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

Commit
1fce6426365d6b979283f3c82096829b9cff4b4f
Parent
aa6137d89836bdd213677a1638d5e7af8d3e6a33
Author
Pablo <pablo-escobar@riseup.net>
Date

Completed the proof of complete reducibility

Replaced the use of the universal Casimir element with that of the Casimir element of a given representation

Its easier to prove that the Casimir element of a given irreducible representation acts as a non-zero scalar in such a representation than to prove that the universal Casimir element does so

Diffstat

1 file changed, 72 insertions, 46 deletions

Status File Name N° Changes Insertions Deletions
Modified sections/semisimple-algebras.tex 118 72 46
diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -146,10 +146,24 @@ Another interesting characterization of semisimple Lie algebras, which will
 come in handy later on, is the following.
 
 % TODO: Define the Killing form beforehand
+% TODO: Define invariant forms beforehand
 \begin{proposition}
-  A Lie algebra \(\mathfrak{g}\) is semisimple if, and only if its Killing form
-  \(B\) is non-degenerate -- i.e. if \(X \in \mathfrak{g}\) is such that \(B(X,
-  Y)\) for all \(Y \in \mathfrak{g}\) then \(X = 0\).
+  Let \(\mathfrak{g}\) be a Lie algebra. The following statements are
+  equivalent.
+  \begin{enumerate}
+    \item \(\mathfrak{g}\) is semisimple.
+    \item For each finite-dimensional representation \(V\) of \(\mathfrak{g}\),
+      the \(\mathfrak{g}\)-invariant bilinear form
+      \begin{align*}
+        B_V : \mathfrak{g} \times \mathfrak{g} & \to K \\
+        (X, Y) & 
+        \mapsto \operatorname{Tr}(X\!\restriction_V \circ Y\!\restriction_V)
+      \end{align*}
+      is non-degenerate\footnote{A simmetric bilinear form $B : \mathfrak{g}
+      \times \mathfrak{g} \to K$ is called non-degenerate if $B(X, Y) = 0$ for
+      all $Y \in \mathfrak{g}$ implies $X = 0$.}.
+    \item The Killing form \(B\) is non-degenerate.
+  \end{enumerate}
 \end{proposition}
 
 We refer the reader for \cite[ch. 5]{humphreys} for a proof of this last
@@ -394,42 +408,44 @@ we will not prove it in its full force. Namely, we will show that
 fact that \(H^1(\mathfrak{g}, \operatorname{Hom}(W, U)) = 0\) for all
 finite-dimensional \(W\) and \(U\) implies complete reducibility. To that end,
 we introduce a distinguished element of \(\mathcal{U}(\mathfrak{g})\), known as
-\emph{the Casimir element}.
+\emph{the Casimir element of a representation}.
 
 \begin{definition}\label{def:casimir-element}
+  Let \(V\) be a finite-dimensional representation of \(\mathfrak{g}\).
   Let \(\{X_i\}_i\) be a basis for \(\mathfrak{g}\) and denote by \(\{X^i\}_i\)
-  its dual basis -- i.e. the unique basis for \(\mathfrak{g}\) satisfying
-  \(B(X_i, X^j) = \delta_{i j}\). We call
+  its dual basis with respect to the form \(B_V\) -- i.e. the unique basis for
+  \(\mathfrak{g}\) satisfying \(B_V(X_i, X^j) = \delta_{i j}\). We call
   \[
-    C = X_1 X^1 + \cdots + X_n X^n \in \mathcal{U}(\mathfrak{g})
+    C_V = X_1 X^1 + \cdots + X_n X^n \in \mathcal{U}(\mathfrak{g})
   \]
-  the \emph{Casimir element of \(\mathcal{U}(\mathfrak{g})\)}.
+  the \emph{Casimir element of \(V\)}.
 \end{definition}
 
 \begin{lemma}
-  The definition of \(C\) is independent of the choice of basis \(\{X_i\}_i\).
+  The definition of \(C_V\) is independent of the choice of basis
+  \(\{X_i\}_i\).
 \end{lemma}
 
