- 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
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