- #1

MathematicalPhysicist

Gold Member

- 4,366

- 242

Now to find out the roots I need to compute:

##[H,X]=\alpha(H) X##

For every ##H## in the above Cartan sublagebra, for some ##X \in sp(4,\mathbb{R})##

Now, I know that ##X## is of the form:

##\left[ {\begin{array}{ccccc} a_{11} & a_{12} & a_{13} & a_{14}\\

a_{21} & a_{22} & a_{23} & a_{13} \\

a_{31} & a_{32} & -a_{22} & -a_{12}\\

a_{41} & a_{31} & -a_{21} & -a_{11}\\

\end{array}} \right]##

So if I take ##H=diag(h_{11},h_{22},h_{33},h_{44})##, I am getting the next equality:

##\left[ {\begin{array}{ccccc} a_{11} & a_{12} & a_{13} & a_{14}\\

a_{21} & a_{22} & a_{23} & a_{13} \\

a_{31} & a_{32} & -a_{22} & -a_{12}\\

a_{41} & a_{31} & -a_{21} & -a_{11}\\

\end{array}} \right] = \left[ {\begin{array}{ccccc} 0 & (h_{11}-h_{22})a_{12} & (h_{11}-h_{33})a_{13} & (h_{11}-h_{44})a_{14}\\

(h_{22}-h_{11})a_{21} & 0 & (h_{22}-h_{33})a_{23} &(h_{22}-h_{44}) a_{13} \\

(h_{33}-h_{11})a_{31} & (h_{33}-h_{22})a_{32} & 0 & (h_{44}-h_{33})a_{12}\\

(h_{44}-h_{11})a_{41} & (h_{44}-h_{22})a_{31} & (h_{33}-h_{44})a_{21} & 0\\

\end{array}} \right]##

Which means that the roots should be ##h_{11}-h_{44} , h_{22}-h_{33} , h_{33}-h_{22}, h_{44}-h_{11}##, and accodingly the root vectors are:

##\left[ {\begin{array}{ccccc}0 & 0 & 0 & a_{14}\\

0 & 0 & 0 & 0 \\

0 & 0 & 0 & 0\\

0 & 0 & 0 & 0\\

\end{array}} \right],\left[ {\begin{array}{ccccc}0 & 0 & 0 & 0\\

0 & 0 & a_{23} & 0 \\

0 & 0 & 0 & 0\\

0 & 0 & 0 & 0\\

\end{array}} \right],\left[ {\begin{array}{ccccc}0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 \\

0 & a_{32} & 0 & 0\\

0 & 0 & 0 & 0\\

\end{array}} \right], \left[ {\begin{array}{ccccc}0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 \\

0 & 0 & 0 & 0\\

a_{41} & 0 & 0 & 0\\

\end{array}} \right]## respectively.

Is this right, or did I forget something?

Thanks in advance.