Triangular Decomposition of the Crystal Lattice of Quantized Function Algebras: Exceptional Types

For $\g$ of type $G_2$, $F_4$, or $E_8$, let $G$ be the connected simply connected complex Lie group with $\mathrm{Lie}(G)=\g$, and compact real form $K$. We prove the triangular decomposition $\OAztG=A_0\text{-alg}\!<\RAzp \cup \RAzm>$ of the crystal lattice of $\OtG$. Together with~\cite{DDPa}, which treats types $A_n$--$D_n$, $E_6$, $E_7$, this settles the triangular decomposition for all simple complex Lie algebras. As a consequence, we obtain the inclusion $\OAztG\subseteq\OAztK$ conjectured by Matassa--Yuncken in full generality, and the crystal limit $\CpKo$ is a compact quantum semigroup with a bounded counit and a unique bi-invariant (Haar) state.