Crystal base

A crystal base for a representation of a quantum group on a ${\displaystyle \mathbb {Q} (v)}$-vector space is not a base of that vector space but rather a ${\displaystyle \mathbb {Q} }$-base of ${\displaystyle L/vL}$ where ${\displaystyle L}$ is a ${\displaystyle \mathbb {Q} (v)}$-lattice in that vector space. Crystal bases appeared in the work of Kashiwara (1990) and also in the work of Lusztig (1990). They can be viewed as specializations as ${\displaystyle v\to 0}$ of the canonical basis defined by Lusztig (1990).