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
git clone: git://git.pablopie.xyz/lie-algebras-and-their-representations
Log
Files
Refs
README
LICENSE
Commit: 1ae25bac7f4cc95b595ccf6f089d22d2e1198df7
Parent: f4eec662cf14c7a07397a3ae8d5897554d12db5c
Author: Pablo <pablo-escobar@riseup.net>
Date: Fri, 18 Nov 2022 17:10:36 +0000
Reordered the contents of the introduction
Moved the section on representations to after the section on the universal envoloping algebra, and also moved the section on induced representations to after the section on restrictions of representations
Diffstat

1 file changed, 165 insertions, 164 deletions
Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/introduction.tex	329	165	164
diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -409,120 +409,6 @@ ready to dive deeper into them.
   \mathfrak{sl}_n(K) \oplus K\).
 \end{example}
 
-\section{Representations}
-
-\begin{definition}
-  Given a Lie algebra \(\mathfrak{g}\) over \(K\), \emph{a representation \(V\)
-  of \(\mathfrak{g}\)}, or \emph{\(\mathfrak{g}\)-module}, is a \(K\)-vector
-  space endowed with a homomorphism of Lie algebras \(\rho : \mathfrak{g} \to
-  \mathfrak{gl}(V)\).
-\end{definition}
-
-\begin{example}
-  Given a Lie algebra \(\mathfrak{g}\), consider the homomorphism
-  \(\operatorname{ad} : \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g})\) given by
-  \(\operatorname{ad}(X) Y = [X, Y]\). This gives \(\mathfrak{g}\) the
-  structure of a representation of \(\mathfrak{g}\), known as \emph{the adjoint
-  representation}.
-\end{example}
-
-\begin{example}
-  Given a Lie algebra \(\mathfrak{g}\) and \(\mathfrak{g}\)-modules \(V\) and
-  \(W\), the the spaces \(V \oplus W\), \(V^*\), \(V \otimes W\) and
-  \(\operatorname{Hom}(V, W)\) are all \(\mathfrak{g}\)-modules -- where the
-  action of \(\mathfrak{g}\) is given by
-  \begin{align*}
-    X (v + w)       & = X v + X w                     &
-    X \cdot f       & = - f \circ X                   \\
-    X (v \otimes w) & = X v \otimes w + v \otimes X w &
-    (X \cdot T) v   & = X T v - T X v,
-  \end{align*}
-  respectively.
-\end{example}
-
-\begin{definition}
-  Given a Lie algebra \(\mathfrak{g}\) and two representations \(V\) and \(W\)
-  of \(\mathfrak{g}\), we call a linear map \(T : V \to W\) \emph{an
-  intertwiner} or \emph{a homomorphism of representations} if it cummutes with
-  the action of \(\mathfrak{g}\) in \(V\) and \(W\), in the sence that the
-  diagram
-  \begin{center}
-    \begin{tikzcd}
-      V \rar{T} \dar[swap]{X} & W \dar{X} \\
-      V \rar[swap]{T}         & W
-    \end{tikzcd}
-  \end{center}
-  commutes for all \(X \in \mathfrak{g}\). We denote the space of all
-  intertwiners \(V \to W\) by \(\operatorname{Hom}_{\mathfrak{g}}(V, W)\).
-\end{definition}
-
-% TODO: Point out g-Mod is indeed a category
-
-\begin{example}
-  Let \(\mathfrak{g}\) be a Lie algebra and \(\mathfrak{h}\) be a subalgebra.
-  Given a representation \(V\) of \(\mathfrak{g}\), denote by
-  \(\operatorname{Res}_{\mathfrak{h}}^{\mathfrak{g}} V = V\) the representation
-  of \(\mathfrak{h}\) where the action of \(\mathfrak{h}\) is given by
-  restricting the map \(\mathfrak{g} \to \mathfrak{gl}(V)\) to
-  \(\mathfrak{g}\). Any homomorphism of \(\mathfrak{g}\)-modules \(V \to W\) is
-  also a homomorphism of \(\mathfrak{h}\)-modules and this construction is
-  clearly functorial.
