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
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sections/sl2-sl3.tex | 55845B | -rw-r--r-- |
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\chapter{Representations of \(\mathfrak{sl}_2(K)\) \& \(\mathfrak{sl}_3(K)\)}\label{ch:sl3} We are, once again, faced with the daunting task of classifying the finite-dimensional modules of a given (semisimple) algebra \(\mathfrak{g}\). Having reduced the problem a great deal, all its left is classifying the simple \(\mathfrak{g}\)-modules. We have encountered numerous examples of simple \(\mathfrak{g}\)-modules over the previous chapter, but we have yet to subject them to any serious scrutiny. In this chapter we begin a systematic investigation of simple modules by looking at concrete examples. Specifically, we will classify the simple finite-dimensional modules of certain low-dimensional semisimple Lie algebras: \(\mathfrak{sl}_2(K)\) and \(\mathfrak{sl}_3(K)\). The reason why we chose \(\mathfrak{sl}_2(K)\) is a simple one: throughout the previous chapters \(\mathfrak{sl}_2(K)\) has afforded us surprisingly illuminating examples. We begin our analysis by recalling that the elements \begin{align*} e & = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} & f & = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} & h & = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \end{align*} form a basis for \(\mathfrak{sl}_2(K)\) and satisfy \begin{align*} [e, f] & = h & [h, f] & = -2 f & [h, e] = 2 e \end{align*} Let \(M\) be a finite-dimensional simple \(\mathfrak{sl}_2(K)\)-module. We now turn our attention to the action of \(h\) on \(M\), in particular, we investigate the subspace \(\bigoplus_{\lambda} M_\lambda \subset M\) -- where \(\lambda\) ranges over the eigenvalues of \(h\!\restriction_M\) and \(M_\lambda\) is the corresponding eigenspace. At this point, this is nothing short of a gamble: why look at the eigenvalues of \(h\)? The short answer is that, as we shall see, this will pay off. We will postpone the discussion about the real reason of why we chose \(h\), but for now we may notice that, perhaps surprisingly, the action \(h\!\restriction_M\) of \(h\) on a finite-dimensional simple \(\mathfrak{sl}_2(K)\)-module \(M\) is always a diagonalizable operator. Let \(\lambda\) be any eigenvalue of \(h\!\restriction_M\). Notice \(M_\lambda\) is in general not a \(\mathfrak{sl}_2(K)\)-submodule of \(M\). Indeed, if \(m \in M_\lambda\) then the identities \begin{align*} h \cdot (e \cdot m) &= 2e \cdot m + e h \cdot m = (\lambda + 2) e \cdot m \\ h \cdot (f \cdot m) &= -2f \cdot m + f h \cdot m = (\lambda - 2) f \cdot m \end{align*} follow. In other words, \(e\) sends an element of \(M_\lambda\) to an element of \(M_{\lambda + 2}\), while \(f\) sends it to an element of \(M_{\lambda - 2}\). Visually, we may draw \begin{center} \begin{tikzcd} \cdots \rar[bend left=60] & M_{\lambda - 2} \rar[bend left=60]{e} \lar[bend left=60] & M_{\lambda} \rar[bend left=60]{e} \lar[bend left=60]{f} & M_{\lambda + 2} \rar[bend left=60] \lar[bend left=60]{f} & \cdots \lar[bend left=60] \end{tikzcd} \end{center} This implies \(\bigoplus_\lambda M_\lambda\) is a \(\mathfrak{sl}_2(K)\)-submodule, so that \(\bigoplus_\lambda M_\lambda\) is either \(0\) or the entirety of \(M\) -- recall that \(M\) is simple. Since \(M\) is finite dimensional, \(h\!\restriction_M\) has at least one eigenvalue and therefore \[ M = \bigoplus_\lambda M_\lambda \] Even more so, we have seen that for any eigenvalue \(\lambda \in K\) of \(h\!\restriction_M\), \(\bigoplus_{k \in \mathbb{Z}} M_{\lambda - 2 k}\) is a \(\mathfrak{sl}_2(K)\)-invariant subspace, which goes to show \[ M = \bigoplus_{k \in \mathbb{Z}} M_{\lambda - 2 k}, \] and the eigenvalues of \(h\) all have the form \(\lambda - 2 k\) for some \(k\). By the same token, if \(a\) is the greatest \(k \in \mathbb{Z}\) such that \(V_{\lambda - 2 k} \ne 0\) and, likewise, \(b\) is the smallest \(k \in \mathbb{Z}\) such that \(V_{\lambda - 2 k} \ne 0\) then \[ M = \bigoplus_{\substack{k \in \mathbb{Z} \\ a \le k \le b}} M_{\lambda - 2 k} \] The eigenvalues of \(h\) thus form an unbroken string \[ \ldots, \lambda - 4, \lambda - 2, \lambda, \lambda + 2, \lambda + 4, \ldots \] around \(\lambda\). Our main objective is to show \(M\) is determined by this string of eigenvalues. To do so, we suppose without any loss in generality that \(\lambda\) is the right-most eigenvalue of \(h\), fix some nonzero \(m \in M_\lambda\) and consider the set \(\{m, f \cdot m, f^2 \cdot m, \ldots\}\). \begin{proposition}\label{thm:basis-of-irr-rep} The set \(\{m, f \cdot m, f^2 \cdot m, \ldots\}\) is a basis for \(M\). In addition, the action of \(\mathfrak{sl}_2(K)\) on \(M\) is given by the formulas \begin{equation}\label{eq:irr-rep-of-sl2} \begin{aligned} f^k \cdot m & \overset{e}{\mapsto} k(\lambda + 1 - k) f^{k - 1} \cdot m & f^k \cdot m & \overset{f}{\mapsto} f^{k + 1} \cdot m & f^k \cdot m & \overset{h}{\mapsto} (\lambda - 2 k) f^k \cdot m \end{aligned} \end{equation} \end{proposition} \begin{proof} First of all, notice \(f^k \cdot m\) lies in \(M_{\lambda - 2 k}\), so that \(\{m, f \cdot m, f^2 \cdot m, \ldots\}\) is a set of linearly independent vectors. Hence it suffices to show \(M = K \langle m, f \cdot m, f^2 \cdot m, \ldots \rangle\), which in light of the fact that \(M\) is simple is the same as showing \(K \langle m, f \cdot m, f^2 \cdot m, \ldots \rangle\) is invariant under the action of \(\mathfrak{sl}_2(K)\). The fact that \(h \cdot (f^k \cdot m) \in K \langle m, f \cdot m, f^2 \cdot m, \ldots \rangle\) follows immediately from our previous assertion that \(f^k \cdot m \in M_{\lambda - 2 k}\) -- indeed, \(h \cdot (f^k \cdot m) = (\lambda - 2 k) f^k \cdot m \in K \langle m, f \cdot m, f^2 \cdot m, \ldots \rangle\), which also goes to show one of the formulas in (\ref{eq:irr-rep-of-sl2}). Seeing \(e \cdot (f^k \cdot m) \in K \langle m, f \cdot m, f^2 \cdot m, \ldots \rangle\) is a bit more complex. Clearly, \[ \begin{split} e \cdot (f \cdot m) & = h \cdot m + f \cdot (e \cdot m) \\ \text{(since \(\lambda\) is the right-most eigenvalue)} & = h \cdot m + f \cdot 0 \\ & = \lambda m \end{split} \] Next we compute \[ \begin{split} e \cdot (f^2 \cdot m) & = (h + fe) \cdot (f \cdot m) \\ & = h \cdot (f \cdot m) + f \cdot (\lambda m) \\ & = 2 (\lambda - 1) f \cdot m \end{split} \] The