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\def\ba{{\mathbf a}}
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\begin{document}
\parindent 0in

%\parbox[b]{4in}{
%  {\tiny Dror Bar-Natan: MAT 1350 Knot Theory: \LARGE\bf Tentative Plan}
%  }
\parbox[b]{1.5in}{\tiny
  \href{http://www.math.toronto.edu/~drorbn}{Dror Bar-Natan}: \href{http://www.math.toronto.edu/~drorbn/classes}{Classes}:
  \newline \href{http://www.math.toronto.edu/~drorbn/classes/\#2122}{2021-22}: \href{http://drorbn.net/21-1350}{MAT 1350 Knot Theory}:
}~{\LARGE\bf Homework Assignment 6}
\hfill\parbox[b]{2in}{\tiny
  \null\hfill\sheeturl
  \newline\null\hfill Due December 15 at midnight
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  \qrcode[height=1.2em,level=L,nolink]{drorbn.net/21-1350}
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\vskip -3mm
\rule{\linewidth}{1pt}
\vskip 5mm

\begin{center}\begin{minipage}{6in}

Over the base field $\bbQ$, recall that $\sleps$ is the Lie algebra with generators $\{\by,\bb,\ba,\bx\}$ and with brackets
\[
  [\ba,\bx]=\bx,\quad
  [\bb,\by]=-\epsilon\by,\quad
  [\ba,\bb]=0,\quad
  [\ba,\by]=-\by,
\]
\[
  [\bb,\bx]=\epsilon\bx,\quad
  [\bx,\by]=\bb+\epsilon\ba.
\]
Recall also that $sl_2$ is the Lie algebra of $2\times 2$ matrices whose trace is $0$ and with $[A,B]\coloneqq AB-BA$. Note that if $\fraka$ and $\frakb$ are Lie algebras, then so is their direct sum $\fraka\oplus\frakb$, taken with the obvious bracket. Finally, a 1-dimensional vector space $\langle t\rangle$ is a Lie algebra (in only one way), with $[t,t]=0$.

\vskip 5mm{\bf Problem 1.} Show that for $\eps\neq 0$ the Lie algebra $\sleps$ is isomorphic to the direct sum $sl_2\oplus\langle t\rangle$. (Hint. What properties would the image of $t$ in $\sleps$ need to have?)

\vskip 5mm{\bf Problem 2.} Show that for $\eps=0$ the Lie algebra $sl_{2+}^0$ is not a direct sum of smaller algebras.

\vskip 10mm
(That's kind of unexpected, isn't it? You'd think that on a one-parameter path of Lie algebras some miracle could occur, and some Lie algebra in the middle of the path might have a decomposition into smaller ones even if the generic case is indecomposable. But what we see here is the opposite!)

\end{minipage}\end{center}

\end{document}