-  \[
-    \operatorname{Res}_{\mathfrak{h}}^{\mathfrak{g}} :
-    \mathfrak{g}\text{-}\mathbf{Mod} \to
-    \mathfrak{h}\text{-}\mathbf{Mod}
-  \]
-\end{example}
-
-\begin{definition}
-  Given a Lie algebra \(\mathfrak{g}\) and a representation \(V\) of
-  \(\mathfrak{g}\), a subspace \(W \subset V\) is called \emph{a
-  subrepresentation} if it is stable under the action of \(\mathfrak{g}\) --
-  i.e. \(X w \in W\) for all \(w \in W\) and \(X \in \mathfrak{g}\).
-\end{definition}
-
-\begin{example}
-  Given a Lie algebra \(\mathfrak{g}\), a representation \(V\) of
-  \(\mathfrak{g}\) and a subrepresentation \(W \subset V\), the space
-  \(\mfrac{V}{W}\) has the natural structure of a \(\mathfrak{g}\)-module where
-  \(X (v + W) = X v + W\). The projection \(V \to \mfrac{V}{W}\) is an
-  intertwiner.
-\end{example}
-
-\begin{example}
-  Given a Lie algebra \(\mathfrak{g}\) and representations \(V\) and \(W\) of
-  \(\mathfrak{g}\), the spaces \(V \wedge W\) and \(V \odot W\) are both
-  representations of \(G\): they are both quotients of \(V \otimes W\).
-\end{example}
-
-\begin{definition}
-  A representation of \(\mathfrak{g}\) is called \emph{indecomposable} if it is
-  not isomorphic to the direct sum of two non-zero representations.
-\end{definition}
-
-\begin{definition}
-  A representation of \(\mathfrak{g}\) is called \emph{irreducible} if it has
-  no non-zero subrepresentations.
-\end{definition}
-
-\begin{lemma}[Schur]
-  Let \(\mathfrak{g}\) be a Lie algebra over a field \(K\). If \(V\) and \(W\)
-  irreducible representations of \(\mathfrak{g}\). and \(T : V \to W\) be an
-  intertwiner then \(T\) is either \(0\) or an isomorphism. Furtheremore, if
-  \(K\) is algebraicly closed and \(V = W\) then \(T\) is a scalar operator.
-\end{lemma}
-
-\begin{proof}
-  For the first statement, it suffices to notice that \(\ker T\) and
-  \(\operatorname{im} T\) are both subrepresentations. In particular, either
-  \(\ker T = 0\) and \(\operatorname{im} T = W\) or \(\ker T = V\) and
-  \(\operatorname{im} T = 0\). Now suppose \(K\) is algebraicly closed and \(V
-  = W\). Let \(\lambda \in K\) be an eigenvalue of \(T\) and \(V_\lambda\) be
-  its corresponding eigenspace. Given \(v \in V_\lambda\), \(T X v = X T v =
-  \lambda \cdot X v\). In other words, \(V_\lambda\) is a subrepresentation.
-  It then follows \(V_\lambda = V\), given that \(V_\lambda \ne 0\).
-\end{proof}
-
 \section{The Universal Enveloping Algebra}
 
 \begin{definition}
@@ -607,6 +493,107 @@ ready to dive deeper into them.
   \(\operatorname{Lie} \vdash \mathcal{U}\).
 \end{corollary}
 
+% TODO: Define the algebra of differential operators of a given algebra
+\begin{proposition}
+  Let \(G\) be a Lie group and \(\mathfrak{g}\) be its Lie algebra. Denote by
+  \(\operatorname{Diff}(G)^G\) the algebra of \(G\)-invariant differential
+  operators in \(G\) -- i.e. the algebra of all differential operators \(L :
+  C^\infty(G) \to C^\infty(G)\) such that \((L(f \circ \ell_g)) \circ
+  \ell_{g^{-1}} = L f\) for all \(f \in C^\infty(G)\) and \(g \in G\). There is
+  a canonical isomorphism of algebras \(\mathcal{U}(\mathfrak{g}) \isoto
+  \operatorname{Diff}(G)^G\).
