A triangular decomposition for the crystal lattice of quantized function algebras

We prove a triangular decomposition theorem for the lower crystal lattice $\mathcal{O}_{t}^{A_{0}}(G)$ of the quantized function algebra $\mathcal{O}_{t}(G)$, where $G$ is a connected simply connected complex Lie group with Lie algebra $\mathfrak {g}$ of type $A_{n}$, $B_{n}$, $C_{n}$, $D_{n}$, $E_{6}$ or $E_{7}$. As a consequence, we prove the inclusion $\mathcal{O}_{t}^{A_{0}}(G)\subseteq\mathcal{O}_{t}^{A_{0}}(K)$ conjectured by Matassa \& Yuncken in these cases. We also give a precise definition of the specialization map used by Matassa \& Yuncken, which helps simplify their description of the crystallized algebra. This allows us to prove that the crystallized algebra $C(K_{0})$ is a compact quantum semigroup for the above mentioned cases, thus extending an earlier result for type $A_{n}$ compact quantum groups. As another consequence of the triangular decomposition, we prove that the notions of crystallized quantized function algebra given by Matassa \& Yuncken coincide with that of Giri \& Pal in the type $A_{n}$ case.