Crystal basis

In algebra, a crystal base or canonical base is a base of a representation, such that generators of a quantum group or semisimple Lie algebra have a particularly simple action on it. Crystal bases were introduced by Kashiwara (1990) and Lusztig (1990) (under the name of canonical bases).

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

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