lie-algebras-and-their-representations

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -761,23 +761,22 @@ combinatorial structures -- such as groups, quivers and associative algebras.
 An introductory exploration of some of this structures can be found in
 \cite{etingof}.
 
-We begin by noting that any \(\mathcal{U}(\mathfrak{g})\)-module \(V\) may be
+We begin by noting that any \(\mathcal{U}(\mathfrak{g})\)-module \(M\) may be
 regarded as a \(K\)-vector space endowed with a ``linear action'' of
 \(\mathfrak{g}\). Indeed, by restricting the action map
-\(\mathcal{U}(\mathfrak{g}) \to \operatorname{End}(V)\) to \(\mathfrak{g}
+\(\mathcal{U}(\mathfrak{g}) \to \operatorname{End}(M)\) to \(\mathfrak{g}
 \subset \mathcal{U}(\mathfrak{g})\) yields a homomorphism of Lie algebras
-\(\mathfrak{g} \to \mathfrak{gl}(V) = \operatorname{End}(V)\). In fact
+\(\mathfrak{g} \to \mathfrak{gl}(M) = \operatorname{End}(M)\). In fact
 Proposition~\ref{thm:universal-env-uni-prop} implies that given a vector space
-\(V\) there is a one-to-one correspondence between
-\(\mathcal{U}(\mathfrak{g})\)-module structures for \(V\) and homomorphisms
-\(\mathfrak{g} \to \mathfrak{gl}(V)\). This leads us to the following
+\(M\) there is a one-to-one correspondence between
+\(\mathcal{U}(\mathfrak{g})\)-module structures for \(M\) and homomorphisms
+\(\mathfrak{g} \to \mathfrak{gl}(M)\). This leads us to the following
 definition.
 
 \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 \(\rho : \mathfrak{g} \to
-  \mathfrak{gl}(V)\).
+  of \(\mathfrak{g}\)} is a \(K\)-vector space endowed with a homomorphism of
+  Lie algebras \(\rho : \mathfrak{g} \to \mathfrak{gl}(V)\).
 \end{definition}
 
 Hence there is a one-to-one correspondence between representations of
@@ -798,63 +797,45 @@ Hence there is a one-to-one correspondence between representations of
 \end{example}
 
 \begin{example}
-  Given a subalgebra \(\mathfrak{g} \subset \mathfrak{gl}_n(K)\), the inclusion
-  \(\mathfrak{g} \to \mathfrak{gl}_n(K)\) endows \(K^n\) with the structure of
-  a representation of \(\mathfrak{g}\), known as \emph{the natural
-  representation of \(\mathfrak{g}\)}.
-\end{example}
-
-It is usual practice to write simply \(X \cdot v\) or \(X v\) for \(\rho(X) v\)
-when the map \(\rho\) is clear from the context. For instance, one might
-say\dots
-
-\begin{example}\label{ex:sl2-polynomial-rep}
-  The space \(K[x, y]\) is a representation of \(\mathfrak{sl}_2(K)\) with
-  \begin{align*}
-    e \cdot p & = x \frac{\mathrm{d}}{\mathrm{d}y} p &
-    f \cdot p & = y \frac{\mathrm{d}}{\mathrm{d}x} p &
-    h \cdot p & =
-    \left(
-      x \frac{\mathrm{d}}{\mathrm{d}x} - y \frac{\mathrm{d}}{\mathrm{d}y}
-    \right) p
-  \end{align*}
-\end{example}
-
-\begin{example}
-  Given a Lie algebra \(\mathfrak{g}\), the algebra
-  \(\mathcal{U}(\mathfrak{g})\) is a \(\mathfrak{g}\)-module, where the action
-  of \(\mathfrak{g}\) on \(\mathcal{U}(\mathfrak{g})\) is given by left
-  multiplication. This is known as \emph{the regular representation of
+  Given a Lie algebra \(\mathfrak{g}\), the map \(\rho : \mathfrak{g} \to
+  \mathfrak{gl}(\mathcal{U}(\mathfrak{g}))\) given by left multiplication
+  endows \(\mathcal{U}(\mathfrak{g})\) with the structure of a representation
+  of \(\mathfrak{g}\), known as \emph{the regular representation of
   \(\mathfrak{g}\)}.
+  \[
+    \arraycolsep=1.4pt
+    \begin{array}[t]{rl}
+      \rho : \mathfrak{g} & \to \mathfrak{gl}(\mathcal{U}(\mathfrak{g})) \\
+      X & \mapsto
+      \begin{array}[t]{rl}
+        \rho(X) : \mathcal{U}(\mathfrak{g}) & \to \mathcal{U}(\mathfrak{g}) \\
+        u & \mapsto X \cdot u
+      \end{array}
+    \end{array}
+  \]
 \end{example}
 
