Triangular Decomposition of the Crystal Lattice of Quantized Function Algebras: Revisited

Let $\text{g}$ be a simple complex Lie algebra of type $G_2$, $F_4$, or $E_8$, and let $G$ be the unique connected simply connected Lie group with $\mathrm{L}\mathrm{i}\mathrm{e}(G)=\text{g}$ with compact real form $K$. We prove a triangular decomposition theorem for the lower crystal lattice $\text{OAztG}$ of the quantized function algebra $\text{OtG}$, establishing that $\text{OAztG}=\text{RAzm}\cdot \text{RAzp}$. This extends the triangular decomposition recently obtained for types $A_n,B_n,C_n,D_n,E_6$, and $E_7$ in~\cite{DDPa} to all complex simple Lie algebras. As a consequence, we obtain: (i) the inclusion $\text{OAztG}\subseteq \text{OAztK}$ conjectured by Matassa-Yuncken and (ii) the crystal limit $\text{CpKo}$ is a compact quantum semigroup for all connected, simply connected, compact simple Lie groups $K$.