A triangular decomposition for the crystal lattice of quantized function algebras

Abstract

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. Using this inclusion, we then prove that the notions of crystallized quantized function algebra given by Matassa \& Yuncken coincide with that of Giri \& Pal.