 \begin{example}
-  Given a Lie algebra \(\mathfrak{g}\) and \(\mathfrak{g}\)-modules \(V\) and
-  \(W\), 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)(v)  & = - f (X v)                     \\
-    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.
+  Given a subalgebra \(\mathfrak{g} \subset \mathfrak{gl}_n(K)\), the inclusion
+  \(\mathfrak{g} \to \mathfrak{gl}_n(K)\) endows \(K^n\) with the structure of
+  a representation of \(\mathfrak{g}\), known as \emph{the natural
+  representation of \(\mathfrak{g}\)}.
 \end{example}
 
-Of course, there is also a natural notion of \emph{transformations} between
-representations.
+When the map \(\rho : \mathfrak{g} \to \mathfrak{gl}(V)\) is clear from context
+it is usual practive to denote the \(K\)-endomorphism \(\rho(X) : V \to V\),
+\(X \in \mathfrak{g}\), simply by \(X\!\restriction_V\). This leads us to the
+natural notion of \emph{transformations} between representations.
 
 \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 commutes with
-  the action of \(\mathfrak{g}\) on \(V\) and \(W\), in the sense that the
-  diagram
+  of \(\mathfrak{g}\), we call a \(K\)-linear map \(f : V \to W\) \emph{an
+  intertwining operator}, or \emph{an intertwiner}, if it commutes with the
+  action of \(\mathfrak{g}\) on \(V\) and \(W\), in the sense that the diagram
   \begin{center}
     \begin{tikzcd}
-      V \rar{T} \dar[swap]{X} & W \dar{X} \\
-      V \rar[swap]{T}         & W
+      V \rar{f} \dar[swap]{X\!\restriction_V} & W \dar{X\!\restriction_W} \\
+      V \rar[swap]{f}                         & W
     \end{tikzcd}
   \end{center}
   commutes for all \(X \in \mathfrak{g}\). We denote the space of all
@@ -864,101 +845,140 @@ representations.
 \end{definition}
 
 The collection of representations of a fixed Lie algebra \(\mathfrak{g}\) thus
-forms a category, which we call \(\mathfrak{g}\text{-}\mathbf{Mod}\). This
-allow us formulate the correspondence between representations of
-\(\mathfrak{g}\) and \(\mathcal{U}(\mathfrak{g})\)-modules in more precise
-terms.
+forms a category, which we call \(\mathbf{Rep}(\mathfrak{g})\). This allow us
+formulate the correspondence between representations of \(\mathfrak{g}\) and
+\(\mathcal{U}(\mathfrak{g})\)-modules in more precise terms.
 
 \begin{proposition}
-  There is a natural equivalence of categories
-  \(\mathfrak{g}\text{-}\mathbf{Mod} \isoto
-  \mathcal{U}(\mathfrak{g})\text{-}\mathbf{Mod}\).
+  There is a natural equivalence of categories \(\mathbf{Rep}(\mathfrak{g})
+  \isoto \mathcal{U}(\mathfrak{g})\text{-}\mathbf{Mod}\).
 \end{proposition}
 
