- Commit
- 255321a75dfa82ccd3b2b5ffb94ebb22f4d552c2
- Parent
- e7eac0dc9f80e09802b396761357ca6f07e53c79
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Added further comments to the section on cohomology
Emphasized the fact that cohomology only vanishes for semisimple algebras
Also noted that the Ext functor are characterized by the conditions of
the theorem they are first mentioned in. This provides a way to justify
the fact that the Lie algebra cohomology groups with coefficients in Hom
classify obstructions to complete reducibility
diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -365,17 +365,18 @@ further ado, we may proceed to our\dots
\section{Proof of Complete Reducibility}
-Let \(\mathfrak{g}\) be a finite-dimensional semisimple Lie algebra over \(K\).
-We want to establish that all finite-dimensional \(\mathfrak{g}\)-modules are
-semisimple. Historically, this was first proved by Herman Weyl for \(K =
-\mathbb{C}\), using his knowledge of smooth representations of compact Lie
-groups. Namely, Weyl showed that any finite-dimensional semisimple complex Lie
-algebra is (isomorphic to) the complexification of the Lie algebra of a unique
-simply connected compact Lie group, known as its \emph{compact form}. Hence the
-category of the finite-dimensional modules 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
-because of Maschke's Theorem -- see \cite[ch. 3]{serganova} for instance.
+Let \(\mathfrak{g}\) be a finite-dimensional Lie algebra over \(K\). We want to
+establish that if \(\mathfrak{g}\) is semisimple then all finite-dimensional
+\(\mathfrak{g}\)-modules are semisimple. Historically, this was first proved by
+Herman Weyl for \(K = \mathbb{C}\), using his knowledge of smooth
+representations of compact Lie groups. Namely, Weyl showed that any
+finite-dimensional semisimple complex Lie algebra is (isomorphic to) the
+complexification of the Lie algebra of a unique simply connected compact Lie
+group, known as its \emph{compact form}. Hence the category of the
+finite-dimensional modules 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 because of Maschke's
+Theorem -- see \cite[ch. 3]{serganova} for instance.
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
@@ -464,7 +465,7 @@ basic}. In fact, all we need to know is\dots
\end{theorem}
\begin{theorem}\label{thm:ext-1-classify-short-seqs}
- Given \(\mathfrak{g}\)-modules \(N\) and \(N\), there is a one-to-one
+ Given \(\mathfrak{g}\)-modules \(N\) and \(L\), there is a one-to-one
correspondence between elements of \(\operatorname{Ext}^1(L, N)\) and
isomorphism classes of short exact sequences
\begin{center}
@@ -478,20 +479,22 @@ basic}. In fact, all we need to know is\dots
extremes splits.
\end{theorem}
-\begin{note}
- 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 a complete exposition, or to part II of \cite{ribeiro} for
- a more modern account using derived categories.
-\end{note}
+We should point out that, although we have not provided an explicit definition
+of the bifunctors \(\operatorname{Ext}^i\), they are uniquely determined by
+the conditions of Theorem~\ref{thm:ext-exacts-seqs} and some additional
+minimality constraints. 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 a complete exposition, or to part II of
+\cite{ribeiro} for a more modern account using derived categories.
We are particularly interested in the case where \(L' = K\) is the trivial
\(\mathfrak{g}\)-module. Namely, we may define\dots
\begin{definition}\index{Lie algebra!cohomology}\index{cohomology of Lie algebras}
- Given a \(\mathfrak{g}\)-module \(M\), we refer to the Abelian group
- \(H^i(\mathfrak{g}, M) = \operatorname{Ext}^i(K, M)\) as \emph{the \(i\)-th
- Lie algebra cohomology group of \(\mathfrak{g}\) with coefficients in \(M\)}.
+ Given a Lie algebra \(\mathfrak{g}\) and a \(\mathfrak{g}\)-module \(M\), we
+ refer to the Abelian group \(H^i(\mathfrak{g}, M) = \operatorname{Ext}^i(K,
+ M)\) as \emph{the \(i\)-th Lie algebra cohomology group of \(\mathfrak{g}\)
+ with coefficients in \(M\)}.
\end{definition}
\begin{definition}\index{cohomology of Lie algebras!invariants}
@@ -505,6 +508,28 @@ We are particularly interested in the case where \(L' = K\) is the trivial
K\text{-}\mathbf{Vect}\).
\end{definition}
+\begin{example}
+ Let \(M\) be a \(\mathfrak{g}\)-module. Then \(M\) is a direct sum of copies
+ of the trivial \(\mathfrak{g}\)-module if, and only if \(M =
+ M^{\mathfrak{g}}\).
