---- 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]]. Recall that, in general, if $A$ is an [[algebra|associative algebra]] and $I \subset A$ is a left ideal ($x \cdot I \subset I$ for all $x \in A$), then the [[quotient module|quotient]] [[vector space]] $A / I$ has the structure of an $A$-[[representation of an algebra|representation]] via $x \cdot (y+I)=x \cdot y + I$. > [!definition] Definition. ([[Verma module]]) > Let $\lambda \in \mathfrak{t}^{*}$. > > The **Verma module for $\lambda$**, denoted $(M(\lambda), m_{\lambda})$ is the 'biggest' [[highest weight module]] with highest weight $\lambda$, in the sense that it satisfies the following [[universal property]]. For any [[highest weight module]] $V$ [[Lie algebra subrepresentation generated by a vector|generated by]] [[on the weights of a representation|highest weight vector]] $v \in V_{\lambda}$, there is a unique [[surjection|surjective]] [[morphism of Lie algebra representations|morphism]] of $\mathfrak{g}$-[[Lie algebra representation|representations]] $M(\lambda) \to V$ sending $m_{\lambda} \mapsto v$. > > > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage{amsfonts} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBoBGAXVJADcBDAGwFcYkQAKAWVIAIBbAPoAdYY3r8ARlHoBKEAF9S6TLnyEU5CtTpNW7DgDVSteUpXY8BImQBMOhizaIQo-vRwALAGYAnegDWwADmCorKIBiW6jakxA56ziBCouJSMuEWataacQlO7LSKOjBQwfBEoH4Q-Ei2NDgQSMTmINW1iPUgjUjkre3NDU2IWroFLqIwAB5YcDhwvACEvKIMvmieWCA0njD0UOw4AO4Qu-sI-b41SADMQ0gALAqUCkA > \begin{tikzcd} > m_\lambda \arrow[r] & v \\ > {(M(\lambda), m_\lambda)} \arrow[r, "\exists ! \varphi", two heads] & {(V,v)} \\ > \mathfrak{g} \arrow[u] \arrow[ru] & > \end{tikzcd} > \end{document} > ``` > > > > > > Thus, $M(\lambda)$ is the biggest highest weight module with highest weight $\lambda$, in the sense that any other [[first isomorphism theorem for modules|is a quotient of it]]. This defines $M(\lambda)$ [[terminal objects are unique up to a unique isomorphism|up to isomorphism]], if it exists. > > Indeed, exist it does: let $J(\lambda)$ be the left ideal of $\mathcal{U}(\mathfrak{g})$ generated by > - $\{ e_{\beta}: \beta \in \Phi^{+} , e_{\beta} \in \mathfrak{g}_{\beta} \text{ choice of generator}\}$ > - $\{t-\lambda(t) \cdot 1: t \in \mathfrak{t}\}$ > > where $e_{\beta},t$ are viewed as elements of $\mathfrak{g} \hookrightarrow \mathcal{U}(\mathfrak{g})$. Then $M(\lambda)$ is instantiated by the quotient $M(\lambda)= \frac{\mathcal{U}(\mathfrak{g})}{J(\lambda)}$, with $m_{\lambda}:=[1]=1+J(\lambda)$. $M(\lambda)$ is an [[representation of an algebra|algebra representation]] of $\mathcal{U}(\mathfrak{g})$; [[universal enveloping algebra|equivalently]] (by restriction) a [[Lie algebra representation]] of $\mathfrak{g}$. > > Note that we have for all $t \in \mathfrak{t}$ that $(t - \lambda(t) \cdot 1) \cdot m_{\lambda}=0, \text{ hence } t \cdot m_{\lambda}=\lambda(t)m_{\lambda}$ > and for all $\beta \in \Phi^{+}$: $e_{\beta} \cdot m_{\lambda}=0$ > which confirms that $m_{\lambda}$ is indeed a [[on the weights of a representation|highest weight vector]].[^1] > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage{amsfonts} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBoBmAXVJADcBDAGwFcYkQAdDgW3pwAsAZgCd6Aa2ABzAL4hppdJlz5CKMgCZqdJq3ZdeAgMZNgAVWkAKfXyGiJMgJRyFIDNjwEiZAIxaGLNkQQCwBZKw5Gem4AIyh6B1IAAm4AfS5ImLineUV3FSJvUl8af10giwA1UlpslzdlTzVSYj8dQJBU9KjY+mdchtVkQpaStvZaOS0YKEl4IlARCG4kQpAcCCR1UYC9DmF+CBTgLhwYAA8cYDRhCAAraVkafhh6KHYcAHcIZ9eEHJBFstEGQ1htEKtSu0uPgcL1-oCkCD1khyNsypw9gcQDRItEYIwAApKDyqEDCLCSfg4PoAm5AraglFPF5vIKfb4shA4rBgdpxODPN5oqEcc5YOA4OAAQkSXAYwjQ-CwNIRiAALDRkYgAKya+hYRjsXhoCUbaSUaRAA > \begin{tikzcd} > m_\lambda \arrow[r, maps to] & v \\ > {(M(\lambda), m_\lambda)} \arrow[r, "\exists! \varphi", two heads, dashed] & {(V,v)} \\ > \mathcal{U}(\mathfrak{g}) \arrow[u, "\rho_{\text{proj}}", two heads] & \\ > \mathfrak{g} \arrow[u, "\iota"] \arrow[ruu, "\rho"'] & > \end{tikzcd} > \end{document} > ``` > > > [!definition] Definition. ($V(\lambda)$) > By property $5$ in [[highest weight module]], $M(\lambda)$ has a unique maximal proper [[Lie algebra subrepresentation|subrepresentation]], and therefore a unique [[irreducible Lie algebra representation|irreducible]] [[quotient representation|quotient]]. We denote this quotient $V(\lambda)$. > > Slogan: "$M(\lambda)$ is the biggest highest weight module, and $V(\lambda)$ is the smallest". > ^definition ---- #### - [ ] bring over justification that $M(\lambda)=\frac{\mathcal{U}(\mathfrak{g})}{J(\lambda)}$ satisfies the [[universal property]] [^1]: Actually, we need to check also that $m_{\lambda}$ is nonzero. This follows from the [[PBW Theorem]] (see handwritten notes). ---- #### 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 > ```