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).

As a consequence of its defining relations, the quantum group ${\displaystyle U_{q}(G)}$ can be regarded as a Hopf algebra over the field of all rational functions of an indeterminate q over ${\displaystyle \mathbb {Q} }$, denoted ${\displaystyle \mathbb {Q} (q)}$.

For simple root ${\displaystyle \alpha _{i}}$ and non-negative integer ${\displaystyle n}$, define

In an integrable module ${\displaystyle M}$, and for weight ${\displaystyle \lambda }$, a vector ${\displaystyle u\in M_{\lambda }}$ (i.e. a vector ${\displaystyle u}$ in ${\displaystyle M}$ with weight ${\displaystyle \lambda }$) can be uniquely decomposed into the sums

where ${\displaystyle u_{n}\in \ker(e_{i})\cap M_{\lambda +n\alpha _{i}}}$, ${\displaystyle v_{n}\in \ker(f_{i})\cap M_{\lambda -n\alpha _{i}}}$, ${\displaystyle u_{n}\neq 0}$ only if ${\displaystyle n+{\frac {2(\lambda ,\alpha _{i})}{(\alpha _{i},\alpha _{i})}}\geq 0}$, and ${\displaystyle v_{n}\neq 0}$ only if ${\displaystyle n-{\frac {2(\lambda ,\alpha _{i})}{(\alpha _{i},\alpha _{i})}}\geq 0}$.

Linear mappings ${\displaystyle {\tilde {e}}_{i},{\tilde {f}}_{i}:M\to M}$ can be defined on ${\displaystyle M_{\lambda }}$ by

Let ${\displaystyle A}$ be the integral domain of all rational functions in ${\displaystyle \mathbb {Q} (q)}$ which are regular at ${\displaystyle q=0}$ (i.e. a rational function ${\displaystyle f(q)}$ is an element of ${\displaystyle A}$ if and only if there exist polynomials ${\displaystyle g(q)}$ and ${\displaystyle h(q)}$ in the polynomial ring ${\displaystyle \mathbb {Q} [q]}$ such that ${\displaystyle h(0)\neq 0}$, and ${\displaystyle f(q)=g(q)/h(q)}$).

A crystal base for ${\displaystyle M}$ is an ordered pair ${\displaystyle (L,B)}$, such that