 \begin{proof}
-  We have seen that given a \(K\)-vector space \(V\) there is a one-to-one
-  correspondence between \(\mathfrak{g}\)-module structures for \(V\) -- i.e.
-  homomorphisms \(\mathfrak{g} \to \mathfrak{gl}(V)\) -- and
-  \(\mathcal{U}(\mathfrak{g})\)-module structures for \(V\) -- i.e.
-  homomorphisms \(\mathcal{U}(\mathfrak{g}) \to \operatorname{End}(V)\). This
-  gives us a map that takes objects in \(\mathfrak{g}\text{-}\mathbf{Mod}\) to
-  objects in \(\mathcal{U}(\mathfrak{g})\text{-}\mathbf{Mod}\).
-
-  As for the corresponding maps \(\operatorname{Hom}_{\mathfrak{g}}(V, W) \to
-  \operatorname{Hom}_{\mathcal{U}(\mathfrak{g})}(V, W)\), it suffices to note
-  that a \(K\)-linear map between representations \(V\) and \(W\) is an
+  We have seen that given a \(K\)-vector space \(M\) there is a one-to-one
+  correspondence between \(\mathfrak{g}\)-representation structures for \(M\)
+  -- i.e. homomorphisms \(\mathfrak{g} \to \mathfrak{gl}(M)\) -- and
+  \(\mathcal{U}(\mathfrak{g})\)-module structures for \(M\) -- i.e.
+  homomorphisms \(\mathcal{U}(\mathfrak{g}) \to \operatorname{End}(M)\). This
+  gives us a surjective map that takes objects in
+  \(\mathbf{Rep}(\mathfrak{g})\) to objects in
+  \(\mathcal{U}(\mathfrak{g})\text{-}\mathbf{Mod}\).
+
+  As for the corresponding maps \(\operatorname{Hom}_{\mathfrak{g}}(M, N) \to
+  \operatorname{Hom}_{\mathcal{U}(\mathfrak{g})}(M, N)\), it suffices to note
+  that a \(K\)-linear map between representations \(M\) and \(N\) is an
   intertwiner if, and only if it is a homomorphism of
   \(\mathcal{U}(\mathfrak{g})\)-modules. Our functor thus takes an intertwiner
-  \(V \to W\) to itself.
+  \(M \to N\) to itself. In particular, our functor
+  \(\mathbf{Rep}(\mathfrak{g}) \to \mathfrak{g}\text{-}\mathbf{Mod}\) is fully
+  faithfull.
 \end{proof}
 
-\begin{note}
-  We should point out that the monoidal structure of
-  \(\mathfrak{g}\text{-}\mathbf{Mod}\) is \emph{not} the same as that of
-  \(\mathcal{U}(\mathfrak{g})\text{-}\mathbf{Mod}\). In other words, \(V
-  \otimes W\) is not the same thing as \(V \otimes_{\mathcal{U}(\mathfrak{g})}
-  W\) and hence the equivalence \(\mathfrak{g}\text{-}\mathbf{Mod} \isoto
-  \mathcal{U}(\mathfrak{g})\text{-}\mathbf{Mod}\) does not preserve tensor
-  products.
-\end{note}
+The language of representation is thus equivalent to that of
+\(\mathcal{U}(\mathfrak{g})\)-modules, which we call
+\emph{\(\mathfrak{g}\)-modules}. Correspondly, we refer to the category
+\(\mathcal{U}(\mathfrak{g})\text{-}\mathbf{Mod}\) as
+\(\mathfrak{g}\text{-}\mathbf{Mod}\). The terms
+\emph{\(\mathfrak{g}\)-submodule} and \emph{\(\mathfrak{g}\)-homomorphism}
+should also be self-explanatory. To avoid any confusion, we will, for the most
+part, exclusively use the language of \(\mathfrak{g}\)-modules. It should be
+noted, however, that both points of view are profitable.
+
+For starters, the notation for \(\mathfrak{g}\)-modules is much cleaner than
+that of representations: it is much easier to write ``\(X \cdot m\)'' than
+``\((\rho(X))(m)\)'' or even ``\(X\!\restriction_M(m)\)''. By using the
+language of \(\mathfrak{g}\)-modules we can also rely on the general theory of
+modules over associative algebras -- which we assume the reader is already
+familiarized with. On the other hand, it is usually easier to express geometric
+considerations in terms of the language representations, particularly in group
+representation theory.
+
+Often times it is easier to define a \(\mathfrak{g}\)-module \(M\) in terms of
+the corresponding map \(\mathfrak{g} \to \mathfrak{gl}(M)\) -- this is
+technique we will use throughout the text. In general, the equivalence between
+both languages makes it clear that to understand the action of
+\(\mathcal{U}(\mathfrak{g})\) on \(M\) it suffices to undertand the action of
+\(\mathfrak{g} \subset \mathcal{U}(\mathfrak{g})\). For instance, for defining
+a \(\mathfrak{g}\)-module \(M\) it suffices to define the action of each \(X
+\in \mathfrak{g}\) and verify this action respects the commutator relations of
+\(\mathfrak{g}\) -- indeed, \(\mathfrak{g}\) generates
+\(\mathcal{U}(\mathfrak{g})\) as an algebra, and the only relations between
+elements of \(\mathfrak{g}\) are the ones derived from the commutator
+relations.
+
+\begin{example}\label{ex:sl2-polynomial-rep}
+  The space \(K[x, y]\) is a \(\mathfrak{sl}_2(K)\)-module with
+  \begin{align*}
+    e \cdot p & = x \frac{\mathrm{d}}{\mathrm{d}y} p &
+    f \cdot p & = y \frac{\mathrm{d}}{\mathrm{d}x} p &
+    h \cdot p & =
+    \left(
+      x \frac{\mathrm{d}}{\mathrm{d}x} - y \frac{\mathrm{d}}{\mathrm{d}y}
+    \right) p
+  \end{align*}
+\end{example}
+
+\begin{example}
+  Given a Lie algebra \(\mathfrak{g}\) and \(\mathfrak{g}\)-modules \(M\) and
+  \(N\), the space \(\operatorname{Hom}(M, N)\) of \(K\)-linear maps \(M \to
+  N\) is a \(\mathfrak{g}\)-module where \((X \cdot f)(m) = X \cdot f(m) - f(X
+  \cdot m)\) for all \(X \in \mathfrak{g}\) and \(f \in \operatorname{Hom}(M,
+  N)\). In particular, if we take \(N = K\) the trivial
+  \(\mathfrak{g}\)-module, we can view \(M^*\) -- the dual of \(M\) in the
+  category of \(K\)-vector spaces -- as a \(\mathfrak{g}\)-module where \((X
+  \cdot f)(m) = - f(X \cdot m)\) for all \(f : M \to K\).
+\end{example}
 
