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\parbox{2.25in}{\bf Math 4010/5530}
\hfill {\Large \bf Homework \#1} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Sept. $6^\mathrm{th}$, 2017}

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\begin{description}
\item[\#1 General Algebra] \quad Let $\mathcal{A}$ be an algebra over a field $\mathbb{F}$ (i.e. $\mathcal{A}$ is a vector space over $\mathbb{F}$ equipped with a bilinear multiplication). Denote the multiplication in $\mathcal{A}$ by juxtaposition: $(a,b) \mapsto ab$. Suppose $a,b \in \mathcal{A}$. Define $L_a, R_a : \mathcal{A} \to \mathcal{A}$ by $L_a(b)=ab$ and $R_a(b)=ba$ (i.e. $L_a$ is left multiplication by $a$ and $R_a$ is right multiplication by $a$).

\begin{enumerate}[(a)]
\item Show that $L_a$ and $R_a$ are linear operators (i.e. linear endomorphisms). 
\item Show that $\mathcal{A}$ is an associative algebra if and only if $L_x\circ R_y = R_y \circ L_x$ for all $x,y \in \mathcal{A}$.
\end{enumerate}

\item[\#2 Identity Crisis] Suppose that $L$ is a Lie algebra (over some field $\mathbb{F}$). In addition, suppose that $L$ is a unital algebra (i.e. there is some $\mathbbold{1} \in L$ such that $[\mathbbold{1},x]=x=[x,\mathbbold{1}]$ for all $x \in L$). Show that $L$ Abelian (i.e. $[x,y]=\B 0$ for all $x,y \in L$). [{\it Hint:} Consider $[\mathbbold{1},[x,y]]$. Use the Jacobi identity.]

\item[\#3 Linearly Independent] Let $L$ be a Lie algebra (over some field $\mathbb{F}$). Let $x,y \in L$ and suppose that $[x,y] \not= \B 0$. Show that $S=\{x,y\}$ is linearly independent. What does say about $1$-dimensional Lie algebras?

\item[\#4 Basis Problem] Let $L$ be a vector space (over some field $\mathbb{F}$) with basis $\beta = \{e,f,h\}$. Suppose that $[e,f]=h$, $[h,e]=2e$, and $[h,f]=-2f$. In addition, suppose that the bracket is bilinear, alternating, and skew-symmetric (i.e. extend the definition of $[\,\cdot\,,\,\cdot\,]$ to $L \times L$ using bilinearity, alternation, and skew-symmetry). 

\begin{enumerate}[(a)]
\item What is $[e,e]$? What is $[f,h]$? 
\item Show that this bracket makes $L$ a Lie algebra.\\ \force \hfill  [{\it Hint:} Use Misra's Theorem 1.8 on page 2 to minimize your work.]
\item Let $E = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$, $F = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}$, and $H = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$. Also, let $\alpha = \{E,F,H\}$. 

Prove that $\alpha$ is a basis for $\mathfrak{sl}_2(\mathbb{F}) = \left\{ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \;|\; a,b,c,d \in \mathbb{F} \mbox{ and } a+d=0 \right\}$.
\item Using the commutator bracket, show that $[E,F]=H$, $[H,E]=2E$, and $[H,F]=-2F$.
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