---- > [!definition] Definition. ([[weight space for sl2(C)]]) > > View $\mathfrak{sl}_{2}(\mathbb{C})$ as follows: > ![[special linear Lie subalgebra#^basic-example]] > > Let $V$ be an [[irreducible Lie algebra representation|irrep]] of $\mathfrak{sl}_{2}(\mathbb{C})$. For $\lambda \in \mathbb{C}$ define $V_{\lambda}:=\{ v \in V:h \cdot v = \lambda v \}.$ If $\lambda$ is an [[eigenvalue]] of $\rho(h)$ then $V_{\lambda}$ is the corresponding $\lambda$-[[eigenspace]] and (I think) we call $\lambda$ a **weight** of $V$ and $V_{\lambda}$ the $\lambda$-**weight space**. Else it's zero. > > An element $\lambda \in \mathbb{C}$ with $V_{\lambda} \neq \{ 0 \}$ but $V_{\lambda+2}=\{ 0 \}$ is called a **highest weight**, and a nonzero element $v \in V_{\lambda}$ a **highest weight vector**. > > We may hop between weight spaces as follows: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRADUB9YAHR4boBbAEZQ6AAgC04gEwBfEHNLpMufIRQBGclVqMWbLr35DRdBUpXY8BIjJ3V6zVog7c+AkWPEBqWReUQDGt1IgBmBz1nQ3cTLwk-ABYLXRgoAHN4IlAAMwAnCEEkMhAcCCRtEGEYMCgkMJKnA1c+AGMoCBxxVksQfMKK6jKkeyqausQGx30XEDaOrp7A-qLEUeHJ6mra+saZtnnO7sVlgtWI0vK1rfHd6eiWnnajnJPcs5Ghq8rtiamo5pzJ4LcSvXorQaXYrUATVBgABVUNg0IAYMByOBANx2kz2DyBzy6YIociAA > \begin{tikzcd} > V_{\lambda - 2} \arrow[r, "\cdot e", bend left] & V_{\lambda} \arrow[r, "\cdot e", bend left] \arrow[l, "\cdot f", bend left] & V_{\lambda + 2} \arrow[r, "\cdot e", bend left] \arrow[l, "\cdot f", bend left] & V_{\lambda + 4} \arrow[l, "\cdot f", bend left] > \end{tikzcd} > \end{document} > > ``` > > I.e., $e \cdot V_{\lambda} \subset V_{\lambda+2}$ and $f \cdot V_{\lambda} \subset V_{\lambda-2}$ for any $\lambda$. (of course $h \cdot V_{\lambda}=V_{\lambda}$). > > More generally, for *any* $\mathfrak{sl}_{2}(\mathbb{C})$-rep $V$ we can speak of its **multiset of weights** $\{ \{ \lambda \in \mathbb{Z} : V_{\lambda} \neq (0) \} \}$, where each $\lambda$ has multiplicity $\dim V_{\lambda}$ (geometric multiplicity?) [[weights characterize any representation of sl2(C)|These turn out to characterize]] $V$. ^b1d691 > [!generalization] > - [[on factorizing a Lie algebra representation into weight spaces]] > - [[on the weights of a representation]] > Note, however, that a great deal of the general theory boils down to the present note. ^generalization > [!justification]+ > We must show correctness of the hopping diagram., i.e., that $e \cdot V_{\lambda} \subset V_{\lambda+2}$ and $f \cdot V_{\lambda} \subset V_{\lambda-2}$. > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRADUB9YAHR4boBbAEZQ6AAgC04gEwBfEHNLpMufIRQBGclVqMWbLr35DRdBUpXY8BIjJ3V6zVog7c+AkWPEBqWReUQDGt1IgBmBz1nQ3cTLwk-ABYLXRgoAHN4IlAAMwAnCEEkMhAcCCRtEGEYMCgkMJKnA1c+AGMoCBxxVksQfMKK6jKkeyqausQGx30XEDaOrp7A-qLEUeHJ6mra+saZtnnO7sVlgtWI0vK1rfHd6eiWnnajnJPcs5Ghq8rtiamo5pzJ4LcSvXorQaXYrUATVBgABVUNg0IAYMByOBANx2kz2DyBzy6YIociAA > \begin{tikzcd} > V_{\lambda - 2} \arrow[r, "\cdot e", bend left] & V_{\lambda} \arrow[r, "\cdot e", bend left] \arrow[l, "\cdot f", bend left] & V_{\lambda + 2} \arrow[r, "\cdot e", bend left] \arrow[l, "\cdot f", bend left] & V_{\lambda + 4} \arrow[l, "\cdot f", bend left] > \end{tikzcd} > \end{document} > ``` > Recalling the defining property of $\rho$ as a [[Lie algebra representation]] and that $[h,e]=2e$, we have > > $\begin{align} > h \cdot (e \cdot v) &= [h,e] \cdot v + e \cdot (h \cdot v) \\ > &= 2e \cdot v + e \cdot (\lambda v) \\ > &= (e \cdot v) 2 + (e \cdot v) \lambda \\ > &= (\lambda+2) (e \cdot v). > \end{align}$ > A similar calculation shows the result for $f$. > > > ---- #### ---- #### 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 > ```