 The fundamental problem of representation theory is a simple one: classifying
 all representations of a given Lie algebra up to isomorphism. However,
 understanding the relationship between representations is also of huge
 importance. In other words, to understand the whole of
 \(\mathfrak{g}\text{-}\mathbf{Mod}\) we need to study the collective behavior
-of representations -- as opposed to individual examples. To that end, we
-define\dots
-
-\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}, or \emph{a \(\mathfrak{g}\)-submodule}, 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}
+of representations -- as opposed to individual examples. For instance, we may
+consider \(\mathfrak{g}\)-submodules, quotients and tensor products.
 
 \begin{example}\label{ex:sl2-polynomial-subrep}
   Let \(K[x, y]\) be the \(\mathfrak{sl}_2(K)\)-module as in
   Example~\ref{ex:sl2-polynomial-rep}. Since \(e\), \(f\) and \(h\) all
-  preserve the degree of monomials, the space \(K[x, y]^{(n)} = \bigoplus_{k +
-  \ell = n} K x^k y^\ell\) of homogeneous polynomials of degree \(n\) is a
-  finite-dimensional subrepresentation of \(K[x, y]\).
+  preserve the degree of monomials, the space \(K[x, y]^{(d)} = \bigoplus_{k +
+  \ell = d} K x^k y^\ell\) of homogeneous polynomials of degree \(d\) is a
+  finite-dimensional \(\mathfrak{sl}_2(K)\)-submodule of \(K[x, y]\).
 \end{example}
 
 \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.
+  Given a Lie algebra \(\mathfrak{g}\), a \(\mathfrak{g}\)-module \(M\) and \(m
+  \in M\), the subspace \(\mathcal{U}(\mathfrak{g}) \cdot m = \{ u \cdot m : u
+  \in \mathcal{U}(\mathfrak{g}) \}\) is a \(\mathfrak{g}\)-submodule of \(M\),
+  which we call \emph{the submodule generated by \(m\)}.
 \end{example}
 
