Topology
《Topology》 · 第 190 页 · Chapter 9，Section 9.4
DEFINITION. A crystal base of $M$ is a pair $(\mathcal{L}, B)$ where $\mathcal{L}$ is an admissible lattice in $M$ and $B$ is a basis of the vector space $\mathcal{L}/q\mathcal{L}$ over $\mathbb{Q}$ such that (C1) $B = \bigcup_{\lambda \in \Lambda} B_\lambda$ where $B_\lambda = B \cap ((\mathcal{L}/q\mathcal{L})_\lambda)$, (C2) $\tilde{F}_\alpha B \subset B \cup \{0\}$ and $\tilde{E}_\alpha B \subset B \cup \{0\}$ for all $\alpha \in \Pi$, (C3) for any $b, b' \in B$ and $\alpha \in \Pi$: $\tilde{F}_\alpha b = b' \iff \tilde{E}_\alpha b' = b$.