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.

v1: 21 pages. Comments welcome; v2: 26 pages, substantially rewritten, two sections and some references added; v3: 27 pages, Proof of the triangular decomposition rewritten due to an error in the earlier version, some other small reorganization of the content; v4: 25 pages, Introduction part rewritten, one subsection removed that will be included in a separate article by the second author