Branched Crystals and Category O

The theory of crystal bases introduced by Kashiwara in [6] to study the category of integrable representations of quantized Kac–Moody Lie algebras has been a major development in the combinatorial approach to representation theory. In particular Kashiwara defined the tensor product of crystal bases and showed that it corresponded to the tensor product of representations. Later, in [7] he defined the abstract notion of a crystal, the tensor product of crystals and showed that the tensor product was commutative and associative.