---- Let $\mathfrak{g}$ be a (finite-dimensional) [[semisimple Lie algebra|semisimple]] [[Lie algebra]] with [[Cartan subalgebra|CSA]] $\mathfrak{t}$, [[root system of a Lie algebra|corresponding]] [[root system]] $\Phi$, and let $\Delta=\{ \alpha_{1},\dots, \alpha_{\ell} \} \subset \Phi$ be a choice of [[root basis|simple roots]]. Let $V$ be a (not necessarily finite-dimensional) [[Lie algebra representation|representation]] of $\mathfrak{g}$. For such a [[Lie algebra representation|representation]] $V$, we can still define the [[simultaneously diagonalizable|simultaneous eigenspaces]] $V_{\lambda}$ ($\lambda \in \mathfrak{t}^{*}$). Analogously to [[on the weights of a representation|the finite-dimensional case]], we say $v \in V$ is a **highest weight vector** if $v \in V_{\lambda} - \{0 \}$ for some $\lambda \in \mathfrak{t}^{*}$ and $e_{\alpha} \cdot v=0$ for all $\alpha \in \Phi^{+}$, and in that case call $\lambda$ a **highest weight**. > [!definition] Definition. ([[highest weight module]]) > A [[Lie algebra representation|representation]] $V$ of $\mathfrak{g}$ is a **highest weight module** if $V$ is [[Lie algebra subrepresentation generated by a vector|generated by]] a highest weight vector $v$, in the sense that $V=\mathcal{U}(\mathfrak{g}) \cdot v$ for $\mathcal{U}(\mathfrak{g})$ the [[universal enveloping algebra]] corresponding to $\mathfrak{g}$. ^definition > [!note] Remark. > There is a proper [[inclusion map|inclusion]] $\{ \text{finite-dimensional } \mathfrak{g}\text{-irreps} \} \subsetneq \{ \text{highest weight modules} \}.$ Example 1 and Example 2 below show this. While one is generally most interested in the LHS, it is convenient to study the more general RHS. ^note > [!basicproperties] Properties. ("The five-part proposition") > Define [[Lie subalgebra|subalgebras]] $\mathfrak{n}^{+}=\bigoplus_{\alpha \in \Phi^{+}}\mathfrak{g}_{\alpha}$ and $\mathfrak{n}^{-}=\bigoplus_{\alpha \in \Phi^{-}}\mathfrak{g}_{\alpha}$ of $\mathfrak{g}$, so that $\mathfrak{g}=\mathfrak{n}^{-} \oplus \mathfrak{t} \oplus \mathfrak{n}^{+}$. Note that if $V$ is a highest weight module generated by highest weight vector $v \in V_{\lambda}$, then in fact[^3] $V=\mathcal{U}(\mathfrak{n}^{-}) \cdot v.$ Here are some consequences: > > > 1. Consider the [[cone and boundary of an element in a poset|cone]] $D(\lambda)=\{ \mu \in \mathfrak{t}^{*} : \mu \leq \lambda\}=\left\{ \lambda-\sum_{i}k_{i}\alpha_{i} : k_{i} \geq 0 \right\}$. Then $V=\bigoplus_{\mu \in D(\lambda)} V_{\mu}.$ > 2. If $W \subset V$ is a [[Lie algebra subrepresentation|subrepresentation]], then $W=\bigoplus_{\mu}W_{\mu}$. In other words, $W$ is a direct sum of its weight spaces > 3. $\text{dim }V_{\lambda}=1$ and $\text{dim }V_{\mu}$ is finite for all $\mu$ > 4. $V$ is [[irreducible Lie algebra representation|irreducible]] iff every highest weight vector lies in $V_{\lambda}$. By $(3)$, this means an irrep $V$ has a unique highest weight vector up to scaling > 5. $V$ contains a unique maximal proper subrepresentation, and therefore a unique irreducible [[quotient representation|quotient]]. > > > Here, the notation $\mu \leq \lambda$ is a [[poset|partial order]] on $\mathfrak{t}^{*}$ meaning $\lambda-\mu \in \Phi^{+}$. For example, $\mu \leq \lambda$ in the picture below. > > ![[Pasted image 20250517165118.png|500]] > Here is an example of what $D(\lambda)$ might look like: > ![[Pasted image 