----- > [!proposition] Proposition. ([[root space decomposition of sp4(C)]]) > > > Put $\mathfrak{g}:=\mathfrak{sp}_{4}(\mathbb{C})$. > > By [[diagonal matrices form Cartan subalgebras for special linear, orthogonal, symplectic Lie algebras]] we know that the [[Lie subalgebra|subalgebra]] $\mathfrak{t}$ of diagonal matrices is a [[Cartan subalgebra]] with [[basis]] $\{e_{1}= E_{11}-E_{33}, e_{2}=E_{22}-E_{44} \}.$ Let $\{ e^{1},e^{2} \} \subset \mathfrak{t}^{*}$ be the corresponding [[dual basis]]. > > We want to find the set of $\alpha \in \mathfrak{t}^{*}$ [[root space decomposition of a Lie algebra|such that]] > $\mathfrak{g}=\mathfrak{t} \oplus \bigoplus_{\alpha \in \Phi}\mathfrak{g}_{\alpha},$ > where $\mathfrak{g}_{\alpha}$ denotes the simultaneous eigenspace/weight space/root space $\mathfrak{g}_{\alpha}=\{ x \in \mathfrak{g}: [t, x] = \alpha(t) x \text{ for all } t \in \mathfrak{t} \}.$ > We know $\dim \mathfrak{g}=10$; since $\text{dim }\mathfrak{t}=2$ and for every root $\alpha$ its negation $-\alpha$ is also a root, we are looking for four roots $\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}.$ > > We need to find a [[simultaneously diagonalizable|simultaneous eigenbasis]] for the $\text{ad }t$, $t \in \mathfrak{t}$. Each eigenvector of this eigenbasis will give a weight. > > Let $t=c_{1}e_{1}+c_{2}e_{2} \in \mathfrak{t}$ be arbitrary. We have $\text{ad}_{t}(x)=[t,x]=tx-xt.$ > A general element of $\mathfrak{g}$ has the form $x=\begin{bmatrix} > P & Q \\ > R & -P^{\top} > \end{bmatrix}= \begin{bmatrix} > p_{11} & p_{12} & q_{11} & q_{12} \\ > p_{21} & p_{22} & q_{12} & q_{22} \\ > r_{11} & r_{12} & -p_{11} & -p_{21} \\ > r_{12} & r_{22} & -p_{12} & -p_{22} > \end{bmatrix}$ > where $Q$ and $R$ are symmetric. We may compute $[t,x]=tx - xt=\begin{bmatrix} 0 & c_1 p_{12} - c_2 p_{12} & 2c_1 q_{11} & c_1 q_{12} + c_2 q_{12} \\ -c_1 p_{21} + c_2 p_{21} & 0 & c_1 q_{12} + c_2 q_{12} & 2c_2 q_{22} \\ -2c_1 r_{11} & -c_1 r_{12} - c_2 r_{12} & 0 & c_1 p_{21} - c_2 p_{21} \\ -c_1 r_{12} - c_2 r_{12} & -2c_2 r_{22} & -c_1 p_{12} + c_2 p_{12} & 0 \end{bmatrix}$ > From this we can see the usual basis for $\mathfrak{sp}_{2 \ell}(\mathbb{C})$ (cf. example sheet 1): > $\begin{align} > E_{13} \\ > E_{14} + E_{23} \\ > E_{24} \\ > \ \\ > E_{31} \\ > E_{32} + E_{41} \\ > E_{42} \\ > \ \\ > E_{11}-E_{33} \\ > E_{22} - E_{44} \\ > E_{12} - E_{43} \\ > E_{21} - E_{34} > \end{align}$ > > is a simultaneous eigenbasis: $\begin{align*} > [t, E_{13}] & = 2c_{1}E_{13} & & = 2e^{1}(t) \ E_{13} \\ > [t, E_{14}+E_{23}] & = (c_{1} + c_{2}) (E_{14}+E_{23}) & & =(e^{1}+e^{2})(t) \ (E_{14}+E_{23}) \\ > [t, E_{24}] & = 2c_{2}E_{24} & & = 2e^{2}(t) \ E_{24} \\ > [t, E_{31}] & = -2c_{1}E_{31} & & = -2e^{1}(t) \ E_{31} \\ > [t, E_{32} + E_{41}] & = -(c_{1} + c_{2}) (E_{32} + E_{41}) & & = -(e^{1}+e^{2})(t) \ (E_{32}+E_{41}) \\ > [t, E_{42}] & = -2c_{2}E_{42} & & = -2e^{2}(t) \ E_{42} \\ > [t, E_{11}- E_{33}] & = 0 & & = 0 \\ > [t, E_{22}-E_{44}] & = 0 & & = 0 \\ > [t, E_{12}- E_{43}] & = (c_{1}-c_{2}) (E_{12} - E_{43}) & & = (e^{1}-e^{2})(t) \ (E_{12}-E_{43}) \\ > [t, E_{21}-E_{34}] & = -(c_{1}-c_{2})(E_{21}-E_{34}) & & = -(e^{1}-e^{2})(t) \ (E_{21}-E_{34}) > \end{align*}$ > So we have nonzero roots $\begin{align} > \pm 2 e^{1}, \pm 2 e^{2}, \pm (e^{1}+e^{2}), \pm (e^{1}-e^{2}) > \end{align}$whose corresponding root spaces are just the span of the provided basis elements for $\mathfrak{sp}_{4}(\mathbb{C})$. The zero root space $\mathfrak{t}$ is two-dimensional with basis $\{ E_{11} - E_{33}, E_{22}-E_{44} \}$ (as we already knew.) Thus, $\mathfrak{sp}_{4}(\mathbb{C}) = \mathfrak{t} \oplus \mathfrak{g}_{\pm e^{1}} \oplus \mathfrak{g}_{\pm e^{2}} \oplus \mathfrak{g}_{\pm(e^{1} + e^{2})} \oplus \mathfrak{g}_{\pm (e^{1}-e^{2})}.$ > ----- #### ---- #### 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 > ```