---- > [!definition] Definition. ([[root string in a Lie algebra]]) > Let $\mathfrak{g}$ be a [[Lie algebra]] with [[root space decomposition of a Lie algebra|roots]] $\Phi$. Let $\alpha, \beta \in \Phi$ and assume $\beta \neq \pm \alpha$ ([[root spaces are one-dimensional|and hence]] $\beta \neq c \alpha$ for any $c \in \mathbb{C}$). Consider the [[vector space]] $V := \bigoplus_{k \in \mathbb{Z}} \mathfrak{g}_{\beta+k\alpha} \subset \mathfrak{g}$ note that $\beta + k\alpha$ is never zero and so no element from $\mathfrak{g}_{0}=\mathfrak{t}$ lives in $V$. > Since $V$ is stable under the [[adjoint representation|adjoint action]] of $\mathfrak{m}_{\alpha}$[^1], it is a [[Lie algebra representation|representation]] [[finding sl2-triples|of]] $\mathfrak{m}_{\alpha} \cong \mathfrak{sl}_{2}(\mathbb{C})$.[^2] > Thus, as a representation of $\mathfrak{sl}_{2}(\mathbb{C})$, $V$ [[weights characterize any representation of sl2(C)|is characterized]] by its [[weight space for sl2(C)|weights]], i.e., the multiset of [[eigenvalue|eigenvalues]] of (the image under $\text{ad}$ of) $h_{\alpha} \in \mathfrak{m}_{\alpha}$. It suffices to look for eigenvalues on each summand $\mathfrak{g}_{\beta + k\alpha}$. These are precisely given by $(\beta+k \alpha)(h_{\alpha})=\beta(h_{\alpha})+k \alpha(h_{\alpha})=\beta(h_{\alpha})+2k$,[^3] each with multiplicity $1$ since [[root spaces are one-dimensional]]. > So the weights are all integers of the same parity. (*why is $\beta(h_{\alpha})$ an integer?*) and they all have multiplicity $1$. [[classification of the irreps of sl2 over C|This implies]] that $V$ is [[irreducible Lie algebra representation|irreducible]], $V \cong V(n)$ for some integer $n \in \mathbb{Z}_{\geq 0}$, with weights $\{ n,n-2,\dots,-n \}$. > The set $\{ \beta + k\alpha: k \in \mathbb{Z} \} \cap \Phi$ is called the **$\alpha$-root string through $\beta$**. ^definition Let $r \geq 0$ be the largest integer such that $\beta-r\alpha \in \Phi$. Let $q \geq 0$ be the largest integer such that $\beta+q\alpha \in \Phi$. Then **1.** The root string has everything in-between: $\{ \beta + k \alpha: k \in \mathbb{Z} \} \cap \Phi = \{ \beta + k \alpha : -r \leq k \leq q \}$ and $\beta(h_{\alpha})=r-q$ (in particular, is an integer.) ```tikz \usepackage{tikz-cd} \usepackage{amsmath} \begin{document} % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12AjGHOgAgC0-AE79OjNAAs6IAL6l0mXPkIoAjOSq1GLNpwDGUCDgQKl2PASIAmLdXrNWiDt16zzIDJdVEAzPY6TvpufADUEgzSHopeylZqyAAsgY56LobGpvKx3irWKACsqbrOrjzhAI6R0fLaMFAA5vBEoABmIhAAtkhkIDgQSJpB6SBh4uySMiDUPGBQSH7Enh3dQ9QDSHYjZeM107Mw84vLsas9iNubiAE7bBGTUQcgcws3p+2dF7fXKS9HbyWDlK932MU+a0Qf2uxX+x3ewOCLgeU3BIHOSFhv0O8KBdxcgjBOQhF2hgxuiNGhMetRxgI+6K+iw25O2aTK1NRMzh9JWTMuLPW+JAnKesjpJz5kOG1z67LYotpPMlFDkQA \begin{tikzcd} \beta - r \alpha \arrow[r, "+ \alpha", bend left] & \cdots \arrow[r, "+ \alpha", bend left] \arrow[l, "-\alpha", bend left] & \beta \arrow[r, "+\alpha", bend left] \arrow[l, "-\alpha", bend left] & \beta+\alpha \arrow[r, "+\alpha", bend left] \arrow[l, "-\alpha", bend left] & \cdots \arrow[r, "+\alpha", bend left] \arrow[l, "-\alpha", bend left] & \beta+q\alpha \arrow[l, "-\alpha", bend left] \end{tikzcd} \end{document} ``` **2. (The reflection along $\alpha$ is a root)** $\beta - \beta(h_{\alpha})\alpha \in \Phi$ **3.** If $\alpha+\beta \in \Phi$, then $[\mathfrak{g}_{\alpha}, \mathfrak{g}_{\beta}]=\mathfrak{g}_{\alpha+b}$ (not just a subset of). ---- #### [^1]: This follows from the [[root space decomposition of a Lie algebra|general fact]] that $[\mathfrak{g}_{\alpha}, \mathfrak{g}_{\beta}]\subset \mathfrak{g}_{\alpha+\beta}$. For example, if take some $x \in \mathfrak{g}_{\beta + k \alpha}$ and bracket it with, say, $e_{\alpha} \in \mathfrak{m}_{\alpha}$, the result $\text{ad}_{e_{\alpha}}(x)=[e_{\alpha}, x]$ will live in $\mathfrak{g}_{\beta + (k+1)\alpha} \subset V$. Similarly, $[f_{\alpha}, x] \in \mathfrak{g}_{\beta+(k-1) \alpha} \subset V$ and $[h_{\alpha}, x] \in \mathfrak{g}_{\beta + k \alpha + 0} \subset V$. [^2]: As it is a [[Lie algebra subrepresentation|subrepresentation]] of the [[adjoint representation]] $\text{ad }\mathfrak{m}_{\alpha}$. [^3]: Where we have used that $\alpha(h_{\alpha})=2\cancel{\frac{\alpha(t_{\alpha})}{\kappa(t_{\alpha},t_{\alpha})}}^{=1}=2.$ ---- #### 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 > ```