 \begin{proof}
-  Whatever basis \(\{X_i\}_i\) we choose, the image of \(C\) under the
+  Whatever basis \(\{X_i\}_i\) we choose, the image of \(C_V\) under the
   canonical isomorphism \(\mathfrak{g} \otimes \mathfrak{g} \isoto \mathfrak{g}
   \otimes \mathfrak{g}^* \isoto \operatorname{End}(\mathfrak{g})\) is the
-  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 \(B\).}.
+  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 form $B_V$.}.
 \end{proof}
 
 \begin{proposition}
-  The Casimir element \(C \in \mathcal{U}(\mathfrak{g})\) is central, so that
-  \(C : V \to V\) is an intertwining operator for any \(\mathfrak{g}\)-module
-  \(V\). Furthermore, \(C\) acts as a non-zero scalar operator whenever \(V\)
-  is a non-trivial finite-dimensional irreducible representation of
-  \(\mathfrak{g}\).
+  The Casimir element \(C_V \in \mathcal{U}(\mathfrak{g})\) is central, so that
+  \(C_V : W \to W\) is an intertwining operator for any \(\mathfrak{g}\)-module
+  \(W\). Furthermore, \(C_V\) acts in \(V\) as a non-zero scalar operator
+  whenever \(V\) is a non-trivial finite-dimensional irreducible representation
+  of \(\mathfrak{g}\).
 \end{proposition}
 
 \begin{proof}
-  To see that \(C\) is central fix a basis \(\{X_i\}_i\) for \(\mathfrak{g}\)
+  To see that \(C_V\) is central fix a basis \(\{X_i\}_i\) for \(\mathfrak{g}\)
   and denote by \(\{X^i\}_i\) its dual basis as in
   definition~\ref{def:casimir-element}. Let \(X \in \mathfrak{g}\) and denote
   by \(\lambda_{i j}, \mu_{i j} \in K\) the coefficients of \(X_j\) and \(X^j\)
@@ -439,33 +455,42 @@ we introduce a distinguished element of \(\mathcal{U}(\mathfrak{g})\), known as
   The invariance of \(B\) implies
   \[
     \lambda_{i k}
-    = B([X, X_i], X^k)
-    = B(-[X_i, X], X^k)
-    = B(X_i, -[X, X^k])
+    = B_V([X, X_i], X^k)
+    = B_V(-[X_i, X], X^k)
+    = B_V(X_i, -[X, X^k])
     = - \mu_{k i}
   \]
 
   Hence
   \[
     \begin{split}
-      [X, C]
+      [X, C_V]
       & = \sum_i [X, X_i X^i] \\
       & = \sum_i [X, X_i] X^i + \sum_i X_i [X, X^i] \\
       & = \sum_{i j} \lambda_{i j} X_j X^i + \sum_{i j} \mu_{i j} X_i X^j \\
       & = 0
     \end{split},
   \]
-  and \(C\) is central. This implies that \(C : V \to V\) is an intertwiner for
-  all representations \(V\) of \(\mathfrak{g}\): its action commutes with the
-  action of any other element of \(\mathfrak{g}\).
+  and \(C_V\) is central. This implies that \(C_V : W \to W\) is an intertwiner
+  for all representations \(W\) of \(\mathfrak{g}\): its action commutes with
+  the action of any other element of \(\mathfrak{g}\).
 
-  % TODOO: Prove that the action is not zero when V is non-trivial
   In particular, it follows from Schur's lemma that if \(V\) is
-  finite-dimensional and irreducible then \(C\) acts in \(V\) as a scalar
-  operator.
+  finite-dimensional and irreducible then \(C_V\) acts in \(V\) as a scalar
+  operator. To see that this scalar is nonzero we compute
+  \[
+    \operatorname{Tr}(C_V\!\restriction_V)
+    = \operatorname{Tr}(X_1\!\restriction_V X^1\!\restriction_V)
+    + \cdots
+    + \operatorname{Tr}(X_n\!\restriction_V X^n\!\restriction_V)
+    = \dim \mathfrak{g},
+  \]
+  so that \(C_V\!\restriction_V = \lambda \operatorname{Id}\) for \(\lambda =
+  \frac{\dim \mathfrak{g}}{\dim V} \ne 0\).
 \end{proof}
 
-As promised, the Casimir element can be used to establish\dots
+As promised, the Casimir element of a representation can be used to
+establish\dots
 
 \begin{proposition}\label{thm:first-cohomology-vanishes}
   Let \(V\) be a finite-dimensional representation of \(\mathfrak{g}\). Then
@@ -503,37 +528,38 @@ As promised, the Casimir element can be used to establish\dots
   \(1\) to \(\sfrac{w}{\pi(w)}\) is a splitting of
   (\ref{eq:trivial-extrems-exact-seq}).
 