20250517170805.png|500]] > > > [!basicexample] Example 1. (Fin-dim $\mathfrak{g}$-irrep $\implies$ highest weight module) > If $V$ is finite-dimensional and [[irreducible Lie algebra representation|irreducible]], then $V$ is a highest weight module. Indeed, [[on the weights of a representation|any finite-dimensional representation of]] $\mathfrak{g}$ has a [[on the weights of a representation|highest weight vector]] $v$. Since $V$ is [[irreducible Lie algebra representation|irreducible]], the [[Lie algebra subrepresentation|subrepresentation]] $\mathcal{U}(\mathfrak{g}) \cdot v$ must in fact equal $V$ itself. ^basic-example > [!basicexample] Example 2. (Highest weight modules can be reducible and have multiple highest weight vectors) > Let $V$ be the infinite-dimensional [[vector space]] with [[basis]] $\{ v_{0},v_{1},\dots \}$ and define an $\mathfrak{sl}_{2}$-action on $V$ by the rules $e \cdot v_{0}=0, h \cdot v_{0}=0, f \cdot v_{i}=v_{i+1}$ and extending linearly using the bracket relation. Then we can calculate $h \cdot v_{n}=-2n v_{n}$ and $e \cdot v_{n}=n(1-n)v_{n-1}$.[^1] We see that $v_{0}$ is a highest weight vector (with highest weight $0$) [[Lie algebra subrepresentation generated by a vector|generating]] $V$. However, $v_{1}$ is also a highest weight vector: $h \cdot v_{1}=-2v_{1} \in V_{-2}$, $e \cdot v_{1}=1(1-1)v_{0}=0$. And $\mathcal{U}(\mathfrak{g}) \cdot v_{1}=\text{span}(v_{1},v_{2},\dots)$[^2]. This example shows the highest weight modules can be irreducuble ($\mathcal{U}(\mathfrak{g}) \cdot v_{1} \subsetneq V$) and have multiple highest weight vectors (here, $v_{0}$ and $v_{1}$). ^basic-example ---- #### [^1]: $v_{n}$ is gotten by hopping left $n$ times with $f$. [[weight space for sl2(C)|Each hop left]] decrements the weight by $2$. Since we started from $\lambda=0$ ($h \cdot v_{0}=0$), the weight lands at $-2n$. (The linked note currently assumes $V$ is an irrep, but that isn't necessary.) He omitted discussion of $e \cdot v_{n}$, so I omit too for now. [^2]: $\mathfrak{g}$ has [[basis]] $e,h,f$. $e \cdot v_{1}=0$, $h \cdot v_{1}=-2 v_{1}$, $f \cdot v_{1}=v_{2}$. Thus there is no way to apply an element from $\mathfrak{g}$ to $v_{1}$ and 'go backward' to $v_{0}$. Thus $\mathfrak{g} \cdot v_{1}$ doesn't contain $v_{0}$, but does contain $v_{1},v_{2},v_{3},\dots$ Arguing then that $\mathcal{U}(\mathfrak{g})$ *still* doesn't contain $v_{0}$ can be done e.g. with [[PBW Theorem]]. [^3]: The reasoning is an easy application of the [[PBW Theorem]]. Choose, for each root $\beta \in \Phi$, a nonzero element $e_{\beta} \in \mathfrak{g}_{\beta}$. Let $t_{1},\dots,t_{\ell}$ be a [[basis]] of $\mathfrak{t}$. By (the easy part of) the [[PBW Theorem]], $V=\mathcal{U}(\mathfrak{g}) \cdot v$ is [[submodule generated by a subset|spanned]] by elements of the form $e^{a_{1}}_{-\beta_{1}} \cdots e^{a_{n}}_{-\beta_{n}} t_{1}^{b_{1}} \cdots t_{\ell}^{b_{\ell}} e_{\beta_{1}}^{c_{1}} \cdots e^{c_{n}}_{\beta_{n}} \cdot v$where $\beta_{1},\dots,\beta_{n} \in \Phi^{+}$ are [[root basis|positive roots]] and $a_{i},b_{i},c_{i} \in \mathbb{Z}_{\geq 0}$. Since $v$ is a highest weight vector, $\mathfrak{n}^{+} \cdot v=0$ and $t \cdot v \in \text{span }v$ for all $t \in \mathfrak{t}$, so it suffices to only consider such expressions with $b_{i}=c_{i}=0$ for all $i$. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` [[dual vector space|dual vector]] [[special linear Lie subalgebra]]