For any semisimple Lie algebra, there is a family of maximal
commutative subalgebras of its universal envelopping algebras. These can be used to construct special bases of representations,
generalizing the Gelfand-Zetlin basis for gl_n. By varying in this
family, we obtain an action of the cactus group on these bases. This action of the cactus group matches an action defined combinatorially using crystals.