+\end{proposition}
+
+\begin{proof}
+  An order \(1\) \(G\)-invariant differential operator in \(G\) is simply a
+  left invariant derivation \(C^\infty(G) \to C^\infty(G)\). All other
+  \(G\)-invariant differetial operators are generated by such derivations. Now
+  recall that there is a canonical isomorphism of Lie algebras
+  \(\mathfrak{X}(G) \isoto \operatorname{Der}(G)\). This isomorphism takes left
+  invariant fields to left invariant derivations, so it restricts to an
+  isomorphism \(f : \mathfrak{g} \isoto \operatorname{Der}(G)^G \subset
+  \operatorname{Diff}(G)^G\) -- here \(\operatorname{Der}(G)^G \subset
+  \operatorname{Der}(G)\) denotes the Lie subalgebra of invariant derivations.
+
+  Since \(f\) is a homomorphism of Lie algebras, it can be extended to an
+  algebra homomorphism \(g : \mathcal{U}(\mathfrak{g}) \to
+  \operatorname{Diff}(G)^G\). We claim \(g\) is an isomorphim. To see that
+  \(g\) is injective, it suffices to notice
+  \[
+    g(X_1 \cdots X_n)
+    = g(X_1) \cdots g(X_n)
+    = f(X_1) \cdots f(X_n)
+    \ne 0
+  \]
+  for all nonzero \(X_1, \cdots, X_n \in \mathfrak{g}\) --
+  \(\operatorname{Diff}(G)^G\) is a domain. Since \(\mathcal{U}(\mathfrak{g})\)
+  is generated by the image of the inclusion \(\mathfrak{g} \to
+  \mathcal{U}(\mathfrak{g})\), this implies \(\ker g = 0\). Given that
+  \(\operatorname{Diff}(G)^G\) is generated by \(\operatorname{Der}(G)^G\),
+  this also goes to show \(g\) is surjective.
+\end{proof}
+
+% TODO: Comment on the fact this holds for algebraic groups too
+
+\begin{theorem}[Poincaré-Birkoff-Witt]
+  Let \(\mathfrak{g}\) be a finite-dimensional Lie algebra over \(K\) and
+  \(\{X_i\}_i \subset \mathfrak{g}\) be a basis for \(\mathfrak{g}\). Then
+  \(\{X_{i_1} \cdot X_{i_2} \cdots X_{i_n} : n \ge 0, i_1 \le i_2 \le \cdots
+  \le i_n\}\) is a basis for \(\mathcal{U}(\mathfrak{g})\).
+\end{theorem}
+
+% TODO: Comment on the fact that modules of invariant differential operators
+% over G are precisely the same as representations of g
+
+\section{Representations}
+
+\begin{definition}
+  Given a Lie algebra \(\mathfrak{g}\) over \(K\), \emph{a representation \(V\)
+  of \(\mathfrak{g}\)}, or \emph{\(\mathfrak{g}\)-module}, is a \(K\)-vector
+  space endowed with a homomorphism of Lie algebras \(\rho : \mathfrak{g} \to
+  \mathfrak{gl}(V)\).
+\end{definition}
+
+\begin{example}
+  Given a Lie algebra \(\mathfrak{g}\), consider the homomorphism
+  \(\operatorname{ad} : \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g})\) given by
+  \(\operatorname{ad}(X) Y = [X, Y]\). This gives \(\mathfrak{g}\) the
+  structure of a representation of \(\mathfrak{g}\), known as \emph{the adjoint
+  representation}.
+\end{example}
+
+\begin{example}
+  Given a Lie algebra \(\mathfrak{g}\) and \(\mathfrak{g}\)-modules \(V\) and
+  \(W\), the the spaces \(V \oplus W\), \(V^*\), \(V \otimes W\) and
+  \(\operatorname{Hom}(V, W)\) are all \(\mathfrak{g}\)-modules -- where the
+  action of \(\mathfrak{g}\) is given by
+  \begin{align*}
+    X (v + w)       & = X v + X w                     &
+    X \cdot f       & = - f \circ X                   \\
+    X (v \otimes w) & = X v \otimes w + v \otimes X w &
+    (X \cdot T) v   & = X T v - T X v,
+  \end{align*}
+  respectively.