pattern is starting to become clear: \(e\) sends \(f^k \cdot m\) to a multiple of \(f^{k - 1} \cdot m\). Explicitly, it is not hard to check by induction that \[ e \cdot (f^k \cdot m) = k (\lambda + 1 - k) \cdot f^{k - 1} m, \] which which is the first formula of (\ref{eq:irr-rep-of-sl2}). \end{proof} The significance of Proposition~\ref{thm:basis-of-irr-rep} should be self-evident: we have just provided a complete description of the action of \(\mathfrak{sl}_2(K)\) on \(M\). In particular, this goes to show\dots \begin{corollary} Every eigenspace of the action of \(h\) on \(M\) is \(1\)-dimensional. \end{corollary} \begin{proof} It suffices to note \(\{m, f \cdot m, f^2 \cdot m, \ldots \}\) is a basis for \(M\) consisting of eigenvalues of \(h\) and whose only element in \(M_{\lambda - 2 k}\) is \(f^k \cdot m\). \end{proof} \begin{corollary}\label{thm:sl2-find-weights} The eigenvalues of \(h\) in \(M\) form a symmetric, unbroken string of integers separated by intervals of length \(2\) whose right-most value is \(\dim M - 1\). \end{corollary} \begin{proof} If \(f^r\) is the lowest power of \(f\) that annihilates \(m\), it follows from the formulas in (\ref{eq:irr-rep-of-sl2}) that \[ 0 = e \cdot 0 = e \cdot (f^r \cdot m) = r (\lambda + 1 - r) f^{r - 1} \cdot m \] This implies \(\lambda + 1 - r = 0\) -- i.e. \(\lambda = r - 1 \in \mathbb{Z}\). Now since \(\{m, f \cdot m, f^2 \cdot m, \ldots, f^{r - 1} \cdot m\}\) is a basis for \(M\), \(r = \dim V\). Hence if \(\lambda = \dim V - 1\) then the eigenvalues of \(h\) are \[ \ldots, \lambda - 6, \lambda - 4, \lambda - 2, \lambda \] To see that this string is symmetric around \(0\), simply note that the left-most eigenvalue of \(h\) is precisely \(\lambda - 2 (r - 1) = -\lambda\). \end{proof} Visually, the situation it thus \begin{center} \begin{tikzcd} M_{-\lambda} \rar[bend left=60]{e} & M_{- \lambda + 2} \rar[bend left=60]{e} \lar[bend left=60]{f} & M_{- \lambda + 4} \rar[bend left=60] \lar[bend left=60]{f} & \cdots \rar[bend left=60] \lar[bend left=60] & M_{\lambda - 4} \rar[bend left=60]{e} \lar[bend left=60] & M_{\lambda - 2} \rar[bend left=60]{e} \lar[bend left=60]{f} & M_\lambda \lar[bend left=60]{f} \end{tikzcd} \end{center} Corollary~\ref{thm:sl2-find-weights} can be used to find the eigenvalues of the action of \(h\) on an arbitrary finite-dimensional \(\mathfrak{sl}_2(K)\)-module. Namely, if \(M\) and \(N\) are \(\mathfrak{sl}_2(K)\)-modules, \(m \in M_\mu\) and \(n \in N_\mu\) then by computing \[ h \cdot (m + n) = h \cdot m + h \cdot n = \mu (m + n) \] we can see that \((M \oplus N)_\mu = M_\mu + N_\mu\). Hence the set of eigenvalues of \(h\) in a \(\mathfrak{sl}_2(K)\)-module \(M\) is the union of the sets of eigenvalues in its simple components, and the corresponding eigenspaces are the direct sums of the eigenspaces of such simple components. In particular, if the eigenvalues of \(M\) all have the same parity -- i.e. they are either all even integers or all odd integers -- and the dimension of each eigenspace is no greater than \(1\) then \(M\) must be simple, for if \(N, L \subset M\) are submodules with \(M = N \oplus L\) then either \(N_\lambda = 0\) for all \(\lambda\) or \(L_\lambda = 0\) for all \(\lambda \in \mathfrak{h}^*\). To conclude our analysis all it is left is to show that for each \(\lambda \in \mathbb{Z}\) with \(\lambda \ge 0\) there is some finite-dimensional simple \(M\) whose highest weight is \(\lambda\). Surprisingly, we have already encountered such a \(M\). \begin{theorem}\label{thm:sl2-exist-unique} For each \(\lambda \ge 0\), \(\lambda \in \mathbb{Z}\), there exists a unique simple \(\mathfrak{sl}_2(K)\)-module whose left-most eigenvalue of \(h\) is \(\lambda\). \end{theorem} \begin{proof} Let \(M = K[x, y]^{(\lambda)}\) be the \(\mathfrak{sl}_2(K)\)-module of homogeneous polynomials of degree \(\lambda\) in two variables, as in Example~\ref{ex:sl2-polynomial-subrep}. A simple calculation shows \(M_{n - 2 k} = K x^{\lambda - k} y^k\) for \(k = 0, \ldots, \lambda\) and \(M_\mu = 0\) otherwise. In particular, the right-most eigenvalue of \(M\) is \(\lambda\). Alternatively, one can readily check that if \(K^2\) is the natural \(\mathfrak{sl}_2(K)\)-module, then \(M = \operatorname{Sym}^\lambda K^2\) satisfies the relations of (\ref{eq:irr-rep-of-sl2}). Indeed, the map \begin{align*} K[x, y]^{(\lambda)} & \to \operatorname{Sym}^\lambda K^2 \\ x^k y^\ell & \mapsto e_1^k \cdot e_2^\ell \end{align*} is an isomorphism. Either way, by the previous observation that a finite-dimensional \(\mathfrak{sl}_2(K)\)-module whose eigenvalues all have the same parity and whose corresponding eigenspace are all \(1\)-dimensional must be simple, \(M\) is simple. As for the uniqueness of \(M\), it suffices to notice that if \(N\) is a finite-dimensional simple \(\mathfrak{sl}_2(K)\)-module with right-most eigenvalue \(\lambda\) and \(n \in N_\lambda\) is nonzero then relations (\ref{eq:irr-rep-of-sl2}) imply the map \begin{align*} M & \to N \\ f^k \cdot m & \mapsto f^k \cdot n \end{align*} is an isomorphism -- this is, in effect, precisely how the isomorphism \(K[x, y]^{(\lambda)} \isoto \operatorname{Sym}^\lambda K^2\) was constructed. \end{proof} Our initial gamble of studying the eigenvalues of \(h\) may have seemed arbitrary at first, but it payed off: we have \emph{completely} described \emph{all} simple \(\mathfrak{sl}_2(K)\)-modules. It is not yet clear, however, if any of this can be adapted to a general setting. In the following section we shall double down on our gamble by trying to reproduce some of these results for \(\mathfrak{sl}_3(K)\), hoping this will somehow lead us to a general solution. In the process of doing so we will find some important clues on why \(h\) was a sure bet and the race was fixed all along. \section{Representations of \(\mathfrak{sl}_{2 + 1}(K)\)}\label{sec:sl3-reps} The study of representations of \(\mathfrak{sl}_2(K)\) reminds me of the difference between the derivative of a function \(\mathbb{R} \to \mathbb{R}\) and that of a smooth map between manifolds: it is a simpler case of something greater, but in some sense it is 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 distinctly remember my Calculus I teacher telling the class ``the derivative of the composition of two functions is not the composition of their derivatives'' -- which is, of course, the \emph{correct} formulation of