+\end{example}
+
+\begin{example}\label{ex:hom-invariants-are-g-homs}
+ Let \(M\) and \(N\) be \(\mathfrak{g}\)-modules. Then \(\operatorname{Hom}(M,
+ N)^{\mathfrak{g}} = \operatorname{Hom}_{\mathfrak{g}}(M, N)\). Indeed, given
+ a \(K\)-linear map \(f : M \to N\) we find
+ \[
+ \begin{split}
+ f \in \operatorname{Hom}(M, N)^{\mathfrak{g}}
+ & \iff X \cdot f(m) - f(X \cdot m) = (X \cdot f)(m) = 0
+ \; \forall X \in \mathfrak{g}, m \in M \\
+ & \iff X \cdot f(m) = f(X \cdot m)
+ \; \forall X \in \mathfrak{g}, m \in M \\
+ & \iff f \in \operatorname{Hom}_{\mathfrak{g}}(M, N)
+ \end{split}
+ \]
+\end{example}
+
The Lie algebra cohomology groups are very much related to invariants of
\(\mathfrak{g}\)-modules. Namely, constructing a \(\mathfrak{g}\)-homomorphism
\(f : K \to M\) is precisely the same as fixing an invariant of \(M\) --
@@ -555,6 +580,8 @@ This implies\dots
\end{center}
\end{corollary}
+\newpage
+
\begin{proof}
We have an isomorphism of sequences
\begin{center}
@@ -596,8 +623,12 @@ Explicitly\dots
\end{center}
\end{theorem}
-For the readers already familiar with homological algebra: this correspondence
-can be computed very concretely by considering a canonical acyclic resolution
+This is essentially a consequence of
+Theorem~\ref{thm:ext-1-classify-short-seqs} and
+Example~\ref{ex:hom-invariants-are-g-homs}, as well as the minimality
+conditions that characterize \(\operatorname{Ext}^i\). For the readers already
+familiar with homological algebra: this correspondence can be computed very
+concretely by considering a canonical projective resolution
\begin{center}
\begin{tikzcd}
\cdots \rar[dashed] &
@@ -611,21 +642,23 @@ can be computed very concretely by considering a canonical acyclic resolution
of the trivial \(\mathfrak{g}\)-module \(K\), known as \emph{the
Chevalley-Eilenberg resolution}, which provides an explicit construction of the
cohomology groups -- see \cite[sec.~9]{lie-groups-serganova-student} or
-\cite[sec.~24]{symplectic-physics} for further details. We will use the
-previous result implicitly in our proof, but we will not prove it in its full
-force. Namely, we will show that \(H^1(\mathfrak{g}, M) = 0\) for all
-finite-dimensional \(M\), and that the fact that \(H^1(\mathfrak{g},
-\operatorname{Hom}(N, L)) = 0\) for all finite-dimensional \(N\) and \(L\)
-implies complete reducibility. To that end, we introduce a distinguished
-element of \(\mathcal{U}(\mathfrak{g})\), known as \emph{the Casimir element of
-a \(\mathfrak{g}\)-module}.
+\cite[sec.~24]{symplectic-physics} for further details.
+
+We will use the previous result implicitly in our proof, but we will not prove
+it in its full force. Namely, we will show that if \(\mathfrak{g}\) is
+semisimple then \(H^1(\mathfrak{g}, M) = 0\) for all finite-dimensional \(M\),
+and that the fact that \(H^1(\mathfrak{g}, \operatorname{Hom}(N, L)) = 0\) for
+all finite-dimensional \(N\) and \(L\) implies complete reducibility. To that
+end, we introduce a distinguished element of \(\mathcal{U}(\mathfrak{g})\),
+known as \emph{the Casimir element of a \(\mathfrak{g}\)-module}.
