\include{preamble}


\begin{document}
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\paragraph{\bf Exercise 1:} Some things about the classical Lie algebras.   
\begin{enumerate}
\item For each of the following types, give a basis $B$ which has exactly $r$ diagonal matrices and is otherwise as workable as possible. In particular, keep symmetry, so that if $x \in B$ then $x^T \in B$.  Express elements as sums of elementary matrices $E_{ij}$ (the matrix with a 1 in the $(i,j)$ position and 0's elsewhere. Clearly, we're choosing a basis for $V$ in the process. A form on $V$ defined by a matrix $J$ is defined by $\<u, v\> = u^TJ v$. Let $I_r$ be the $r \times r$ identity matrix.
\begin{enumerate}
	\item {\bf Type $A_r$.} For $r \geq 1$, give a  basis for $\fsl_{r+1}$, and verify that $\dim(\fsl_{r+1})=r(r+2)$.
	\item {\bf Type $C_r$.} For $r \geq 1$, put the form on $V = \CC^{2r}$ given by $J = \begin{pmatrix}0 & I_r\\-I_r &0\end{pmatrix}$.
	\begin{enumerate}[($*$)]
		\item Verify that $\<,\>$ is skew symmetric, i.e.\ $\<u,v\> = - \<v,u\>$. 
		\item Verify that if $\fsp_{2r} = \{x \in \fgl(V) ~|~ \<xu,v\> = - \<u,xv\>\}$, then $\fsp_{2r}$ is in fact closed ($\<,\>$ is bilinear, so you only need to check $[,]$.)
		\item Give a basis for $\fsp_{2r}$, and verify that $\dim(\fsp_{2r})=r(2r+1)$. (Break each $x \in \fsp_{2r}$ into the four $r \times r$ matrices that $J$ effect independently, (see below) and get conditions on each of them) 
	\end{enumerate}
	
	\item {\bf Type $D_r$.} For $r \geq 2$, put the form on $V = \CC^{2r}$ given by
	$J = \begin{pmatrix}0 & I_r\\I_r &0\end{pmatrix}$.
	\begin{enumerate}[($*$)]		
		\item Verify that $\<,\>$ is symmetric, i.e.\ $\<u,v\> = \<v,u\>$. 
		\item	Verify that if $\fso_{2r} = \{x \in \fgl(V) ~|~ \<xu,v\> = - \<u,xv\>\}$, then $\fso_{2r}$ is closed.
		\item Give a basis for $\fso_{2r}$, and verify that $\dim(\fso_{2r})=r(2r-1)$. (Break each $x \in \fso_{2r}$ into the four $r \times r$ matrices that $J$ effects independently, (see below) and get conditions on each of them) 
	\end{enumerate}
	\item {\bf Type $B_r$.} For $r \geq 1$, put the form on $V = \CC^{2r+1}$ given by
	$
		J = \begin{pmatrix}1&0&0\\0&0 & I_r\\0&I_r &0\end{pmatrix}.$
	Give a basis for $\fso_{2r+1}$, and verify that $\dim(\fso_{2r+1})=r(2r+1)$. (Break each $x \in \fso_{2r+1}$ into the nine blocks that $J$ effects independently (see below) and get conditions on each of them.) 
\end{enumerate}
\item As mentioned in class, $B_1$, $C_1$, $C_2$, $D_1$, $D_2$, and $D_3$ are either not distinct from, or decompose into direct sums of Lie algebras from amongst. 
$$\{A_r\}_{r \geq 1} \sqcup \{B_r\}_{r \geq 2}  \sqcup \{C_r\}_{r \geq 3}  \sqcup \{D_r\}_{r \geq 4} $$
Verify this for any 4 of these 6 Lie algebras by expressing them in terms of the others. 
\end{enumerate}
\vfill

\hrule

\medskip

\centerline{Decompositions of elements for $\fg$ of each type.}
$$\begin{array}{cc}
C_r, D_r & B_r\\
	\begin{pmatrix}
	\begin{tikzpicture}[yscale=-.5, xscale=.5]
	\begin{scope}[black!70, fill=black!5]
		\draw (1.1,1.1)--(1.1,3.9)--(3.9,3.9)--(3.9,1.1)--(1.1,1.1);
		\draw (4.1,1.1)--(4.1,3.9)--(6.9,3.9)--(6.9,1.1)--(4.1,1.1);
		\draw (1.1,4.1)--(1.1,6.9)--(3.9,6.9)--(3.9,4.1)--(1.1,4.1);
		\draw (4.1,4.1)--(4.1,6.9)--(6.9,6.9)--(6.9,4.1)--(4.1,4.1);
	\end{scope}
		\node at (2.5,2.5) {\tiny$X$};
	 	\node at (5.5,2.5) {\tiny$Y$};
	 	\node at (2.5,5.5) {\tiny$Y$'};
	 	\node at (5.5,5.5) {\tiny$Z$};
	\end{tikzpicture}
	\end{pmatrix}&
	\begin{pmatrix}
	\begin{tikzpicture}[yscale=-.5, xscale=.5]
	\begin{scope}[black!70, fill=black!5]
		\draw (.1,.1)--(.1,.9)--(.9,.9)--(.9,.1)--(.1,.1);
		\draw (1.1,.1)--(1.1,.9)--(3.9,.9)--(3.9,.1)--(1.1,.1);
		\draw (4.1,.1)--(4.1,.9)--(6.9,.9)--(6.9,.1)--(4.1,.1);
		\draw (1.1,1.1)--(1.1,3.9)--(3.9,3.9)--(3.9,1.1)--(1.1,1.1);
		\draw (4.1,1.1)--(4.1,3.9)--(6.9,3.9)--(6.9,1.1)--(4.1,1.1);
		\draw (1.1,4.1)--(1.1,6.9)--(3.9,6.9)--(3.9,4.1)--(1.1,4.1);
		\draw (4.1,4.1)--(4.1,6.9)--(6.9,6.9)--(6.9,4.1)--(4.1,4.1);
		\draw (.1,1.1)--(.1,3.9)--(.9,3.9)--(.9,1.1)--(.1,1.1);
		\draw (.1,4.1)--(.1,6.9)--(.9,6.9)--(.9,4.1)--(.1,4.1);
	\end{scope}
		 \node at (.5,.5) {\tiny$a$};
		 \node at (2.5,.5) {\tiny$\mathbf{b}$};
		 \node at (.5,2.5) {\tiny$\mathbf{b}'$};
		 \node at (5.5,.5) {\tiny$\mathbf{c}$};
		 \node at (.5,5.5) {\tiny$\mathbf{c}'$};
		 \node at (2.5,2.5) {\tiny$X$};
		 \node at (5.5,2.5) {\tiny$Y$};
		 \node at (2.5,5.5) {\tiny$Y$'};
		 \node at (5.5,5.5) {\tiny$Z$};
		\end{tikzpicture}
		\end{pmatrix}
	\end{array}
	$$ 
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