matrices of Lie algebra of Dynkin diagram B2 - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-25T08:18:56Z http://mathoverflow.net/feeds/question/52259 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/52259/matrices-of-lie-algebra-of-dynkin-diagram-b2 matrices of Lie algebra of Dynkin diagram B2 garretstar 2011-01-16T19:33:00Z 2011-01-31T01:22:13Z <p>The Lie algebra $so_5$ has 10 elements and its root structure is given by the Dynkin diagram B2. I have been having trouble creating an explicit $5 \times 5$ complex matrix representation of its 10 elements from its Cartan matrix. I would greatly appreciate help with this.</p> http://mathoverflow.net/questions/52259/matrices-of-lie-algebra-of-dynkin-diagram-b2/52277#52277 Answer by Allen Knutson for matrices of Lie algebra of Dynkin diagram B2 Allen Knutson 2011-01-17T01:06:53Z 2011-01-17T01:06:53Z <p>While I'm not sure this question is appropriate for this site, here goes.</p> <p>First, you need a maximal torus. Inside SO(5) we have SO(4) and then SO(2) x SO(2).</p> <p>Write your antisymmetric matrices as</p> <p>$$\begin{matrix} aJ &amp; C &amp; v \end{matrix}$$ $$\begin{matrix} -C^T &amp; bJ &amp; w \end{matrix}$$ $$\begin{matrix} -v^T &amp; -w^T &amp; 0 \end{matrix}$$ where $J = \left( {0\atop -1}{1\atop 0} \right)$, $C$ is square, and $v$ and $w$ are columns. Then the $a$ and $b$ parts are the torus, the $v$ gets you the $\pm x_1$ weights, the $w$ gets you the $\pm x_2$, and the $C$ gets you the $\pm x_1\pm x_2$. </p> <p>Taking $x_1$ and $x_2 - x_1$ as simple roots, the $e_{x_1}$ is $$0 0 0 0 1$$ $$0 0 0 0 i$$ $$0 0 0 0 0$$ $$0 0 0 0 0$$ $$-1 -i 0 0 0$$ and the $e_{x_2-x_1}$ is $$0 0 1 i 0$$ $$0 0 i -1 0$$ $$-1 -i 0 0 0$$ $$-i 1 0 0 0$$ $$0 0 0 0 0$$ Sorry for the ugly matrices -- I'm having trouble getting the matrix environment working here.</p>