-  Now suppose that \(V\) is non-trivial, so that \(C\) acts on \(V\) as
+  Now suppose that \(V\) is non-trivial, so that \(C_V\) acts on \(V\) as
   \(\lambda \operatorname{Id}\) for some \(\lambda \ne 0\). Given an eigenvalue
   \(\mu \in K\) of the action of \(C\) in \(W\), denote by \(W^\mu\) its
   associated generalized eigenspace. We claim \(W^0\) is the image of the
-  inclusion \(K \to W\). Since \(C\) acts as zero in \(K\), this image is
+  inclusion \(K \to W\). Since \(C_V\) acts as zero in \(K\), this image is
   clearly contained in \(W^0\). On the other hand, if \(w \in W\) is such that
-  \(C^n w = 0\) then
+  \(C_V^n w = 0\) then
   \[
     \lambda^n \pi(w)
-    = C^n \pi(w)
-    = \pi(C^n w)
+    = C_V^n \pi(w)
+    = \pi(C_V^n w)
     = 0,
   \]
   so that \(w \in \ker \pi\) -- because \(\lambda^n \ne 0\). The exactness of
   (\ref{eq:exact-seq-h1-vanishes}) then implies the desired conclusion.
 
-  We furthermore claim that the only eigenvalues of \(C\) in \(W\) are \(0\)
+  We furthermore claim that the only eigenvalues of \(C_V\) in \(W\) are \(0\)
   and \(\lambda\). Indeed, if \(\mu \ne 0\) is eigenvalue and \(w\) is an
   associated eigenvector, then
   \[
-    \mu \pi(w) = \pi(C w) = C \pi(w) = \lambda \pi(w)
+    \mu \pi(w) = \pi(C_V w) = C_V \pi(w) = \lambda \pi(w)
   \]
 
   Since \(w \notin W^0\), \(\pi(w) \ne 0\) and therefore \(\mu = \lambda\).
-  Hence \(W = W^0 \oplus W^\lambda\). The fact that \(C\) is central implies
-  \((C - \lambda \operatorname{Id})^n X v = X (C - \lambda \operatorname{Id})^n
-  v\) for all \(v \in V\), \(X \in \mathfrak{g}\) and \(n > 0\). In particular,
-  \(W^\lambda\) is stable under the action of \(\mathfrak{g}\) -- i.e.
-  \(W^\lambda\) is a subrepresentation. Since \(W^0\) is precisely the kernel
-  of \(\pi\), we have an isomorphism of representations \(W^\lambda \cong
-  \sfrac{W}{W^0} \isoto V\), which induces a splitting \(W \cong K \oplus V\).
+  Hence \(W = W^0 \oplus W^\lambda\) as vector space. The fact that \(C_V\) is
+  central implies \((C_V - \lambda \operatorname{Id})^n X v = X (C_V - \lambda
+  \operatorname{Id})^n v\) for all \(v \in V\), \(X \in \mathfrak{g}\) and \(n
+  > 0\). In particular, \(W^\lambda\) is stable under the action of
+  \(\mathfrak{g}\) -- i.e. \(W^\lambda\) is a subrepresentation. Since \(W^0\)
+  is precisely the kernel of \(\pi\), we have an isomorphism of representations
+  \(W^\lambda \cong \sfrac{W}{W^0} \isoto V\), which induces a splitting \(W
+  \cong K \oplus V\).
 
   Finally, we consider the case where \(V\) is not irreducible. Suppose
   \(H^1(\mathfrak{g}, W) = 0\) for all \(\mathfrak{g}\)-modules with \(\dim W <
@@ -668,7 +694,7 @@ We are now finally ready to prove\dots
 
 We should point out that this last results are just the beginning of a well
 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
+Casimir elements 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 \(\mathfrak{g} =
 \mathbb{C} \otimes \operatorname{Lie}(G)\) are intimately related with the