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.