the chain rule in the context of smooth manifolds. The same applies to \(\mathfrak{sl}_2(K)\). It is a simple and beautiful example, but unfortunately the general picture, modules of arbitrary semisimple algebras, lacks its simplicity. The general purpose of this section is to investigate to which extent the framework we developed for \(\mathfrak{sl}_2(K)\) can be generalized to other semisimple Lie algebras. Of course, the algebra \(\mathfrak{sl}_3(K)\) stands as a natural candidate for potential generalizations: \(\mathfrak{sl}_3(K) = \mathfrak{sl}_{2 + 1}(K)\) after all. Our approach is very straightforward: we will fix some simple \(\mathfrak{sl}_3(K)\)-module \(M\) and proceed step by step, at each point asking ourselves how we could possibly adapt the framework we laid out for \(\mathfrak{sl}_2(K)\). The first obvious question is one we have already asked ourselves: why \(h\)? More specifically, why did we choose to study its eigenvalues and is there an analogue of \(h\) in \(\mathfrak{sl}_3(K)\)? The answer to the former question is one we will discuss at length in the next chapter, but for now we note that perhaps the most fundamental property of \(h\) is that \emph{there exists an eigenvector \(m\) of \(h\) that is annihilated by \(e\)} -- that being the generator of the right-most eigenspace of \(h\). This was instrumental to our explicit description of the simple \(\mathfrak{sl}_2(K)\)-modules culminating in Theorem~\ref{thm:sl2-exist-unique}. Our first task is to find some analogue of \(h\) in \(\mathfrak{sl}_3(K)\), but it is still unclear what exactly we are looking for. We could say we are looking for an element of \(M\) that is annihilated by some analogue of \(e\), but the meaning of \emph{some analogue of \(e\)} is again unclear. In fact, as we shall see, no such analogue exists and neither does such element. Instead, the actual way to proceed is to consider the subalgebra \[ \mathfrak{h} = \left\{ X \in \begin{pmatrix} K & 0 & 0 \\ 0 & K & 0 \\ 0 & 0 & K \end{pmatrix} : \operatorname{Tr}(X) = 0 \right\} \] The choice of \(\mathfrak{h}\) may seem like an odd choice at the moment, but the point is we will later show that there exists some \(m \in M\) that is simultaneously an eigenvector of each \(H \in \mathfrak{h}\) and annihilated by half of the remaining elements of \(\mathfrak{sl}_3(K)\). This is exactly analogous to the situation we found in \(\mathfrak{sl}_2(K)\): \(h\) corresponds to the subalgebra \(\mathfrak{h}\), and the eigenvalues of \(h\) in turn correspond to linear functions \(\lambda : \mathfrak{h} \to k\) such that \(H \cdot m = \lambda(H) m\) for each \(H \in \mathfrak{h}\) and some nonzero \(m \in M\). We call such functionals \(\lambda\) \emph{eigenvalues of \(\mathfrak{h}\)}, and we say \emph{\(m\) is an eigenvector of \(\mathfrak{h}\)}. Once again, we will pay special attention to the eigenvalue decomposition \begin{equation}\label{eq:weight-module} M = \bigoplus_\lambda M_\lambda \end{equation} where \(\lambda\) ranges over all eigenvalues of \(\mathfrak{h}\) and \(M_\lambda = \{ m \in M : H \cdot m = \lambda(H) m, \forall H \in \mathfrak{h} \}\). We should note that the fact that (\ref{eq:weight-module}) holds is not at all obvious. This is because in general \(M_\lambda\) is not the eigenspace associated with an eigenvalue of any particular operator \(H \in \mathfrak{h}\), but instead the eigenspace of the action of the entire algebra \(\mathfrak{h}\). Fortunately for us, (\ref{eq:weight-module}) always holds, but we will postpone its proof to the next chapter. Next we turn our attention to the remaining elements of \(\mathfrak{sl}_3(K)\). In our analysis of \(\mathfrak{sl}_2(K)\) we saw that the eigenvalues of \(h\) differed from one another by multiples of \(2\). A possible way to interpret this is to say \emph{the eigenvalues of \(h\) differ from one another by integral linear combinations of the eigenvalues of the adjoint action of \(h\)}. In English, since \begin{align*} \operatorname{ad}(h) e & = 2 e & \operatorname{ad}(h) f & = -2 f & \operatorname{ad}(h) h & = 0, \end{align*} the eigenvalues of the adjoint actions of \(h\) are \(0\) and \(\pm 2\), and the eigenvalues of the action of \(h\) on a simple \(\mathfrak{sl}_2(K)\)-module differ from one another by integral multiples of \(2\). In the case of \(\mathfrak{sl}_3(K)\), a simple calculation shows that if \([H, X]\) is scalar multiple of \(X\) for all \(H \in \mathfrak{h}\) then all but one entry of \(X\) are zero. Hence the eigenvectors of the adjoint action of \(\mathfrak{h}\) are \(E_{i j}\) and its eigenvalues are \(\epsilon_i - \epsilon_j\), where \[ \epsilon_i \begin{pmatrix} a_1 & 0 & 0 \\ 0 & a_2 & 0 \\ 0 & 0 & a_3 \end{pmatrix} = a_i \] Visually we may draw \begin{figure}[h] \centering \begin{tikzpicture}[scale=2] \begin{rootSystem}{A} \filldraw[black] \weight{0}{0} circle (.5pt); \node[black, above right] at \weight{0}{0} {\small$0$}; \wt[black]{-1}{2} \wt[black]{-2}{1} \wt[black]{1}{1} \wt[black]{-1}{-1} \wt[black]{2}{-1} \wt[black]{1}{-2} \node[above] at \weight{-1}{2} {$\epsilon_2 - \epsilon_3$}; \node[left] at \weight{-2}{1} {$\epsilon_2 - \epsilon_1$}; \node[right] at \weight{1}{1} {$\epsilon_1 - \epsilon_3$}; \node[left] at \weight{-1}{-1} {$\epsilon_3 - \epsilon_1$}; \node[right] at \weight{2}{-1} {$\epsilon_1 - \epsilon_2$}; \node[below] at \weight{1}{-2} {$\epsilon_3 - \epsilon_1$}; \node[black, above] at \weight{1}{0} {$\epsilon_1$}; \node[black, above] at \weight{-1}{1} {$\epsilon_2$}; \node[black, above] at \weight{0}{-1} {$\epsilon_3$}; \filldraw[black] \weight{1}{0} circle (.5pt); \filldraw[black] \weight{-1}{1} circle (.5pt); \filldraw[black] \weight{0}{-1} circle (.5pt); \end{rootSystem} \end{tikzpicture} \end{figure} If we denote the eigenspace of the adjoint action of \(\mathfrak{h}\) on \(\mathfrak{sl}_3(K)\) associated to \(\alpha\) by \(\mathfrak{sl}_3(K)_\alpha\) and fix some \(X \in \mathfrak{sl}_3(K)_\alpha\), \(H \in \mathfrak{h}\) and \(m \in M_\lambda\) then \[ \begin{split} H \cdot (X \cdot m) & = X \cdot (H \cdot m) + [H, X] \cdot m \\ & = X \cdot (\lambda(H) m) + \alpha(H) X \cdot m \\ & = (\lambda + \alpha)(H) X \cdot m \end{split} \] so that \(X\) carries \(m\) to \(M_{\lambda + \alpha}\). In other words, \(\mathfrak{sl}_3(k)_\alpha\) \emph{acts on \(M\) by translating vectors between eigenspaces}. For instance \(\mathfrak{sl}_3(K)_{\epsilon_1 - \epsilon_3}\) will act on the adjoint \(\mathfrak{sl}_3(K)\)-modules via \begin{figure}[h] \centering \begin{tikzpicture}[scale=2] \begin{rootSystem}{A} \wt[black]{0}{0} \wt[black]{-1}{2} \wt[black]{-2}{1} \wt[black]{1}{1} \wt[black]{-1}{-1} \wt[black]{2}{-1} \wt[black]{1}{-2} \draw[-latex, black] \weight{-1.9}{1.1} -- \weight{-1.1}{1.9}; \draw[-latex, black] \weight{-.9}{-.9} -- \weight{-.1}{-.1}; \draw[-latex, black] \weight{0.1}{0.1} -- \weight{.9}{.9}; \draw[-latex, black] \weight{1.1}{-1.9} -- \weight{1.9}{-1.1}; \end{rootSystem} \end{tikzpicture} \end{figure} This is again entirely analogous to the situation we observed in \(\mathfrak{sl}_2(K)\). In fact, we may once more conclude\dots \begin{theorem}\label{thm:sl3-weights-congruent-mod-root} The eigenvalues of the action of \(\mathfrak{h}\) on a simple \(\mathfrak{sl}_3(K)\)-module \(M\) differ from one another by integral linear combinations of the eigenvalues \(\epsilon_i - \epsilon_j\) of the adjoint action of \(\mathfrak{h}\) on \(\mathfrak{sl}_3(K)\). \end{theorem} \begin{proof} This proof goes exactly as that of the analogous statement for \(\mathfrak{sl}_2(K)\): it suffices to note that if we fix some eigenvalue \(\lambda\) of \(\mathfrak{h}\) and let \(i\) and \(j\) vary then \[ \bigoplus_{i j} M_{\lambda + \epsilon_i - \epsilon_j} \] is an invariant subspace of \(M\). \end{proof} To avoid confusion we better introduce some notation to differentiate between eigenvalues of the action of \(\mathfrak{h}\) on \(M\) and eigenvalues of the adjoint action of \(\mathfrak{h}\). \begin{definition}\index{weights} Given a \(\mathfrak{sl}_3(K)\)-module \(M\), we will call the \emph{nonzero} eigenvalues of the action of \(\mathfrak{h}\) on \(M\) \emph{weights of \(M\)}. As you might have guessed, we will correspondingly refer to eigenvectors and eigenspaces of a given weight by \emph{weight vectors} and \emph{weight spaces}. \end{definition} It is clear from our previous discussion that the weights of the adjoint \(\mathfrak{sl}_3(K)\)-module deserve some special attention. \begin{definition}\index{weights!roots} The weights of the adjoint \(\mathfrak{sl}_3(K)\)-module are called \emph{roots of \(\mathfrak{sl}_3(K)\)}. Once again, the expressions \emph{root vector} and \emph{root space} are self-explanatory. \end{definition} Theorem~\ref{thm:sl3-weights-congruent-mod-root} can thus be restated as\dots \begin{definition}\index{weights!root lattice} The lattice \(Q = \mathbb{Z} \langle \epsilon_i - \epsilon_j : i, j = 1, 2, 3 \rangle\) is called \emph{the root lattice of \(\mathfrak{sl}_3(K)\)}. \end{definition} \begin{corollary} The weights of a simple \(\mathfrak{sl}_3(K)\)-module \(M\) are all congruent modulo the root lattice \(Q\). In other words, the weights of \(M\) all lie in a single \(Q\)-coset \(\xi \in \mfrac{\mathfrak{h}^*}{Q}\). \end{corollary} At this point we could keep playing the tedious game of reproducing the arguments from the previous section in the context of \(\mathfrak{sl}_3(K)\). However, it is more profitable to use our knowledge of \(\mathfrak{sl}_2(K)\)-modules instead. Notice that the canonical inclusion \(\mathfrak{gl}_2(K) \to \mathfrak{gl}_3(K)\) -- as described in Example~\ref{ex:gln-inclusions} -- restricts to an injective homomorphism \(\mathfrak{sl}_2(K) \to \mathfrak{sl}_3(K)\). In other words, \(\mathfrak{sl}_2(K)\) is isomorphic to the image \(\mathfrak{s}_{1 2} = K \langle E_{1 2}, E_{2 1}, [E_{1 2}, E_{2 1}] \rangle \subset \mathfrak{sl}_3(K)\) of the inclusion \(\mathfrak{sl}_2(K) \to \mathfrak{sl}_3(K)\). We may thus regard \(M\) as a \(\mathfrak{sl}_2(K)\)-module by restricting to \(\mathfrak{s}_{1 2}\). Our first observation is that, since the root spaces act by translation, the subspace \[ \bigoplus_{k \in \mathbb{Z}} M_{\lambda - k (\epsilon_1 - \epsilon_2)}, \] must be invariant under the action of \(E_{1 2}\) and \(E_{2 1}\) for all \(\lambda \in \mathfrak{h}^*\). This goes to show \(\bigoplus_k M_{\lambda - k (\epsilon_1 - \epsilon_2)}\) is a \(\mathfrak{sl}_2(K)\)-submodule of \(M\) for all weights \(\lambda\) of \(M\). Furthermore, one can easily see that the eigenspace of the action of \(h\) on \(\bigoplus_{k \in \mathbb{Z}} M_{\lambda - k (\epsilon_1 - \epsilon_2)}\) associated with the eigenvalue \(\lambda(H) - 2k\) is precisely the weight space \(M_{\lambda - k (\epsilon_2 - \epsilon_1)}\). Visually, \begin{center} \begin{tikzpicture} \begin{rootSystem}{A} \node at \weight{-4}{2} (l) {}; \node at \weight{-2}{1} (a) {}; \node at \weight{0}{0} (b) {}; \node at \weight{2}{-1} (c) {}; \node at \weight{4}{-2} (r) {}; \draw \weight{-3}{1.5} -- \weight{3}{-1.5}; \draw[dotted] \weight{-3}{1.5} -- (l); \draw[dotted] \weight{3}{-1.5} -- (r); \foreach \i in {-1, 0, 1}{\wt[black]{-2*\i}{\i}} \draw[-latex] (l) to[bend left=40] (a); \draw[-latex] (a) to[bend left=40] (b); \draw[-latex] (b) to[bend left=40] (c); \draw[-latex] (c) to[bend left=40] (r); \draw[-latex] (r) to[bend left=40] (c); \draw[-latex] (c) to[bend left=40] (b); \draw[-latex] (b) to[bend left=40] (a); \draw[-latex] (a) to[bend left=40] (l); \node[above right] at (b) {\small\(\lambda\)}; \node[above right=2pt] at \weight{-3}{1.5} {\small\(E_{1 2}\)}; \node[below left=2pt] at \weight{-3}{1.5} {\small\(E_{2 1}\)}; \end{rootSystem} \end{tikzpicture} \end{center} In general, we find\dots \begin{proposition} Given \(i < j\), the subalgebra \(\mathfrak{s}_{i j} = K \langle E_{i j}, E_{j i}, [E_{i j}, E_{j i}] \rangle\) is isomorphic to \(\mathfrak{sl}_2(K)\). In addition, given a weight \(\lambda \in \mathfrak{h}^*\) of \(M\), the space \[ N = \bigoplus_{k \in \mathbb{Z}} M_{\lambda - k (\epsilon_i - \epsilon_j)} \] is invariant under the action of \(\mathfrak{s}_{i j}\) and \[ M_{\lambda - k (\epsilon_i - \epsilon_j)} = N_{\lambda([E_{i j}, E_{j i}]) - 2k} \] \end{proposition} \begin{proof} In effect, if \(i \ne k \ne j\) then \(\mathfrak{s}_{i j}\) is the subalgebra of matrices whose \(k\)-th row and \(k\)-th column are nil. For instance, if \(i = 1\) and \(j = 3\) then \[ \mathfrak{s}_{1 3} = \begin{pmatrix} K & 0 & K \\ 0 & 0 & 0 \\ K & 0 & K \end{pmatrix} \cap \mathfrak{sl}_3(K) \] In this case, the map \begin{align*} \mathfrak{s}_{1 3} & \to \mathfrak{sl}_2(K) \\ \begin{pmatrix} a & 0 & b \\ 0 & 0 & 0 \\ c & 0 & -a \end{pmatrix} & \mapsto \begin{pmatrix} a & \tm{topA}{0} & b \\ \tm{leftA}{0} & 0 & \tm{rightA}{0} \\ c & \tm{bottomA}{0} & -a \end{pmatrix} = \begin{pmatrix} a & b \\ c & -a \end{pmatrix} \DrawVLine[black]{topA}{bottomA} \DrawHLine[black]{leftA}{rightA} \end{align*} is an isomorphism of Lie algebras. In general, the map \begin{align*} \mathfrak{s}_{i j} & \to \mathfrak{sl}_2(K) \\ E_{i j} & \mapsto e \\ E_{j i} & \mapsto f \\ [E_{i j}, E_{j i}] & \mapsto h \end{align*} which ``erases the \(k\)-th row and the \(k\)-th column'' of a matrix is an isomorphism. To see that \(N\) is invariant under the action of \(\mathfrak{s}_{i j}\), it suffices to notice \(E_{i j}\) and \(E_{j i}\) map \(m \in M_{\lambda - k (\epsilon_i - \epsilon_j)}\) to \(E_{i j} \cdot m \in M_{\lambda - (k - 1) (\epsilon_i - \epsilon_j)}\) and \(E_{j i} \cdot m \in M_{\lambda - (k + 1) (\epsilon_i - \epsilon_j)}\), respectively. Moreover, \[ (\lambda - k (\epsilon_i - \epsilon_j))([E_{i j}, E_{j i}]) = \lambda([E_{i j}, E_{j i}]) - k (1 - (-1)) = \lambda([E_{i j}, E_{j i}]) - 2 k, \] which goes to show \(M_{\lambda - k (\epsilon_i - \epsilon_j)} \subset N_{\lambda([E_{i j}, E_{j i}]) - 2k}\). On the other hand, if we suppose \(0 < \dim M_{\lambda - k (\epsilon_i - \epsilon_j)} < \dim N_{\lambda([E_{i j}, E_{j i}]) - 2 k}\) for some \(k\) we arrive at \[ \dim N = \sum_k \dim M_{\lambda - k (\epsilon_i - \epsilon_j)} < \sum_k \dim N_{\lambda([E_{i j}, E_{j i}]) - 2k} = \dim N, \] a contradiction. \end{proof} As a first consequence of this, we show\dots \begin{definition}\index{weights!weight lattice} The lattice \(P = \mathbb{Z} \langle \epsilon_1, \epsilon_2, \epsilon_3 \rangle\) is called \emph{the weight lattice of \(\mathfrak{sl}_3(K)\)}. \end{definition} \begin{corollary}\label{thm:sl3-weights-fit-in-weight-lattice} Every weight \(\lambda\) of \(M\) lies in the weight lattice \(P\). \end{corollary} \begin{proof} It suffices to note \(\lambda([E_{i j}, E_{j i}])\) is an eigenvalue of \(h\) in a finite-dimensional \(\mathfrak{sl}_2(K)\)-module, so it must be an integer. Now since \[ \lambda \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & -a -b \end{pmatrix} = \lambda \begin{pmatrix} a & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -a \end{pmatrix} + \lambda \begin{pmatrix} 0 & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & -b \end{pmatrix} = a \lambda([E_{1 3}, E_{3 1}]) + b \lambda([E_{2 3}, E_{3 2}]), \] which is to say \(\lambda = \lambda([E_{1 3}, E_{3 1}]) \epsilon_1 + \lambda([E_{2 3}, E_{3 2}]) \epsilon_2 \in P\). \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 \(\{\epsilon_1, \epsilon_2, \epsilon_3\}\), we may as well draw them in the Cartesian plane. In fact, the attentive reader may notice that \(\kappa(E_{1 2}, E_{2 3}) = - \sfrac{1}{2}\), so that the angle -- with respect to the Killing form \(\kappa\) -- between the root vectors \(E_{1 2}\) and \(E_{2 3}\) is precisely the same as the angle between the points representing their roots \(\epsilon_1 - \epsilon_2\) and \(\epsilon_2 - \epsilon_3\) in the Cartesian plane. Since \(\epsilon_1 - \epsilon_2\) and \(\epsilon_2 - \epsilon_3\) 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 \((\epsilon_i - \epsilon_j, \epsilon_k - \alpha_\ell) \mapsto \kappa(E_{i j}, E_{k \ell})\) -- which we denote by \(\kappa\) 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 around \(0\). To prove this we analyzed the right-most eigenvalues of \(h\) and their eigenvectors, providing an explicit description of the simple \(\mathfrak{sl}_2(K)\)-modules in terms of these vectors. We may reproduce these steps in the context of \(\mathfrak{sl}_3(K)\) by fixing a direction in the plane an considering the weight lying the furthest in that direction. For instance, let's say we fix the direction \begin{center} \begin{tikzpicture}[scale=2] \begin{rootSystem}{A} \wt[black]{0}{0} \wt[black]{-1}{2} \wt[black]{-2}{1} \wt[black]{1}{1} \wt[black]{-1}{-1} \wt[black]{2}{-1} \wt[black]{1}{-2} \draw[-latex, black, thick] \weight{-1.5}{-.5} -- \weight{1.5}{.5}; \end{rootSystem} \end{tikzpicture} \end{center} 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 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 lies to the right of the direction we chose. Otherwise we say \(\alpha\) is negative. Notice that \(f(\alpha) \ne 0\) since by definition \(\alpha \ne 0\) and \(f\) is irrational with respect to the lattice \(Q\). \end{definition} The next observation we make is that all others weights of \(M\) must lie in a sort of \(\frac{1}{3}\)-cone with apex at \(\lambda\), as shown in \begin{center} \begin{tikzpicture} \AutoSizeWeightLatticefalse \begin{rootSystem}{A} \weightLattice{3} \fill[gray!50,opacity=.2] (hex cs:x=5,y=-7) -- (hex cs:x=1,y=1) -- (hex cs:x=-7,y=5) arc (150:270:{7*\weightLength}); \draw[black, thick] (hex cs:x=5,y=-7) -- (hex cs:x=1,y=1) -- (hex cs:x=-7,y=5); \filldraw[black] (hex cs:x=1,y=1) circle (1pt); \node[above right=-2pt] at (hex cs:x=1,y=1) {\small\(\lambda\)}; \end{rootSystem} \end{tikzpicture} \end{center} Indeed, if this is not the case then, by definition, \(\lambda\) is not the weight placed the furthest in the direction we chose. Given our previous assertion that the root spaces of \(\mathfrak{sl}_3(K)\) act on the weight spaces of \(M\) via translation, this implies that \(E_{1 2}\), \(E_{1 3}\) and \(E_{2 3}\) all annihilate \(M_\lambda\), or otherwise one of \(M_{\lambda + \epsilon_1 - \epsilon_2}\), \(M_{\lambda + \epsilon_1 - \epsilon_3}\) and \(M_{\lambda + \epsilon_2 - \epsilon_3}\) would be nonzero -- which contradicts the hypothesis that \(\lambda\) lies the furthest in the direction we chose. In other words\dots \begin{proposition}\label{thm:sl3-mod-is-highest-weight} There is a weight vector \(m \in M\) that is annihilated by all positive root spaces of \(\mathfrak{sl}_3(K)\). \end{proposition} \begin{proof} It suffices to note that the positive roots of \(\mathfrak{sl}_3(K)\) are precisely \(\epsilon_1 - \epsilon_2\), \(\epsilon_1 - \epsilon_3\) and \(\epsilon_2 - \epsilon_3\), with root vectors \(E_{1 2}\), \(E_{1 3}\) and \(E_{2 3}\), respectively. \end{proof} \index{weights!highest weight} We call \(\lambda\) \emph{the highest weight of \(M\)}, and we call any nonzero \(m \in M_\lambda\) \emph{a highest weight vector}. Going back to the case of \(\mathfrak{sl}_2(K)\), we then constructed an explicit basis for our simple module in terms of a highest weight vector, which allowed us to provide an explicit description of the action