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\subsection*{PHYS 570/MA 595: Unit 6 Exercises, Due November 25, 2025, Accepted until December 11}
\textcolor{red}{\textbf{Note:}} our November 20th class is an asynchronous video lecture so I will not be physically collecting homeworks that day. Instead, you may slide HWs under the door of my office, place in my mailbox in MATH 427, or bring to one of our class meetings after November 25.\\
Exercises will be added to this list but no new problems will be added after Tuesday November 18. Thank you for your patience while I catch up on TeXing lecture notes.\\

%\textbf{Tuesday November 4 and Thursday November 6} 


\textbf{Tuesday November 11}
 \begin{enumerate}
\item[(?)] There are several equivalent criteria that detect abelianness of anyons in a UMTC. A usual let $L$ be the label set of anyon types.

An anyon $a$ is abelian 
\begin{itemize}
\item[(I)] if $a \otimes a^* = 1$
\item[(II)] if its quantum dimension $d_a=1$. 
\item[(III)] if there exists an anyon $c$ such that $a \otimes b = c$ for all anyons $b \in L$
\end{itemize}

\textbf{Prove} that (I) $\implies$ (II) $\implies$ (III) $\implies$ (I). \hfill (3 points)\\
\noindent \textit{Hint:} for (II) $\implies$ (III) use that the quantum dimension is equal to the Frobenius-Perron dimension by unitarity.
\end{enumerate}
%\textbf{Thursday November 13} 


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