---- > [!theorem] Theorem. ([[the classification of complex irreducible semisimple Lie algebra representations]]) > Let $\mathfrak{g}$ be a [[semisimple Lie algebra|semisimple]] $\mathbb{C}$-[[Lie algebra]] [[classification of complex semisimple Lie algebras|of]] [[root system of a Lie algebra|root system]] $\Phi$, with [[Cartan subalgebra|CSA]] $\mathfrak{t}$ and [[root space decomposition of a Lie algebra|root space decomposition]] $\mathfrak{g}=\mathfrak{t}\oplus \bigoplus_{\alpha \in \Phi \subset \mathfrak{t}^{*}}\mathfrak{g}_{\alpha}.$ > This data classifies the [[irreducible Lie algebra representation|irreducible]] [[Lie algebra representation|representations]] $V$ of $\mathfrak{g}$ (up to [[morphism of Lie algebra representations|isomorphism]]). Specifically: any $\lambda \in \mathfrak{t^{*}}$ gives rise to a [[Verma module|unique irreducible]] [[highest weight module]] $V(\lambda)$, and the map $\lambda \mapsto V(\lambda)$ is in fact a [[bijection]]. This map restricts in turn to a [[bijection]] between finite-dimensional $\mathfrak{g}$-irreps and [[dominant weight in a root system|dominant]] elements of the [[root and weight lattice of a root system|weight lattice]] $X$. > > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage{amsfonts} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB13hOcYAPHMABmWMADoABFCwBbSVgBOCmGjgBfTmoD0nAMYEA5iDWl0mXPkIoyARiq1GLNp27teA4IuVQmurACMGGAkACywDEPgcCQB3GHCQ6JloJiD1TR12fTAjEzNsPAIiG1I7anpmVkQOdhk6HBChBToAa2AcNQA9ACpjUxAMAsti8nsKp2qXHn5BKAgZUTowaLUJTgY6GX8oOjX2UQkADU1jexgoA3giUCb5pDIQHAgkEpBIuig2SDBWagZRKogOY4XifcqOQF6OhoEB-Oj+GAMAAK5kKVhAomwsD6NwUd0QAGZqE8kAAmajvT7Vb6-ED-H5sYGg2EOSrOLLQlkbBHI1HDaqYrDYvIgW4yF7E56IclvGAfJBgVIMEVismSl4qvHixAPEmEzX4omPKXENQUNRAA > \begin{tikzcd} > \{\text{fin. dim. irreps}\}/\cong \arrow[d, "\cap" description, no head, dotted] \arrow[r] & \{\text{dominant } \lambda \in X\} \arrow[d, "\cap" description, no head, dotted] \arrow[l] \\ > \{\text{irred. highest wt. modules}\}/\cong \arrow[r] & \mathfrak{t}^* \arrow[l] > \end{tikzcd} > \end{document} > ``` > > The lower [[bijection]] is very easy to prove with tools developed thus far, as is one direction of the upper [[bijection]]. Showing ($\lambda \in X$ dominant $\implies$ $V(\lambda)$ fin. dim.) constitutes a majority of the proof. > > [!basicexample] > - $\mathfrak{sl}_{2}(\mathbb{C})$ has [[root system of a Lie algebra|root system]] $\Phi$ of type $A_{1}$. Its sole fundamental weight $\omega_{1}$, root basis $\{ 2e_{1} \} \subset \mathbb{R}$, and [[dominant weight in a root system|dominant weights]] are pictured below, each dominant weight labeled with the [[classification of the irreps of sl2 over C|corresponding]] [[irreducible Lie algebra representation|irrep]] $V(n)$, $n \in \mathbb{Z}_{\geq 0}$. > ![[Pasted image 20250515173957.png]] > > - $\mathfrak{sl}_{3}(\mathbb{C})$ has [[root system]] $\Phi$ of type $A_{2}$. The finite-dimensional irreps are depicted as the blue dots below (positive $\mathbb{Z}$-linear combinations of the [[root and weight lattice of a root system|fundamental weights]] $\omega_{\alpha},\omega_{\beta}$) > > ![[Pasted image 20250518162927.png|500]] > > $\mathfrak{so}_{5}$ has [[root system]] $B_{2}$. $\mathfrak{sp}_{4}$ has [[root system]] $C_{2}$. $B_{2}\cong C_{2}$; hence the finite-dimensional irreps of $\mathfrak{so}_{5}$ and $\mathfrak{sp}_{4}$ are depicted below > > ![[Pasted image 20250518164055.png|500]] > > > [!note] What this doesn't tell us. > We now have a [[classification of complex semisimple Lie algebras]] and a [[the classification of complex irreducible semisimple Lie algebra representations|classification of complex semisimple Lie algebra representations]], but the story is not over! See [[character of a Lie algebra representation]] and the ensuing discussions. ^note ^basic-example > [!proof]- Proof. ([[the classification of complex irreducible semisimple Lie algebra representations]]) > **Proof of the lower bijection $\lambda \mapsto V(\lambda)$.** Surjectivity: Let $V$ be an irreducible [[highest weight module]] generated by highest weight vector $v_{\lambda} \in V_{\lambda}$. By the [[universal property]] of [[Verma module|Verma modules]], $\mathfrak{g}$ is a quotient of $M(\lambda)$ for some [[Verma module]] $M(\lambda)$. But $M(\lambda)$ has a unique irreducible quotient, namely $V(\lambda)$. So $V=V(\lambda)$. Injectivity: we need to show $\lambda$ is unique. This follows from [[highest weight module|the five-part proposition on highest weight modules]], which says as such. > **Proof that $V(\lambda)$ finite-dimensional $\implies$ $\lambda$ $\in$ $X$ is dominant.** This has all been said before: a highest weight of any finite-dimensional representation is always dominant (a consequence of $\mathfrak{sl}_{2}$ theory): [[on the weights of a representation]]. > **Proof that $\lambda \in X$ dominant $\implies$ $V(\lambda)$ finite-dimensional.** This fact is thus where the main task lies. (Written notes) ---- #### [[on the weights of a representation]] [[special linear Lie subalgebra]] [[orthogonal Lie algebra]] ----- #### 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 > ```