+\end{example}
+
+\begin{definition}
+  Given a Lie algebra \(\mathfrak{g}\) and two representations \(V\) and \(W\)
+  of \(\mathfrak{g}\), we call a linear map \(T : V \to W\) \emph{an
+  intertwiner} or \emph{a homomorphism of representations} if it cummutes with
+  the action of \(\mathfrak{g}\) in \(V\) and \(W\), in the sence that the
+  diagram
+  \begin{center}
+    \begin{tikzcd}
+      V \rar{T} \dar[swap]{X} & W \dar{X} \\
+      V \rar[swap]{T}         & W
+    \end{tikzcd}
+  \end{center}
+  commutes for all \(X \in \mathfrak{g}\). We denote the space of all
+  intertwiners \(V \to W\) by \(\operatorname{Hom}_{\mathfrak{g}}(V, W)\).
+\end{definition}
+
+% TODO: Point out g-Mod is indeed a category
+
 % TODO: Point out Hom(U(g), End(V)) ≃ Hom(g, gl(V))
 
 \begin{proposition}
@@ -616,6 +603,43 @@ ready to dive deeper into them.
   finite-dimensional repesentations to finitely generated modules.
 \end{proposition}
 
+\begin{definition}
+  Given a Lie algebra \(\mathfrak{g}\) and a representation \(V\) of
+  \(\mathfrak{g}\), a subspace \(W \subset V\) is called \emph{a
+  subrepresentation} if it is stable under the action of \(\mathfrak{g}\) --
+  i.e. \(X w \in W\) for all \(w \in W\) and \(X \in \mathfrak{g}\).
+\end{definition}
+
+\begin{example}
+  Given a Lie algebra \(\mathfrak{g}\), a representation \(V\) of
+  \(\mathfrak{g}\) and a subrepresentation \(W \subset V\), the space
+  \(\mfrac{V}{W}\) has the natural structure of a \(\mathfrak{g}\)-module where
+  \(X (v + W) = X v + W\). The projection \(V \to \mfrac{V}{W}\) is an
+  intertwiner.
+\end{example}
+
+\begin{example}
+  Given a Lie algebra \(\mathfrak{g}\) and representations \(V\) and \(W\) of
+  \(\mathfrak{g}\), the spaces \(V \wedge W\) and \(V \odot W\) are both
+  representations of \(G\): they are both quotients of \(V \otimes W\).
+\end{example}
+
+\begin{example}
+  Let \(\mathfrak{g}\) be a Lie algebra and \(\mathfrak{h}\) be a subalgebra.
+  Given a representation \(V\) of \(\mathfrak{g}\), denote by
+  \(\operatorname{Res}_{\mathfrak{h}}^{\mathfrak{g}} V = V\) the representation
+  of \(\mathfrak{h}\) where the action of \(\mathfrak{h}\) is given by
+  restricting the map \(\mathfrak{g} \to \mathfrak{gl}(V)\) to
+  \(\mathfrak{g}\). Any homomorphism of \(\mathfrak{g}\)-modules \(V \to W\) is
+  also a homomorphism of \(\mathfrak{h}\)-modules and this construction is
+  clearly functorial.
+  \[
+    \operatorname{Res}_{\mathfrak{h}}^{\mathfrak{g}} :
+    \mathfrak{g}\text{-}\mathbf{Mod} \to
+    \mathfrak{h}\text{-}\mathbf{Mod}
+  \]
+\end{example}
+
 \begin{example}
   Let \(\mathfrak{g}\) be a Lie algebra and \(\mathfrak{h}\) be a subalgebra.
   Given a representation \(V\) of \(\mathfrak{h}\), denote by
@@ -663,54 +687,31 @@ ready to dive deeper into them.
   \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}}\).
 \end{proposition}
 
-% TODO: Define the algebra of differential operators of a given algebra
-\begin{proposition}
-  Let \(G\) be a Lie group and \(\mathfrak{g}\) be its Lie algebra. Denote by
-  \(\operatorname{Diff}(G)^G\) the algebra of \(G\)-invariant differential
-  operators in \(G\) -- i.e. the algebra of all differential operators \(L :
-  C^\infty(G) \to C^\infty(G)\) such that \((L(f \circ \ell_g)) \circ
-  \ell_{g^{-1}} = L f\) for all \(f \in C^\infty(G)\) and \(g \in G\). There is
-  a canonical isomorphism of algebras \(\mathcal{U}(\mathfrak{g}) \isoto
-  \operatorname{Diff}(G)^G\).