\begin{definition}\label{def:casimir-element}\index{Casimir element}
- Let \(M\) be a finite-dimensional \(\mathfrak{g}\)-module. Let
- \(\{X_i\}_i\) be a basis for \(\mathfrak{g}\) and denote by \(\{X^i\}_i
- \subset \mathfrak{g}\) its dual basis with respect to the form \(\kappa_M\) --
- i.e. the unique basis for \(\mathfrak{g}\) satisfying \(\kappa_M(X_i, X^j) =
- \delta_{i j}\). We call
+ Let \(\mathfrak{g}\) be a finite-dimensional semisimple Lie algebra and \(M\)
+ be a finite-dimensional \(\mathfrak{g}\)-module. Let \(\{X_i\}_i\) be a basis
+ for \(\mathfrak{g}\) and denote by \(\{X^i\}_i \subset \mathfrak{g}\) its
+ dual basis with respect to the form \(\kappa_M\) -- i.e. the unique basis for
+ \(\mathfrak{g}\) satisfying \(\kappa_M(X_i, X^j) = \delta_{i j}\), whose
+ existence is a consequence of the non-degeneracy of \(\kappa_M\). We call
\[
\Omega_M = X_1 X^1 + \cdots + X_r X^r \in \mathcal{U}(\mathfrak{g})
\]
@@ -704,8 +737,8 @@ As promised, the Casimir element of a \(\mathfrak{g}\)-module can be used to
establish\dots
\begin{proposition}\label{thm:first-cohomology-vanishes}
- Let \(M\) be a finite-dimensional \(\mathfrak{g}\)-module. Then
- \(H^1(\mathfrak{g}, M) = 0\).
+ Suppose \(\mathfrak{g}\) is semisimple and let \(M\) be a finite-dimensional
+ \(\mathfrak{g}\)-module. Then \(H^1(\mathfrak{g}, M) = 0\).
\end{proposition}
\begin{proof}
@@ -875,32 +908,13 @@ We are now finally ready to prove\dots
\cdots
\end{tikzcd}
\end{center}
- of vector spaces. But \(H^1(\mathfrak{g}, \operatorname{Hom}(L, N))\)
- vanishes because of Proposition~\ref{thm:first-cohomology-vanishes}. Hence we
- have an exact sequence
- \begin{center}
- \begin{tikzcd}
- 0 \rar &
- \operatorname{Hom}(L, N)^{\mathfrak{g}} \rar{f \circ -} &
- \operatorname{Hom}(L, M)^{\mathfrak{g}} \rar{g \circ -} &
- \operatorname{Hom}(L, L)^{\mathfrak{g}} \rar &
- 0
- \end{tikzcd}
- \end{center}
-
- Now notice \(\operatorname{Hom}(L, L')^{\mathfrak{g}} =
- \operatorname{Hom}_{\mathfrak{g}}(L, L')\) for all \(\mathfrak{g}\)-modules
- \(L'\). Indeed, given a \(K\)-linear map \(f : L \to L'\)
- \[
- \begin{split}
- f \in \operatorname{Hom}(L, L')^{\mathfrak{g}}
- & \iff X f - f X = X \cdot f = 0 \quad \forall X \in \mathfrak{g} \\
- & \iff X f = f X \quad \forall X \in \mathfrak{g} \\
- & \iff f \in \operatorname{Hom}_{\mathfrak{g}}(L, L')
- \end{split}
- \]
+ of vector spaces.
- We thus have a short exact sequence
+ But \(H^1(\mathfrak{g}, \operatorname{Hom}(L, N))\) vanishes because of
+ Proposition~\ref{thm:first-cohomology-vanishes}. In addition, recall from
+ Example~\ref{ex:hom-invariants-are-g-homs} that \(\operatorname{Hom}(L,
+ L')^{\mathfrak{g}} = \operatorname{Hom}_{\mathfrak{g}}(L, L')\). We thus have
+ a short exact sequence
\begin{center}
\begin{tikzcd}
0 \rar &
@@ -924,13 +938,13 @@ We are now finally ready to prove\dots
We should point out that these last results are just the beginning of a well
developed cohomology theory. For example, a similar argument involving the
Casimir elements can be used to show that \(H^i(\mathfrak{g}, M) = 0\) for all
-non-trivial finite-dimensional simple \(M\), \(i > 0\). For \(K =
-\mathbb{C}\), the Lie algebra cohomology groups of the 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 \(G\) with coefficients in \(\mathbb{C}\). We refer the reader to
-\cite{cohomologies-lie} and \cite[sec.~24]{symplectic-physics} for further
-details.
+semisimple \(\mathfrak{g}\) and all non-trivial finite-dimensional simple
+\(M\), \(i > 0\). For \(K = \mathbb{C}\), the Lie algebra cohomology groups of
+the 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 \(G\) with coefficients in
+\(\mathbb{C}\). We refer the reader to \cite{cohomologies-lie} and
+\cite[sec.~24]{symplectic-physics} for further details.
Complete reducibility can be generalized for arbitrary -- not necessarily
semisimple -- \(\mathfrak{g}\), to a certain extent, by considering the exact