Kashiwara introduced the concept of crystal bases (Lusztig introduced canonical basis), see for example the article. Kashiwara also introduces the concept of crystals (Section 7).

What is the relation between crystals and crystal bases? Thank you very much.

  • $\begingroup$ These are comments, not answers. I'll note that I often hear the terms "crystal" and "crystal base" referred to interchangeably, so be careful there. A crystal basis in the quantum group sense is very linear-algebraic, and is some kind of very nice choice of lattice and basis inside a representation of that quantum group. The basis then satisfies some nice combinatorial properties. I guess you could say the notion of "crystal" kind of divorces this from the quantum group, and considers crystals as abstract directed edge-coloured graphs satisfying some properties. $\endgroup$ – Joppy May 6 '18 at 16:31

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