-\end{proposition}
+%\begin{definition}
+%  A representation of \(\mathfrak{g}\) is called \emph{indecomposable} if it is
+%  not isomorphic to the direct sum of two non-zero representations.
+%\end{definition}
+%
+%\begin{definition}
+%  A representation of \(\mathfrak{g}\) is called \emph{irreducible} if it has
+%  no non-zero subrepresentations.
+%\end{definition}
+%
+%\begin{lemma}[Schur]
+%  Let \(\mathfrak{g}\) be a Lie algebra over a field \(K\). If \(V\) and \(W\)
+%  irreducible representations of \(\mathfrak{g}\). and \(T : V \to W\) be an
+%  intertwiner then \(T\) is either \(0\) or an isomorphism. Furtheremore, if
+%  \(K\) is algebraicly closed and \(V = W\) then \(T\) is a scalar operator.
+%\end{lemma}
+%
+%\begin{proof}
+%  For the first statement, it suffices to notice that \(\ker T\) and
+%  \(\operatorname{im} T\) are both subrepresentations. In particular, either
+%  \(\ker T = 0\) and \(\operatorname{im} T = W\) or \(\ker T = V\) and
+%  \(\operatorname{im} T = 0\). Now suppose \(K\) is algebraicly closed and \(V
+%  = W\). Let \(\lambda \in K\) be an eigenvalue of \(T\) and \(V_\lambda\) be
+%  its corresponding eigenspace. Given \(v \in V_\lambda\), \(T X v = X T v =
+%  \lambda \cdot X v\). In other words, \(V_\lambda\) is a subrepresentation.
+%  It then follows \(V_\lambda = V\), given that \(V_\lambda \ne 0\).
+%\end{proof}
 
-\begin{proof}
-  An order \(1\) \(G\)-invariant differential operator in \(G\) is simply a
-  left invariant derivation \(C^\infty(G) \to C^\infty(G)\). All other
-  \(G\)-invariant differetial operators are generated by such derivations. Now
-  recall that there is a canonical isomorphism of Lie algebras
-  \(\mathfrak{X}(G) \isoto \operatorname{Der}(G)\). This isomorphism takes left
-  invariant fields to left invariant derivations, so it restricts to an
-  isomorphism \(f : \mathfrak{g} \isoto \operatorname{Der}(G)^G \subset
-  \operatorname{Diff}(G)^G\) -- here \(\operatorname{Der}(G)^G \subset
-  \operatorname{Der}(G)\) denotes the Lie subalgebra of invariant derivations.
-
-  Since \(f\) is a homomorphism of Lie algebras, it can be extended to an
-  algebra homomorphism \(g : \mathcal{U}(\mathfrak{g}) \to
-  \operatorname{Diff}(G)^G\). We claim \(g\) is an isomorphim. To see that
-  \(g\) is injective, it suffices to notice
-  \[
-    g(X_1 \cdots X_n)
-    = g(X_1) \cdots g(X_n)
-    = f(X_1) \cdots f(X_n)
-    \ne 0
-  \]
-  for all nonzero \(X_1, \cdots, X_n \in \mathfrak{g}\) --
-  \(\operatorname{Diff}(G)^G\) is a domain. Since \(\mathcal{U}(\mathfrak{g})\)
-  is generated by the image of the inclusion \(\mathfrak{g} \to
-  \mathcal{U}(\mathfrak{g})\), this implies \(\ker g = 0\). Given that
-  \(\operatorname{Diff}(G)^G\) is generated by \(\operatorname{Der}(G)^G\),
-  this also goes to show \(g\) is surjective.
-\end{proof}
-
-% TODO: Comment on the fact this holds for algebraic groups too
-
-% TODO: Comment on the fact that modules of invariant differential operators
-% over G are precisely the same as representations of g
-
-\begin{theorem}[Poincaré-Birkoff-Witt]
-  Let \(\mathfrak{g}\) be a finite-dimensional Lie algebra over \(K\) and
-  \(\{X_i\}_i \subset \mathfrak{g}\) be a basis for \(\mathfrak{g}\). Then
-  \(\{X_{i_1} \cdot X_{i_2} \cdots X_{i_n} : n \ge 0, i_1 \le i_2 \le \cdots
-  \le i_n\}\) is a basis for \(\mathcal{U}(\mathfrak{g})\).
-\end{theorem}