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%\input{MyDefns}

\def\ad{\operatorname{ad}\nolimits}
\def\Lie{\operatorname{Lie}\nolimits}
\def\Der{\operatorname{Der}\nolimits}


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{\centerline {\bf Homework 1 for ``Algebraic Structures: Lie algebras", Autumn 2016}}
{\centerline {due Friday, October 21} }
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\noindent
{\bf Problem 1.} Let $A=k[x_1, \ldots, x_n]$. 
\begin{enumerate} \item
Show that any derivation of $A$ has the form $f_1(\underline x) \frac{\partial}{\partial x_1} + \ldots + f_n(\underline x) \frac{\partial}{\partial x_n}$ where $f_1(\underline x), \ldots, f_n(\underline x) \in k[x_1, \ldots, x_n]$.
\item  Let $D_f = f_1(\underline x) \frac{\partial}{\partial x_1} + \ldots + f_n(\underline x)\frac{\partial}{\partial x_n}, D_g = g_1(\underline x) \frac{\partial}{\partial x_1} + \ldots + g_n(\underline x)\frac{\partial}{\partial x_n}$. Find the formula for $[D_f, D_g]$. 
\end{enumerate}


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\noindent
{\bf Problem 2.}  Show that $\Lie(\mathbb G_m) \simeq g_a$. 

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\noindent
{\bf Problem 3.}  Let $e,f,h$ be the standard basis of $sl_2$, and let  $\ad: sl_2 \to gl_3$ be the adjoint representation of $sl_2$ with respect to the standard basis. Calculate $ \ad e, \ad f$ and $\ad h$.  Is this representation faithful? irreducible? If not, how does it decompose? 

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\noindent
{\bf Problem 4.} Let $A = k[x_1, \ldots, x_n]$. Consider an embedding of $gl_n$ into $\Der_k(A)$ as {\it linear derivations}: 
\[ (a_{ij}) \mapsto \Sigma a_{ij} x_i  \frac{\partial}{\partial x_j} .\]
 Since the Lie algebra $\Der_k(A)$ acts on $A$, this defines a representation of $gl_n$ on $A$; moreover, this representation preserves degrees. Show that the induced representation on   $k[x_1, \ldots, x_n]_{(d)}$, homogeneous polynomials  of degree $d$, coincides with the representation of $gl_n$ on $S^d(\bigoplus kx_i)$ under the standard identification $k[x_1, \ldots, x_n]_{(d)} \simeq S^d(\bigoplus kx_i)$.
 
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\noindent
{\bf Problem 5.} 
Let $k$ be an algebraically closed field of characteristic $p>0$. Show that the highest weight  modules $V(0), V(1), \ldots, V(p-1)$ form a complete list of irreducible representations of $sl_2$ over $k$.
 
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