \(\mathfrak{sl}_3(K)\) and that of
 lattice \(P = \ZZ\) and were congruent modulo the sublattice \(Q = 2 \ZZ\).
 Among other things, this last result goes to show that the diagrams we've been
 drawing are in fact consistant with the theory we've developed. Namely, since
-all weights lie in the rational span of \(\{\alpha_1, \alpha_2, \alpha_3\}\) in
-\(\mathfrak{h}^*\) we may as well draw them in the Cartesia plane.
+all weights lie in the rational span of \(\{\alpha_1, \alpha_2, \alpha_3\}\),
+we may as well draw them in the Cartesia plane.
 
 To proceed we once more refer to the previously established framework: next we
 saw that the eigenvalues of \(h\) formed an unbroken string of integers
@@ -939,7 +939,7 @@ Finally\dots
 
 Having found all of the weights of \(V\), the only thing we're missing is an
 existence and uniqueness theorem analogous to
-theorem~\ref{thm:irr-rep-of-sl2-exists}. In other words, our next goal is
+theorem~\ref{thm:sl2-exist-unique}. In other words, our next goal is
 establishing\dots
 
 \begin{theorem}\label{thm:sl3-existence-uniqueness}