  A conjecture by Gekhtman, Shapiro and Vainshtein suggests a correspondence
between the Belavin - Drinfeld classification of solutions of the classical
Yang - Baxter equation and cluster structures on simple Lie groups. This paper
confirms the conjecture for $SL_{5}$. Given a Belavin - Drinfeld class, we
construct the corresponding cluster structure in
$\mathcal{O}\left(SL_{5}\right)$, and show that it satisfies all parts of the
conjecture.