of \(\mathfrak{sl}_2(K)\) in terms of its standard basis, and finally we concluded that the eigenvalues of \(h\) must be symmetrical around \(0\). An analogous procedure could be implemented for \(\mathfrak{sl}_3(K)\) -- and indeed that's what we will do later down the line -- but instead we would like to focus on the problem of finding the weights of \(M\) in the first place. We will start out by trying to understand the weights in the boundary of previously drawn cone. As we have just seen, we can get to other weight spaces from \(M_\lambda\) by successively applying \(E_{2 1}\). \begin{center} \begin{tikzpicture} \begin{rootSystem}{A} \node at \weight{3}{1} (a) {}; \node at \weight{1}{2} (b) {}; \node at \weight{-1}{3} (c) {}; \node at \weight{-3}{4} (d) {}; \node at \weight{-5}{5} (e) {}; \draw \weight{3}{1} -- \weight{-4}{4.5}; \draw[dotted] \weight{-4}{4.5} -- \weight{-5}{5}; \foreach \i in {1,...,4}{\wt[black]{5-2*\i}{\i}} \node[above right=-2pt] at (hex cs:x=3,y=1){\small\(\lambda\)}; \draw[-latex] (a) to[bend left=40] (b); \draw[-latex] (b) to[bend left=40] (c); \draw[-latex] (c) to[bend left=40] (d); \draw[-latex] (d) to[bend left=40] (e); \end{rootSystem} \end{tikzpicture} \end{center} Notice that \(\lambda([E_{1 2}, E_{2 1}]) \in \mathbb{Z}\) is the right-most eigenvalue of the \(\mathfrak{sl}_2(K)\)-module \(\bigoplus_{k \in \mathbb{Z}} M_{\lambda - k (\epsilon_1 - \epsilon_2)}\). In particular, \(\lambda([E_{1 2}, E_{2 1}])\) must be positive. In addition, since the eigenspace of the eigenvalue \(\lambda([E_{1 2}, E_{2 1}]) - 2k\) of the action of \(h\) on \(\bigoplus_{k \in \mathbb{N}} M_{\lambda - k (\epsilon_1 - \epsilon_2)}\) is \(M_{\lambda - k (\epsilon_1 - \epsilon_2)}\), the weights of \(M\) appearing the string \(\lambda, \lambda + (\epsilon_1 - \epsilon_2), \ldots, \lambda + k (\epsilon_1 - \epsilon_2), \ldots\) must be symmetric with respect to the line \(\kappa(\epsilon_1 - \epsilon_2, \alpha) = 0\). The picture is thus \begin{center} \begin{tikzpicture} \AutoSizeWeightLatticefalse \begin{rootSystem}{A} \setlength{\weightRadius}{2pt} \weightLattice{4} \draw[thick] \weight{3}{1} -- \weight{-3}{4}; \wt[black]{0}{0} \node[above left] at \weight{0}{0} {\small\(0\)}; \foreach \i in {1,...,4}{\wt[black]{5-2*\i}{\i}} \node[above right=-2pt] at (hex cs:x=3,y=1){\small\(\lambda\)}; \draw[very thick] \weight{0}{-4} -- \weight{0}{4} node[above]{\small\(\kappa(\epsilon_1 - \epsilon_2, \alpha) = 0\)}; \end{rootSystem} \end{tikzpicture} \end{center} We could apply this same argument to the subspace \(\bigoplus_k M_{\lambda - k (\epsilon_2 - \epsilon_3)}\), so that the weights in this subspace must be symmetric with respect to the line \(\kappa(\epsilon_2 - \epsilon_3, \alpha) = 0\). The picture is now \begin{center} \begin{tikzpicture} \AutoSizeWeightLatticefalse \begin{rootSystem}{A} \setlength{\weightRadius}{2pt} \weightLattice{4} \draw[thick] \weight{3}{1} -- \weight{-3}{4}; \draw[thick] \weight{3}{1} -- \weight{4}{-1}; \wt[black]{0}{0} \wt[black]{4}{-1} \node[above left] at \weight{0}{0} {\small\(0\)}; \foreach \i in {1,...,4}{\wt[black]{5-2*\i}{\i}} \node[above right=-2pt] at (hex cs:x=3,y=1){\small\(\lambda\)}; \draw[very thick] \weight{0}{-4} -- \weight{0}{4} node[above]{\small\(\kappa(\epsilon_1 - \epsilon_2, \alpha) = 0\)}; \draw[very thick] \weight{-4}{0} -- \weight{4}{0} node[right]{\small\(\kappa(\epsilon_2 - \epsilon_3, \alpha) = 0\)}; \end{rootSystem} \end{tikzpicture} \end{center} Needless to say, we could keep applying this method to the weights at the ends of our string, arriving at \begin{center} \begin{tikzpicture} \AutoSizeWeightLatticefalse \begin{rootSystem}{A} \setlength{\weightRadius}{2pt} \weightLattice{5} \draw[thick] \weight{3}{1} -- \weight{-3}{4}; \draw[thick] \weight{3}{1} -- \weight{4}{-1}; \draw[thick] \weight{-3}{4} -- \weight{-4}{3}; \draw[thick] \weight{-4}{3} -- \weight{-1}{-3}; \draw[thick] \weight{1}{-4} -- \weight{4}{-1}; \draw[thick] \weight{-1}{-3} -- \weight{1}{-4}; \wt[black]{-4}{3} \wt[black]{-3}{1} \wt[black]{-2}{-1} \wt[black]{-1}{-3} \wt[black]{1}{-4} \wt[black]{2}{-3} \wt[black]{3}{-2} \wt[black]{4}{-1} \foreach \i in {1,...,4}{\wt[black]{5-2*\i}{\i}} \node[above right=-2pt] at (hex cs:x=3,y=1){\small\(\lambda\)}; \draw[very thick] \weight{-5}{5} -- \weight{5}{-5}; \draw[very thick] \weight{0}{-5} -- \weight{0}{5}; \draw[very thick] \weight{-5}{0} -- \weight{5}{0}; \end{rootSystem} \end{tikzpicture} \end{center} We claim all dots \(\mu\) lying inside the hexagon we have drawn must also be weights -- i.e. \(M_\mu \ne 0\). Indeed, by applying the same argument to an arbitrary weight \(\nu\) in the boundary of the hexagon we get a \(\mathfrak{sl}_2(K)\)-module whose weights correspond to weights of \(M\) lying in a string inside the hexagon, and whose right-most weight is precisely the weight of \(M\) we started with. \begin{center} \begin{tikzpicture} \AutoSizeWeightLatticefalse \begin{rootSystem}{A} \setlength{\weightRadius}{2pt} \weightLattice{5} \draw[thick] \weight{3}{1} -- \weight{-3}{4}; \draw[thick] \weight{3}{1} -- \weight{4}{-1}; \draw[thick] \weight{-3}{4} -- \weight{-4}{3}; \draw[thick] \weight{-4}{3} -- \weight{-1}{-3}; \draw[thick] \weight{1}{-4} -- \weight{4}{-1}; \draw[thick] \weight{-1}{-3} -- \weight{1}{-4}; \wt[black]{-4}{3} \wt[black]{-3}{1} \wt[black]{-2}{-1} \wt[black]{-1}{-3} \wt[black]{1}{-4} \wt[black]{2}{-3} \wt[black]{3}{-2} \wt[black]{4}{-1} \foreach \i in {1,...,4}{\wt[black]{5-2*\i}{\i}} \node[above right=-2pt] at \weight{1}{2} {\small\(\nu\)}; \node[above right=-2pt] at (hex cs:x=3,y=1){\small\(\lambda\)}; \draw[very thick] \weight{-5}{5} -- \weight{5}{-5}; \draw[very thick] \weight{0}{-5} -- \weight{0}{5}; \draw[very thick] \weight{-5}{0} -- \weight{5}{0}; \draw[dashed, thick] \weight{1}{2} -- \weight{-2}{-1}; \wt[black]{1}{2} \wt[black]{-2}{-1} \wt[black]{0}{1} \wt[black]{-1}{0} \end{rootSystem} \end{tikzpicture} \end{center} By construction, \(\nu\) corresponds to the right-most weight of a \(\mathfrak{sl}_2(K)\)-module, so that all dots lying on the dashed string must occur in \(\mathfrak{sl}_2(K)\)-module. Hence they must also be weights of \(M\). The final picture is thus \begin{center} \begin{tikzpicture} \AutoSizeWeightLatticefalse \begin{rootSystem}{A} \setlength{\weightRadius}{2pt} \weightLattice{5} \draw[thick] \weight{3}{1} -- \weight{-3}{4}; \draw[thick] \weight{3}{1} -- \weight{4}{-1}; \draw[thick] \weight{-3}{4} -- \weight{-4}{3}; \draw[thick] \weight{-4}{3} -- \weight{-1}{-3}; \draw[thick] \weight{1}{-4} -- \weight{4}{-1}; \draw[thick] \weight{-1}{-3} -- \weight{1}{-4}; \wt[black]{-4}{3} \wt[black]{-3}{1} \wt[black]{-2}{-1} \wt[black]{-1}{-3} \wt[black]{1}{-4} \wt[black]{2}{-3} \wt[black]{3}{-2} \wt[black]{4}{-1} \foreach \i in {1,...,4}{\wt[black]{5-2*\i}{\i}} \node[above right=-2pt] at (hex cs:x=3,y=1){\small\(\lambda\)}; \draw[very thick] \weight{-5}{5} -- \weight{5}{-5}; \draw[very thick] \weight{0}{-5} -- \weight{0}{5}; \draw[very thick] \weight{-5}{0} -- \weight{5}{0}; \wt[black]{-2}{2} \wt[black]{0}{1} \wt[black]{-1}{0} \wt[black]{0}{-2} \wt[black]{1}{-1} \wt[black]{2}{0} \end{rootSystem} \end{tikzpicture} \end{center} \index{weights!weight diagrams} This final picture is known as \emph{the weight diagram of \(M\)}. Finally\dots \begin{theorem}\label{thm:sl3-irr-weights-class} The weights of \(M\) are precisely the elements of the weight lattice \(P\) congruent to \(\lambda\) module the sublattice \(Q\) and lying inside hexagon with vertices the images of \(\lambda\) under the group generated by reflections across the lines \(\kappa(\epsilon_i - \epsilon_j, \alpha) = 0\). \end{theorem} Having found all of the weights of \(M\), the only thing we are missing is an existence and uniqueness theorem analogous to Theorem~\ref{thm:sl2-exist-unique}. It is clear from the symmetries of the locus of weights found in Theorem~\ref{thm:sl3-irr-weights-class} that if \(\lambda \in P\) is the highest weight of some finite-dimensional simple \(\mathfrak{sl}_3(K)\)-module \(M\) then \(\lambda\) lies in the cone \(\mathbb{N} \langle \epsilon_1, - \epsilon_3 \rangle\). What's perhaps more surprising is the fact that this condition is sufficient for the existence of such a \(M\). In other words, our next goal is establishing\dots \begin{definition}\index{weights!dominant weight} An element \(\lambda \in P\) is called \emph{dominant} if it lies in the cone \(\mathbb{N} \langle \epsilon_1, - \epsilon_3 \rangle\). \end{definition} \begin{theorem}\label{thm:sl3-existence-uniqueness} For each dominant \(\lambda \in P\), there exists precisely one finite-dimensional simple \(\mathfrak{sl}_3(K)\)-module \(M\) whose highest weight is \(\lambda\). \end{theorem} To proceed further we once again refer to the approach we employed in the case of \(\mathfrak{sl}_2(K)\): next we showed in Proposition~\ref{thm:basis-of-irr-rep} that any simple \(\mathfrak{sl}_2(K)\)-module is spanned by the images of its highest weight vector under \(f\). A more abstract way of putting it is to say that a simple \(\mathfrak{sl}_2(K)\)-module \(M\) of is spanned by the images of its highest weight vector under successive applications of the action of half of the root spaces of \(\mathfrak{sl}_2(K)\). The advantage of this alternative formulation is, of course, that the same holds for \(\mathfrak{sl}_3(K)\). Specifically\dots \begin{proposition}\label{thm:sl3-positive-roots-span-all-irr-rep} Given a simple \(\mathfrak{sl}_3(K)\)-module \(M\) and a highest weight vector \(m \in M\), \(M\) is spanned by the images of \(m\) under successive applications of \(E_{2 1}\), \(E_{3 1}\) and \(E_{3 2}\). \end{proposition} \begin{proof} Given the fact \(M\) is simple, it suffices to show that the subspace \(N\) spanned by successive applications of \(E_{2 1}\), \(E_{3 1}\) and \(E_{3 2}\) to \(m\) is stable under the action of \(\mathfrak{sl}_3(K)\). In addition, since \([E_{2 1}, E_{3 1}] = [E_{3 1}, E_{3 2}] = 0\) and \([E_{2 1}, E_{3 2}] = - E_{3 1}\), all successive product of \(E_{2 1}\), \(E_{3 1}\) and \(E_{3 2}\) in \(\mathcal{U}(\mathfrak{sl}_3(K))\) can be written as \(E_{2 1}^a E_{3 1}^b E_{3 1}^c\) for some \(a\), \(b\) and \(c\), so that \(N\) is spanned by the elements \(E_{2 1}^a E_{3 1}^b E_{3 1}^c \cdot m\). Recall that \(E_{i j}\) maps \(M_\mu\) to \(M_{\mu + \epsilon_i - \epsilon_j}\). In particular, \(E_{2 1}^a E_{3 1}^b E_{3 1}^c \cdot m \in M_{\lambda - a (\epsilon_1 - \epsilon_2) - b (\epsilon_1 - \epsilon_3) - c (\epsilon_2 - \epsilon_3)}\). In other words, \[ H E_{2 1}^a E_{3 1}^b E_{3 1}^c \cdot m = (\lambda - a (\epsilon_1 - \epsilon_2) - b (\epsilon_1 - \epsilon_3) - c (\epsilon_2 - \epsilon_3))(H) E_{2 1}^a E_{3 1}^b E_{3 1}^c \cdot m \in N \] for all \(H \in \mathfrak{h}\) and \(N\) is stable under the action of \(\mathfrak{h}\). On the other hand, \(N\) is clearly stable under the action of \(E_{2 1}\), \(E_{3 1}\) and \(E_{3 2}\). All it is left is to show \(N\) is stable under the action of \(E_{1 2}\), \(E_{1 3}\) and \(E_{2 3}\). We begin by analyzing the case of \(E_{1 2}\). We have \[ \begin{split} E_{1 2} E_{2 1}^a E_{3 1}^b E_{3 2}^c \cdot m & = ([E_{1 2}, E_{2 1}] + E_{2 1} E_{1 2}) E_{2 1}^{a - 1} E_{3 1}^b E_{3 2}^c \cdot m \\ & = E_{2 1} ([E_{1 2}, E_{2 1}] + E_{2 1} E_{1 2}) E_{2 1}^{a - 2} E_{3 1}^b E_{3 2}^c \cdot m \\ & \phantom{=} \; + (\lambda - (a - 1) (\epsilon_1 - \epsilon_2) - b (\epsilon_1 - \epsilon_3) - c (\epsilon_2 - \epsilon_3)) ([E_{1 2}, E_{2 1}]) E_{2 1}^{a - 1} E_{3 1}^b E_{3 2}^c \cdot m \\ & = E_{2 1}^2 ([E_{1 2}, E_{2 1}] + E_{2 1} E_{1 2}) E_{2 1}^{a - 3} E_{3 1}^b E_{3 2}^c \cdot m \\ & \phantom{=} \; + (\lambda - (a - 1) (\epsilon_1 - \epsilon_2) - b (\epsilon_1 - \epsilon_3) - c (\epsilon_2 - \epsilon_3)) ([E_{1 2}, E_{2 1}]) E_{2 1}^{a - 1} E_{3 1}^b E_{3 2}^c \cdot m \\ & \phantom{=} \; + (\lambda - (a - 2) (\epsilon_1 - \epsilon_2) - b (\epsilon_1 - \epsilon_3) - c (\epsilon_2 - \epsilon_3)) ([E_{1 2}, E_{2 1}]) E_{2 1}^{a - 2} E_{3 1}^b E_{3 2}^c \cdot m \\ & \; \; \vdots \\ & = E_{2 1}^a E_{1 2} E_{3 1}^b E_{3 2}^c \cdot m \\ & \phantom{=} \; + (\lambda - (a - 1) (\epsilon_1 - \epsilon_2) - b (\epsilon_1 - \epsilon_3) - c (\epsilon_2 - \epsilon_3)) ([E_{1 2}, E_{2 1}]) E_{2 1}^{a - 1} E_{3 1}^b E_{3 2}^c \cdot m \\ & \phantom{=} \; + (\lambda - (a - 2) (\epsilon_1 - \epsilon_2) - b (\epsilon_1 - \epsilon_3) - c (\epsilon_2 - \epsilon_3)) ([E_{1 2}, E_{2 1}]) E_{2 1}^{a - 2} E_{3 1}^b E_{3 2}^c \cdot m \\ & \phantom{=} \; \; \, \vdots \\ & \phantom{=} \; + (\lambda - (a - a) (\epsilon_1 - \epsilon_2) - b (\epsilon_1 - \epsilon_3) - c (\epsilon_2 - \epsilon_3)) ([E_{1 2}, E_{2 1}]) E_{2 1}^{a - a} E_{3 1}^b E_{3 2}^c \cdot m \\ \end{split} \] Since \((\lambda - (a - k) (\epsilon_1 - \epsilon_2) - b (\epsilon_1 - \epsilon_3) - c (\epsilon_2 - \epsilon_3)) ([E_{1 