 \begin{example}
-  Given a Lie algebra \(\mathfrak{g}\) and representations \(V\) and \(W\) of
-  \(\mathfrak{g}\), the exterior and symmetric products \(V \wedge W\) and \(V
-  \odot W\) are both representations of \(\mathfrak{g}\): they are quotients of
-  \(V \otimes W\). In particular, the exterior and symmetric powers \(\wedge^n
-  V\) and \(\operatorname{Sym}^n V\) are \(\mathfrak{g}\)-modules.
+  Given a Lie algebra \(\mathfrak{g}\) and \(\mathfrak{g}\)-modules \(M\) and
+  \(N\), the space \(M \otimes N = M \otimes_K N\) -- the tensor product over
+  \(K\) -- is a \(\mathfrak{g}\)-module where \(X \cdot (m \otimes n) = X \cdot
+  m \otimes n + m \otimes X \cdot n\). The exterior and symmetric products \(M
+  \wedge N\) and \(M \odot N\) are both quotients of \(M \otimes N\) by
+  \(\mathfrak{g}\)-submodules. In particular, the exterior and symmetric powers
+  \(\wedge^r M\) and \(\operatorname{Sym}^r M\) are \(\mathfrak{g}\)-modules.
 \end{example}
 
-\begin{example}
-  Given a Lie algebra \(\mathfrak{g}\), a \(\mathfrak{g}\)-module \(V\) and \(v
-  \in V\), the subspace \(\mathcal{U}(\mathfrak{g}) \cdot v = \{ u v : u \in
-  \mathcal{U}(\mathfrak{g}) \}\) is a subrepresentation of \(V\), which we call
-  \emph{the subrepresentation generated by \(v\)}.
-\end{example}
+\begin{note}
+  We should point out that the monoidal structure of
+  \(\mathfrak{g}\text{-}\mathbf{Mod}\) we've just described is \emph{not} the
+  usual one. In other words, \(M \otimes N\) is not the same thing as \(M
+  \otimes_{\mathcal{U}(\mathfrak{g})} N\).
+\end{note}
 
 It is also interesting to consider the relationship between representations of
 separate algebras. In particular, we may define\dots
 
 \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
-  \(\mathfrak{h}\). Any homomorphism of \(\mathfrak{g}\)-modules \(V \to W\) is
+  Given a \(\mathfrak{g}\)-module \(M\), denote by
+  \(\operatorname{Res}_{\mathfrak{h}}^{\mathfrak{g}} M = M\) the
+  \(\mathfrak{h}\)-module where the action of \(\mathfrak{h}\) is given by
+  restricting the map \(\mathfrak{g} \to \mathfrak{gl}(M)\) to
+  \(\mathfrak{h}\). Any homomorphism of \(\mathfrak{g}\)-modules \(M \to N\) is
   also a homomorphism of \(\mathfrak{h}\)-modules and this construction is
   clearly functorial.
   \[
@@ -969,26 +989,24 @@ separate algebras. In particular, we may define\dots
 \end{example}
 
 \begin{example}
-  Given a Lie algebra \(\mathfrak{g}\), the adjoint representation of
-  \(\mathfrak{g}\) is a subrepresentation of the restriction of the adjoint
-  representation of \(\mathcal{U}(\mathfrak{g})\) to \(\mathfrak{g}\).
+  Given a Lie algebra \(\mathfrak{g}\), the adjoint \(\mathfrak{g}\)-module is
+  a submodule of the restriction of the adjoint
+  \(\mathcal{U}(\mathfrak{g})\)-module to \(\mathfrak{g}\).
 \end{example}
 
 Surprisingly, this functor has a right adjoint.
 
 \begin{example}
   Let \(\mathfrak{g}\) be a Lie algebra and \(\mathfrak{h}\) be a subalgebra.
-  Given a representation \(V\) of \(\mathfrak{h}\), denote by
-  \(\operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} V\) the representation of
-  \(\mathfrak{h}\) corresponding to the \(\mathcal{U}(\mathfrak{g})\)-module
-  \(\mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{U}(\mathfrak{h})} V\) -- where
-  the right \(\mathcal{U}(\mathfrak{h})\)-module structure of
-  \(\mathcal{U}(\mathfrak{g})\) is given by right multiplication. Any
-  homomorphism of \(\mathfrak{h}\)-modules \(T : V \to W\) induces a
-  homomorphism \(\operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} T =
-  \operatorname{Id} \otimes T :
-  \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} V \to
-  \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} W\) and this construction is
+  Given a \(\mathfrak{h}\)-module \(M\), let
+  \(\operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} M =
+  \mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{U}(\mathfrak{h})} M\) -- where
+  the right \(\mathfrak{h}\)-module structure of \(\mathcal{U}(\mathfrak{g})\)
+  is given by right multiplication. Any \(\mathfrak{h}\)-homomorphism \(f : M
+  \to N\) induces a \(\mathfrak{g}\)-homomorphism
+  \(\operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} f = \operatorname{id}
+  \otimes f : \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} M \to
+  \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} N\) and this construction is
   clearly functorial.
   \[
     \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} :
@@ -998,24 +1016,24 @@ Surprisingly, this functor has a right adjoint.
 
