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

TL;DR

This paper extends the triangular decomposition of the lower crystal lattice of quantized function algebras to all complex simple Lie algebras, confirming conjectures and establishing new structural properties of quantum groups.

Contribution

It generalizes the triangular decomposition of the lower crystal lattice to types G_2, F_4, and E_8, beyond previously known types, and proves related conjectures.

Findings

Triangular decomposition holds for all simple Lie algebra types.

Confirmed the inclusion conjectured by Matassa-Yuncken.

Established the crystal limit as a compact quantum semigroup.

Abstract

Let $\g$ 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 $\mathrm{Lie}(G)=\g$ with compact real form $K$. We prove a triangular decomposition theorem for the lower crystal lattice $\OAztG$ of the quantized function algebra $\OtG$, establishing that $\OAztG = \RAzm \cdot \RAzp$. 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 $\OAztG\subseteq\OAztK$ conjectured by Matassa-Yuncken and (ii) the crystal limit $\CpKo$ is a compact quantum semigroup for all connected, simply connected, compact simple Lie groups $K$.