Crystal (awaiting review)

A ($U_q(\mathfrak{g})$-)crystal is a nonempty set $B$ together with maps \[ \begin{aligned} &\mathrm{wt}: B \to P\\ &e_i, f_i : B \to B \cup \{0\} \quad \text{for all } i\in I. \end{aligned} \] Here $\mathfrak{g}$ is a Lie or Kac-Moody algebra of a particular type with index set $I$.

The map $\mathrm{wt}$ is called the weight function which maps a crystal element to a weight in the weight lattice $P$. The maps $e_i$ and $f_i$ are called Kashiwara raising and lowering operators.