 \begin{proposition}
   Given a Lie algebra \(\mathfrak{g}\), a subalgebra \(\mathfrak{h} \subset
-  \mathfrak{g}\), a representation \(V\) of \(\mathfrak{h}\) and a
-  representation \(W\) of \(\mathfrak{g}\), the map
+  \mathfrak{g}\), a \(\mathfrak{h}\)-module \(M\) and a \(\mathfrak{g}\)-module
+  \(N\), the map
   \[
     \arraycolsep=1.4pt
     \begin{array}[t]{rl}
     \alpha :
     \operatorname{Hom}_{\mathfrak{g}}(
-      \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} V,
-      W
+      \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} M,
+      N
     ) & \to
     \operatorname{Hom}_{\mathfrak{h}}(
-      V,
-      \operatorname{Res}_{\mathfrak{h}}^{\mathfrak{g}} W
+      M,
+      \operatorname{Res}_{\mathfrak{h}}^{\mathfrak{g}} N
     ) \\
-    T & \mapsto
+    f & \mapsto
     \begin{array}[t]{rl}
-    \alpha(T) : V & \to \operatorname{Res}_{\mathfrak{h}}^{\mathfrak{g}} W \\
-                v & \mapsto T (1 \otimes v)
+    \alpha(f) : M & \to \operatorname{Res}_{\mathfrak{h}}^{\mathfrak{g}} N \\
+                m & \mapsto f(1 \otimes m)
     \end{array}
     \end{array}
   \]
@@ -1031,17 +1049,17 @@ Surprisingly, this functor has a right adjoint.
     \begin{array}[t]{rl}
     \beta :
     \operatorname{Hom}_{\mathfrak{h}}(
-      V,
-      \operatorname{Res}_{\mathfrak{h}}^{\mathfrak{g}} W
+      M,
+      \operatorname{Res}_{\mathfrak{h}}^{\mathfrak{g}} N
     ) & \to
     \operatorname{Hom}_{\mathfrak{g}}(
-      \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} V,
-      W
+      \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} M,
+      N
     ) \\
-    T & \mapsto
+    f & \mapsto
     \begin{array}[t]{rl}
-    \beta(T) : \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} V & \to W \\
-               u \otimes v & \mapsto u \cdot T v
+    \beta(f) : \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} M & \to N \\
+               u \otimes m & \mapsto u \cdot f(m)
     \end{array}
     \end{array}
   \]
@@ -1054,15 +1072,16 @@ interesting construction is\dots
 
 \begin{example}
   Let \(\mathfrak{g}\) and \(\mathfrak{h}\) be Lie algebras. Given a
-  \(\mathfrak{g}\)-module \(V\) and a \(\mathfrak{h}\)-module \(W\), the space
-  \(V \boxtimes W = V \otimes W\) has the natural structure of a \(\mathfrak{g}
-  \oplus \mathfrak{h}\)-module, where the action of \(\mathfrak{g} \oplus
-  \mathfrak{h}\) is given by
+  \(\mathfrak{g}\)-module \(M\) and a \(\mathfrak{h}\)-module \(N\), the space
+  \(M \boxtimes N = M \otimes_K N\) has the natural structure of a
+  \(\mathfrak{g} \oplus \mathfrak{h}\)-module, where the action of
+  \(\mathfrak{g} \oplus \mathfrak{h}\) is given by
   \[
-    (X + Y)(v \otimes w) = X v \otimes w + v \otimes Y w
+    (X + Y) \cdot (m \otimes n) = X \cdot m \otimes n + m \otimes Y \cdot n
   \]
 \end{example}
 
-This concludes our initial remarks on representations. In the following
-chapters we will explore the fundamental problem of representation theory: that
-of classifying all representations of a given algebra up to isomorphism.
+This concludes our initial remarks on \(\mathfrak{g}\)-modules. In the
+following chapters we will explore the fundamental problem of representation
+theory: that of classifying all representations of a given algebra up to
+isomorphism.