2}, E_{2 1}]) E_{2 1}^{a - k} E_{3 1}^b E_{3 2}^c \cdot m \in N\) for all \(k\), it suffices to show \(E_{2 1}^a E_{1 2} E_{3 1}^b E_{3 2}^c \cdot m \in N\). But \[ \begin{split} E_{1 2} E_{3 1}^b & = (E_{3 1} E_{1 2} - E_{3 2}) E_{3 1}^{b - 1} \\ & = E_{3 1} E_{1 2} E_{3 1}^{b - 1} - E_{3 1} E_{3 2} E_{3 1}^{b - 1} \\ & = E_{3 1} (E_{3 1} E_{1 2} - E_{3 2}) E_{3 1}^{b - 2} - E_{3 2} E_{3 1}^b \\ & \; \; \vdots \\ & = E_{3 1}^b E_{1 2} - b E_{3 2} E_{3 1}^b \\ \end{split}, \] given \([E_{1 2}, E_{3 1}] = - E_{3 2}\) and \([E_{3 2}, E_{3 1}] = 0\). It then follows from the fact \(E_{1 2} \cdot m = 0\) that \[ E_{2 1}^a E_{1 2} E_{3 1}^b E_{3 2}^c \cdot m = E_{2 1}^a E_{3 1}^b E_{3 2}^c E_{1 2} \cdot m - b E_{2 1}^a E_{3 1}^b E_{3 2}^{c + 1} \cdot m = - b E_{2 1}^a E_{3 1}^b E_{3 2}^{c + 1} \cdot m \in N, \] given that \(E_{1 2}\) and \(E_{3 2}\) commute. Hence \(E_{1 2} \cdot (E_{2 1}^a E_{3 1}^b E_{3 2}^c \cdot m) \in N\). Similarly, \[ E_{1 3} \cdot (E_{2 1}^a E_{3 1}^b E_{3 2}^c \cdot m), E_{2 3} \cdot (E_{2 1}^a E_{3 1}^b E_{3 2}^c \cdot m) \in N \] \end{proof} The same argument also goes to show\dots \begin{corollary}\label{thm:irr-component-of-high-vec} Given a finite-dimensional \(\mathfrak{sl}_3(K)\)-module \(M\) with highest weight \(\lambda\) and \(m \in M_\lambda\), the subspace spanned by successive applications of \(E_{2 1}\), \(E_{3 1}\) and \(E_{3 2}\) to \(m\) is a simple submodule whose highest weight is \(\lambda\). \end{corollary} This is very interesting to us since it implies that finding \emph{any} finite-dimensional module whose highest weight is \(\lambda\) is enough for establishing the ``existence'' part of Theorem~\ref{thm:sl3-existence-uniqueness}. Moreover, constructing such a module turns out to be quite simple. \begin{proof}[Proof of existence] Take \(\lambda = k \epsilon_1 - \ell \epsilon_3 \in P\) with \(k, \ell \ge 0\), so that \(\lambda\) is dominant. Consider the natural \(\mathfrak{sl}_3(K)\)-module \(K^3\). We claim that the highest weight of \(\operatorname{Sym}^k K^3 \otimes \operatorname{Sym}^\ell (K^3)^*\) is \(\lambda\). First of all, notice that the weight vector of \(K^3\) are the canonical basis elements \(e_1\), \(e_2\) and \(e_3\), whose corresponding weights are \(\epsilon_1\), \(\epsilon_2\) and \(\epsilon_3\) respectively. Hence the weight diagram of \(K^3\) is \begin{center} \begin{tikzpicture}[scale=2] \AutoSizeWeightLatticefalse \begin{rootSystem}{A} \weightLattice{2} \wt[black]{1}{0} \wt[black]{-1}{1} \wt[black]{0}{-1} \node[right] at \weight{1}{0} {$\epsilon_1$}; \node[above left] at \weight{-1}{1} {$\epsilon_2$}; \node[below left] at \weight{0}{-1} {$\epsilon_3$}; \end{rootSystem} \end{tikzpicture} \end{center} and \(\epsilon_1\) is the highest weight of \(K^3\). On the one hand, if \(\{f_1, f_2, f_3\}\) is the dual basis for \(\{e_1, e_2, e_3\}\) then \(H \cdot f_i = - \epsilon_i(H) f_i\) for each \(H \in \mathfrak{h}\), so that the weights of \((K^3)^*\) are precisely the opposites of the weights of \(K^3\). In other words, \begin{center} \begin{tikzpicture}[scale=2] \AutoSizeWeightLatticefalse \begin{rootSystem}{A} \weightLattice{2} \wt[black]{-1}{0} \wt[black]{1}{-1} \wt[black]{0}{1} \node[left] at \weight{-1}{0} {$-\epsilon_1$}; \node[below right] at \weight{1}{-1} {$-\epsilon_2$}; \node[above right] at \weight{0}{1} {$-\epsilon_3$}; \end{rootSystem} \end{tikzpicture} \end{center} is the weight diagram of \((K^3)^*\) and \(\epsilon_3\) is the highest weight of \((K^3)^*\). On the other hand if we fix two \(\mathfrak{sl}_3(K)\)-modules \(N\) and \(L\), by computing \[ \begin{split} H \cdot (n \otimes l) & = H \cdot n \otimes l + n \otimes H \cdot l \\ & = \lambda(H) n \otimes l + n \otimes \mu(H) l \\ & = (\lambda + \mu)(H) \, (n \otimes l) \end{split} \] for each \(H \in \mathfrak{h}\), \(n \in N_\lambda\) and \(l \in L_\mu\) we can see that the weights of \(N \otimes L\) are precisely the sums of the weights of \(N\) with the weights of \(L\). This implies that the highest weights of \(\operatorname{Sym}^k K^3\) and \(\operatorname{Sym}^\ell (K^3)^*\) are \(k \epsilon_1\) and \(- \ell \epsilon_3\) respectively -- with highest weight vectors \(e_1^k\) and \(f_3^\ell\). Furthermore, by the same token the highest weight of \(\operatorname{Sym}^k K^3 \otimes \operatorname{Sym}^\ell (K^3)^*\) must be \(\lambda = k e_1 - \ell e_3\) -- with highest weight vector \(e_1^k \otimes f_3^\ell\). \end{proof} The ``uniqueness'' part of Theorem~\ref{thm:sl3-existence-uniqueness} is even simpler than that. \begin{proof}[Proof of uniqueness] Let \(M\) and \(N\) be two simple \(\mathfrak{sl}_3(K)\)-modules with highest weight \(\lambda\). By Theorem~\ref{thm:sl3-irr-weights-class}, the weights of \(M\) are precisely the same as those of \(N\). Now by computing \[ H \cdot (m + n) = H \cdot m + H \cdot n = \mu(H) m + \mu(H) n = \mu(H) (m + n) \] for each \(H \in \mathfrak{h}\), \(m \in M_\mu\) and \(n \in N_\mu\), we can see that the weights of \(M \oplus N\) are same as those of \(M\) and \(N\). Hence the highest weight of \(M \oplus N\) is \(\lambda\) -- with highest weight vectors given by the sum of highest weight vectors of \(M\) and \(N\). Fix some \(m \in M_\lambda\) and \(n \in N_\lambda\) and consider the submodule \(L = \mathcal{U}(\mathfrak{sl}_3(K)) \cdot m + n \subset M \oplus N\) generated by \(m + n\). Since \(m + n\) is a highest weight of \(M \oplus N\), it follows from corollary~\ref{thm:irr-component-of-high-vec} that \(L\) is simple. The projection maps \(\pi_1 : L \to M\), \(\pi_2 : L \to N\), being nonzero homomorphism between simple \(\mathfrak{sl}_3(K)\)-modules, must be isomorphism. Finally, \[ M \cong L \cong N \] \end{proof} We have been very successful in our pursue for a classification of the simple modules of \(\mathfrak{sl}_2(K)\) and \(\mathfrak{sl}_3(K)\), but so far we have mostly postponed the discussion on the motivation behind our methods. In particular, we did not explain why we chose \(h\) and \(\mathfrak{h}\), and neither why we chose to look at their eigenvalues. Apart from the obvious fact we already knew it would work a priory, why did we do all that? In the following chapter we will attempt to answer this question by looking at what we did in the last chapter through more abstract lenses and studying the representations of an arbitrary finite-dimensional semisimple Lie algebra \(\mathfrak{g}\).